37

# Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

  • Upload
    others

  • View
    1

  • Download
    0

Embed Size (px)

Citation preview

Page 1: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

�������������� ���������������������������� !���#"��$����&%'���(�$�#�#�)*�$�����,+.-0/�+�+*+2143�563 7

8:9<;�=�>�?�@BA�?DCFE�?G>�9<;�=�>�?�@BHI?�@BJLK�M.>�NOAQPR>TSU>�@�VDK�CGW

XZY\[�]_^�`Z[�]Zacb�dfeg`ihj[2k�h'lnmIXZo2prqsl\]Zk*tia2uDvglxwyaz]|{}a'~,��������c��0�2���(�����2�}�y���z�c�2�������!�2�c�2�c�g�!�n���z�c�z���z���(�R�n�!�!��������!�2�

���0�����#�z�����22�!�n�) ������¡ )����(���¢�y¡ ���6£¤�¥ �¦���2�!�z�§�������(�����2���y���¨����©\���c�����$���\ª&�����$������2����«�����*�¨�!�!�����c���!�n���z�c�z�n�z����0�����#�z�\�(¬�­2����������������¡ (®��z¡ ���2£

¯ �����j����2���(�����2���y�n�z�c� � �&���$�2�j�2�c�§�!�n���z�c�z�n�z���(�R���$�!��������!�2������Q���#�z�\­2°��#�#£x������¡ )����(���¢�y¡ ���6£

�����(��-z�����������±\������ !�²\��(�!1��� !�g����§®n��Q�$ !������£� !���4³}������#��g��(���G�(�$�����#��)�����������§®�£n��(����������������®���²n�#�#����j����¡.%c�f���n����� !����$���±�³��g�(��£��n­� !�����#�0�������-c���$£n (��(����´1§�(���� !�(�f��������2���²\������£��(�Q������­R"��4�����g����®���²n�#�#����j������������(��³&�$�#�z¡������(�4�����*��£��(�µ !�����#�0��������(���� !�¢�f�Q�����n�2�n���®x�$ !���(�4³}����.��¶*�$ !­µ�#��g��(���,�2��2�����������������§®\�! (�z� !�·����­�¶��$ !­Q����§®n��4�®\�! !���������³�����n�#�¸���n����2��2�0�������*�i¡¸¹}����­G��º����§�������»�(º�Q���#���(���#�|�f����®¼����������(�$ ( !���#�¼��� ����g®n£��(����*�§®n����(�§®��*������¡¾½��*�����! �±Z������­���n�­�����¶*�§�(R�����¶����Q�� (�2�#��)����#�D�������(�! ( !���#�y±�(�$���(�§���������� (�2�#��)����*�������$�# 4���!�)��2²\� (�#��)��2������������±y�������(������)*�§�����§�Q�� (�2�#��),��������!�# �$£� ( !�$���O�2���*�����y¡�¹}������������§�#��� ¾�(�³����*���(������#���(�����(����n¡�¿f���(��£��n­R������²\�$����¶��� g�������(�O�'� ����¶*�$ (�#��)Q�(�$ ( !���#�y±��*�O !��À�£��# !�����z� &����$�0�������� ���������#�2£��£���¶��*�$£n£��Á�$������n�#��)R� ���³}�Q���³��#��)�¡�¿º���(��£���­�±�²\*����������� !����������#�­R��������°�®x�$ (����$���(���#�­�±��0³������§®���� !�����#�0�������(���� !�(�D��������2�n�g�����*���6�#Âx�$ ���n�­��#�G���³Ã���������� (�2�#��)*�§�� !�§£�®\�n�*�(���\¡RÄ�����f !�����#�0�������(���� !�(�D��������2�n�g�� !�����)�� (������§�������­s����g�#��� �±������G��°�®\�! (����$���(���& !����£n��(���#���6����*�(��������������!�# ����¶��$ }��������������§�#��� ��#�.�(����4�(�$ ( !���#����¡�Å�£� ��������­2�������������$ !�$�z� !����£� (®n (����#��)n¡�¿f������³·�����*�§���������¶*�$ �����Q����4�����(�������*�µ£��(�R�������������� !�����#�0��������(���� !�(�¼�Q������2���§��)�£��� !�����(�������(�²\��®\��­���*�§������#�f�������2£���²\�$ ¾����2���*��������±Z³����$ !�����4���������¶*�$ �����Q�§�������(�4�����*��£��(�g�����Q������$ � !�����#�0��������(���� !�¢�D��������2�º�����f²\�§��°�®\����$��������&�#�¼-z�����§�(À�£��� !� !2������1��������2£���²\�$ 4��\�2���*����������¶*�$���#�.����§®���§�(�! ( !���#�����������&��� ( !����®x������(R-c®n������� �1£n���2�# !�����(���f�� !������¡�¹�����������³����������������(�4�����*��£��(�§�0³������§�#��� § !�����#�0��������(���� !�¢�f�Q�����n�2�n���'� ����¶*�$ !��)*�R�nR��*�����³���­2��®\�$ (�'� !�Æ����§�#��� (�­�±�³�����¢�D��*���¶��*�(���g�����§��g®\� !�(���������r������� !����������� !����£n��(��¡Ç�È2É\ʧË\Ì�Í�Î�Ï4�����g½��²\*�(��±&Ð}�¶*�$ R¹������±&Ñ� !��®��¸Ð��¶*�$ !��)*��±�ÒZ��³��Ó ��³��#��)�±�ÒZ���� (���#��)½������#�0¹�����g��Ô6±�Õ42�n�QÐ��£������#��)�±\½������#�0¹�����§Ö4�$£� (��(����R×6���� !�(�Z±�ØO�*�$£n£��ÙÐO��������#��)

Ú�]\Û�kÝÜ�Û�Þ�l�Û4ack�ßZl\]iÛ¾à�`Zá�`\Û�k�â4l\à´[Qk�a'qsãZhj[,ä�à�[�l�Û*~�à�[2k�m,ajÛ�ÞfhcazqGacÛ�[2å¼k�[�]Zk�az]ZbDli]ZåDä2`Zqºæã�~ZÛ*lxÛ�ac`y]�l�h�ä�l\ã�liáZajhcacÛ�ac[2k�ç,Úè]iÛ�k.Þ�lnY\[�Û*ÞZ[flxå�Y�l\]iÛ*l�b\[�Û�Þ�l�ÛRÛ*ÞZ[2p_l\à�[�k�azqÓãZhc[GÛ�`Óå�[2k�ajbZ]�ß[�l�k�p_Û�`»ã�à�`\bZà�l\q_ß�l\]Zåéä ÞZ[�l\ãêÛ�`»á�~Zachcå�ç�ëRÞZackèqÓlit�[2kRacÛRì([�l�k�azáZhc[ÓÛ�`så�[�ãZhj`�p_byà´`y~�ãZk.`xìl\]iÛ�k�l\]Zå|Û�l\t�[fl�å�Y�l\]\Û�l�b\[f`xìgÛ*ÞZ[Dà�[2k�~ZhjÛ�a']Zb¼ì!l\~ZhcÛ�Û�`\hc[�à�l\]Zä2[Ól\]Zåíã�l\à�l�hchc[2hcajk�qïîðnñ�ç�ò:[k�Û�~Zå�p»Û*ÞZ[Gá�[�Þ�lnYZaj`Zà�`�ìgl\]\Û�k�ì¦`Zàgä2`�Y\[�à´a']ZbsÛ�[�à�à�l�az]�ß�lxk.à�[2ó�~Za'à´[2å_ì(`yà�Y�lxä�~�~�qôä2hc[�l\]Zaz]Zb�ßhzlnm.]|qõ`�YZaz]Zb�ß�qGaz]Z[Gk�m�[2[�ãZa']ZbOß�l\]Zå|k�~�à�Y\[2achch'li]Zä2[\ç�ögl�ä�~�~�qôä2hc[�l\]Zaz]Zb|l\]Zå_hzlnm.]éqG`�Y\æaz]Zb�ßZì(`yà§[2w�l\qÓãZhc[\ß�l\à�[�à�[�l�hcack�Û�acäGl\ã�ãZhcajä�lxÛ�ac`y]ZkRì(`yà§l\]\Û�kQbiajYi[�]¼Û*Þ�l�Û�Û*ÞZ[,÷&à�k�Û�ä ÞZ[�liã_l\]Zåk*qsl�hch�Y�l�ä�~�~�qºæ!ä2hc[�l\]Zaz]ZbÓl\]ZåGhzlnm.]fqG`�YZaz]ZbfÞZ`y~Zk�[�ÞZ`\hcå»à�`Zá�`\Û�k�l\à�[�l�hzà�[�l�å�pº`y]ºÛ*ÞZ[�ä2`y]iæ

Bolek
Text Box
Annals of Mathematics and Artificial Intelligence, vol. 31, no. 1-4, May 2001, pp. 41-76
Page 2: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

/ � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

k*~�qG[�àRqsl\à�t�[2Û§`yà�l\à�[�[2w�ã�[2ä2Û�[2å_Û�`Óá�[�`y]¼Û*ÞZ[�ä2`y]Zk*~�qõ[�àQqsl\à�t�[2Û�k�`Z`y]�ßZaz]Zä2h'~Zå�az]Zb_Û*ÞZ[^�`yl�hzlÓà�`Zá�`\Û�ß������iðDà´`yá�`\Ûnß��µhc[2ä2Û*à�`\hz~Zw|à�`yá�`\Û�ß���p\[fà´`yá�`\Ûnß��§~�à�[�t�lsà�`yá�`iÛ�ß���`yá�`ZqG`�mà�`yá�`\Û�ß�XZ`\hzl\à! í`�mQ[�à�à�`Zá�`\Û�ß�l\]Zå"��`yã�ÞZaz]|à�`Zá�`\Û�çµò:[fk�Û�~Zå�p|á�`\Û*Þ¼`Z]Z[�æ!Û�azqG[fä2`�Y\[�à�lxb\[l\]Zå:ä2`y]iÛ�az]Z~Z`y~Zk�ä2`�Y\[�à�l�b\[\ßOm.ÞZ[�à�[¼l\]\Û�k.ä2`y]iÛ�az]Z~Z`y~Zk�hcpíä2`�Yi[�à.l¼hzl\à�bi[DÛ�[�à�à�l�az]émRacÛ*ÞZ`y~ZÛb\[2Û�Û�az]Zb�k�mRacÛ�ä2ÞZ[2åG`�#gß�k�`�Û*Þ�l�Û�[2Y\[�à�pfã�lià�Û4`xìrÛ*ÞZ[�Û�[�à�à�l�az]�b\[2Û�k4YZack�acÛ�[2åG`y]Zä2[�[2Y\[�à�p�m.ÞZachc[\çÚè]iÛ�k�ä�l\]�]Z`\ÛR~Zk�[�ä2`Z]\Y\[�]iÛ�ac`y]�l�h�ãZhzl\]�]Zaz]Zb_qG[2Û*ÞZ`Zå�kQå�~Z[�Û�`DÛ�ÞZ[2a'àÝhjazqGacÛ�[2åék�[�]Zk�az]Zb»l\]Zåä2`yqÓã�~ZÛ*lxÛ�ac`y]�l�h�ä�liã�l\áZachjacÛ�ac[2k�m.ÞZacä ÞshcazqGacÛRÛ*ÞZ[2azà�ãZh'li]�]Za']Zbsä�l\ã�l\áZachcajÛ�aj[2k�[2Y\[�]õì¦`Zà�k�azqÓãZhc[ãZhzl\]�]Zaz]Zb·Û*lxk*t\k�k�~Zä Þ l�kÓã�l�Û*Þ:ãZhzl\]�]Zaz]Zb:`Zà�Û*ÞZ[¼ä2`�Y\[�à�l�b\[s`�ìRÛ�[�à�à�l�az]�ç%$O`yà�[2w�l\qÓãZhc[\ßl\]iÛ�kDqGacbyÞiÛÓ]Z`\ÛÓá�[_l\áZhc[_Û�` qsl�az]\Û*lxa'] qÓl\ãZkÓl\]ZåÁ~Zk�[¼Û*ÞZ[�q ì(`yàGã�l�Û*ÞÃãZh'li]�]Za']ZbOç'&!]ì¢lxä2Û�ßZÛ*ÞZ[2p»qGacbyÞiÛR]Z`\ÛRá�[fl\áZhc[�Û�`GãZh'li]¼ä2`yqÓãZhc[2Û�[fã�l�Û*ÞZkRl�ÛRl�hch(çgëRÞZ~Zk�ßZÛ*ÞZ[2p_qGacbyÞiÛÝ]Z`iÛá�[»liáZhj[sÛ�`_ä2`�Y\[�àRÛ�[�à�à�l�az]íl�k�[)(»ä2ac[�]iÛ�hcp lxkfà�`yá�`iÛ�k�mRacÛ*ÞíqG`yà�[Óã�`�m�[�à$ì!~Zh�k�[�]Zk�az]Zb l\]Zåä2`yqÓã�~ZÛ*lxÛ�ac`y]�l�hZä�l\ã�l\áZachcajÛ�aj[2knç+*�]ºÛ*ÞZ[�`\Û*ÞZ[�à�Þ�l\]Zå�ß�byà´`y~�ãZk4`xì&l\]\Û�k4ä�l\] Û*l\t�[�l�å�Y�li]\Û*lxb\[`�ìQá�`iÛ*Þ_Û*ÞZ[2azà,ì!l\~ZhcÛ.Û�`\hc[�à�l\]Zä2[¸Ü¦Û*ÞZ[2p¼ì¢lxajh�byà�lxä2[�ì¢~Zhchcp|[2Y\[�]êa ìQk�`yqõ[fl\]\Û�kèqÓlxhì!~�]Zä2Û�ac`y]�âl\]ZåíÛ*ÞZ[2azà.ã�lià�l�hchc[2hjack*q Ü�byà�`Z~�ãZk.`�ìQli]\Û�k,ä�l\]_ä2`�Y\[�à,Û�[�à�à�lxa']¸ì¢lxk�Û�[�à�Û*Þ�li]|lsk�az]Zb\hc[»l\]iÛ*â2çò:[Ók�Û�~Zå�p·l\]iÛ�k�Û*Þ�l�Ûf~Zk�[_à�[�l�h æ!Û�azqG[¼k�[�l\à´ä ÞíqG[2Û*ÞZ`Zå�k�ç%��[�l�h æ!Û�azqG[·Ü¢ÞZ[�~�à�ack�Û�acä�âºk�[�l\à�ä2ÞqG[2Û*ÞZ`Zå�k»î�,nðnñ4az]\Û�[�à�hc[�l�Yi[»ãZhzl\]�]Zaz]Zb li]Zå ãZhzl\] [2wy[2ä�~ZÛ�ac`y]�çGëRÞZ[2péä�l\]:á�[G[2wZ[2ä�~ZÛ�[2å·áipl\]iÛ�k�mRacÛ*ÞZ`y~ZÛ.qG[�qõ`yà�p\ß�à�[2ó�~Za'à´[�`y]Zhcp|lfYi[�à�pÓhcazqGacÛ�[2åíhj`Z`ytxæ�l\ÞZ[�l�å|l\]Zå_ä2`ZqÓã�~ZÛ*l�Û�ac`y]�lxhä�l\ã�l\áZachcacÛ�ac[2k�ß&li]Zå»ä�l\]¼á�[f~Zk�[2å¸áipÓk�az]Zb\hc[fl\]iÛ�kQlxkQm�[2hch&lxkQbZà�`y~�ãZkg`xì�li]\Û�knç¾ë,Þy~Zk�ßZÛ*ÞZ[2azàã�à�`yã�[�à�Û�ac[2kRqÓl�Û�ä2ÞGÛ*ÞZ[RhcazqGacÛ�[2å_k�[�]Zk�a']ZbÓl\]Zå¼ä2`yqÓã�~ZÛ�l�Û�ac`y]�l�h�ä�l\ã�liáZajhcacÛ�ac[2k�`�ì4l\]\Û�k�ç4ëRÞZ[l\]iÛ�k_`y]Zhcp Þ�l�Yi[ Û�`·hc[�lnY\[·qÓlià�t\az]Zb\k_az]ÆÛ�ÞZ[:Û�[�à�à�lxa']�ß�k�[�]Zk�[ Û�ÞZ[·qÓl\à�tiaz]Zb\k|l�Û_Û*ÞZ[2azà]Z[2acbyÞyá�`yà�az]Zbíhc`yä�l�Û�aj`Z]Zk�ß4l\]Zåíä2Þ�l\]Zb\[GÛ*ÞZ[ÓqÓl\à�t\az]Zb_`�ìgÛ*ÞZ[2azà ä�~�à�à�[�]iÛ�hj`Zä�l�Û�ac`y]�çGëRÞZ[2k�[qÓl\à�tiaz]Zb\k.l\à´[.k*Þ�lià�[2å|l\qG`y]ZbÓl�hch4li]\Û�kRl\]Zå|lxhjhc`�m Û*ÞZ[�q Û�`Gä2`�Yi[�àgÛ�[�à�à�l�az]¼[2Y\[�]¼a ì�Û*ÞZ[2på�`¸]Z`iÛ.Þ�lnY\[fl\]ip¸tiaz]Zå|`�ìgqG[�qG`yà´p\ß�ä�l\]�]Z`\Û.qsl�az]\Û*lxa']éqÓl\ãZk�`�ì�Û*ÞZ[�Û�[�à�à�lxa']�ßO]Z`yà.ãZhzl\]ä2`yqÓãZhc[2Û�[|ã�l�Û�ÞZk�ç|ëRÞZackGackDm�Þ�l�ÛGk�`yqG[Óaz]Zk�[2ä2Û�kDå�`Áî�, ñ�ç%-.]iì(`yà�Û*~�]�lxÛ�[2hcp\ßg]Z`\ÛDqG~Zä2Þ:ackty]Z`�m.]Ãl\á�`y~ZÛºÛ*ÞZ[»ã�à�`Zã�[�à�Û�ac[2kG`�ì.à�[�l�h æ!Û�azqG[sk�[�l\à�ä2Þ:qG[2Û*ÞZ`Zå�kfk�a']Zä2[_Û�ÞZ[»l\à�[�l¼`�ì.à�[�lxhæÛ�azqG[Gk�[�l\à�ä2ÞéackfqG`\k�Û�hcp|l\]é[2w�ã�[�à�azqG[�]iÛ*l�h�`Z]Z[õli]Zå ]Z`iÛèqÓli]\p_Û*ÞZ[2`Zà�[2Û�acä�l�hgà�[2k*~ZhcÛ�kflià�[ty]Z`�m.]�ç.��[�l�h æ!Û�azqG[_k�[�lià�ä ÞÃqG[2Û*ÞZ`Zå�kfå�a/#�[�à�az]:Û�ÞZajkGà�[2k*ã�[2ä2Ûfì!à�`yq Û�à�l�å�acÛ�ac`y]�l�h§k�[�l\à�ä2ÞqG[2Û*ÞZ`Zå�k�ßZm.ÞZ`\k�[Rã�à�`Zã�[�à�Û�ac[2kRÞ�lnY\[.á�[2[�]»à�[2k�[�l\à�ä2ÞZ[2åÓ[2wyÛ�[�]Zk�acY\[2hcp\çgò:[,Û*ÞZ[�à�[�ì(`yà�[�k�Û*~Zå�p\ßá�`\Û�Þ|Û�ÞZ[2`yà�[2Û�acä�l�hchcp:l\]Zåí[2wrã�[�à�azqG[�]\Û�l�hchjp\ß�Û*ÞZ[Óá�[�Þ�l�YZac`yà�`�ìQl\]iÛ�k�Û*Þ�l�Û�~Zk�[GÛ�mQ`¼k�azqÓãZhc[à�[�l�h æ!Û�azqG[¼k�[�l\à�ä2Þ:qG[2Û�ÞZ`yå�kºÛ*Þ�l�Û�ä�l\]:á�[»~Zk�[2å Û�`_ä2`�Y\[�à�Û�[�à�à�l�az]�ç»ëRÞZ[ÓÛ�m�`_à�[�l�h æ!Û�azqG[k�[�l\à�ä2Þ_qG[2Û*ÞZ`Zå�kRå�a/#�[�àR`Z]Zhjp¼az]|ÞZ`�m Û*ÞZ[2p_~�ã�å�lxÛ�[�Û*ÞZ[fqÓl\à�t\az]Zb\k�ç§ëRÞZ[2p¸lià�[fl�hcb\`yà�acÛ*ÞiæqGacä�l�hchcpÓk�azqGachzl\à2ß�l\]ZåÓ[2w�ã�[�à�azqG[�]iÛ*l�h�à�[2k*~ZhcÛ�k�az]Zå�acä�l�Û�[�Û*Þ�l�Û�Û*ÞZ[,ä2`�Yi[�à�Û�azqG[�`�ì}l\]iÛ�k�Û*Þ�lxÛ~Zk�[�Û*ÞZ[�q ackR`xì¦Û�[�]sk�azqGachzl\à,`y]ÓbZà�acå�k�ç�*�~�àRl\]�lxhjpZk�ackRackRÛ*ÞZ[�à´[�ì¦`yà´[.k*~�à�ã�à�ack�az]Zb�çgòÃ[�k*ÞZ`�mÛ*Þ�l�Û�Û*ÞZ[�ä2`�Y\[�à§Û�azqG[º`�ì¾l\]iÛ�k�Û*Þ�l�ÛR~Zk�[�`y]Z[�`xì4Û*ÞZ[fà�[�l�h æ!Û�azqG[ºk�[�l\à�ä2Þ¼qG[2Û*ÞZ`Zå�kRack.by~�lià$æl\]iÛ�[2[2å:Û�`íá�[»ã�`\hcpr]Z`yqõa'lxhQaz]ÃÛ*ÞZ[»]Z~�qGá�[�à�`xìQhc`yä�lxÛ�ac`y]Zk�ß�m.ÞZ[�à�[�l�kºÛ*ÞZ[Óä2`�Y\[�à�Û�azqG[¼`�ìl\]iÛ�k�Û*Þ�l�Û�~Zk�[RÛ*ÞZ[�`\Û�ÞZ[�à�à�[�l�h æ!Û�azqG[Rk�[�lià�ä ÞGqG[2Û�ÞZ`yåÓä�li]Dá�[R[2w�ã�`y]Z[�]iÛ�azl�h�az]|Ü�Û*ÞZ[�k�ó�~�l\à�[à�`Z`\Û�`�ì â§Û*ÞZ[�]y~�qGá�[�àÝ`�ì4hc`yä�lxÛ�ac`y]Zk.[2Yi[�]¼a']¼k�a'qsãZhj[ºÛ�[�à�à�lxa']Zk,Û*Þ�l�Û�ä2`yà�à�[2k�ã�`y]Zå_Û�`|Ü(ãZhzlnæ]�l\à�â�~�]Zå�azà�[2ä2Û�[2åéÛ*à�[2[2k�çgë,ÞZ[2k�[èà´[2k*~ZhcÛ�kRå�[�qG`y]Zk�Û*à�l�Û�[�Û*Þ�l�Û�[2w�ã�[�à�azqG[�]iÛ*l�h�ä2`yqsã�l\à�ack�`y]Zk`�ìRÛ*ÞZ[»á�[�Þ�l�YZac`yàº`�ìÝli]\Û�kºÛ*Þ�l�Ûf~Zk�[¼à�[�l�h æ!Û�azqG[¼k�[�l\à�ä2Þ:qG[2Û*ÞZ`Zå�kDlià�[Ó`�ì(Û�[�]íaz]Zk*~0(»ä2ac[�]iÛÛ�`_[2Y�lxh'~�lxÛ�[»ÞZ`�mIm�[2hjh�Û�ÞZ[2p k�ä�l�hc[»~�ã:á�[2ä�li~Zk�[ÓÛ*ÞZ[Gä2`�Y\[�à�Û�azqG[Ó`xìÝl\]iÛ�k�Û*Þ�l�Ûf~Zk�[ÓÛ�m�`k�azqGachzl\àgà�[�lxhæ!Û�a'qõ[Rk�[�l\à�ä2Þfqõ[2Û*ÞZ`yå�k�ä�li]fá�[,Y\[�à�p�å�a/#�[�à�[�]\Û�[2Yi[�]fa ì�[2wrã�[�à�azqG[�]\Û�l�h�à�[2k*~ZhcÛ�kk�[2[�q Û�`_a']Zå�acä�l�Û�[»`\Û�ÞZ[�à�mRack�[\çGÚ�] li]�l�hcpyk�ackGä�l\]íÞZ[2hzã:Û�`_å�[2Û�[2ä2Û�Û*ÞZ[2k�[»ã�à´`yáZhc[�qGkDl\]Zåã�à�[2Y\[�]iÛGk*~�à�ã�à�ack�[2kGhzl�Û�[�àº`y]�ß�l�kGm�[2hchQl�kÓã�à�`�YZajå�[_lék�`\hcacå·Û�ÞZ[2`yà�[2Û�acä�l�h§ì(`y~�]Zå�l�Û�ac`y] ì¦`Zà

Page 3: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� �

34

3 3

445

4

5

34

3 3

445

4

5

�Z�)�£n !�R7�¡������(��������½������#��¹�����Q×2���� !�(�ä2`�Y\[�à�lxb\[.m,ajÛ�Þ|l\]iÛ�k�Û*Þ�l�Û.~Zk�[fà�[�l�h æ!Û�azqG[fk�[�l\à�ä2Þ_qG[2Û*ÞZ`Zå�k�ç

��� ����� ���������������������

ë�[�à�à�l�az]ZkÓä�li]Ùá�[íqG`Zå�[2hj[2å l�ksbyà�l\ã�ÞZkÓm,ajÛ�Þ Û*ÞZ[|hc`Zä�l�Û�ac`y]Zk¼`�ì l Û�[�à�à�l�az] ä2`yà�à�[�æk*ã�`y]Zå�az]Zb_Û�`¼Û*ÞZ[fY\[�à´Û�acä2[2k�ß�ì(`yà,[2wrliqÓãZhc[õáip_azqÓã�`ik�az]Zb|l»à�[2bZ~Zh'liàRbyà´ajåé`y]_Û�ÞZ[fÛ�[�à�à�l�az]�ç$�acby~�à�[", Ü�hc[�ì(Û*â2ß�ì(`yà�[2w�l\qÓãZhc[\ß�k*ÞZ`�mRkfl|à´[2by~Zhzl\à�ì(`y~�à$æ!ä2`y]�]Z[2ä2Û�[2å·byà´ajå�ß�l\]Zå $�acby~�à�[",Ü(à´ajbZÞ\Û*âºk*ÞZ`�mRksÛ*ÞZ[_ä2`yà�à�[2k�ã�`y]Zå�az]Zb byà�l\ã�Þ�ç ò:[éå�`·]Z`\ÛGazqÓã�`\k�[ à�[2k�Û�à�acä2Û�ac`y]Zk»`Z] Û*ÞZ[Û�`Zã�`\hc`\b\pÃ`�ì.Û*ÞZ[_byà�l\ã�ÞZk|Ü(~�]Zhc[2k�kÓk�Û*lxÛ�[2å:`\Û*ÞZ[�à´mRack�[�â2ß�[2wZä2[�ãZÛÓÛ*Þ�l�ÛGm�[_à�[2ór~Zazà�[_Û�ÞZ[�qÛ�`¼á�[fk�Û*à�`y]Zb\hcp¼ä2`y]�]Z[2ä2Û�[2åÙÜ�[2Y\[�à´p»Y\[�à´Û�[2w»ä�li]_á�[Dà´[�l�ä ÞZ[2åêì¢à´`yq [2Y\[�à�p¼`\Û*ÞZ[�àÝY\[�à�Û�[2w�â2çXZÛ*à´`y]Zb\hcp|ä2`y]�]Z[2ä2Û�[2å bZà�l\ã�ÞZk�by~�l\à�l\]iÛ�[2[ºÛ*Þ�l�Û�l\]\Û�k.l\à�[Ól�hcmRlnpyk.liáZhj[sÛ�`»à�[�l�ä2Þíl�hch4Yi[�à$æÛ�acä2[2k�ß�]Z`_qÓl�Û�Û�[�à�m.ÞZacä2Þ_[2å�b\[2k�Û*ÞZ[2p|Þ�lnY\[ºÛ*à�lnY\[�à�k�[2å_a']éÛ*ÞZ[Gã�l�k�Û�ç $O`yà�k�azqÓãZhcacä2acÛ�p\ß�m�[l�k�k*~�qG[�Û*Þ�l�Û�k�[2Y\[�à�lxh&l\]iÛ�k�ä�l\]¼á�[fl�Û§Û*ÞZ[�k*l\qG[�Y\[�à�Û�[2wõlxÛQÛ�ÞZ[.k*liqG[.Û�a'qõ[\çgëRÞZ[2azà�Û*l�k�tackGÛ�`_ä2`�Y\[�à�Û*ÞZ[ÓbZà�l\ã�ÞÙÜ�YZack�acÛDl�hch§Y\[�à�Û�acä2[2k*â `y]Zä2[Ó`yà�à�[�ã�[�l�Û�[2å�hcp\ç_ò:[¼~Zk�[ÓÛ*ÞZ[Gì(`\hchc`�m�æaz]Zb·]Z`\Û�l�Û�ac`y] az] Û*ÞZack»l\à�Û�ajä2hc[���� å�[�]Z`iÛ�[2kÓÛ*ÞZ[s÷&]ZacÛ�[|k�[2ÛG`�ì�Y\[�à�Û�ajä2[2kíÜ�k�Û*l�Û�[2k*â�`�ì�Û*ÞZ[byà�l\ã�Þ�ßyl\]Zå! �"$#&%(')#+*�� å�[�]Z`\Û�[2k�Û*ÞZ[Rk�Û�l\à�Û�Y\[�à�Û�[2w&ç�ëRÞZ[.]Z~�qGá�[�à§`�ì�Y\[�à�Û�ajä2[2k§ack-,�.0/1�2/cç3GÜ$ �â54.76Óack�Û*ÞZ[º÷&]ZacÛ�[\ß4]Z`Z]Z[�qÓãZÛ�p_k�[2Û.`xìQ[2å�b\[2k�hc[�l�YZaz]Zb|Y\[�à´Û�[2w� 8*9�IÜ(l�ä2Û�aj`Z]Zk�Û*Þ�l�Ûä�l\]íá�[D[2wZ[2ä�~ZÛ�[2åíaz]ík�Û*lxÛ�[: �â2ç; =<>�>\Ü$ @?BAZâ�å�[�]Z`\Û�[2k�Û*ÞZ[ºk*~Zä2ä2[2k�k�`Zà�Y\[�à�Û�[2w_Û*Þ�lxÛèà�[2k�~ZhjÛ�kì!à�`yq Û*ÞZ[_Û*à�lnY\[�à�k�l�h4`xì.[2å�b\[�AC*D3DÜE �â�az] Y\[�à´Û�[2wF G*D�Rçéë�`ík�azqÓãZhca ì¦pÙqsl�Û�Û�[�à�knß�m�[qG[�lxk*~�à�[ÓÛ*ÞZ[Gå�ack�Û*li]Zä2[2kDaz]í[2å�bi[ÓÛ*à�lnY\[�à�k�l�hck.Û*Þ�à´`y~ZbyÞZ`y~ZÛ�Û*ÞZackGl\à�Û�acä2hc[\ß�m.ÞZacä ÞÃackIH�~Zk�Û�a æ÷�[2åÁa ì�Û*ÞZ[»ä2`ik�Û�kG`�ìèl�hch�[2å�b\[2kÓl\à�[_à�`y~ZbZÞZhjp Û*ÞZ[¼k*l\qG[\çíòÃ[¸l�hck�` ~Zk�[»Û�m�`|`Zã�[�à�l�Û�`yà�kmRacÛ*Þ Û�ÞZ[_ì(`\hchj`�mRaz]Zb:k�[�qÓl\]iÛ�acä2k���J�acY\[�]Ál ÷&]ZajÛ�[|k�[2Û8KÁß�Û�ÞZ[_[2wrã�à�[2k�k�ac`y]7L�`Z]Z[�æ!`�ìMKGNà�[2Û�~�à�]ZkÓl\]í[2hc[�qG[�]iÛÓ`�ìIK l�ä2ä2`yà�å�az]ZbíÛ�`íl\]·lià�áZacÛ*à�l\à�p:à�~Zhc[\çéÚ k*~�áZk�[2ó�~Z[�]\ÛÓaz]iY\`Zä�lnæÛ�ac`y]Ã`�ì8L�`y]Z[�æ!`xìOKGN_ä�l\] à�[2Û*~�à�]:Û*ÞZ[sk*l\qG[¼`yàflêå�a/#}[�à´[�]\ÛG[2hc[�qG[�]iÛ�ç ë,ÞZ[»[2w�ã�à�[2k�k�ac`y]L*lià�bgqGaz]QPSRMTGU�Ü$V�â)N_à�[2Û*~�à�]Zk�Û*ÞZ[G[2hj[�qõ[�]\Û�kWV *9KïÛ*Þ�lxÛèqGaz]ZazqGaco2[�U�Ü$V�â6ß�Û*Þ�lxÛ.ack�ß�Û*ÞZ[k�[2ÛYXMV�*�KZ/1U�Ü$VOâ[. qõa'] P�\]RST U�Ü$V_^jâa`yç

b�� cedf�gih(�:�EjkdmlndQ�oSp�q

��[�lxhæ!Û�a'qõ[ Ü(ÞZ[�~�à´ajk�Û�acä�â�k�[�l\à´ä ÞíqG[2Û*ÞZ`Zå�kÓî�,�r�ñ¾az]iÛ�[�à�hc[�lnY\[»ãZhzl\]�]Zaz]Zb:l\]Zå·ãZhzl\]í[2wiæ[2ä�~ZÛ�ac`y]�ߧl\]Zå·l�hchc`�mIì(`yà�÷&]Z[�æ!byà�l�az]Z[2åÁä2`y]\Û�à�`\h�`�Y\[�à�ÞZ`�m qG~Zä2Þ:ãZhzl\]�]Zaz]Zb:Û�`|ã�[�à�ì¦`yà�q

Page 4: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

�������������� ����������������������������� ���!��#"���!�$�%&$�!'����(�*),+.-/0-1�32546� 7�8&9;:�8�-<=-1>?2 4@$0���;��$�%A�!CBEDF��� 9HGHIKJ�7�L �(�&�M�ON�N��&��P;>����Q-R=-ASUTWVO������(�&�M�.���C����BX�����Y��Z�[�=�;�\�=TWVWZ���Y!��=�[�0-]O-A^K!; ����!C���_��V=B��_>O-`=-1�32546� �ON�N����0P;>a�Q-b=-1c*$d�C$F<�-

�Z�)�£n !�Q/6¡�½������#�0¹�����Q×6���� !�(�f³}����fÒy2��2�0�������*�fÅ����

á�[2Û�m�[2[�]ÙãZhzl\] [2wy[2ä�~ZÛ�ac`y]Zknç ëRÞZ[2pÁl\à�[|à´[2h'lxÛ�[2åÙÛ�`:à�[2az]iì(`yà�ä2[�qG[�]iÛ�æ!hc[�l\à�]Zaz]ZbÙqG[2Û�ÞZ`yå�kîfe�ßfgih�ñ�ßylxkÝã�`\az]\Û�[2å_`y~ZÛ�a']íîkjnñ�ç�l�hzl\]�]Zaz]Zb¼ajk,å�`y]Z[.YZazlGhc`yä�lxh�k�[�l\à�ä2ÞZ[2k�ßZÛ*Þ�l�Û�ack�ßZk�[�l\à´ä ÞZ[2kÛ*Þ�l�Û|l\à´[ à´[2k�Û*à�acä2Û�[2åÆÛ�`ÁÛ*ÞZ[:ã�l\à´Û_`�ìfÛ*ÞZ[:byà�l\ã�ÞÙl\à´`y~�]Zå Û*ÞZ[íä�~�à�à�[�]iÛ_Y\[�à�Û�[2wÙ`�ìDl\]l�b\[�]iÛ�çgëRÞZ[ºm.ÞZacÛ�[Dlià�[�lG`�ì $�acby~�à�[ , ajhchz~Zk�Û*à�l�Û�[2k�Û*ÞZ[fhcazqGacÛ�[2åíhc`y`ytxæ�l\ÞZ[�l�åé`�ìgà�[�l�h æ!Û�azqG[k�[�l\à�ä2ÞÆqG[2Û*ÞZ`Zå�k�ç �µwrã�[�à�azqG[�]\Û�l�h�[2Yyacå�[�]Zä2[ k*~Zb\b\[2k�Û�k»Û�Þ�l�Û|à�[�l�h æ!Û�azqG[ík�[�l\à´ä ÞÆqG[2Û*Þiæ`yå�k�l\à�[R[)(¼ä2aj[�]iÛ�å�`yqÓl�az]iæ!az]Zå�[�ã�[�]Zå�[�]iÛRk�[�lià�ä ÞsqG[2Û*ÞZ`Zå�kgÛ*Þ�l�Û�`Z~ZÛ*ã�[�à$ì(`yà�q Û*à�l�å�acÛ�ac`y]�lxhk�[�l\à�ä2ÞÓqG[2Û�ÞZ`yå�k§k*~Zä Þsl�kgÚnmDî�,0onñr`Z]Ól�Y�l\à�ac[2Û�pD`xì}k�[�l\à�ä2ÞDã�à´`yáZhc[�qGk�ç���[�lxhæ!Û�a'qõ[�k�[�l\à�ä2ÞqG[2Û*ÞZ`Zå�kèÞ�lnY\[\ßiì(`yà§[2wrliqÓãZhc[\ß�k*~Zä2ä2[2k�k�ì¢~Zhchcp á�[2[�]|l\ã�ãZhcac[2åíÛ�`ÓÛ*à�l�å�acÛ�ac`y]�l�h�k�[�l\à�ä2Þ_ã�à�`yáiæhc[�qGk î ,�ðnñ�ßgqõ`�YZaz]Zb�æ!Û*l\à´b\[2ÛGk�[�l\à�ä2ÞÁã�à�`yáZhc[�qGk|îkpnñ�ßgX�ë��!&�lgXiæ!Û�p�ã�[:ãZhzl\]�]Zaz]ZbÙã�à�`yáZhc[�qGkî rnñ�ßZà�`yá�`\ÛQ]�lnYZajbZl�Û�ac`y]¼l\]Zå»hc`Zä�l�hcaco�l�Û�ac`y]|ã�à´`yáZhc[�qGkRmRacÛ*Þsaz]ZajÛ�a'lxh4ã�`\k�[è~�]Zä2[�à�Û�l�az]\Û�p|î�,0jnñ�ßÛ�`\Û*lxhjhcpÁ`yáZk�[�à�Y�l\áZhc[" lià�t�`�Yíå�[2ä2ajk�aj`Z] ã�à´`yä2[2k�k_ã�à�`yáZhc[�qGkíîfjnñ�ßgl\]ZåÙã�l\à�Û�a'lxhjhcpÁ`yáZk�[�à�Y\æl\áZhc[% :l\à�t�`�Y|å�[2ä2ack�ac`y]Áã�à´`yä2[2k�kÓã�à�`ZáZhj[�qõk|î5h�ñ�ß�l\qG`y]Zbé`\Û*ÞZ[�à�knç_Úôb\`Z`yåÃ`�Yi[�à�Yyac[2m `�ìà�[�l�h æ!Û�azqG[ºk�[�l\à�ä2Þ»qõ[2Û*ÞZ`yå�k�ackRb\acY\[�]¼az]íîfo�ñ}l\]Zå|]Z[2m�[�à§å�[2Y\[2hc`yã�qG[�]iÛ�kRä�l\]_á�[�ì¦`Z~�]Zå»az]î�, �nñ�çRò:[Gk�Û*~Zå�péÛ�mQ`sk�azqGach'liàfà�[�l�h æ!Û�azqG[sk�[�l\à�ä2Þ|qG[2Û�ÞZ`yå�k�Û*Þ�lxÛèl\]iÛ�k�ä�l\]í~Zk�[GÛ�`¼ä2`�Yi[�àbyà�l\ã�ÞZknßy]�l\qG[2hcpGY�lià�azl\]\Û�kg`�ì4{�[�l\à�]Za']Zb ��[�l�h æ�ë�azqG[�Únm.l\]Zårq,`yå�[��`y~�]iÛ�az]ZbfÛ�Þ�l�ÛgÞ�lnY\[léhj`Z`ytxæ�l\ÞZ[�l�å `�ìR`y]Zhcp:`y]Z[Ó[2å�bi[»Û*à�l�Yi[�à�k*l�hgl\]Zå ÷�ÛDÛ�ÞZ[»l�hcb\`yà�acÛ*Þ�qõajä_k�t�[2hc[2Û�`y]Ãk*ÞZ`�m.]az] $�acby~�à�[sg�ç»e§`\Û*ÞÃà�[�l�h æ!Û�azqG[¼k�[�l\à�ä2Þ:qG[2Û*ÞZ`Zå�kDlxk�k�`Zä2a'lxÛ�[»l�<ræ!Y�l�hz~Z[�<4Ü$ nâ�mRacÛ*Þ [�l�ä ÞY\[�à�Û�[2w� e*9�Æl\]Zåía']ZacÛ�azl�hcaco2[¼Û*ÞZ[�q mRacÛ*Þ_o2[�à´`y[2k�ç�ëRÞZ[Gk�[�qsl\]\Û�ajä2k�`�ìgÛ*ÞZ[8<ræ!Y�l�hz~Z[2kfå�[�æã�[�]Zåí`y]¼Û*ÞZ[fà´[�l�h æ!Û�azqG[Gk�[�l\à�ä2Þ_qG[2Û*ÞZ`Zå|á�~ZÛèlià�[�k�azqÓãZhc[Dl\]Zåíà�[2k*~ZhcÛ�a']êaz]\Û*~ZacÛ�acY\[Óà�~Zhc[2k`�ì�Û*Þy~�qGáõì¦`Zà�m.ÞZacä Þ¼[2å�bi[2k.Û�`GÛ*à�l�Yi[�à�k�[\ç4ë,ÞZ[ <Zæ!Y�l�hz~Z[2k.ä2`Zà�à�[2k*ã�`y]Zå_Û�`ÓqÓl\à�tiaz]Zb\kRÛ*Þ�lxÛÛ*ÞZ[Dli]\Û�k�hc[�lnY\[Dl�Û,Û*ÞZ[GY\[�à�Û�acä2[2k�`�ìgÛ*ÞZ[fbZà�l\ã�ÞZk�ß�k�[2[ $�acby~�à´[ ,\çRëRÞZack�ack�m.Þ�l�Û�k�`yqG[�az]iæk�[2ä2Û�k�å�`�ç���[�l�h�l\]\Û�k�ßZì(`yà�[2w�l\qÓãZhc[Ó~Zk�[fä2ÞZ[�qGacä�l�hRÜ(ã�ÞZ[�à�`yqG`Z]Z[�âRÛ*à�l�ä2[2kRÛ�`ÓbZ~Zajå�[GÛ*ÞZ[2azà]�lnYyacbyl�Û�aj`Z]:î�,�ñ�ç í`yáZachj[Óà�`Zá�`\Û�k�Þ�l�Yi[Dá�[2[�]éá�~ZachjÛ�Û�Þ�l�ÛRhc[�lnY\[DqÓlià�t\az]Zb\k�az]_Û*ÞZ[ºÛ�[�à�à�lxa']á�~ZÛ.k�`fì!l\à�`Z]Zhjp_k�ÞZ`yà�Û�æ!hcacY\[2å qsl\à�tia']ZbikRk*~Zä2Þ_l�kR`Zå�`yà�Û*à�lxä2[2kèîkg�,�ñ�ß�ÞZ[�l�ÛRÛ�à�l�ä2[2k�îfg��nñ�ßy`Zàl�hcä2`yÞZ`\h�Û*à�l�ä2[2kºîfg�jnñ�ç *�]Z[fä2Þ�l�hchj[�]Zbi[Dack�Û�`»á�~Zachcå·qG`yáZachc[Óà�`yá�`\Û�kRÛ*Þ�lxÛRhc[�l�Yi[�hj`Z]Zb�æ!hcajYi[2åqÓl\à�tiaz]Zb\kRaz]¼Û*ÞZ[�Û�[�à�à�lxa']�ßZ[2k*ã�[2ä2azl�hchjpéhc`y]Zb�æ!hcacY\[2å qÓlià�t\az]Zb\k�Û*Þ�lxÛQå�`Ó]Z`\Û�b\[2Û�å�[2k�Û�à�`�p\[2åá\p_l�ä2Û�acYZacÛ�ac[2k�k*~Zä Þ_lxkRY�l�ä�~�~�qTä2hc[�l\]Zaz]Zb_`yà�hzlnm.]_qG`�YZa']ZbOç

Úè]iÛ�kºÛ*Þ�l�Ûf~Zk�[_à�[�l�h æ!Û�azqG[¼k�[�l\à�ä2Þ:qG[2Û�ÞZ`yå�kºå�` ]Z`\ÛG]Z[2[2å·Û�`|tZ]Z`�m `yà hc[�l\à�]íÛ*ÞZ[byà�l\ã�Þ:Ü¢l\]Zå_ä�l\]¼Û*ÞZ~ZkR`yã�[�à�l�Û�[�`y]_tZ]Z`�m�]_l\]Zå|~�]�ty]Z`�m.]¼bZà�l\ã�ÞZk*âR]Z`Zàgä2`ZqÓã�~ZÛ�[�ä2`yqºæãZhc[2Û�[ ã�l�Û�ÞZk�ç �Rl�Û*ÞZ[�à2ßµÛ*ÞZ[2p lxhjmRlnpZkDå�[2ä2acå�[ím.ÞZacä2ÞÁ]Z[2ajbZÞyá�`yà�az]Zb·Yi[�à�Û�[2w Û�`:qG`�Y\[»Û�`á�l�k�[2å `y]Zhcp·`Z]íÛ*ÞZ[�<Zæ!Y�lxh'~Z[2ks`�ì.Û*ÞZ[_]Z[2acbyÞyá�`yà�az]Zb:Y\[�à´Û�acä2[2k�ç e§[�ì(`yà�[»qõ`�YZaz]ZbíÛ�`|Û*Þ�lxÛY\[�à�Û�[2w�ß�Û�ÞZ[»l\]iÛ�k�ä�l\]éä Þ�li]Zb\[ÓÛ*ÞZ[ <ræ!Y�l�hz~Z[¼`�ìQÛ�ÞZ[2a'à ä�~�à�à�[�]iÛfY\[�à´Û�[2w&çÓe§`\Û�Þ {}[�l\à�]Zaz]Zb��[�l�h æ�ë�azqG[_Únm¸l\]ZåtqR`Zå�[ �Ý`y~�]\Û�a']Zbê÷&à�k�Û�å�[2ä2acå�[_m.ÞZacä2Þ:[2å�b\[sÛ*ÞZ[¸li]\Û�k�k�ÞZ`y~Zhcå·Û*à�lnæ

Page 5: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� �

Y\[�à´k�[Raz]¼Û*ÞZ[2azà§ä�~�à�à�[�]iÛgY\[�à�Û�[2wêÜ¢{�az]Z[ g\â6ç¾ë,ÞZ[2pÓhc`y`ytº`y]Z[�[2å�b\[�Û*à�lnY\[�à�k�l�hrliÞZ[�l�å ܦh'lià�b\[�àhc`Z`yt�æ�liÞZ[�l�å�kèlià�[fã�`\k�k�azáZhj[�â�l\]Zåíl�hcmÝlnpZk�byà�[2[2å�achcp|hc[2ÛRÛ*ÞZ[Gl\]\Û�kQÛ�à�lnY\[�à�k�[�Û*ÞZ[�[2å�b\[ºÛ*Þ�l�Ûhc[�l�å�k§Û�`fl�]Z[2acbyÞyá�`yà�az]ZbfYi[�à�Û�[2wfm,ajÛ�ÞÓlRk�qÓl�hchc[2k�Û <ræ!Y�l�hz~Z[ÓÜ�Û�ac[2k�ä�l\]Óá�[Ýá�à´`yt�[�]Ól\à�áZajÛ�à�l\à$æachcprâ6çQëRÞZ[�]�ßOÛ*ÞZ[2p|~�ã�å�l�Û�[�Û*ÞZ[W<Zæ!Y�lxh'~Z[G`�ì�Û*ÞZ[2azà,ä�~�à�à�[�]iÛRY\[�à�Û�[2w_~Zk�az]Zb_lGY�l�hz~Z[�æ�~�ã�å�l�Û�[à�~Zhc[ºÛ*Þ�l�Û�å�[�ã�[�]Zå�k�`y]sÛ*ÞZ[�k�[�qÓl\]iÛ�acä2kR`�ì�Û*ÞZ[:<Zæ!Y�l�hz~Z[2kèl\]Zå_Û�Þy~Zk�Û*ÞZ[fà�[�lxhæ!Û�a'qõ[�k�[�l\à�ä2ÞqG[2Û�ÞZ`yå·Ü({}a']Z[Fjiâ2ç�ò:[èlxk�k*~�qG[�Û*Þ�l�Û�Û*ÞZ[�l\]iÛ�kg[2wZ[2ä�~ZÛ�[f{�az]Z[2kXgDli]Zå jGl�Û�`ZqGacä�l�hchjp:ܨì(`yà[2w�l\qÓãZhc[\ß�á�[2ä�li~Zk�[GÛ*ÞZ[Óá�`Zå�p|`�ìgÛ*ÞZ[Gl\]iÛ.ä2`�Y\[�à�k,Û*ÞZ[Gä�~�à�à�[�]iÛ.hc`Zä�l�Û�ac`y]íl\]Zå|acÛ�kf]Z[2acbyÞiæá�`yà�k*â2ç $�az]�l�hchcp\ßgá�`\Û�Þ à´[�l�h æ!Û�azqG[Ók�[�l\à´ä Þ:qõ[2Û*ÞZ`yå�k�hc[2ÛºÛ*ÞZ[Ól\]\Û�k�Û*à�lnY\[�à�k�[ºÛ*ÞZ[Ók�[2hc[2ä2Û�[2å[2å�b\[ÙÜ({}a']Z[ hyâ2ßR~�ã�å�lxÛ�[:Û*ÞZ[íä�~�à�à�[�]iÛ¼Y\[�à�Û�[2wIÜ({�az]Z[9r\â2ß.li]Zå acÛ�[�à�l�Û�[|Û�ÞZ[ ã�à´`yä2[2å�~�à�[Ü({}a']Z[Óð\â6ç¾e§`\Û�Þ_à�[�l�h æ!Û�azqG[fk�[�l\à�ä2Þ_qG[2Û*ÞZ`Zå�k.ä�l\]_á�[f[2wZ[2ä�~ZÛ�[2å áip_l\]\Û�kRmRacÛ*ÞZ`y~ZÛ.qõ[�qºæ`yà´p\ß�à�[2ór~Zazà�[s`y]Zhcp l¼Y\[�à�péhca'qõajÛ�[2å:hc`y`ytxæ�l\ÞZ[�l�å l\]Zåíä2`yqÓã�~ZÛ*l�Û�aj`Z]�l�h4ä�liã�l\áZachjacÛ�ac[2k�ߧl\]Zåä�l\]íá�[õ~Zk�[2å áip|k�az]Zb\hc[Ól\]iÛ�kèlxk.m�[2hch¾lxk.byà´`y~�ãZk.`xìQl\]\Û�kRÛ*Þ�l�Û�k*Þ�l\à�[ºÛ*ÞZ[ <ræ!Y�l�hz~Z[2kGá�~ZÛå�`|]Z`\Û�å�azà�[2ä2Û�hjpÃä2`yqÓqG~�]Zacä�l�Û�[GmRacÛ*Þí[�l�ä2Þé`\Û*ÞZ[�à2çºëRÞZ[2k�[»ã�à�`Zã�[�à�Û�ac[2kGqÓl�Û�ä2ÞéÛ*ÞZ[GhjazqºæacÛ�[2å_k�[�]Zk�a']ZbÓl\]Zå¼ä2`yqÓã�~ZÛ�l�Û�ac`y]�l�hOä�l\ã�l\áZachcajÛ�aj[2k�`�ì¾li]\Û�knç &!]¼Û�ÞZ[Rì(`\hchc`�m,a']ZbOßym�[Rå�[2k�ä�à�azá�[{�[�lià�]Zaz]Zb ��[�lxhæ�ëÝa'qõ[DÚnmDl\]Zå q,`Zå�[ ��`y~�]iÛ�az]ZbÓaz]_å�[2Û�l�ach¦ç

������������ ������������������������ �"!ØO���#£��!��#�®\�����(�g½}£���g���ÒZ½�¹��¾ÔÝ-zÒ\�#��� � �#�:�Z�)�£n !�Q/�1

$&% "('*) +�,�- $&% " $/.0.1% "32 %�'�'54{�[�lià�]Zaz]Zb ��[�l�h æ�ë�azqG[GÚFm|Ü({ ��ë�ÚFm�â,ä�l\]éá�[Gå�[2k�ä�à�azá�[2å l�k�ì(`\hchj`�mRknç.ëRÞZ[W<Zæ!Y�lxh'~Z[

<�Ü$ �â�`�ì�Yi[�à�Û�[2w ,l\ã�ã�à�`�wyazqÓlxÛ�[2k4Û�ÞZ[Qk�~�q `�ì&Û*ÞZ[�k*qsl�hchj[2k�Û+<Zæ!Y�lxh'~Z[�`�ì&l\]ipfk�Û*l�Û�[�l\]ZåGÛ*ÞZ[å�ack�Û*li]Zä2[Rì!à�`yq Y\[�à�Û�[2w8 �Û�`fl ä2hj`ik�[2k�Û�Y\[�à�Û�[2wGmRacÛ*Þ¼l�k�qÓl�hchc[2k�ÛI<Zæ!Y�lxh'~Z[\ç§ë�`Ó~�]Zå�[�à�k�Û�l\]ZåÛ*ÞZ[ék�[�qÓl\]iÛ�acä2k¼`�ìfÛ*ÞZ[k<ræ!Y�l�hz~Z[2k�ß,ä2`y]Zk�acå�[�à¼Û*ÞZ[G<Zæ!Y�lxh'~Z[2k|~�]iÛ�ach�Û*ÞZ[|byà�l\ã�ÞÙÞ�l�k_á�[2[�]ä2`�Y\[�à�[2å ì¦`yà Û*ÞZ[G÷&à�k�ÛfÛ�azqG[\ç%&!]ÃÛ*ÞZackfä�lxk�[\ß�l_Y\[�à�Û�[2w m,ajÛ�Þ9<Zæ!Y�l�hz~Z[_o2[�à�`|ä2`yà�à�[2k*ã�`Z]Zå�kÛ�`:l\] ~�]iYyack�acÛ�[2åLY\[�à�Û�[2w ܦÛ�` á�[|ã�à�[2ä2ack�[��|léY\[�à�Û�[2wíÛ*Þ�lxÛõÞ�lxkõ]Z[2Yi[�àÓá�[2[�] hc[�ì(Û»á\p l\]l\]iÛ*â2çGëRÞZ[ <ræ!Y�l�hz~Z[�<4Ü$ nâ�`�ìRY\[�à�Û�[2wG ÓÛ�ÞZ[�] liã�ã�à�`�wyazqÓl�Û�[2kfÛ*ÞZ[Gå�ack�Û*l\]Zä2[sì!à�`yq Y\[�à�Û�[2w éÛ�`Álíä2hc`\k�[2k�Û¼Y\[�à�Û�[2wÁmRacÛ*ÞZ<ræ!Y�l�hz~Z[:o2[�à�`OßgÛ*Þ�lxÛ»ack�ßRlÃä2hc`\k�[2k�Û¼Y\[�à�Û�[2w Û*Þ�l�Û_Þ�l�k»]Z`iÛp\[2Ûfá�[2[�]íYZajk�ajÛ�[2å�çD{ ��ë�ÚFm_l�hcmÝlnpZkfqG`�Y\[2k�Û*ÞZ[Ól\]\Û�k.Û�`_l_]Z[2acbyÞZá�`yà�az]ZbíY\[�à�Û�[2w_mRacÛ*Þílk*qsl�hchj[2k�Û <Zæ!Y�l�hz~Z[_Û�` à�[�lxä Þ léY\[�à�Û�[2w:Û�Þ�l�ÛDÞ�l�kÓ]Z`\Ûºp\[2ÛÓá�[2[�] YZack�acÛ�[2å�ßgÛ�`|[2Y\[�]iÛ*~�l�hchcpYZack�acÛ.l�hch�Y\[�à�Û�acä2[2kRl\]Zå¼ä2`�Yi[�àgÛ�ÞZ[.byà�l\ã�Þ�ç�{ �4ë�Únmfk�[2Û�k�Û*ÞZ[�]Z[2mD<Zæ!Y�lxh'~Z[º`�ì�Û*ÞZ[�ä�~�à�à�[�]iÛY\[�à´Û�[2w_Û�`Ó`y]Z[GãZhz~Zk�Û*ÞZ[DqGaz]ZazqG~�qô`xìgÛ*ÞZ[ <ræ!Y�l�hz~Z[2k�`�ì�acÛ�kè]Z[2acbyÞZá�`Zà�az]Zb|Y\[�à�Û�ajä2[2knçRëRÞZ[à�[�lxk�`y]êì¦`ZàRÛ*ÞZack�Y�l�hz~Z[�æ�~�ã�å�l�Û�[_à�~Zhc[ Ü�Û�Þ�l�Û.ä�li]|[�lxk�achjp:á�[Dbi[�]Z[�à�l�hcaco2[2å·Û�`¼byà�liã�ÞZk�mRacÛ*Þ]Z`y]iæ�~�]Za ì(`yà�q [2å�bi[»ä2`\k�Û�k*â�ackÓÛ�Þ�l�ÛfÛ*ÞZ[¼k*ÞZ`yà´Û�[2k�ÛÓã�l�Û*ÞÃÛ�` léä2hc`\k�[2k�Ûõ~�]iYZajk�ajÛ�[2åÙY\[�à�Û�[2wÞ�l�kGÛ�`éb\`|Û�Þ�à�`y~ZbyÞÃ`y]Z[Ó`�ì�Û*ÞZ[¸]Z[2acbyÞZá�`Zà�az]Zb:Y\[�à�Û�acä2[2k�çíëRÞZ~Zk�ß�l\] l\ã�ã�à�`�wyazqÓlxÛ�ac`y]:`xìÛ*ÞZ[�å�ack�Û�l\]Zä2[�ì!à�`yq Û�ÞZ[�ä�~�à�à�[�]iÛRY\[�à�Û�[2wsÛ�`»lGä2hc`\k�[2k�Û.~�]\YZack�acÛ�[2å|Yi[�à�Û�[2w¼ajk�`y]Z[fãZhz~Zk�Û*ÞZ[qGaz]ZazqG~�qï`xìèÛ*ÞZ[_[2k�Û�azqÓl�Û�[2kÓ`xìèÛ*ÞZ[_å�ack�Û�l\]Zä2[2kÓì!à�`yq Û*ÞZ[|]Z[2acbyÞyá�`yà�az]Zb Y\[�à�Û�acä2[2k¼Û�`:lä2hc`\k�[2k�Ûè~�]iYyack�acÛ�[2åéY\[�à�Û�[2w�ç

{ ��ë�ÚFm_mRl�k�÷&à´k�ÛGå�[2k�ä�à�azá�[2åÁaz]Æî�,�ð�ñÝl\]Zå ackõã�à´`yá�l\áZhcp:Û*ÞZ[_qG`\k�ÛGã�`yã�~Zhzl\àsà�[�l�h æÛ�azqG[_k�[�l\à�ä2Þ:qG[2Û*ÞZ`Zå�ç &(ÛGackÓ`�ì(Û�[�] ~Zk�[2å·Û�`|÷&]ZåÙã�lxÛ*ÞZkGÛ�` léb\`ylxhgY\[�à�Û�[2w `Z]·tZ]Z`�m.]`yà�~�]�tZ]Z`�m�]|bZà�l\ã�ÞZk�ç &!]éÛ*ÞZack�ä�l�k�[\ß�`y]Z[Gä�l\]êaz]ZajÛ�a'lxhjaco2[_Û*ÞZ[W<Zæ!Y�l�hz~Z[Ó`�ìQl¼Y\[�à�Û�[2w_mRacÛ*Þl\]_liã�ã�à�`�wyazqÓl�Û�aj`Z]¼`�ì¾acÛ�k,b\`yl�hOå�ajk�Û*l\]Zä2[\ßOm.ÞZacä Þ¸ì¦`Zä�~Zk�[2kRÛ*ÞZ[�k�[�l\à�ä2Þ¼Û�`�mÝlià�å�kgÛ*ÞZ[�bi`yl�hY\[�à´Û�acä2[2k�ç�$&~�à�Û*ÞZ[�à�qG`yà�[\ßna ì�{ �4ë�ÚFmRackgl�hcmRlnpyk�à�[2k�Û�l\à�Û�[2åºm.ÞZ[�]ºajÛgà´[�l�ä ÞZ[2k�l�b\`ylxh\Y\[�à�Û�[2w�ß

Page 6: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

1 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

1 1 0 0 0

1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

1 1 1 0 0

1 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 2 0 0

1 1 1 0 0

1 1 0 0

0 1 0 0 0 0

0 0 0 0 0 0

1 1 2 0 0

1 1 2 0 0

1 1 0 0

0 1 1 0 0 0

0 1 0 0 0 0

1 1 2 0 0

1 2 2 0 0

1 1 0 0

0 1 1 0 0 0

1 1 1 0 0 0

Three Ants

0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

0 1 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

0 1 1 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 2 0 0

0 1 1 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 2 0 0

0 1 2 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 2 0 0

0 1 2 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

One Ant

�Z�)�£� !� � ¡O��°2���g®n����&ÒZ½�¹��¾Ô

Û*ÞZ[�] acÛÓmRachchR[2Y\[�]iÛ*~�l�hchcp:ì(`\hchc`�m lék*ÞZ`yà�Û�[2k�ÛDã�l�Û*Þ Û�` léb\`Zl�hgY\[�à´Û�[2w&ç &$] Û*ÞZack»l\à�Û�ajä2hc[\ßÞZ`�mQ[2Yi[�à2ß�m�[Ól\ã�ãZhcp·{ ��ë�Únm»Û�`_byà�l\ã�Þíä2`�Y\[�à�l�b\[\ç $�ajbZ~�à�[rj_å�[�qõ`y]Zk�Û*à�lxÛ�[2kfÞZ`�m l\]\Û�kÛ*Þ�l�ÛR~Zk�[G{ �4ë�Únmfä2`�Y\[�àRlGà�[2by~Zhzl\à§ì(`y~�à$æ!ä2`Z]�]Z[2ä2Û�[2å|byà´ajå�ç§ëRÞZ[fl\]iÛ�k�ä�l\]_qG`�Y\[�Û�`G[�l�ä Þ`�ìRÛ*ÞZ[sì¦`y~�àº]Z[2acbyÞyá�`yà�az]Zb:ä2[2hchckÓ`�ìRÛ*ÞZ[2azàºä�~�à�à�[�]iÛGä2[2hjhRã�à�`�YZacå�[2å·Û*Þ�lxÛfÛ*ÞZ[Óå�[2k�Û�az]�l�Û�ac`y]ä2[2hch�ajk�Û�à�lnY\[�à�k*l\áZhc[_Ü�m.ÞZacÛ�[�â2ç�ëRÞZ[Ól\]iÛ�k.qG`�Y\[ºaz] l¼biajYi[�]|k�[2ór~Z[�]iÛ�azl�h�`yà�å�[�àGÜ(l�hcÛ*ÞZ`y~ZbyÞÛ*ÞZackÓackÓ]Z`\ÛÓ]Z[2ä2[2k�k*l\à�pÃa'] bi[�]Z[�à�l�hzâ2ç:ëRÞZ[¼ä2[2hchjk_l\à�[_qÓl\à�t�[2å:mRacÛ*Þ Û*ÞZ[2azàW<Zæ!Y�l�hz~Z[2k�ç'&¦ìl¼ä2[2hchgä2`y]iÛ*l�az]ZkDli] li]\Û�ß�`Z]Z[Ó`�ìQacÛ�kºä2`yà�]Z[�à�k�ackGqÓl\à�t�[2å�ç ��a/#�[�à�[�]\Û�ä2`Zà�]Z[�à�kfà�[�ã�à�[2k�[�]\Ûå�a/#}[�à´[�]\Û�l\]\Û�k�ç $�acby~�à�[_jÓÜ�Û�`yã�â�å�[�qG`y]Zk�Û�à�l�Û�[2k�ÞZ`�mÁk�az]Zb\hc[.l\]iÛ�k�Û*Þ�l�Û�~Zk�[.{ ��ë�ÚFm�ä2`�Yi[�àÛ*ÞZ[¼byà�acå�ß4l\]Zå $�acby~�à�[rj Ü(á�`iÛ�Û�`yqÓâ�å�[�qõ`y]Zk�Û*à�lxÛ�[2kfÞZ`�m Û*Þ�à�[2[¼l\]\Û�k�Û*Þ�l�ÛD~Zk�[»{ ��ë�Únmä2`�Y\[�à�acÛ�ç

��� �������������� � �0�����ØO���#£��$��#�®\�n�*�(�§½�£���§���Õ�2���gÐ��£������#��)�-zÒi�#��� � �#�:�Z�)�£� !��/*1

$&% "('*) +�, - $&% "('54ò:[¼ä2`yqÓã�l\à�[_{ ��ë�Únm|Û�`3qR`Zå�[%��`y~�]iÛ�az]Zb�ß�Û*Þ�l�Ûºä�l\] á�[_å�[2k�ä�à�azá�[2å l�kºì(`\hchj`�mRknç

ëRÞZ[�<Zæ!Y�l�hz~Z[�<4Ü$ nâº`�ì.Y\[�à�Û�[2wF ¼ä2`yà�à�[2k�ã�`y]Zå�kÓÛ�`|Û*ÞZ[_]y~�qGá�[�àG`�ì.Û�azqG[2kGY\[�à´Û�[2wF _Þ�l�ká�[2[�]íYZack�acÛ�[2å:á\p|li]\Û�knçRë�`_~�]Zå�[�à�k�Û*li]Zå Û�ÞZ[fk�[�qÓli]\Û�acä2k�`�ì�Û*ÞZ[8<Zæ!Y�l�hz~Z[2k�ß�ä2`y]Zk�acå�[�à�Û*ÞZ[<Zæ!Y�l�hz~Z[2kè~�]iÛ�ach�Û*ÞZ[.bZà�l\ã�Þ¼Þ�l�kRá�[2[�]¼ä2`�Y\[�à�[2åsì¦`Zà�Û*ÞZ[,÷&à�k�Û�Û�azqG[\ç�&$]¼Û*ÞZack�ä�l�k�[\ßZY\[�à�Û�acä2[2kmRacÛ*Þ9<ræ!Y�l�hz~Z[2kfo2[�à´`_ä2`yà�à�[2k*ã�`y]ZåíÛ�`|~�]iYyack�acÛ�[2å:Yi[�à�Û�acä2[2k»Ü�Û�`_á�[»ã�à´[2ä2ajk�[���Y\[�à�Û�acä2[2k�Û*Þ�lxÛ

Page 7: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� �

Three Ants

One Ant

0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

3 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

3 1 0 0 0

2 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

3 1 0 0 0

2 2 0 0 0

1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

3 1 0 0 0

2 2 1 0 0

1 2 0 0

0 0 0 0 0 0

0 0 0 0 0 0

3 1 1 0 0

2 2 1 0 0

1 2 0 0

0 2 0 0 0 0

0 0 0 0 0 0

3 1 1 0 0

2 2 2 0 0

1 2 0 0

0 2 1 0 0 0

0 1 0 0 0 0

3 1 2 0 0

2 2 2 0 0

1 2 0 0

0 2 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

0 1 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0

0 1 1 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 1 0 0

0 1 1 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 1 0 0

0 1 2 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 1 0 0

0 2 2 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

�Z�)�£� !� � ¡��O°2�*�§®������Õ�2���gÐ��£������#��)

Þ�lnY\[|]Z[2Y\[�à¼á�[2[�]Lhc[�ì(Û¸áipÙl\]Ál\]iÛ*â2ç qR`Zå�[ �Ý`y~�]\Û�a']ZbÁl�hcmRlnpyk»qõ`�Yi[2kÓÛ*ÞZ[|l\]iÛ�kÓÛ�` l]Z[2acbyÞZá�`yà´a']Zb Y\[�à�Û�[2w·mRacÛ*ÞÁl k*qÓl�hchc[2k�Û <ræ!Y�l�hz~Z[íÛ�`:à�[�l�ä2ÞÁléY\[�à�Û�[2w Û*Þ�l�ÛÓÞ�l�k»]Z`iÛÓp\[2Ûá�[2[�]_YZajk�ajÛ�[2å�ß�Û�`G[2Y\[�]iÛ*~�l�hchcp_YZajk�ajÛ�l�hch�Y\[�à�Û�acä2[2k.li]Zå_ä2`�Yi[�àgÛ�ÞZ[�byà�l\ã�Þ�ç'q,`Zå�[���`y~�]iÛ�az]Zbk�[2Û�k»Û*ÞZ[í]Z[2m <ræ!Y�l�hz~Z[í`�ìfÛ*ÞZ[éä�~�à�à�[�]iÛ»Yi[�à�Û�[2w Û�`·`Z]Z[ ãZhz~Zk_Û*ÞZ[|`ihjå�<Zæ!Y�l�hz~Z[ `xìfÛ*ÞZ[ä�~�à�à´[�]\ÛRYi[�à�Û�[2w:Ü(qG[�l\]Zaz]ZbÓÛ*Þ�lxÛRÛ*ÞZ[�Y\[�à�Û�[2w¸Þ�l�k�á�[2[�]_YZack�acÛ�[2å:l\]Z`\Û*ÞZ[�àÝÛ�azqG[�â2ç

ögl\à´a'li]\Û�k�`�ì_qR`Zå�[ ��`y~�]iÛ�az]Zb|Þ�l�Yi[Dá�[2[�]Ã~Zk�[2åía'] Û*ÞZ[GhjacÛ�[�à�l�Û*~�à�[GÛ�`¼[2wrãZhc`yà´[õ~�]iætZ]Z`�m.]:Û�[�à�à�l�az]�ß�[2acÛ*ÞZ[�àG`Z]:Û*ÞZ[2azàG`�m.] î�,�enñ�`yàºÛ�`:l�ä2ä2[2hc[�à�l�Û�[_à�[2az]iì¦`Zà�ä2[�qG[�]iÛ�æ!hc[�l\à�]Zaz]ZbqG[2Û�ÞZ`yå�k_îfg�o�ñ�ç qR`Zå�[%��`y~�]iÛ�az]Zb|ackÓl�hck�`|Û*ÞZ[õì¦`y~�]Zå�lxÛ�ac`y]:`xì5L�Ú.Y\`iajå�az]ZbíÛ*ÞZ[rl�lxk�Û���ÚXZazqÓãZhc[»á�~ZÛ�� #}[2ä2Û�ajYi[DXZÛ*à�lxÛ�[2b\p¼ì(`yà �Ý[�l�ä2Û�acY\[_î ��`yá�`\Û�ñ1q�lnYyacbylxÛ�ac`y]QN îkgnñ�ç &¢Û.ackfã�à�`Zá�lnæáZhcp_[�l�k�ac[�à�Û�`DazqÓãZhc[�qG[�]iÛ�l\]\Û�kRÛ*Þ�l�ÛR~Zk�[Fq,`yå�[��Ý`y~�]\Û�a']ZbsÛ*Þ�l\]_l\]iÛ�k�Û*Þ�l�ÛR~Zk�[f{ ��ë�ÚFmá�[2ä�l\~Zk�[Fq,`yå�[��`y~�]iÛ�az]ZbÓl�hcmÝlnpZkgaz]Zä�à�[�l�k�[2kRÛ*ÞZ[;<Zæ!Y�lxh'~Z[�`xì4Û*ÞZ[�ä�~�à�à�[�]iÛQYi[�à�Û�[2wÓá\psÛ*ÞZ[k*liqG[fä2`y]Zk�Û*l\]iÛ�ß�m�ÞZ[�à�[�l�k�{ �4ë�ÚnmÓa']Zä�à´[�l�k�[2k�Û*ÞZ[8<Zæ!Y�l�hz~Z[Ó`�ì�Û*ÞZ[Gä�~�à�à�[�]iÛ.Yi[�à�Û�[2w|áip|lY�l�hz~Z[fÛ*Þ�l�Û�å�[�ã�[�]Zå�k�`y]¼Û*ÞZ[:<Zæ!Y�l�hz~Z[2k.`xì¾Û*ÞZ[f]Z[2acbyÞZá�`Zà�az]Zb»Y\[�à´Û�acä2[2k�ç $�acby~�à�[XhÓå�[�qõ`y]iæk�Û�à�l�Û�[2kÓÞZ`�m l\]iÛ�kGÛ*Þ�l�ÛÓ~Zk�[3qR`Zå�[ ��`y~�]iÛ�az]Zb:ä2`�Y\[�àÓlíà�[2by~Zhzl\àDì¦`y~�à�æ!ä2`y]�]Z[2ä2Û�[2åÙbZà�acå�ç$�acby~�à�[ h·Ü�Û�`yã�â,å�[�qG`y]Zk�Û�à�l�Û�[2k�ÞZ`�mIk�az]Zb\hc[¸l\]iÛ�k�Û*Þ�l�Ûf~Zk�[ q,`yå�[.��`y~�]iÛ�az]Zb_ä2`�Yi[�à�Û*ÞZ[byà´ajå�ß\li]Zå $�acby~�à´[ hÓÜ(á�`\Û�Û�`yqsâ�å�[�qG`y]Zk�Û�à�l�Û�[2k�ÞZ`�m·Û*Þ�à´[2[Ýli]\Û�k�Û*Þ�l�Û�~Zk�[_qR`Zå�[���`y~�]iÛ�az]Zbä2`�Y\[�à�acÛ�ç

Page 8: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

��� � ����:����d o=����d� ���SqZcedf�gih(�:�EjkdmlndQ�oSp�q

&!]êÛ*ÞZack.k�[2ä2Û�aj`Z]�ß�m�[fk�ÞZ`�m Û*Þ�l�Û.l\]iÛ�kRÛ�Þ�l�Û.~Zk�[G{ �4ë�Únmõl\]Zåíl\]iÛ�kRÛ*Þ�l�Û�~Zk�[ q,`yå�[��`y~�]iÛ�az]Zb:ä2`�Y\[�à_Ü�k�Û*à´`y]Zb\hcp:ä2`y]�]Z[2ä2Û�[2å�âõbyà�l\ã�ÞZkÓà�[�ã�[�l�Û�[2å�hcpÙl\]ZåÁÛ*ÞZ~ZkÓl�Yi`\acåÁä2pyä2hcaz]Zbì¦`Zà�[2Y\[�à�az] ã�l\à�Û�kº`�ìRÛ*ÞZ[¼byà�l\ã�ÞZk�ß�å�[2k*ãZacÛ�[_Û*ÞZ[2azàºhca'qõajÛ�[2åÁk�[�]Zk�az]Zb:l\]Zå·ä2`ZqÓã�~ZÛ*l�Û�ac`y]�lxhä�l\ã�l\áZachcacÛ�ac[2k�çíò:[s÷&à�k�Ûfk�ÞZ`�m Û*Þ�l�Ûfl\]iÛ�kºÛ*Þ�l�ÛD~Zk�[»{ ��ë�Únm|ä2`�Y\[�à�byà�liã�ÞZkDà�[�ã�[�l�Û�[2å�hcp\ßì¦`ihjhc`�mRaz]ZbÁî�,�enñgl\]ZåÙîkg�gnñ�çGëRÞZackfä�l\]Ãá�[Ók*ÞZ`�m.]íáip:l�k�k*~�qGaz]ZbéÛ*Þ�l�Û�ß�l�Û�k�`yqG[Óã�`\az]iÛfaz]Û�azqG[\ßOÛ*ÞZ[2p_å�`»]Z`\ÛRä2`�Y\[�àÝÛ*ÞZ[fbyà�l\ã�ÞZkDÜ(l�bZl�az]�â2çQë,ÞZ[�]�ß�Û*ÞZ[�à´[�ajk�k�`yqG[»Ü(ã�`\k�k�azáZhcp|hzl�Û�[�à�âã�`\az]iÛ�az]íÛ�azqG[Gm.ÞZ[�]éÛ*ÞZ[2p_`y]Zhcp|YZack�acÛfÛ*ÞZ`ik�[fY\[�à´Û�acä2[2kfl�byl�az]_Û�Þ�l�Û.Û�ÞZ[2p|YZajk�ajÛ�az]i÷&]ZacÛ�[2hcp`�ì(Û�[�]��Û�ÞZ[2p|ä2pZä2hj[G`y]éã�l\à�Û�`�ìgÛ*ÞZ[ºbyà�l\ã�ÞZk�çRë,ÞZ[5<ræ!Y�l�hz~Z[2k�`�ìQlxhjh�Y\[�à�Û�ajä2[2k�az]éÛ*ÞZ[ºä2pyä2hc[Û*ÞZ[�]¼az]Zä�à�[�l�k�[fá�[2p\`y]Zåé[2Y\[�à�p»á�`y~�]Zå_k�az]Zä2[D{ ��ë�ÚFmfaz]Zä�à�[�lxk�[2kRÛ*ÞZ[�k*qsl�hchj[2k�Û2<Zæ!Y�l�hz~Z[f`�ìÛ*ÞZ[�Y\[�à�Û�acä2[2k�az]ÓÛ*ÞZ[�ä2pZä2hj[fáip»l�Û�hc[�l�k�ÛQ`Z]Z[.[2Y\[�à´pDÛ�a'qõ[èl\]¼l\]iÛ�kghc[�lnY\[2k�Û*ÞZ[RYi[�à�Û�[2w�ç¾e�~ZÛÛ*ÞZ[�]sÛ*ÞZ[2<Zæ!Y�l�hz~Z[2k�`�ì}l�hchOY\[�à�Û�acä2[2k�az]GÛ*ÞZ[Rä2pZä2hc[.az]Zä�à�[�l�k�[èl\á�`�Y\[RÛ*ÞZ[2<ræ!Y�l�hz~Z[2k�`�ì4l�hch�Y\[�à$æÛ�acä2[2k�Û*Þ�l�Ûgá�`Zà�å�[�à�Û*ÞZ[,ä2pyä2hc[\çgX�~Zä2ÞÓYi[�à�Û�acä2[2kg[2wZack�ÛQl�ä2ä2`Zà�å�az]ZbfÛ�`�`y~�à§l�k�k*~�qsãZÛ�ac`y]ZkgÛ*Þ�lxÛÛ*ÞZ[Óbyà�l\ã�Þíackfk�Û�à�`y]Zb\hcp|ä2`Z]�]Z[2ä2Û�[2å á�~ZÛºmRachchQ]Z`\ÛGá�[Óä2`�Y\[�à�[2å l�byl�az]�çÓëRÞZ[�]�ß�ÞZ`�m�[2Y\[�à2ßl�Û�hj[�lxk�Û.`Z]Z[Dl\]iÛ.ack�ì(`yà�ä2[2åéÛ�`Óhc[�lnY\[GÛ*ÞZ[fä2pZä2hc[\ß�m.ÞZacä ÞéackflÓä2`Z]\Û*à�l�å�acä2Û�ac`y]�ç.ëRÞZ[Gk*liqG[l\à�by~�qõ[�]\Û�l�hck�`fl\ã�ãZhcac[2kR~�]Zä Þ�l\]Zbi[2åDÛ�`èl\]iÛ�k�Û*Þ�l�Û�~Zk�[dq,`yå�[���`Z~�]\Û�az]Zb�ç�ëRÞZackgl\à´by~�qG[�]iÛÞZ`\hcå�kf]Z`¸qsl�Û�Û�[�à�az]êm.ÞZacä Þé`yà�å�[�à,Û*ÞZ[Gl\]\Û�k.qG`�Y\[�[2Y\[�]êaìgk�`yqG[Dli]\Û�k�qG`�Y\[�hc[2k�k�`�ì(Û�[�]Û*Þ�l\]¼`\Û�ÞZ[�à�k�ç�ëRÞZ~Zk�ß�l\]\Û�kgÛ*Þ�l�ÛR~Zk�[f{ �4ë�ÚFmGl\]Zå|l\]iÛ�kgÛ*Þ�lxÛÝ~Zk�[FqR`Zå�[���`y~�]iÛ�az]ZbGä2`�Yi[�àbyà�l\ã�ÞZkfà´[�ã�[�l�Û�[2å�hcp:l\]Zåíä�l\]íá�[Ó~Zk�[2å ì¦`yà,`y]Z[�æ!Û�azqG[s`yàRä2`Z]\Û�az]Z~Z`y~Zk�Y�lxä�~�~�q ä2hc[�l\]Zaz]Zb`yà,h'lnm.]éqG`�mRaz]Zb�ç $&~�à�Û*ÞZ[�à�qG`yà�[\ß�Û�ÞZ[2p|Þ�l�Yi[fl�å�Y�l\]iÛ*l�bi[2kR`�Y\[�à�`\Û*ÞZ[�à,k�[�l\à´ä Þ_qG[2Û�ÞZ`yå�kÛ*Þ�l�Û�l�hck�`�ä2`�Yi[�à�byà�l\ã�ÞZk�à�[�ã�[�l�Û�[2å�hcp\ç $�`Zà�[2w�l\qÓãZhc[\ß�Û*ÞZ[2pfl\à´[�ì!l\à4qõ`yà�[�k�pZk�Û�[�qÓlxÛ�acägÛ*Þ�l\]à�l\]Zå�`yq mRl�hzt\kgl\]Zå�ßiå�a/#}[�à´[�]\Ûµì¢à�`Zq ä Þ�à�`Z]Z`\hc`\b\acä�l�h�á�l�ä2t\Û�à�l�ä2t\az]Zb»Ü�å�[�ãZÛ*Þiæ(÷&à´k�Ûgk�[�lià�ä Þ�â6ßä�l\]_á�[�k*~Zk*ã�[�]Zå�[2å l\]Zå_à�[2k�Û�l\à�Û�[2å¼[2hck�[2m.ÞZ[�à�[\ß�m,ajÛ�ÞZ`y~ZÛ�[2Y\[�]»tZ]Z`�mRaz]ZbGm.ÞZ[�à�[�Û*ÞZ[2pÓbi[2Ûà�[2k�Û*lià�Û�[2å�ç�ëRÞZack�ajk,a'qsã�`yà�Û�l\]\Ûµì¦`Zà�byà�liã�ÞÓä2`�Y\[�à�lxb\[�mRacÛ*Þ_l\]iÛ�kQá�[2ä�l\~Zk�[�k�`yqG[2Û�azqG[2k�Û*ÞZ[l\]iÛ�kèqõajbZÞ\Û�b\[2Û�ã�~Zk*ÞZ[2å·lxä2ä2ajå�[�]iÛ*l�hchcpíÛ�`_l¼å�a #�[�à�[�]iÛ�Y\[�à�Û�[2w&ç í`\k�ÛR`xìgÛ*ÞZ[GÛ�azqG[GÛ*ÞZ[2pmRachch¾]Z`\Û,[2Y\[�]_à�[�l�hcaco2[fÛ*ÞZack�ç§Ú�]\Û�k�Û*Þ�lxÛ.~Zk�[�[2acÛ*ÞZ[�à�{ �4ë�ÚnmD`yàYq,`yå�[ ��`y~�]iÛ�az]Zb»Þ�l\]Zå�hc[Û*ÞZ[2k�[ºk�acÛ*~�l�Û�ac`y]Zk.li~ZÛ�`yqÓl�Û�ajä�lxhjhcp\ç

�[� �:����d oW� �$jGdG�� ;c8dQ�g�h(� �$jGdCl�df�oSp@q�� dQ�Sqn���

ëRÞZ[Rä2`�Y\[�à�Û�azqG[,ajk§Û*ÞZ[RÛ�azqG[,ajÛ�Û�l\t�[2kg~�]iÛ�ach&l\]iÛ�k�ä2`�Y\[�à�l�byà�liã�Þ�ì(`yà�Û*ÞZ[Ý÷&à�k�Û�Û�azqG[\ßÛ*Þ�l�Û�ack�ß�Yyack�acÛfl�hch�`�ìgacÛ�k�Y\[�à�Û�ajä2[2k�l�Û�hc[�l�k�Û�`y]Zä2[_Ü�acÛ�å�`y[2k�]Z`\Û�qÓl�Û�Û�[�à�áip_m.ÞZacä Þíl\]iÛ.`Zàl\]iÛ�kDlêY\[�à�Û�[2wímRl�kfYZack�acÛ�[2å�â2çíëRÞZackGajkGazqÓã�`yà�Û*l\]iÛ�ì(`yà `y]Z[�æ!Û�azqG[¼Y�l�ä�~�~�q ä2hc[�l\]Zaz]Zb:`Zàhzl�m�]|qG`�mRaz]ZbÓk�az]Zä2[º[�l�ä Þêä2[2hch¾Þ�l�k,Û�`»á�[ºY�l�ä�~�~�qõ[2å|`yàRqõ`�m�[2å|l�Û�hc[�l�k�Û�`y]Zä2[\ç�&$]_Û*ÞZ[ì¦`ihjhc`�mRaz]ZbGk�[2ä2Û�ac`y]Zk�ßZm�[Rk�Û�~Zå�pÓÛ*ÞZ[Rä2`�Y\[�àµÛ�azqG[R`�ì4l\]iÛ�kgÛ�Þ�l�Ûg~Zk�[�{ �4ë�Únm li]Zå»l\]iÛ�kgÛ*Þ�lxÛ~Zk�[ q,`Zå�[ ��`y~�]iÛ�az]Zb�ß�á�`iÛ*Þ_Û*ÞZ[2`Zà�[2Û�acä�l�hchcp|l\]Zå|[2w�ã�[�à�azqG[�]iÛ*l�hchcp\ç

� ����� ���� / �������� ���� �� �0��������� �� �� ���&!]LÛ*ÞZack_k�[2ä2Û�ac`y]�ß�m�[ li]�l�hcpyo2[:Û�ÞZ[|ä2`�Y\[�àsÛ�azqG[ÙÜ(l�k_lÃì¢~�]Zä2Û�ac`y]L`�ìfÛ�ÞZ[ ]Z~�qGá�[�à

`�ìfY\[�à�Û�ajä2[2k�â¼Û*Þ�l�Û»l\]iÛ�k¼Û*Þ�l�Û_~Zk�[ { �4ë�ÚFm·li]Zå li]\Û�k¼Û*Þ�l�Û¼~Zk�[ qR`Zå�[ ��`y~�]iÛ�az]Zb·ä�l\]by~�l\à�li]\Û�[2[\çéëRÞZ[¸li]\Û�kºk*Þ�l\à�[sÛ*ÞZ[ <ræ!Y�l�hz~Z[2k»á�~ZÛGå�`:]Z`\Ûºå�a'à´[2ä2Û�hcp·ä2`yqsqD~�]Zacä�l�Û�[»mRacÛ*Þ

Page 9: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� �

[�l�ä2Þí`\Û�ÞZ[�à2ç�ò:[»lxk�k*~�qG[GÛ*Þ�l�Û�[�lxä ÞíqG`�Y\[G`�ìQl\]Ãl\]\Û�Û*l\tx[2k�`y]Z[GÛ�azqG[Ók�Û�[�ãíl\]ZåÃÛ*Þ�l�Û[�l�ä2ÞGl\]iÛ¾qG`�Y\[2k�`y]Zä2[�å�~�à�az]Zb�[�l�ä ÞºÛ�azqG[�k�Û�[�ã�ç $�`Zà�k�azqÓãZhcacä2acÛ�p\ß�m�[Ql�k�k�~�qG[gÛ*Þ�lxÛ4k�[2Yi[�à�l�hl\]iÛ�k�ä�l\]|á�[Ól�ÛRÛ*ÞZ[Gk*l\qõ[�Y\[�à�Û�[2w|l�Û�Û*ÞZ[ºk*l\qG[ºÛ�azqG[Dá�~ZÛ�Û*ÞZ[2p_ä�li]|qG`�Y\[ºä2`yqÓãZhc[2Û�[2hcpl�k�pr]Zä2Þ�à�`y]Z`y~Zk�hjp\ç�XZaz]Zä2[D{ ��ë�ÚFmÓl\]Zå q,`yå�[ ��`y~�]iÛ�az]Zb»l\à�[fl�hcb\`Zà�acÛ*Þ�qGacä�l�hchcp|k�azqGachzl\à2ßOacÛä�l\] á�[¼k*ã�[2ä�~Zh'lxÛ�[2å·Û�Þ�l�ÛDl\]iÛ�k�Û*Þ�lxÛD~Zk�[ qR`Zå�[ ��`Z~�]\Û�az]Zbíby~�l\à�l\]\Û�[2[õlêk�azqGachzl\àGä2`�Y\[�àÛ�azqG[íl�kÓl\]iÛ�kÓÛ*Þ�l�ÛÓ~Zk�[í{ ��ë�ÚnmyçÁò:[_k�ÞZ`�mTÛ*Þ�l�ÛGÛ*ÞZ[_ä2`�Y\[�àGÛ�azqG[_`xì l\]iÛ�kGÛ*Þ�l�Û¼~Zk�[{ ��ë�ÚFm:ack_by~�l\à�l\]\Û�[2[2åÙÛ�`Áá�[:ã�`ihjp�]Z`yqGazl�hºaz]ÙÛ�ÞZ[ ]Z~�qDá�[�à¼`�ìfY\[�à�Û�ajä2[2knç ë,ÞZ[�]�ßRm�[k*ÞZ`�mfß�k*~�à�ã�à�ack�az]Zb\hcp\ߧÛ*Þ�l�ÛºÛ*ÞZ[Óä2`�Y\[�à�Û�a'qõ[Ó`�ì.l\]iÛ�kfÛ*Þ�lxÛf~Zk�[ q,`Zå�[ ��`Z~�]\Û�az]Zbéä�l\]:á�[[2w�ã�`y]Z[�]iÛ�azl�h�az]·Ü¦Û*ÞZ[fk�ó�~�l\à�[Gà�`Z`\ÛR`�ì âÝÛ*ÞZ[D]Z~�qGá�[�à,`�ìgY\[�à�Û�acä2[2k�`y]|á�`\Û*Þêå�a'à´[2ä2Û�[2å:l\]Zå~�]Zå�azà�[2ä2Û�[2åÆbyà�l\ã�ÞZkGa']Zä2hz~Zå�az]Zb Ü¢ãZh'li]�l\à�âG~�]Zå�azà�[2ä2Û�[2åÆÛ*à�[2[2k�ç $�az]�l�hchjp\ß�m�[|l�Û�Û�[�qÓãZÛGÛ�`[2w�ãZhzl�az]|Û�ÞZ[få�a/#�[�à�[�]Zä2[faz]_ä2`�Y\[�àÝÛ�azqG[Dá�[2Û�m�[2[�]_l\]iÛ�kRÛ�Þ�l�Û.~Zk�[f{ ��ë�ÚFm»l\]Zå|li]\Û�k,Û*Þ�l�Û~Zk�[Fq,`Zå�[���`y~�]iÛ�az]Zbõáip_ã�à�[2k�[�]iÛ�az]Zb»lºä2`y]Zå�acÛ�ac`y]_Û*Þ�lxÛgackRk*~0(»ä2ac[�]iÛRì(`yàµÛ*ÞZ[�ä2`�Y\[�à§Û�azqG[Û�`_á�[Óã�`\hcpr]Z`yqõa'lxh¦çºÚ�h'qõ`\k�Û�l�hch4`�ìgÛ�ÞZ[õã�à´`y`�ì(k�l\à�[Dáip|az]Zå�~Zä2Û�ac`y]�çºëRÞZ[Ghj`�m�[�à�á�`y~�]Zå�kà�[2ó�~Zazà�[Ó`y]Z[sÛ�`_ä2`yqG[Ó~�ãímRacÛ*Þ byà�l\ã�ÞéÛ�`yã�`\hc`\b\ac[2kf`Z]ím.ÞZacä ÞÃl\]\Û�k�Û*Þ�l�Û�~Zk�[»à�[�l�h æ!Û�azqG[k�[�lià�ä ÞGqG[2Û�ÞZ`yå�k�ã�[�à$ì(`yà�qIá�l�å�hcp\çgë,ÞZajk§ä�l\]Gá�[RÛ�à�acä tipGk�az]Zä2[.]Z`\ÛgqG~Zä ÞºackQtZ]Z`�m.]Gl\á�`Z~ZÛà�[�lxhæ!Û�a'qõ[�k�[�l\à�ä2Þ»qõ[2Û*ÞZ`yå�knç�&!]Zk�Û�[�l�å_`�ì4k�Û�~Zå�pyaz]ZbÓÛ*ÞZ[�ä2`�Y\[�à§Û�azqG[�`�ì¾l\]iÛ�k�å�azà�[2ä2Û�hcp\ß�m�[k�Û�~Zå�p_Û*ÞZ[fÛ�azqG[GacÛ�Û*l\t�[2kR`Z]Z[Dl\]iÛ.`xìµlsbyà�`y~�ãê`�ìgl\]\Û�kRÛ�`_à�[�l�ä2Þ|lGb\`Zl�h�Y\[�à�Û�[2w¼ì¦`yàÝÛ*ÞZ[÷&à�k�ÛfÛ�azqG[\ß�Û*Þ�l�Ûfack�ß�l\]ip Yi[�à�Û�[2wíaz]:l_b\acY\[�] k�[2Û 6 4.���� � `�ì�b\`yl�h�Y\[�à´Û�acä2[2k�ç�����Ü$ �âå�[�]Z`\Û�[2kDÛ�ÞZ[_b\`yl�h§å�ajk�Û*l\]Zä2[_`�ìRY\[�à�Û�[2wF \ß�Û�Þ�l�Ûfack�ߧÛ*ÞZ[¼å�ajk�Û*l\]Zä2[¼ì!à�`yq êÛ�` léä2hc`\k�[2k�Ûb\`Zl�h�Y\[�à�Û�[2w&ç�XZaz]Zä2[õli]\Û�k,Û*Þ�l�Û.~Zk�[G{ ��ë�Únm»l\]Zå li]\Û�k,Û*Þ�l�Û.~Zk�[ qR`Zå�[ �Ý`y~�]\Û�a']Zb¼ä2`�Y\[�àbyà�l\ã�ÞZk�ß�Û*ÞZ[2p|l�hck�`|à´[�l�ä Þílsb\`yl�h�Y\[�à�Û�[2w&ç�ëRÞZ[2azà�ä2`�Y\[�à,Û�azqG[Gaz]íÛ*ÞZ[ºmQ`Zà�k�Û�ä�l�k�[Gack�Û*ÞZ[k*liqG[»l�kºÛ*ÞZ[»Û�a'qõ[»acÛGÛ*l\t�[2k�az] Û*ÞZ[smQ`Zà�k�Ûfä�lxk�[ÓÛ�`íà�[�l�ä2Þ:léb\`yl�h�Y\[�à´Û�[2wéì¦`yà Û*ÞZ[G÷&à�k�ÛÛ�azqG[\ß�ì(`yà§[2wrl\qsãZhj[\ßOÛ*ÞZ[ºY\[�à�Û�[2w¼Û*Þ�l�Û�mRl�kRYZack�acÛ�[2åíh'lxk�ÛRm.ÞZ[�]êä2`�Yi[�à�az]ZbÓÛ*ÞZ[fbZà�l\ã�Þ�ç�Ú�hch`�ìg`Z~�à�[2wrl\qsãZhj[sbyà�l\ã�ÞZk�l\à�[GãZh'li]�l\à�byà�l\ã�ÞZk�ß�l»ã�à�`Zã�[�à�Û�p|Û�Þ�l�Û.ackfà�[�lxhjack�Û�ajäõì¦`yà,byà�liã�ÞZkÛ*Þ�lxÛ.qG`Zå�[2h�Û�[�à�à�lxa']�ç

� ��������� � �� �� ��0� �������� � � ��� � �� � � � � �"!&$]ÁÛ*ÞZack_k�[2ä2Û�ac`y]�ß�m�[íã�à�`�Y\[_Û*Þ�l�ÛÓÛ�ÞZ[|ä2`�Y\[�àGÛ�azqG[í`�ìfl\]iÛ�kÓÛ*Þ�lxÛ¸~Zk�[í{ ��ë�Únm:ack

by~�lià�l\]iÛ�[2[2å|Û�`_á�[Ól�Û�qG`\k�Û;,��� ,�ßOÛ*Þ�l�Û�ack�ß&ã�`\hcpr]Z`ZqGazl�h�az]|Û�ÞZ[õ]Z~�qGá�[�à,`�ìgY\[�à�Û�ajä2[2knß]Z`ÓqÓl�Û�Û�[�à�m.ÞZ[2Û�ÞZ[�àgÛ�ÞZ[.byà�l\ã�ÞZkQl\à´[Rå�azà�[2ä2Û�[2å_`yà§m.ÞZ[2Û*ÞZ[�à§Û*ÞZ[2p¼l\à�[.ãZhzl\]�l\à6çge�~ZÛg÷&à´k�Û�ßm�[gk�Û*~Zå�pfÞZ`�m·hc`y]Zb�acÛ4Û*lit�[2k¾~�]iÛ�achZ`y]Z[Ql\]iÛ�`�ì&l,byà�`Z~�ã�`�ì&l\]iÛ�k�Û*Þ�l�Û�~Zk�[�{ �4ë�Únm.à�[�l�ä2ÞZ[2klêb\`yl�h�Y\[�à´Û�[2wéì¦`yà Û*ÞZ[G÷&à�k�ÛfÛ�azqG[\ç_òÃ[»]Z[2[2åÁÛ*ÞZ[Gì(`\hchc`�m,a']Zb:ã�à�`Zã�[�à�Û�pí`�ì2<Zæ!Y�l�hz~Z[2kÓÛ�`k�Û�l�Û�[�`y~�àRà�[2k*~ZhcÛ�knç

� d�� ���i�������D�n��� æ!Y�l�hz~Z[2k¼`�ìf{ �4ë�Únm l\à´[ l�å�qGack�k�a'áZhc[ía/# ��� <4ÜE �â������&Ü$ nâ ì¦`yàsl�hch *��Rç

Ú�å�qGack�k�azáZachcajÛ�p_qG[�l\]Zk�Û*Þ�lxÛ4Û*ÞZ[I<ræ!Y�l�hz~Z[2k�`�ì}{ �4ë�ÚFm�]Z[2Y\[�à�`�Y\[�à�[2k�Û�a'qsl�Û�[gÛ�ÞZ[Qbi`yl�hå�ack�Û*li]Zä2[2k�ç $�`yàÝ[2wrl\qsãZhj[\ßOo2[�à�`�æ!az]ZacÛ�azl�hcaco2[�åC<Zæ!Y�l�hz~Z[2kfl\à�[fl�å�qGack�k�a'áZhc[\ç,Ú.å�qGack�k�a'áZachcazÛ�píackl\]ÃazqÓã�`Zà�Û*l\]iÛfä2`y]Zä2[�ãZÛsa']ÃÛ*ÞZ[¼ä2`y]iÛ�[2wyÛº`�ì.Û*à�l�å�acÛ�ac`y]�l�hgk�[�l\à�ä2Þ·qG[2Û�ÞZ`yå�kGk*~Zä2Þ·lxkfÚnmî�,0pnñ�ç�ò:[�ä�l\]_]Z`�mBã�à�`�Y\[�Û�ÞZ[�ì¦`\hchc`�mRaz]Zb»Û�ÞZ[2`yà�[�q_ç

Page 10: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

7�+ � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

�Wq�df��o=dj b�� &$]ZajÛ�a'lxh <Zæ!Y�l�hz~Z[2kGÛ*Þ�l�Ûfl\à�[Ól�å�qõajk�k�azáZhj[_à�[�qÓlxa']Ãl�å�qGack�k�azáZhc[|lnì(Û�[�àR[2Yi[�à�p[2å�b\[�Û*à�lnY\[�à´k*l�h�`�ì¾li]\Û�k§Û*Þ�l�ÛR~Zk�[�{ ��ë�Únm¸Ü(~�]iÛ�ach�lfbi`yl�h�Yi[�à�Û�[2wÓackRà�[�l�ä2ÞZ[2å¼ì¦`Zà�Û*ÞZ[�÷&à�k�ÛÛ�azqG[�â�l\]Zå|l\à�[fqG`Z]Z`\Û�`y]Zacä�l�hchcp|]Z`y]iæ!å�[2ä�à�[�l�k�a']Zb¼`�Y\[�à�Û�azqG[\ç

� � ��� � á\p_az]Zå�~Zä2Û�ac`y]�ßOk�[2[¼îfpnñ�ç

ò:[fä�li]|]Z`�mBã�à�`�Y\[�Û*ÞZ[ ì¦`ihjhc`�mRaz]Zb¼Û*ÞZ[2`yà�[�qGknßZì¦`ihjhc`�mRaz]Zbíî�,�,�ñ�ç�XZazqGachzl\à�Û*ÞZ[2`yà�[�qõkm�[�à�[fl�hck�`Óaz]Zå�[�ã�[�]Zå�[�]iÛ�hcp·ã�à�`�Y\[2å¼az] îfg�enñ�ç

�Wq�df��o=dj ��� ��`Z]Zk�acå�[�àG`y]Z[_ã�l\à�Û�acä�~Zhzl\àÓl\]iÛf`xìèlébyà�`y~�ãÃ`�ìèl\]iÛ�kGÛ*Þ�l�ÛG~Zk�[|{ ��ë�Únmyçò ÞZ[�]Z[2Y\[�àRÛ�ÞZ[Dl\]iÛè~�ã�å�l�Û�[2k�Û*ÞZ[ <ræ!Y�l�hz~Z[G`�ìgacÛ�k�ä�~�à�à�[�]iÛRY\[�à�Û�[2w|l\]Zå|az]Zk�Û*li]\Û*li]Z[2`y~Zk�hcpÛ*à�lnY\[�à�k�[2kQÛ�ÞZ[�ä ÞZ`\k�[�]¼[2å�b\[\ß�Û*ÞZ[fã�`\Û�[�]iÛ�azl�hAg�� ")R�� <4Ü$ nâ 9<4ÜE �#(â�az]Zä�à�[�l�k�[2k�á\p_l�Û�hc[�l�k�Û`y]Z[\ß�m.ÞZ[�à´[ �#�ack�Û*ÞZ[�ä�~�à�à�[�]iÛRY\[�à´Û�[2w»`xì¾Û*ÞZ[fli]\Û�ç

� � ��� � {�[2Û8 �#�å�[�]Z`\Û�[¼Û*ÞZ[Óä�~�à�à�[�]\Û�Yi[�à�Û�[2wí`�ìQÛ�ÞZ[»l\]iÛ�ß�l\]ZåF #�- ,fÛ*ÞZ[ÓY\[�à´Û�[2wí`�ìQÛ*ÞZ[l\]iÛÝl�ì¦Û�[�àµacÛ.Þ�l�k�Û*à�lnY\[�à�k�[2åÓÛ*ÞZ[�ä2ÞZ`\k�[�]s[2å�b\[\çg{�[2Û2<_#*Ü$ �âµå�[�]Z`\Û�[�Û*ÞZ[;<Zæ!Y�l�hz~Z[f`�ì�Y\[�à�Û�[2w _azqÓqG[2å�azl�Û�[2hjpÙá�[�ì¦`Zà�[_Û*ÞZ[|l\]iÛÓ~�ã�å�l�Û�[2ksÛ*ÞZ[�<Zæ!Y�l�hz~Z[|`�ì #�ßgl\]ZåZ<_#�- ,�Ü$ �âºÛ*ÞZ[_k*liqG[<Zæ!Y�l�hz~Z[_a'qsqG[2å�azl�Û�[2hcpÙlnì(Û�[�à�Û*ÞZ[_~�ã�å�lxÛ�[\ç ë,ÞZ[»`y]ZhcpF<ræ!Y�l�hz~Z[_Û*Þ�lxÛfä2Þ�l\]Zb\[2kGack5<�Ü$ �#!â6ç&(Û�az]Zä�à�[�l�k�[2kDl�ä2ä2`yà´å�a']Zb_Û�`¸ëRÞZ[2`Zà�[�q g_áipk< #�- , ÜE # â < # ÜE # â�� ��ç Ú.ä2ä2`yà´å�a']Zb_Û�`»Û*ÞZ[Y�l�hz~Z[�æ�~�ã�å�l�Û�[�à�~Zhc[R`�ì&{ ��ë�ÚnmyßZacÛ�ä Þ�li]Zb\[2k�Û�`;<�#�- ,�Ü$ #!â . , <_#´Ü$ #�- ,*â2ç�òÃ[Qå�ack�Û�a']ZbZ~Zajk�ÞÛ�mQ`Gä�l�k�[2k�ç,\ç�&¦ì �#�- ,;4.D �#�ßZÛ*ÞZ[�]_Û*ÞZ[fã�`\Û�[�]iÛ�azl�h�az]Zä�à�[�l�k�[2kèáip

-0/� �� �����������-���1�����������-���������1(1��¼-0/� �� ������¢-���1��� �¢-�����1(1! -0/"� ������-����01#�$��������-��������!1(1�s-0/%���¢-����01������¢-����01(1! -0/"� ������-����01#�$���¢-��������!1(1��$���¢-�����1! -0/"� ����� -�� � 1#�»-�� ����� -�� � 1��»7�1(1��&� � -�� � 1! 7�'¼-�� ����� -�� � 1��&� � -�� � 1(1( 7")

g�ç�&¦ì �#�- ,+.D �#�ßZÛ*ÞZ[�]_Û*ÞZ[fã�`\Û�[�]iÛ�azl�h�az]Zä�à�[�l�k�[2kèáip

-0/ � �� �� � ����� -���1���� ����� -�� ����� 1(1��¼-0/ � �� �� � � -���1��� � -�� � 1(1! -0/"� ����� -�� � 1#�$� ����� -�� ����� 1(1�s-0/%� � -�� � 1���� � -�� � 1(1! -0/"� ����� -�� � 1#�$� ����� -�� � 1(1��»-0/"� � -�� � 1��&� � -�� � 1(1! � ����� -�� � 1���� � -�� � 1! -(7�'&���¢-���������1(1���� �¢-�����1! -(7�'&���¢-�����1(1�����¢-�����1! 7")

Page 11: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� 7*7

start

goal

s1

...s2 s3 s4 sn-1 sn

�Z�)�£� !� � ¡&× �#��)���Q�����(�������*��£��(��Òy½�¹���Ô§������# !�����(���fÑ� !��®����

...

start goal

s1 s3 s5 sn-3 sn-1

...s2 s4 s6 sn-2 sn

�Z�)�£n !� � ¡&×��#��)���������(����������£��(��ÒZ½�¹���Ôg�� #����2�# !�����(���GÑ� !��®n���

�5qndQ�no=d�j �[� ëRÞZ[¼Û�azqG[¼acÛfÛ*l\tx[2k�`y]Z[¸li]\Û�`�ì.lêbyà�`y~�ãí`xì,li]\Û�k�Û�Þ�l�ÛD~Zk�[_{ ��ë�Únm_Û�`à�[�lxä Þ_lGbi`yl�h�Y\[�à�Û�[2wsì(`yà§Û*ÞZ[�÷&à�k�ÛQÛ�a'qõ[fackèlxÛÝqõ`\k�Ûdg � ")R�� ���&Ü$ nâ2ç� ��� �/� ��`Z]Zk�acå�[�à¼`y]Z[ ã�l\à´Û�acä�~Zh'lià_l\]\Û_l\]ZåLÛ*ÞZ[ ã�`\Û�[�]iÛ�azl�h g � ")R�� <�Ü$ �â �<4Ü$ #!âGì(`yàÛ*ÞZack.li]\Û�ßZm.ÞZ[�à´[: # ackRÛ*ÞZ[�ä�~�à�à´[�]\Û�Y\[�à�Û�[2w»`�ì�Û*ÞZ[fl\]iÛ�ç &!]ZacÛ�azl�hchcp\ß�Û*ÞZ[fã�`\Û�[�]\Û�azl�h�ack�o2[�à�`�çÚ�ä2ä2`yà�å�az]ZbÁÛ�`·ë,ÞZ[2`yà�[�q j�ß�a ìfÛ*ÞZ[:l\]\Û¼~�ã�å�l�Û�[2k¼Û*ÞZ[G<Zæ!Y�lxh'~Z[Ã`�ìèacÛ�k_ä�~�à�à�[�]\Û¼Y\[�à�Û�[2wl\]Zå_Û�à�lnY\[�à�k�[2k§Û*ÞZ[.ä2ÞZ`\k�[�]s[2å�b\[\ß�Û�ÞZ[èã�`\Û�[�]iÛ�azl�h�a']Zä�à´[�l�k�[2k.áip¸lxÛghc[�l�k�ÛR`Z]Z[\ç4Ú�ä2ä2`yà�å�az]ZbÛ�`:ëRÞZ[2`Zà�[�q g�ß�a ì l å�a #�[�à�[�]iÛÓl\]iÛõ~�ã�å�l�Û�[2kÓÛ�ÞZ[�<Zæ!Y�lxh'~Z[é`�ì.acÛ�kÓä�~�à�à�[�]\ÛGY\[�à´Û�[2w&ßµÛ*ÞZ[ã�`\Û�[�]iÛ�azl�h�ä�l\]�]Z`\Û_å�[2ä�à�[�lxk�[\ç $�az]�l�hchjp\ߺl�ä2ä2`yà�å�az]ZbÙÛ�`ÙëRÞZ[2`yà�[�q g�ß,Û*ÞZ[·ã�`\Û�[�]iÛ�azl�h�acká�`y~�]Zå�[2åéì¢à�`Zqôl\á�`�Y\[fáip g � ")R�� ����Ü$ nâ2ç���`y]Zk�[2ór~Z[�]iÛ�hcp\ß&lxÛRhc[�l�k�Û�`y]Z[Dl\]iÛR`�ìglGbyà´`y~�ã`�ìfl\]iÛ�kGÛ*Þ�l�Û»~Zk�[|{ �4ë�Únm à�[�lxä ÞZ[2k»l b\`yl�h�Y\[�à�Û�[2w:ì(`yàºÛ*ÞZ[_÷&à´k�ÛÓÛ�azqG[|lnì(Û�[�àGl�Û»qõ`\k�Ûg � ")R�� ���&Ü$ nâ§[2å�b\[�Û*à�lnY\[�à�k�l�hck�ç

ëRÞZ[2`Zà�[�q h:k�Û�l�Û�[2kÓÛ*Þ�lxÛÓÛ*ÞZ[_Û�azqG[éajÛ¼Û*l\tx[2kD`Z]Z[ l\]iÛÓ`�ìflíbyà�`Z~�ã `�ì li]\Û�ksÛ*Þ�l�Û~Zk�[:{ ��ë�ÚFm:Û�`Áà�[�l�ä2ÞÆl:bi`yl�hRY\[�à´Û�[2w ì¦`ZàÓÛ*ÞZ[é÷&à�k�Û»Û�azqG[éack|l�Û¸qõ`\k�Û g � " R � ����Ü$ �â2çëRÞZ~Zk�ßZÛ*ÞZ[�ì([2m�[�àgYi[�à�Û�acä2[2kRÛ*ÞZ[�à�[flià�[èli]Zå_Û*ÞZ[�k*qÓl�hchc[�à�Û*ÞZ[flnY\[�à�l�b\[Rb\`Zl�h�å�ajk�Û*l\]Zä2[�`�Y\[�àl�hch�Y\[�à�Û�acä2[2k�ß,Û*ÞZ[|k�qÓl�hchc[�à¼Û*ÞZ[ ~�ã�ã�[�à|á�`Z~�]Zå�çBëRÞZack_ä2`yà�à´[2k*ã�`y]Zå�k_Û�`:`y~�àsa']iÛ*~ZacÛ�ac`y]�ç$O`yà¼[2wrliqÓãZhc[\ß.a ì»l�hch�Y\[�à�Û�ajä2[2kíl\à�[íä2hz~Zk�Û�[�à�[2å l\à�`y~�]ZåÆÛ*ÞZ[íb\`ylxh.Y\[�à´Û�acä2[2k�ß�Û*ÞZ[�]Æ[�l�ä2Þl\]iÛ�ä�l\]í]Z[2Y\[�àfqG`�Y\[ ì¢lià.lnmRlnpÓì!à�`yqTÛ*ÞZ[Gb\`yl�h�Y\[�à�Û�ajä2[2kfli]Zå·l�hcmRl�pZk�Þ�l�kèl¼ä2Þ�l\]Zä2[Ó`xìb\[2Û�Û�az]ZbÓÛ�`Ó`y]Z[ºór~Zacä2t\hcpé[2Y\[�]_a ìgacÛ.á�[�Þ�lnY\[2åék*~�á�`ZãZÛ�azqÓl�hchcp az]êÛ*ÞZ[Dã�l�k�Û�ç�&(Û�ÞZ`\hcå�k�Û*Þ�l�Ûg � ")R�� ���&Ü$ nâ � g������ ,� +�� .7, � C, l�ä2ä2`yà�å�az]Zb_Û�`¼`y~�à�l�k�k*~�qÓãZÛ�aj`Z]|Û�Þ�l�Û.Û�ÞZ[fbyà�liã�ÞZkl\à´[ k�Û*à�`y]Zb\hcpLä2`y]�]Z[2ä2Û�[2å�ç &!] Û*ÞZ[íì(`\hchj`�mRaz]Zb�ß�m�[:ã�à�[2k�[�]iÛ|l\]Æ[2w�l\qÓãZhc[:byà�l\ã�ÞLÛ*Þ�l�Ûk*ÞZ`�mRk�Û�Þ�l�Û.Û�ÞZajkf~�ã�ã�[�àGá�`y~�]Zå:ack�Û�acbyÞiÛ�ì(`yàRk�a']Zbihj[_l\]iÛ�k�Û*Þ�l�Ûf~Zk�[Ó{ ��ë�Únmyç ��`Z]Zk�acå�[�àÛ*ÞZ[�byà�liã�ÞÓaz].$�acby~�à�[:r�ç�ëRÞZ[�ì(`\hchc`�mRaz]Zb»ã�à�`\byà�l\qIaz]_ãZk�[�~Zå�`Óä2`Zå�[�k*ÞZ`�mRk�`y]Z[èã�`\k�k�azáZhc[Y\[�à´Û�[2wÁk�[2ó�~Z[�]Zä2[ Û�Þ�l�Û»k�a']Zbihj[:l\]iÛ�k¼Û*Þ�l�Û»~Zk�[ { �4ë�ÚFm:ä�li]ÙÛ�à�lnY\[�à�k�[\çÆëRÞZ[ík�ä2`yã�[ `xìÛ*ÞZ[�ì(`yà$æ!k�Û*l�Û�[�qG[�]iÛ�k§ajk�k*ÞZ`�m.]¼á\p¼az]Zå�[�]\Û�l�Û�ac`y]�ç�ë�ac[2k.l\à�[fá�à�`Zt�[�]_á\p¼à�[�qÓl�az]Zaz]Zb»az]êÛ*ÞZ[ä�~�à�à´[�]\Û§Y\[�à�Û�[2wºaì4ã�`\k�k�azáZhc[èli]ZåÓ`\Û*ÞZ[�à�m,ajk�[Rä ÞZ`Z`\k�az]ZbºÛ*ÞZ[�Y\[�à�Û�[2wºmRacÛ*ÞGÛ*ÞZ[Rhc`�m�[2k�Û�hzl\á�[2h(ç

Page 12: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

7´/ � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

����������� ��������������������������� �������������!

������������"���#��������� �����������$��

� �������%���&�' ëRÞZackGä�l\]:á�[_ã�à�`�Y\[2å·áip:az]Zå�~Zä2Û�ac`y] `y] ,4ç"&(ÛfackGÛ*ÞZ[�]Ã[�l�k�píÛ�`éå�[2Û�[�à�qGaz]Z[sì¢à�`Zq

l\]Ãa']Zk�ã�[2ä2Û�ac`y] `�ìRÛ*ÞZ[¸ãZk�[�~Zå�` ä2`Zå�[¼Û*Þ�l�Ûfk�a']Zbihj[|li]\Û�kºÛ*à�lnY\[�à�k�[ , � ,Á[2å�b\[2kÓá�[�ì(`yà�[Û*ÞZ[2p·à�[�lxä ÞÃÛ*ÞZ[¼b\`yl�h§Y\[�à�Û�[2wíì(`yà Û*ÞZ[Ó÷&à´k�ÛGÛ�azqG[\ß�ã�à�`�Yyaz]Zb:Û*Þ�lxÛfÛ*ÞZ[_~�ã�ã�[�àÓá�`y~�]ZåÁ`�ì, � F, [2å�b\[fÛ*à�l�Yi[�à�k*l�hckRack�Û�acbyÞ\Ûnç $�`yà�[2w�l\qÓãZhc[\ßOì(`yàI, .7r�ß�Û*ÞZ[2p_Û*à�l�Yi[�à�k�[�Û*ÞZ[ºY\[�à�Û�[2wk�[2ó�~Z[�]Zä2[ ,�ß� &,�ß� � ß� � ß� � ß� ,�ß� !(\ß� !(\ß� )(\ß� !(\ß� � ß� &,�ß� !*\ß� )*\ß� !*\ß� )*\ß[ !*\ß� )(\ß� � ß� ,�ßl\]Zå� �+�ßOl\]Zå_Û*ÞZ~Zkdg��º[2å�b\[2k�çgë,ÞZ[�byà�l\ã�Þ_~Zk�[2å_az]êÛ*ÞZ[fl\á�`�Y\[�[2w�l\qÓãZhc[�mRl�kRl\à�Û�a÷�ä2azl�hchcpä2`y]Zk�Û*à�~Zä2Û�[2å�ç-,R`�m�[2Y\[�à2ß�Û*ÞZ[_~�ã�ã�[�àÓá�`y~�]ZåÁ`�ì/._Ü$, � âº[2å�b\[¼Û*à�lnY\[�à�k�l�hckfackGÛ�acbyÞ\ÛG[2Y\[�]ì¦`ZàRqG`yà�[�à�[�l�hcack�Û�acäGbyà�liã�ÞZkRk*~Zä2Þ_l�k.à�[2by~Zhzl\àQì¦`y~�à�æ!ä2`y]�]Z[2ä2Û�[2å|bZà�acå�k�ç���`y]Zk�ajå�[�à,Û*ÞZ[�byà´ajåk*ÞZ`�m.]ºaz] $�acby~�à�[Rð�l\]ZåÓl�k�k*~�qõ[-, � g�mRacÛ*ÞW,_qG`Zå h:. g�ç¾ë,ÞZ[gì(`\hchc`�mRaz]ZbDã�à�`\bZà�l\qÆaz]ãZk�[�~Zå�`Óä2`Zå�[.k�ÞZ`�m,kg`y]Z[�ã�`\k�k�azáZhj[ºY\[�à�Û�[2wGk�[2ó�~Z[�]Zä2[�Û*Þ�l�Û§k�az]Zb\hc[fl\]iÛ�kgÛ*Þ�lxÛQ~Zk�[�{ ��ë�Únmä�l\]_Û�à�lnY\[�à�k�[\ç

�������-�%���!0�"���#������1��2!3�����4 � 3�������5���6����������4 � 3� �����������$��

�������5����7��1"���#������%3�����4 � 3� �����������$��

�������-�8�6��������� ����4 � 3� �������������!

ëRÞZackGä�l\]Áá�[_ã�à�`�Y\[2åÁá\p:az]Zå�~Zä2Û�ac`y] `Z]F,�ç &(ÛGackÓÛ*ÞZ[�]Ã[�l�k�p:Û�`éå�[2Û�[�à�qGaz]Z[_Û*Þ�lxÛk�az]Zb\hc[Ól\]\Û�kQÛ�à�lnY\[�à�k�[Xj�, �!9 ,�ð j 9 hG[2å�b\[2k�á�[�ì(`yà�[�Û*ÞZ[2p_à�[�l�ä2Þ¼Û*ÞZ[�b\`ylxh&Yi[�à�Û�[2wsì¦`yà§Û*ÞZ[÷&à�k�Û§Û�azqG[\ßyã�à�`�YZaz]ZbfÛ*Þ�l�Û�Û�ÞZ[.~�ã�ã�[�à§á�`y~�]Zå¼`�ì:._Ü$, � â�[2å�b\[�Û*à�lnY\[�à�k*lxhjk�ackgÛ�acbyÞiÛg[2Yi[�]ºì¦`Zà~�]Zå�azà�[2ä2Û�[2å:ãZhzl\]�l\à�byà�l\ã�ÞZk�mRacÛ*ÞZ`y~ZÛ�[2å�b\[2kRÛ*Þ�lxÛQhc[�lnY\[�Û*ÞZ[�ä�~�à�à�[�]\Û�Y\[�à´Û�[2w¸~�]Zä2Þ�l\]Zb\[2ål\]Zåsì¦`yàµm.ÞZacä ÞsÛ*ÞZ[.]Z~�qGá�[�à§`�ì�`y~ZÛ�bi`\az]Zbf[2å�b\[2k�`�ì4l\]ipGY\[�à�Û�[2wºajkRá�`y~�]Zå�[2å¼ì¢à´`yq l\á�`�Y\[á\p l_k*qÓlxhjh�ä2`y]Zk�Û*l\]iÛ¸Ü(ÞZ[�à�[��GÛ�Þ�à�[2[�â2ç%$O`yà�[2w�l\qÓãZhc[\ß�ì(`yà:, . , ��ß�Û*ÞZ[2pÃÛ*à�lnY\[�à�k�[GÛ*ÞZ[Y\[�à�Û�[2wék�[2ó�~Z[�]Zä2[ &,�ß !(\ß� �+�ß� <;�ß� !=\ß� )>\ß[ !*\ß� � ß� &,�ß� )(\ß� �+�ß[ )>\ß� !*\ß� � ß� &,�ß� !(\ß �+�ß� <;�ßl\]Zåk )?\ß�l\]Zå_Û*ÞZ~Zk�,�põ[2å�b\[2k�ç

ëRÞZ[2`yà�[�q hÓä�l\]_á�[º[2wyÛ�[�]Zå�[2å|Û�`Gbyà�`Z~�ãZkR`�ìA@_l\]\Û�k Ü0ì¦`yà�l\]ip¼ä2`y]Zk�Û*l\]iÛB@�â2çgXZaz]Zä2[Û*ÞZ[_Û�azqG[¼acÛÓÛ*l\t�[2kG`y]Z[_l\]iÛG`�ìèlébyà�`y~�ãÃ`�ìèli]\Û�kGaz] Û*ÞZ[¼m�`yà�k�Ûºä�l�k�[_Û�`íà�[�l�ä2ÞÁl|bi`yl�hY\[�à�Û�[2w¼ì(`yà,Û*ÞZ[º÷&à�k�Û�Û�azqG[Gackfl\]í~�ã�ã�[�à�á�`Z~�]Zå:`y]_Û�ÞZ[Dä2`�Y\[�à,Û�azqG[ºa'] Û*ÞZ[fm�`yà�k�Û.ä�l�k�[\ßÛ*ÞZ[�ì(`\hchc`�mRaz]Zb»ä2`yà´`\hch'lià�pÓì(`\hchc`�mRk�ç

�:�no=��g gi�o<CED�� ëRÞZ[�ä2`�Y\[�à�Û�azqG[R`xì�li]\Û�k�Û�Þ�l�ÛQ~Zk�[�{ �4ë�Únmè`Z]Gk�Û*à�`Z]Zb\hcpDä2`Z]�]Z[2ä2Û�[2å ܦå�aæà�[2ä2Û�[2åG`yà�~�]Zå�azà�[2ä2Û�[2å�ßrãZhzl\]�lià�`yà�]Z`y]iæ�ãZhzl\]�l\à�â�byà�l\ã�ÞZk4ack�by~�lià�l\]iÛ�[2[2å�Û�`�á�[Rã�`\hcp�]Z`yqGazl�haz]_Û*ÞZ[G]y~�qGá�[�àÝ`�ì¾Yi[�à�Û�acä2[2k�ç

� ����� � � ���� �� �0������������ � ��� � �� �� � �� � ���� �0�����&!]sÛ*ÞZackgk�[2ä2Û�aj`Z]�ßym�[Rã�à�`�Y\[RÛ�Þ�l�Û�Û*ÞZ[Rä2`�Y\[�à�Û�azqG[�`�ì�l\]\Û�k�Û*Þ�l�ÛQ~Zk�[dqR`Zå�[ ��`Z~�]\Û�az]Zb

ä�l\]Gá�[�[2w�ã�`y]Z[�]iÛ�azl�h�az]_Ü�Û*ÞZ[gk�ór~�l\à´[Qà�`Z`\Û�`�ì â�Û*ÞZ[R]Z~�qGá�[�à�`�ì�Y\[�à�Û�acä2[2k�ì(`yà�á�`\Û*Þ å�azà�[2ä2Û�[2å

Page 13: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� 7 �

start goal

r1

number of timesthe vertex has been

visited after termination

g2g2g2 g5g3 g4g1g0

r2

r3

r4

r5

16 16 8 4 2 1

8

4

2

1

0

�Z�)�£� !� � ¡&× �#��)���Q�����(�������*��£��(��Õ42�n�QÐ��£������#��)R�� ���# !�����(���fÑ� !��®n���§-�� ! � ��� ! 7*7�1l\]ZåÆ~�]Zå�azà�[2ä2Û�[2å byà�l\ã�ÞZk�ß�az]Zä2hz~Zå�a']Zb Ü(ãZhzl\]�lià�â»~�]Zå�azà�[2ä2Û�[2å Û*à�[2[2k�ç $�azà�k�Û�ß�m�[|å�ack�ä�~Zk�kå�azà�[2ä2Û�[2å byà�liã�ÞZk�çfëRÞZ[�]�ß�mQ[Gå�ack�ä�~Zk�kG~�]Zå�azà�[2ä2Û�[2å byà�l\ã�ÞZk�ß�l¼k�ã�[2ä2azl�hgä�lxk�[G`�ìQå�azà�[2ä2Û�[2åbyà�l\ã�ÞZkfk�az]Zä2[_[2Y\[�à´p ~�]Zå�azà�[2ä2Û�[2åÙ[2å�b\[Óä�li]·á�[¸qG`Zå�[2hc[2åÁmRacÛ*ÞÃÛ�m�`_å�azà�[2ä2Û�[2åÁ[2å�bi[2kD`xì`yã�ã�`\k�acÛ�[§å�a'à´[2ä2Û�ac`y]Zk�ç�ò:[�÷&à´k�Û�å�ack�ä�~Zk�k�å�azà�[2ä2Û�[2åGbyà�l\ã�ÞZk4á�[2ä�li~Zk�[g`y~�à�å�azà�[2ä2Û�[2åº[2wrl\qsãZhj[byà�l\ã�ÞZkèlià�[Gk�azqÓãZhc[�à�Û*Þ�l\]ê`y~�à�~�]Zå�azà�[2ä2Û�[2å:[2w�l\qÓãZhc[Gbyà�liã�ÞZk�ç&!]:á�`\Û*Þ_ä�lxk�[2k�ß�m�[�÷&à�k�Ûk�Û�~Zå�p»ÞZ`�mÙhc`y]ZbfacÛ�Û*l\tx[2kQ~�]iÛ�ach&l�k�a']Zbihj[fli]\Û§Û*Þ�l�ÛQ~Zk�[2k q,`yå�[��`y~�]iÛ�az]ZbÓà�[�l�ä2ÞZ[2kQl�bi`yl�hY\[�à´Û�[2wºì¦`yà�Û�ÞZ[�÷&à�k�Û§Û�azqG[\çgëRÞZ[�]�ßimQ[,b\[�]Z[�à�l�hcaco2[�Û�`fÞZ`�mÙhc`y]ZbºajÛ§Û*l\t�[2k�~�]\Û�ajhO`y]Z[.l\]iÛg`xìlsbyà�`y~�ã¼`xìQl\]\Û�kRÛ*Þ�l�Û�~Zk�[ q,`Zå�[ ��`y~�]iÛ�az]Zb_à�[�l�ä2ÞZ[2kèlsb\`yl�h�Y\[�à�Û�[2wsì¦`Zà�Û*ÞZ[º÷&à�k�ÛRÛ�a'qõ[\ç$�az]�l�hchcp\ß�m�[�b\[�]Z[�à�l�hcaco2[fÛ�`sÛ*ÞZ[�ä2`�Y\[�à�Û�azqG[�`�ìgl\]iÛ�k�Û*Þ�l�Û.~Zk�[FqR`Zå�[ ��`Z~�]\Û�az]Zb�ç�� �// ����� �� � � � �� ��� � �0����� ����� �� ���0 ��� ��� � � &$]|Û�ÞZajk�k�[2ä2Û�ac`y]�ß�m�[Gã�à�[2k�[�]iÛlså�azà�[2ä2Û�[2å:[2w�l\qÓãZhc[Gbyà�l\ã�ÞêÛ*Þ�l�ÛRk�ÞZ`�m,k.Û*Þ�lxÛRÛ*ÞZ[Ó]y~�qGá�[�àR`�ìg[2å�bi[DÛ�à�lnY\[�à�k*l�hck.~�]iÛ�ach�lk�az]Zb\hc[fl\]iÛ�Û*Þ�l�Û�~Zk�[2k q,`Zå�[ ��`Z~�]\Û�az]ZbGà�[�l�ä2ÞZ[2kÝl b\`yl�h�Yi[�à�Û�[2wºì¦`Zà�Û*ÞZ[R÷&à´k�Û�Û�azqG[�ä�l\]Óá�[[2w�ã�`y]Z[�]iÛ�azl�hOa']õÛ*ÞZ[R]y~�qGá�[�àµ`�ì&Yi[�à�Û�acä2[2k�ç�$�acby~�à�[do�k*ÞZ`�mRk�l\]Gaz]Zk�Û�l\]Zä2[R`�ì�Û*ÞZack�[2wrl\qsãZhj[byà�l\ã�Þ�ç &$]íb\[�]Z[�à�l�h(ßOÛ*ÞZ[Gbyà�l\ã�ÞZk�Þ�lnY\[�� ,ºhc[2Y\[2hck¸Ü¨ì(`yà � � ,�â6ç.ëRÞZ[Ghc[2Y\[2hckfä2`y]Zk�ajk�Û`�ìgYi[�à�Û�acä2[2k.`xìgÛ�m�`Gå�a/#�[�à�[�]\Ûftiaz]Zå�k�� �Zæ!Y\[�à�Û�acä2[2k�l\]Zå��6æ!Y\[�à�Û�ajä2[2knçRÚRÛRhc[2Y\[2h . ��ß�Û�ÞZ[�à�[ackR`y]ZhcpÓ`Z]Z[.Y\[�à´Û�[2w&ß�]�l\qõ[2hjp¼l �Zæ!Y\[�à�Û�[2w � ��çgëRÞZack�Y\[�à�Û�[2wsajk�ä2`y]�]Z[2ä2Û�[2å_Û�` �Zæ!Y\[�à�Û�[2w � ,�çÚRÛ§hc[2Y\[2hck . ,��������íß\Û�ÞZ[�à�[èlià�[.Û�m�`�Y\[�à�Û�ajä2[2knßr]�liqG[2hcp¸l �yæ!Y\[�à´Û�[2w � � l\]Zå¸li]��6æ!Y\[�à�Û�[2w� � çgë,ÞZ[ �Zæ!Y\[�à�Û�[2w�� � ack.ä2`y]�]Z[2ä2Û�[2å|Û�`��Zæ!Y\[�à�Û�[2w�� � - ,DÜ�a ì 4.��_âRl\]Zå��6æ!Y\[�à´Û�[2w�� � ç§ëRÞZ[�6æ!Y\[�à�Û�[2w�� � ajk,ä2`y]�]Z[2ä2Û�[2å¼Û�` �Zæ!Y\[�à�Û�[2w � ��çgëRÞZ[�k�Û*l\à�Û�Yi[�à�Û�[2wGack � �.li]Zå»Û*ÞZ[�b\`ylxh�Y\[�à�Û�[2wack ���fçgëRÞZ[�byà�l\ã�ÞZk�Þ�lnY\[ ,!. g�� ,�Y\[�à�Û�acä2[2k�ç§XZa']Zbihj[Gl\]iÛ�kgÛ*Þ�lxÛQ~Zk�[XqR`Zå�[��`y~�]iÛ�az]Zbä�l\]õÛ*à�lnY\[�à�k�[§Û*ÞZ[�Y\[�à�Û�[2wºk�[2ó�~Z[�]Zä2[.Û*Þ�lxÛ�ajkgã�à´a']iÛ�[2å_á\pWU�Ü��_â�`�ì&Û�ÞZ[gì(`\hchc`�mRaz]ZbDã�à�`\bZà�l\qaz]|ãZk�[�~Zå�`»ä2`Zå�[\çgë�ac[2k�l\à�[fá�à�`ytx[�]¼Û�`�mÝlià�å�k��2æ!Yi[�à�Û�acä2[2k�ç

� ������� ���! 6��������6���4 �� �������! ��"�

4�#!�!4� ��� ���! � �������%� ��� ���! � ��� ���!

� �������$ ���! ëRÞZackºä�l\]·á�[¸ã�à´`�Yi[2å áipíaz]Zå�~Zä2Û�ac`y] `y]%�íç%&(ÛºackDÛ�ÞZ[�]í[�l�k�píÛ�`éå�[2Û�[�à�qGaz]Z[¼Û*Þ�l�Û

k�az]Zb\hc[Rl\]iÛ�k�Û*à�l�Yi[�à�k�['g �'& � - , & � �jR[2å�b\[2k�á�[�ì(`yà�[§Û*ÞZ[2pfà�[�l�ä2Þ�Û�ÞZ[gb\`ylxh�Yi[�à�Û�[2w,ì¦`yà�Û*ÞZ[�÷&à�k�Û

Page 14: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

7 � � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

� ��������� ��� �������� ����� ������������������� �� !#"%$%&('���� )�(�*�+��������+), � ���� ������+�������*+�, �

�����-�#��.�����)������ ����� ���� )�(���+��������+), � ���� )�(�*�+��������+), ������-�#������ ����*��� �����

/ 0 / 1�2 0 / 132 04 5 5 /)2 1 5 /�2 10 6 /�0 /)2 7 /�1 /�2 88 7 4�7 0�2 4 /�9 /�2 :5 /)/ 93/ 5�2 5 4)9 432 89 /�0 /�4�5 7�2 9 0)8 432 96 /�5 4)5�0 /�9�2 7 8)0 432 7: /�6 5)1�7 4�7�2 7 9)1 032 5

;);�; ;�;�; ;�;�; ;�;�; ;);�; ;�;�;4)1 8�/ 4�1)7�6�/�8�7 53/)/�5)132 1 0�4�/ 632 :4�/ 8)0 83/�7�8)0�13/ 7�6)5�8�/�2 7 0�8)8 :32 14)4 8)5 :�0):�:)9�1�5 /�:�9)83/�032 8 0�6): :32 84)0 8)6 /�9�6)6�6)43/�0 0�5�9)7�9)432 1 8�0)0 732 4

¹���²n��R7×���µ£���������Ó½�����£���(���z� 4Õ�2���gÐ��£������#��)R������Òy½�¹���Ô

10

100

1000

10000

100000

1e+06

1e+07

1e+08

1e+09

1e+10

1e+11

1e+12

0 50 100 150 200 250 300 350 400number of vertices (n)

edge traversals until Node Counting reaches a goal vertex for the first time (from simulation or formula)edge traversals until LRTA* reaches a goal vertex for the first time (from simulation)

�Z�)�£n !� � ¡�½�����£���(�§-zÒy�)R×2������.<#1

Û�azqG[\ß�ã�à�`�YZaz]ZbÓÛ*Þ�l�Û�Û*ÞZ[G]y~�qGá�[�àÝ`�ì4[2å�b\[ºÛ*à�lnY\[�à�k*lxhjk�~�]\Û�ajh�Û*ÞZ[�b\`yl�h�Y\[�à´Û�[2w»ack.à´[�l�ä ÞZ[2åì¦`Zà�Û*ÞZ[§÷&à�k�Û4Û�azqG[�ä�l\]Gá�[Q[2w�ã�`Z]Z[�]\Û�azl�h�az]GÛ�ÞZ[Ý]Z~�qGá�[�à�`�ì�Yi[�à�Û�acä2[2k�ç�ëRÞZ[2`yà�[�q ð,ì(`\hchj`�mRknç$�`yà [2wrliqÓãZhc[\ß�ì¦`Zà � . rél\]ZåC,C. ,�,iß�Û*ÞZ[2p:Û�à�lnY\[�à�k�[GÛ*ÞZ[ÓYi[�à�Û�[2wík�[2ó�~Z[�]Zä2[ � � ß � , ß�&,�ß � ��ß � ,2ß � � ß � � ß � �\ß � , ß�� , ß � �\ß � , ß � � ß � (\ß �)(�ß � �\ß � ,2ß�� , ß � �\ß � ,2ß � � ß � � ß � ��ß � ,�ß�&,�ß � ��ß � ,2ß � � ß ��(�ß ��*�ß �)*�ß � ��ß � ,�ß � , ß � ��ß � ,�ß � � ß�� � ß � ��ß � ,2ß � ,�ß � ��ß � ,2ß � � ß ��(�ß �!(�ß� �\ß � ,�ß �&,2ß � ��ß � , ß � � ß � � ß � ��ß � ,2ß �&,2ß � ��ß�� ,�ß�� � ß ��(\ß���*\ß&l\]Zå��'+�ß�l\]ZåíÛ*Þy~Zk�ð�,º[2å�b\[2k�çëRÞZack.ack�ä2`y]Zk�ack�Û�[�]iÛRmRacÛ*Þ.$�acby~�à´[FoGk�az]Zä2[fk*~�qsqGaz]Zb¸~�ã¼Û*ÞZ[G]y~�qGá�[�à�`�ì�Û�azqG[2kRÛ*Þ�l�Û�Û*ÞZ[Y\[�à�Û�acä2[2k�az]_Û*ÞZ[�÷�by~�à´[fÞ�l�Yi[fá�[2[�]_YZack�acÛ�[2å ~�]iÛ�ach4lGbi`yl�h�Y\[�à�Û�[2wsajk�à�[�l�ä2ÞZ[2å¼ì(`yà�Û*ÞZ[�÷&à�k�ÛÛ�azqG[ Ü�[2wZä2[�ãZÛ�ì(`yàRÛ�ÞZ[Dbi`yl�h�Y\[�à�Û�[2w�âRpZac[2hcå�kõð ,\ç�ëRÞZ[Óà�[�l�k�`y]êì(`yà,Û*ÞZ[Gh'lià�b\[Ó]y~�qGá�[�à�`�ì[2å�b\[�Û*à�lnY\[�à�k*lxhjk�ack�Û*ÞZ[Rhzl\à´b\[.]Z~�qDá�[�à�`�ì�Û�ac[2kRá�[2Û�m�[2[�]GÛ*ÞZ[,k*~Zä2ä2[2k�k�`yà �Zæ!Y\[�à�Û�[2wGl\]ZåÓÛ*ÞZ[k*~Zä2ä2[2k�k�`Zà �6æ!Y\[�à�Û�[2wém.ÞZ[�]íÛ*ÞZ[Ól\]iÛ�ackDl�Û�l��Zæ!Y\[�à�Û�[2w�çDXZaz]Zä2[¼Û�ac[2kDl\à´[õá�à´`yt�[�]éÛ�`�mRl\à�å�k�6æ!Y\[�à�Û�acä2[2k�ß�Û�ÞZ[fl\]\ÛRqG`�Y\[2kRlnmRlnpfì!à�`yq Û*ÞZ[�bi`yl�h�Yi[�à�Û�[2w�çgëRÞZ[:<Zæ!Y�l�hz~Z[f`�ì�Û*ÞZ[ �6æ!Y\[�à�Û�[2wack.k*~�áZk�[2ór~Z[�]iÛ�hcp_az]Zä�à�[�l�k�[2åíá\p¼`y]Zhcp_`y]Z[\ßZm.ÞZacä Þêk�`Z`y]_à�[2k*~ZhcÛ�kRaz]él\]Z`\Û*ÞZ[�à�Û�aj[fá�[2Û�mQ[2[�]Û*ÞZ[��Zæ!Y\[�à�Û�[2w|li]Zå:Û*ÞZ[!�6æ!Y\[�à�Û�[2w�çDë,ÞZajkºä2`y~Zhcå á�[»lnY\`\acå�[2å:á\píaz]Zä�à�[�l�k�a']ZbéÛ*ÞZ[ <ræ!Y�l�hz~Z[`�ì�Û*ÞZ[��6æ!Y\[�à�Û�[2w¸áip¸qõ`yà�[�Û*Þ�l\]¼`Z]Z[\ß�m.ÞZacä ÞéackRm.Þ�lxÛ.Þ�l\ã�ã�[�]Zk�a ìgÛ*ÞZ[fli]\Û.~Zk�[2k.{ ��ë�Únmaz]Zk�Û�[�l�åí`�ì'qR`Zå�[ ��`y~�]iÛ�az]Zb�ç�ë4l\áZhc[%,Gl\]Zå $�acby~�à�[ pGk*ÞZ`�mBà�[2k�~ZhjÛ�k�ì¢à�`ZqôlGk�azqG~Zhzl�Û�ac`y]

Page 15: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� 7 �

start

goal

start

goal

�Z�)�£� !� � ¡������(����������£��(��Õ�2���gÐ}�£n�����#��)k�Û�~Zå�p»`�ì�k�az]Zb\hc[Dl\]iÛ�k�Û*Þ�l�Û�~Zk�[FqR`Zå�[���`y~�]iÛ�az]Zb»l\]Zå_k�a']Zbihj[fli]\Û�k�Û*Þ�lxÛÝ~Zk�[f{ �4ë�Únmyç�&!]á�`\Û*Þ¼ä�l�k�[2knß�Û*ÞZ[fl\]iÛ�k.á�à�[�litÓÛ�ac[2k�Û�`�mRl\à�å�k �6æ!Y\[�à´Û�acä2[2k�ç

�5qndQ�no=d�j���� ëRÞZ[Á]Z~�qDá�[�à_`�ìÓ[2å�b\[ÃÛ*à�lnY\[�à�k*lxhjk|~�]iÛ�achfk�a']Zbihj[Ùli]\Û�k_Û�Þ�l�Û|~Zk�[ qR`Zå�[��`Z~�]\Û�az]Zb_à�[�l�ä2ÞílÓb\`Zl�h�Y\[�à�Û�[2w¼ì¦`yàÝÛ*ÞZ[º÷&à�k�ÛRÛ�azqG[G`y]êk�Û*à�`Z]Zb\hcp_ä2`y]�]Z[2ä2Û�[2å Ü�å�azà�[2ä2Û�[2å�âbyà�l\ã�ÞZk�ä�l\]Óá�[R[2wrã�`y]Z[�]iÛ�azl�h�az]ÓÛ*ÞZ[R]Z~�qGá�[�à�`xì&Y\[�à�Û�ajä2[2k�[2Yi[�]Ga ì}Û�ÞZ[QbZà�l\ã�ÞZkgl\à�[RãZhzl\]�l\à6ç

� ��� �/� áip»az]Zå�~Zä2Û�ac`y]é`y]êÛ*ÞZ[f]Z~�qDá�[�à�`�ì�Y\[�à�Û�acä2[2k�ç

ëRÞZack�Û*ÞZ[2`Zà�[�qôä�li]|[�lxk�achjpíá�[G[2wZÛ�[�]Zå�[2å:Û�`¼byà�`y~�ãZk�`�ì @él\]\Û�kõܨì(`yà�l\]ip_ä2`y]Zk�Û*l\]iÛ@�â2ç &$]ÓÛ*ÞZack�ä�l�k�[\ßim�[.à�[�ãZhcacä�l�Û�[�`y~�àµbyà�l\ã�Þ%@sÛ�azqG[2k�ßik�[2[ $�ajbZ~�à�[ e¼Ü�hc[�ì¦Û�â2ç�ò:[Raz]\Û�à�`yå�~Zä2[l_]Z[2m k�Û*l\à�Û�Y\[�à�Û�[2w&ßOmRacÛ*ÞÙÜ�å�azà�[2ä2Û�[2å�â�[2å�b\[2k�b\`ia']Zbêì¢à´`yqôacÛºÛ�`¸lxhjh�`ihjåÃk�Û*l\à�Û�Y\[�à�Û�ajä2[2knçëRÞZ[f]Z[2m b\`yl�h�Y\[�à�Û�[2wÓackDÜ(l\]iprâ§`y]Z[�`xì4Û*ÞZ[�`\hcå_b\`ylxh&Yi[�à�Û�acä2[2k�ç $�az]�l�hchcp\ß�m�[fà�[2å�azà�[2ä2Û�l�hch[2å�b\[2kÝì¢à�`ZqIÛ*ÞZ[�k*~Zä2ä2[2k�k�`yà�Y\[�à´Û�acä2[2kQ`xì4Û*ÞZ[�`\hcå_b\`ylxh�Y\[�à�Û�ajä2[2k,Û�`GÛ*ÞZ[è]Z[2m k�Û*l\à�Û§Y\[�à�Û�[2w�ç&¦ì @Ãl\]\Û�k.Û*Þ�lxÛf~Zk�[rq,`yå�[ �Ý`y~�]\Û�a']Zbíl\à�[Gk�Û*lià�Û�[2åíaz]íÛ*ÞZ[»]Z[2mIk�Û�l\à�Û�Y\[�à�Û�[2w&ß�[�l�ä2Þ:l\]iÛä�l\]sqG`�Y\[�Û�`fl�å�a/#�[�à�[�]iÛg`y]Z[R`xì&Û*ÞZ[R`ihjåsk�Û*l\à�Û�Yi[�à�Û�acä2[2kQl\]ZåsÛ*ÞZ[�]G[2w�ÞZa'áZacÛRÛ�ÞZ[.á�[�Þ�lnYZaj`Zàå�[2k�ä�à´a'á�[2å l\á�`�Y\[.az]sÛ*ÞZ[�ä2`y]iÛ�[2wZÛg`�ì4k�a']Zbihj[fli]\Û�knç¾XZaz]Zä2[�Û*ÞZ[�]Z~�qDá�[�à�`�ì�Y\[�à�Û�ajä2[2k�`�ì�Û*ÞZ[]Z[2mI[2w�l\qÓãZhc[Gbyà�l\ã�Þêack�`y]Zhcp l¼ä2`y]Zk�Û�l\]\ÛÝì¢l�ä2Û�`yàRhzl\à´b\[�àRÛ�Þ�l\]_Û*ÞZ[Ó]Z~�qGá�[�à�`�ìgY\[�à´Û�acä2[2k`�ì�Û*ÞZ[�`\hcåé[2wrl\qsãZhj[ºbyà�l\ã�Þ�ß�Û*ÞZ[�ì¦`ihjhc`�mRaz]Zb¼Û*ÞZ[2`yà�[�q ì(`\hchc`�mRk�ç

�:��o=�ng&g��o<C���� ëRÞZ[�Û�azqG[�ajÛ,Û*l\t�[2k�lfbiajYi[�]_]y~�qGá�[�àÝ`�ì¾l\]iÛ�kgÛ�Þ�l�ÛR~Zk�[ q,`Zå�[���`y~�]iÛ�az]Zb

Page 16: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

7 � � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

start goal

leve

l 1

r2

r1

g2

g1

g0

leve

l 2le

vel 0

pass 0(down pass)

5

1

4

1

3

pass 1(up pass)

5

1

4

5

3

pass 2(down pass)

5

1

4

5

12

pass 3(up pass)

5

7

5

7

12

pass 4(down pass)

5

7

24

16

35

pass 5(up pass)

6

8

25

29

35

number of timesthe vertex has beenvisited at the end

of the pass

1

1

1

0

1

1

1

0

1

1

1

1

1

1

0

1

1

1

3

1

1

1

1

1

1

0

1

1

1

3

5

5

1

1

1

1

5

1

1

1

4

5

5

1

1

1

1

5

7

7

7

12

16

16

1

1

1

1

5

7

7

7

24

16

16

1

1 0 1 0 1 0 1 0 1 0 1 1

number of timesthe vertex has been

visited after termination( )u4,2(g0)

�Z�)�£� !��7�+�¡&× �#��)���������(�������*�}£��(��Õ�2���gÐ��£������#��).�� #����6�# !�����(���GÑ� !��®n���§-�� ! / ��� ! 7 � 1Û�`_à�[�l�ä2Þ_lÓb\`Zl�h�Y\[�à�Û�[2wÓì(`yà�Û*ÞZ[ ÷&à�k�Û�Û�azqG[f`Z]_k�Û*à´`y]Zb\hcp»ä2`Z]�]Z[2ä2Û�[2åÙÜ�å�azà�[2ä2Û�[2å�â�byà�liã�ÞZkä�l\]|á�[f[2w�ã�`y]Z[�]iÛ�azl�h�az]êÛ*ÞZ[D]Z~�qGá�[�à�`�ì�Y\[�à�Û�ajä2[2k,[2Y\[�]_a ìgÛ*ÞZ[�byà�l\ã�ÞZk.l\à�[fãZhzl\]�lià2ç

XZaz]Zä2[|Û*ÞZ[_Û�azqG[_acÛÓÛ�l\t�[2kÓlébyà�`y~�ãÃ`�ìfl\]\Û�kfaz]ÁÛ*ÞZ[_m�`yà´k�Ûfä�l�k�[_Û�`:à�[�l�ä2Þ·líbi`yl�hY\[�à�Û�[2wõì¦`yàµÛ*ÞZ[,÷&à�k�Û�Û�azqG[�ajk�l�hc`�mQ[�à�á�`y~�]Zå_`y]sÛ*ÞZ[.ä2`�Y\[�àµÛ�azqG[�az]¼Û*ÞZ[�mQ`Zà�k�Û�ä�l�k�[\ßZÛ*ÞZ[ì¦`ihjhc`�mRaz]Zb¼ä2`yà�`\hchzl\à�p¼ì(`\hchc`�m,k�ç

�:�no=��g gi�o<C���� ëRÞZ[�ä2`�Y\[�à�Û�a'qõ[.`�ì4lºb\acY\[�]¼]y~�qGá�[�àg`xì�li]\Û�k§Û*Þ�l�Û�~Zk�[ q,`Zå�[ ��`Z~�]\Û�az]Zb`y]Ák�Û*à�`y]Zb\hcpÁä2`y]�]Z[2ä2Û�[2å ܦå�a'à´[2ä2Û�[2å�âÓbZà�l\ã�ÞZk¼ä�l\]Ùá�[ [2w�ã�`Z]Z[�]\Û�azl�h�az]LÛ*ÞZ[ ]Z~�qGá�[�à¼`�ìY\[�à�Û�acä2[2k�[2Y\[�]¼a ìgÛ*ÞZ[�byà�liã�ÞZk.l\à�[fãZhzl\]�l\à6ç

���� / � ���� � � ������ �� �� �0����� ����� � ���� ���0 � � ����� � Jfà�l\ã�ÞZkRÛ*Þ�l�Û�ä2`yà�à´[2k*ã�`y]ZåÛ�`¼Û�[�à�à�l�az]íl\à�[G`�ì(Û�[�]é~�]Zå�a'à´[2ä2Û�[2å�ß�az]Zä2hz~Zå�az]�bíbyà�acå�k�ç�&¢Û�ack�ä2`y]Zä2[2acY�l\áZhc[¼Û*Þ�l�Û�l\]\Û�k.Û*Þ�lxÛ~Zk�[Fq,`yå�[���`Z~�]\Û�az]Zb»lià�[fqG`yà�[�[)(»ä2ac[�]iÛR`y]_~�]Zå�azà�[2ä2Û�[2åébyà�l\ã�ÞZk�Û*Þ�l\]¼å�azà�[2ä2Û�[2å|byà�liã�ÞZkl\]ZåÙÛ�Þy~Zk¼Û*Þ�l�ÛÓÛ�ÞZ[2a'àsä2`�Y\[�àGÛ�azqG[íqGacbyÞiÛ¸l�hcmRlnpyk¼á�[ ã�`\hcpr]Z`ZqGazl�h.az]LÛ*ÞZ[í]y~�qGá�[�às`�ìY\[�à�Û�acä2[2k�ç &!]éÛ*ÞZack�k�[2ä2Û�ac`y]�ß�ÞZ`�m�[2Y\[�à2ß�m�[Dã�à´[2k�[�]\Û�l\]é~�]Zå�a'à´[2ä2Û�[2åí[2wrl\qsãZhj[Gbyà�l\ã�Þ_Û*Þ�lxÛk*ÞZ`�mRk�Û*Þ�l�Û�Û�ÞZ[2a'à ä2`�Y\[�à�Û�azqG[Óä�l\]ék�Û�ajhch�á�[Ó[2w�ã�`y]Z[�]iÛ�azl�h§a']ÆÜ�Û*ÞZ[Gk�ó�~�l\à�[Óà�`Z`\Û�`�ì2âRÛ*ÞZ[]y~�qGá�[�à�`�ì�Û*ÞZ[GY\[�à�Û�acä2[2kf[2Yi[�]éì¦`ZàÓÜ(ãZhzl\]�l\à�â�~�]Zå�azà�[2ä2Û�[2å Û*à�[2[2knç���[2k*ãZacÛ�[sajÛ�kf[2hc[2byl\]Zä2[\ßÛ*ÞZack.[2w�l\qÓãZhc[�byà�l\ã�Þ_ã�à�`�Y\[2å_Û�`Óá�[fà�l�Û*ÞZ[�à�å�a/(»ä�~ZhcÛ�Û�`Gä2`y]Zk�Û*à�~Zä2Ûnç�ò:[�÷&à�k�Û�k�Û�~Zå�p»Û*ÞZ[]y~�qGá�[�àR`�ì�[2å�b\[ºÛ*à�lnY\[�à�k*lxhjk�~�]\Û�ach�k�az]Zb\hc[Dl\]iÛ�kRÛ�Þ�l�Û.~Zk�[ qR`Zå�[ ��`Z~�]\Û�az]Zb_à�[�l�ä2Þ_lÓbi`yl�hY\[�à�Û�[2w¼ì(`yà,Û*ÞZ[�÷&à´k�Û.Û�a'qõ[\ç $�acby~�à´[ , �sk*ÞZ`�mRk.l\]éaz]Zk�Û*li]Zä2[D`xìg`y~�àR[2w�l\qÓãZhc[Gbyà�liã�Þ�ç &!]b\[�]Z[�à�l�h(ß�Û*ÞZ[ÓÛ�à�[2[2kDÞ�lnY\[!� ,Óhc[2Y\[2hck|ܨì¦`Zà � �@g\â2çÓëRÞZ[Óhc[2Y\[2hckGä2`y]Zk�ack�ÛG`�ì�Y\[�à�Û�acä2[2k

Page 17: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� 7 �

`�ì�Û*Þ�à�[2[Gå�a/#�[�à�[�]\Ûftiaz]Zå�k�� �Zæ!k*~�á�à�`Z`\Û�k�ß �6æ!k*~�á�à�`Z`\Û�k�ß�l\]Zåíhc[�lnì(kfÛ*Þ�l�Û�l\à�[Gä2`y]�]Z[2ä2Û�[2å:Û�`Û*ÞZ[sk*~�á�à�`Z`\Û�k�ç �Zæ!k*~�á�à�`Z`\Û�kfl\]Zå �2æ!k�~�á�à�`y`iÛ�kfl�hcÛ�[�à�]�l�Û�[\çGÚRÛ�hc[2Y\[2h . ��ß�Û*ÞZ[�à�[Óackº`y]Z[k*~�á�à´`y`\Ûnß¾]�liqG[2hcp·l �yæ!k*~�á�à´`y`\Û � ��çéÚRÛfhc[2Y\[2hck . ,��������íß�Û*ÞZ[�à�[_l\à�[¼Û�mQ`_k�~�á�à�`y`iÛ�k�ß]�l\qõ[2hjp_l\] �6æ!k*~�á�à�`Z`\Û�� � l\]Zå|l �yæ!k*~�á�à´`y`\Û � � ç¾X�~�á�à´`y`\Û � � Þ�l�k � hc[�lnì(kRä2`y]�]Z[2ä2Û�[2åÛ�`�acÛ�ßyli]ZåDk�~�á�à�`y`iÛ�� � Þ�l�k�`y]Z[�hc[�lnì�ä2`y]�]Z[2ä2Û�[2åÓÛ�`�acÛ�ç�$�az]�l�hchjp\ßZk*~�á�à´`y`\Û ���Áack�ä2`y]�]Z[2ä2Û�[2åÛ�`ºÛ�mQ`Gl�å�å�acÛ�ac`y]�l�h�Y\[�à�Û�ajä2[2knßr]�liqG[2hcpÓÛ*ÞZ[�k�Û*l\à�Û�Yi[�à�Û�[2w»l\]ZåsÛ*ÞZ[.`Z]Zhjpsb\`yl�hOY\[�à�Û�[2w�ç�ëRÞZ[Û*à´[2[2kÝÞ�lnY\[;,�. j 9 g�� � e 9 g � tjºY\[�à�Û�acä2[2k�çgXZaz]Zb\hc[Gl\]\Û�kQÛ�Þ�l�ÛQ~Zk�[Fq,`Zå�[���`y~�]iÛ�az]Zbã�à�`Zä2[2[2å:az]ílÓk�[�à�ac[2kf`�ìQã�l�k�k�[2k.Û�Þ�à�`y~ZbyÞéÛ*ÞZ[GÛ*à�[2[2knç��f~�à�az]Zb_[�l�ä2Þíã�l�k�k�ß�Û*ÞZ[2p_Û�à�lnY\[�à�k�[Û*ÞZ[�k�~�á�à�`y`iÛ�k�a']¼Û�ÞZ[�`yã�ã�`\k�ajÛ�[�`yà�å�[�à§Û*Þ�l\]¼Û*ÞZ[�ã�à�[2YZac`y~Zkèã�lxk�k�ç�ò:[�ä�l�hch4lGã�l�k�k�å�~�à�az]Zbm.ÞZacä2Þ Û*ÞZ[2p:Û*à�l�Yi[�à�k�[GÛ*ÞZ[¼k*~�á�à�`Z`\Û�kfaz] å�[2k�ä2[�]Zå�az]Zb:`Zà�å�[�àGl_å�`�m.] ã�l�k�k�ß�l\]ZåÁl|ã�l�k�kå�~�à�az]ZbÁm.ÞZacä2ÞÁÛ*ÞZ[2pÁÛ�à�lnY\[�à�k�[_Û*ÞZ[�q a']Æl�k�ä2[�]Zå�a']ZbÁ`Zà�å�[�àÓl\]L~�ãÙã�l�k�knçÙò:[í]y~�qGá�[�àã�l�k�k�[2kRì!à�`yq o2[�à�`»`Z]|~�ãimÝlià�å�ß�k�`¼[2Y\[�]|ã�lxk�k�[2k�l\à�[�å�`�m.]|ã�lxk�k�[2k�l\]Zå|`Zå�å:ã�l�k�k�[2k.lià�[~�ãéã�l�k�k�[2k�ç§ÚIã�l�k�k�[�]Zå�k.azqÓqG[2å�azl�Û�[2hjpíá�[�ì(`yà�[ºajÛ�k�mRacÛ�ä2ÞZ[2k�å�a'à´[2ä2Û�ac`y]Zk�ç�ò:[fá�à�[�lit¼Û�ac[2kl�k§ì(`\hchc`�m,k�� �f~�à�az]Zb»ã�l�k�kQo2[�à´`�ßZÛ�ac[2k.l\qG`y]ZbGk*~Zä2ä2[2k�k�`yà�Y\[�à´Û�acä2[2kÝlià�[fá�à�`yt�[�]saz]sì¢lnY\`Zàg`xìhc[�lnìRY\[�à�Û�ajä2[2kº`�ì��yæ!k�~�á�à�`y`iÛ�kõܦmRacÛ*Þ:ÞZacbyÞZ[2k�ÛÓã�à�ac`yà�acÛ�p�â�l\]Zå:Û*ÞZ[�]ík�~�á�à�`y`iÛ�k�ç l�l�k�k�o2[�à´`[�]Zå�kGm.ÞZ[�]ÃÛ*ÞZ[Ók�az]Zb\hc[_l\]iÛ�kDÞ�lnY\[ÓYZack�acÛ�[2å Û*ÞZ[Óhc[�lnì(kD`xìÝk�~�á�à�`y`iÛ � �s`y]Zä2[Ó[�l�ä2Þ�ç¼Ú.kGlà�[2k�~ZhjÛnß&l�Û�Û*ÞZ[º[�]Zå|`�ìQã�l�k�kRo2[�à�`¼Û*ÞZ[2p|Þ�lnY\[fYZack�acÛ�[2åíÛ*ÞZ[fk*~�á�à´`y`\Û�k�� � � ,fÛ�a'qõ[2k[�l�ä2Þ|li]Zå|Û*ÞZ[2azà,hj[�l�ì¦k�`y]Zä2[º[�l�ä2Þ�çRëRÞZ[2p|Þ�lnY\[�Yyack�acÛ�[2åíÛ*ÞZ[ºk*~�á�à�`Z`\Û�k � � `y]Zä2[f[�lxä Þél\]ZåÛ*ÞZ[2azàsk*~�á�à�`Z`\Û�k_]Z`\Û»l�Û_l�hch(ç �f~�à�az]ZbÙl�hchRk*~�áZk�[2ó�~Z[�]iÛ|ã�l�k�k�[2k�ß�Û�aj[2k_l\qG`Z]Zb:k*~Zä2ä2[2k�k�`ZàY\[�à´Û�acä2[2kõlià�[¸á�à´`yt�[�]Ãa']Ãì!lnY\`yà�`�ì�k*~�á�à�`Z`\Û�kfm�ÞZ[�]Z[2Y\[�àÓã�`ik�k�azáZhc[\ç:ÚôÛ�ac[_á�[2Û�m�[2[�]:Û�m�`�6æ!k*~�á�à�`Z`\Û�k|Ü�m�ÞZ[�] Û*ÞZ[_k�az]Zb\hc[ l\]iÛ�kGl\à�[|l�ÛÓl �yæ!k�~�á�à�`y`iÛ*â�ack»à�[2k�`\hcY\[2å áip:ä2`y]iÛ�az]y~Zaz]ZbmRacÛ*ÞsÛ*ÞZ[�ä�~�à�à´[�]\Û�ã�l�k�k�ç�Ú Û�ac[èá�[2Û�m�[2[�]¼Û�m�` �yæ!k�~�á�à�`y`iÛ�kèÜ�m�ÞZ[�]ÓÛ*ÞZ[�k�az]Zb\hc[Dli]\Û�k�l\à�[�l�Ûl\] �6æ!k*~�á�à�`Z`\Û*â§ackRà�[2k�`\hcY\[2å|áipÓÛ�[�à�qõa']�lxÛ�az]ZbÓÛ*ÞZ[�ä�~�à�à�[�]iÛÝã�lxk�kRl\]Zå»k�Û*l\à�Û�az]ZbÓlG]Z[2mÆ`y]Z[az]_Û�ÞZ[�`yã�ã�`\k�ajÛ�[få�azà�[2ä2Û�ac`y]�ç &!]êÛ*ÞZ[�ì¦`ihjhc`�mRaz]Zb�ß�m�[fã�à�`�YZacå�[DlGk*tx[2Û�ä Þs`�ì�Û*ÞZ[fã�à�`Z`�ì�Û*Þ�l�Ûk�az]Zb\hc[Ól\]iÛ�k�Û*Þ�l�Û.~Zk�[FqR`Zå�[ ��`Z~�]\Û�az]ZbÓÛ�à�lnY\[�à�k�[ÓÜ(l�Û�hc[�l�k�Û*â

��- �� -�/

9���� 1�� 1[2å�b\[2ksa'] Û�ÞZajk¼ä�l�k�[\ßµm.ÞZ[�à�[�,LackÓÛ*ÞZ[|]Z~�qDá�[�àG`�ì.Yi[�à�Û�acä2[2k¸li]Zå ����� , 9 ðéajk¼l\]l\à�áZajÛ�à�l\à�achcp_k*qÓl�hch�ä2`Z]Zk�Û*l\]iÛ�ç

ò:[¼~Zk�[»Û�ÞZ[Gì¦`ihjhc`�mRaz]Zb:]Z`\Û*l�Û�aj`Z]éì¦`Zà�Û*ÞZ[»ã�à´`y`�ì�ç_{�[2Û <� /2 �DÜ$ nâ�å�[�]Z`\Û�[ÓÛ*ÞZ[¼Û�`\Û*l�h]Z~�qGá�[�à,`�ìgÛ�azqG[2k�Û*Þ�l�Û�k*~�á�à�`Z`\Û2 ÓÞ�l�k�á�[2[�]í[�]iÛ�[�à�[2åíl�ÛRÛ*ÞZ[º[�]Zå `xìQã�l�k�k��êì¦`Zà.lGÛ�à�[2[mRacÛ*Þ � ,�hc[2Y\[2hck�çQe§pGå�[�÷&]ZacÛ�ac`y]�ß<� /2 �DÜ$ nâ�ajkRlº]Z`y]Zå�[2ä�à�[�l�k�az]Zbºì!~�]Zä2Û�ac`y]Ó`xì���ç *�~�à§Û�ac[á�à�[�lit\az]Zb à�~Zhj[2k�bZ~�l\à�l\]iÛ�[2[fÛ�Þ�l�Ûèlxhjh�hc[�lnì(k�`�ìRlÓk*~�á�à´`y`\ÛfÞ�lnY\[Gá�[2[�]í[�]iÛ�[�à�[2åíÛ*ÞZ[Gk*liqG[]Z~�qGá�[�à�`xìÝÛ�a'qõ[2kDl�Û�Û*ÞZ[s[�]Zå·`�ì�[�l�ä2Þ:ã�l�k�k�ß�k�`|m�[Ghc[2Û��� /2 �fÜ$ �â å�[�]Z`\Û�[ÓÛ�ÞZ[»]y~�qGá�[�à`�ì�Û�azqG[2k�Û*Þ�l�Û�[�l�ä2Þé`�ìgÛ*ÞZ[Ghc[�lnì¦kº`�ìgk*~�á�à�`Z`\Û: ÓÞ�lxkfá�[2[�]í[�]iÛ�[�à�[2å:l�Û�Û*ÞZ[G[�]Zå:`�ìQã�l�k�k� ì¦`yàºléÛ*à�[2[ÓmRacÛ*Þ � ,Óhc[2Y\[2hck�ç"$�az]�l�hchcp\ßghc[2Û V� /2 �DÜ$ nâ�å�[�]Z`\Û�[»Û*ÞZ[¼Û�`\Û�l�hg]y~�qGá�[�àf`xìÛ�azqG[2k�Û�Þ�l�Û.k�~�á�à�`y`iÛ ÓÞ�lxkfá�[2[�]í[�]iÛ�[�à�[2åéì!à�`yq ]Z`y]iæ!hc[�lnì¦kflxÛ.Û*ÞZ[G[�]Zåí`�ìQã�l�k�k��_ì(`yà�lÛ*à´[2[.mRacÛ*Þ � ,�hc[2Y\[2hck�çgëRÞZ[2k�[�Y�l�hz~Z[2k.à�[2hzl�Û�[fl�k§ì¦`\hchc`�mRk���ëRÞZ[�Û�`iÛ*l�h�]Z~�qDá�[�à�`�ì4Û�a'qõ[2kÛ*Þ�lxÛDl¼k*~�á�à�`Z`\ÛºmÝlxkf[�]iÛ�[�à�[2å·l�ÛºÛ*ÞZ[Ó[�]Zå `�ìÝã�lxk�k��ÃajkG[2ó�~�l�hgÛ�`_Û*ÞZ[¸ã�à´`yå�~Zä2Ûº`�ìRÛ*ÞZ[]Z~�qGá�[�à,`�ìgacÛ�k.hc[�lnì(k�l\]Zå|Û*ÞZ[ºÛ�`\Û*lxh}]Z~�qGá�[�à�`�ì�Û�azqG[2k�Û*Þ�l�ÛRacÛ�mRl�kR[�]iÛ�[�à�[2åêì¢à´`yq [�l�ä2Þ`�ì�ajÛ�kÓhc[�lnì¦k_l�ÛGÛ*ÞZ[_[�]ZåÁ`�ì�ã�l�k�k��Bܦm.ÞZacä Þ [2ó�~�l�hckÓÛ*ÞZ[_Û�`iÛ*l�hQ]Z~�qGá�[�àG`xìèÛ�azqG[2ksÛ*Þ�l�Û[�l�ä2Þ¼`�ì4acÛ�k,hj[�l�ì¦k�mRl�kg[�]iÛ�[�à�[2åíl�Û�Û*ÞZ[�[�]Zå_`�ì¾ã�l�k�k���â�ãZhz~Zk.Û*ÞZ[�Û�`\Û*l�h�]y~�qGá�[�àÝ`�ì¾Û�a'qõ[2k

Page 18: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

7 � � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

Û*Þ�l�Û�Û*ÞZ[Gk*~�á�à�`Z`\Û�mÝlxk.[�]iÛ�[�à�[2åéì!à�`yqô]Z`y]iæ!hc[�lnì(kDlxÛ.Û*ÞZ[º[�]Zå:`�ìQã�l�k�k ��ç $O`yàR[2w�l\qÓãZhc[\ß< /2 �GÜ � � â . Ü � â �� 2 � Ü � � â CV /2 �GÜ � � â2çò:[Q]Z`�m:`Z~ZÛ�hca']Z[,Û*ÞZ[�acå�[�l�`�ì�Û*ÞZ[Qã�à´`y`�ì�çAl�lxk�k ��ack�Û*ÞZ[µ÷&à�k�Û4ã�l�k�k4å�~�à�az]Zb�m.ÞZacä ÞºÛ*ÞZ[k�az]Zb\hc[Dl\]iÛ�k�qG`�Y\[�ì!à�`yq k*~�á�à�`Z`\Û � , Û�`fk*~�á�à�`Z`\Û � � ç�ëRÞZ[:<ræ!Y�l�hz~Z[�`�ì�k*~�á�à�`Z`\Û � � l�ÛgÛ*ÞZ[[�]ZåÓ`�ì&ã�l�k�k ��ack+< � 2 �DÜ � � â . � %,\ç &!]ºÛ*ÞZ[§ì(`\hchc`�m,a']ZbOß\m�[gk*ÞZ`�m Û*Þ�l�Û�[2Y\[�à�pºk*~�áZk�[2ó�~Z[�]\Ûã�l�k�kGå�~�à�az]Zbím.ÞZacä Þ Û�ÞZ[Ók�az]Zb\hc[|l\]iÛ�kDqG`�Y\[õì¢à�`Zq k*~�á�à�`Z`\Û$�&,fÛ�`ék*~�á�à�`Z`\Û � �»lxÛfhc[�l�k�Ûå�`y~�áZhc[2k_Û*ÞZ[k<ræ!Y�l�hz~Z[í`�ì�k*~�á�à�`Z`\Û�� ��ç ëRÞZ~Zk�ß�Û*ÞZ[éÛ�`\Û*lxhÝ]Z~�qGá�[�àG`xìf[2å�b\[_Û*à�lnY\[�à�k�l�hckbyà�`�mRk�[2w�ã�`y]Z[�]iÛ�azl�hchcp»az]sÛ*ÞZ[.]Z~�qGá�[�à�`xì&å�`�m.]Óã�l�k�k�[2kgÛ*Þ�lxÛgqG`�Y\[�Û*ÞZ[Rk�az]Zb\hc[�l\]\Û�k�ì¢à�`Zqk*~�á�à�`Z`\Û �&,.Û�`Ók*~�á�à�`Z`\Û � �\ç!$�`yà,Û*ÞZ[Gã�à�`y`xì´ß�l�k�k*~�qõ[fÛ*Þ�l�Û�Û*ÞZ[ºk�az]Zb\hc[»l\]iÛ�kèqõ`�Yi[�ì¢à�`Zqk*~�á�à�`Z`\Û � ,&Û�`,k*~�á�à�`Z`\Û�� �4å�~�à´a']Zb�å�`�m.]�ã�l�k�k � � ��ç�ÚRÛ�Û*ÞZack¾ã�`\az]\Û�a']ºÛ�azqG[\ßnÛ*ÞZ[+<ræ!Y�l�hz~Z[`�ì�k*~�á�à�`Z`\Û�� �fqG~Zk�Û.á�[ºhc[2k�k.Û�Þ�l\]¼`yà�[2ó�~�l�h�Û�`GÛ*ÞZ[W<Zæ!Y�l�hz~Z[D`xì¾Û*ÞZ[�hc[�lnìg`xì¾k*~�á�à´`y`\Û �&,2ß`\Û*ÞZ[�à�m,ajk�[fÛ*ÞZ[ºk�az]Zb\hc[õli]\Û�k,ä2`y~Zhcå ]Z`\Û�qG`�Y\[�Û�`Ók*~�á�à�`Z`\Û � �»Ü$< � , 2 �fÜ � �2â � �� /2 �fÜ�� ,´â�â6ç$&~�à�Û�ÞZ[�à�qG`yà�[\ßOl�Û�Û*ÞZackèã�`\az]\Û�az]_Û�azqG[\ß�Û�ÞZ[ <Zæ!Y�l�hz~Z[G`�ì4Û*ÞZ[ºhj[�l�ìg`�ì4k*~�á�à�`Z`\Û � , qG~Zk�Û.á�[hc[2k�kfÛ*Þ�li]í`yà�[2ó�~�l�h�Û�`_Û*ÞZ[8<Zæ!Y�l�hz~Z[»`xìQk*~�á�à�`Z`\Û � ,�ß�á�[2ä�l\~Zk�[¼Û*ÞZ[Gk�az]Zb\hc[_l\]\Û�kfÞ�l�Yi[DÛ�`qG`�Y\[gÛ�`�k*~�á�à�`Z`\Û � ,4m.ÞZ[�]Z[2Yi[�à�Û*ÞZ[2pGl\à�[gaz]GacÛ�k�hc[�lnìgÜ��� /2 �fÜ�� ,´â �Z< 2 � Ü �&,�â�â2ç4ë,ÞZ[Rå�`�m.]ã�l�k�k�ackg`�Y\[�àµm.ÞZ[�]ÓÛ�ÞZ[Rk�az]Zb\hc[fl\]iÛ�kgk�m,ajÛ�ä ÞGå�azà�[2ä2Û�aj`Z]_l\]Zå¸qõ`�Yi[gì!à�`yq k*~�á�à�`Z`\Û � �Rá�l�ä2tÛ�`Ók*~�á�à´`y`\Û�� ,�ç4Ú,ÛRÛ*ÞZackèã�`\az]\Û,a']êÛ�azqG[\ß�Û*ÞZ[:<ræ!Y�l�hz~Z[G`�ì4k�~�á�à�`y`iÛ �&,RqG~Zk�Û.á�[ºhc[2k�kRÛ*Þ�l\]`yàº[2ór~�lxhQÛ�`|Û*ÞZ[�<ræ!Y�l�hz~Z[_`�ì.[�lxä ÞÃ`�ì.Û*ÞZ[¼hc[�lnì(kÓ`�ìRk*~�á�à�`Z`\Û � � ß4`iÛ*ÞZ[�à�mRack�[_Û*ÞZ[¼k�az]Zb\hc[l\]iÛ�k�ä2`y~Zhcå·]Z`\ÛfqG`�Y\[GÛ�`_k�~�á�à�`y`iÛ �&,õÜE< /2 �DÜ �&,�â � �� /2 �fÜ � ��â�â2ç$�~�à�Û*ÞZ[�à�qõ`yà�[\ß�l�Û�Û*ÞZackã�`\az]iÛRaz]_Û�azqG[\ß � Û�azqG[2k,Û*ÞZ[ <ræ!Y�l�hz~Z[º`�ì4[�l�ä2Þ¼`�ì4Û*ÞZ[�hc[�lnì(k�`�ì�k*~�á�à�`Z`\Û � ��qG~Zk�ÛQá�[fhc[2k�kÛ*Þ�l\]¼`yàÝ[2ór~�l�h�Û�`ÓÛ*ÞZ[:<ræ!Y�l�hz~Z[f`xì¾k*~�á�à´`y`\Û � ��ß�á�[2ä�l\~Zk�[fÛ�ÞZ[fk�az]Zb\hc[Dli]\Û�k�Þ�l�Yi[�Û�`ÓqG`�Y\[Û�`ík*~�á�à�`Z`\Û � �Óm�ÞZ[�]Z[2Y\[�àGÛ*ÞZ[2p·lià�[Óaz] `y]Z[¼`�ì.acÛ�k!�ïhc[�lnì(k|Ü�� �� 2 �èÜ � �2â � < 2 �DÜ � � â6çlg~ZÛ�Û�`\b\[2Û�ÞZ[�à�����< � , 2 � Ü � � â � � � /2 � Ü�� , â � �D< /2 � Ü�� , â � � � /2 � Ü � � â � < /2 � Ü � � âì¦`Zà�� � g�ß�m.ÞZacä2Þ|ã�à�`�Y\[2k�Û*Þ�lxÛR[2Y\[�à�p_ã�l�k�kRå�~�à´a']Zb¼m.ÞZacä2Þ_Û*ÞZ[�k�az]Zb\hc[Gl\]\Û�k.qG`�Y\[,ì¢à�`Zqk*~�á�à�`Z`\Û �&,�Û�`Rk�~�á�à�`y`iÛ � �gl�Û�hc[�l�k�Û�å�`y~�áZhc[2k�Û*ÞZ[ <ræ!Y�l�hz~Z[�`�ì�k*~�á�à�`Z`\Û � ��ç�ëRÞy~Zknß�Û*ÞZ[gÛ�`iÛ*l�h]y~�qGá�[�àg`xì&[2å�b\[RÛ�à�lnY\[�à�k*l�hck�byà´`�m,k4[2w�ã�`y]Z[�]iÛ�azl�hchcp»az]sÛ*ÞZ[.]Z~�qGá�[�à�`xì}å�`�m.]Gã�l�k�k�[2kgÛ*Þ�lxÛqG`�Y\[�Û*ÞZ[ºk�az]Zb\hc[Dl\]iÛ�k�ì!à�`yq k�~�á�à�`y`iÛ �&,�Û�`Ók*~�á�à´`y`\Û � �\ç§ë�`Ó[�]Zk�~�à�[fÛ*Þ�lxÛRÛ*ÞZackè]Z~�qGá�[�àackQhzl\à´b\[\ßZmQ[�]Z[2[2å_Û�`f[�]Zk*~�à´[.Û*Þ�lxÛgÛ*ÞZ[.]Z~�qGá�[�à§`�ì�å�`�m.]¼ã�l�k�k�[2k§ajk�hzl\à�bi[èl\]Zå¼Û*ÞZ~ZkgÛ*Þ�lxÛqÓl\]ip¸~�ãéã�l�k�k�[2k,[�]Zå á�[�ì(`yà�[�Û�ÞZ[fb\`ylxh&Yi[�à�Û�[2wÓack�à�[�l�ä2ÞZ[2å�ç�&!]êÛ*ÞZ[�ì¦`ihjhc`�mRaz]Zb�ßOm�[�k*ÞZ`�mÛ*Þ�l�ÛR[2Yi[�à�p_Û�azqG[Dli]|~�ã|ã�lxk�k�[2k�à�[�l�ä2ÞZ[2kèlsk*~�á�à�`Z`\Û � � - ,�ì(`yà,Û*ÞZ[fk�[2ä2`Z]Zå|Û�azqG[\ßOÛ*ÞZ[D~�ãã�l�k�kº[�]Zå�k�ç l�l�k�k�¼ackfÛ*ÞZ[G÷&à´k�Ûfã�l�k�k�å�~�à�az]Zbém.ÞZacä ÞíÛ�ÞZ[Ók�az]Zb\hc[¸li]\Û�k�YZajk�ajÛGk*~�á�à´`y`\Û � �ì¦`Zà , � � �íçºÚ�k�k*~�qG[GÛ*Þ�l�Û�Û�ÞZ[Dk�[2ä2`y]Zå·ã�l�k�k�å�~�à�az]Zb|m�ÞZajä2ÞíÛ*ÞZ[sk�az]Zb\hc[»l\]iÛ�k�Yyack�acÛk*~�á�à�`Z`\Û � � ackQã�l�k�k ��ç�ëRÞZackRã�l�k�k�qD~Zk�Ûgá�[�l\]Ó~�ãÓã�l�k�kgk�az]Zä2[�ã�l�k�k+� ackQl�å�`�m.]sã�l�k�k�ç &!]Û*ÞZ[�ì(`\hchc`�mRaz]Zb�ß�m�[�k*ÞZ`�m Û�Þ�l�Û�Û*ÞZ[fk�az]Zb\hc[Ól\]iÛ�k�ß�lnì(Û�[�à�Û*ÞZ[2p_Þ�lnY\[fqG`�Y\[2å¼ì!à�`yq k�~�á�à�`y`iÛ� � Û�` k�~�á�à�`y`iÛ � � ì¦`ZàfÛ�ÞZ[Ó÷&à�k�ÛÓÛ�azqG[\ß�ä2`y]iÛ�az]Z~Z[_Û�`:qG`�Y\[»Û�` k�~�á�à�`y`iÛ�� � -,Ól\]ZåÁÛ*ÞZ[�][�]Zå:Û*ÞZ[Ó~�ã:ã�l�k�kèl\]ZåÃk�Û*l\à�Û�l¼å�`�m�] ã�lxk�k�çfëRÞZ~Zk�ß�[2Y\[�à´p|Û�azqG[Ól\]í~�ã:ã�l�k�kfà�[�l�ä2ÞZ[2kflk*~�á�à�`Z`\Û!� � - , ì(`yà�Û*ÞZ[sk�[2ä2`y]ZåÁÛ�azqG[\ß�Û*ÞZ[_~�ã·ã�lxk�kf[�]Zå�k�ç ��`y]Zk�[2ór~Z[�]iÛ�hcp\ß�Û*ÞZ[�à�[¼l\à�[_lhzl\à�b\[f]Z~�qGá�[�à�`�ì�~�ã_l\]Zå_Û*ÞZ~ZkÝlxhjk�`Då�`�m.]_ã�lxk�k�[2k�ç�$�`ZàgÛ�ÞZ[fã�à�`y`xì´ß�]Z`\Û�acä2[�Û*Þ�lxÛ�ßyl�Û�Û*ÞZ[[�]Zå:`�ìQã�l�k�k���ß�Û*ÞZ[8<Zæ!Y�l�hz~Z[2kf`�ì�k*~�á�à�`Z`\Û � � ß�k�~�á�à�`y`iÛ � � - ,�ß�l\]ZåíÛ*ÞZ[Ghc[�lnì(k�`�ìQk�~�á�à�`y`iÛ� � lià�[Ó`y]Z[\ç_ëRÞZ[ <ræ!Y�l�hz~Z[_`�ì�k*~�á�à�`Z`\Û � � ack�� ,\ß�l\]Zå:Û*ÞZ[�<Zæ!Y�l�hz~Z[»`xìÝk�~�á�à�`y`iÛ� � - ,Óack � g�ç qR`�m l�k�k*~�qG[_Û*Þ�l�ÛGÛ*ÞZ[ék�az]Zb\hc[ l\]iÛ�k»qõ`�Yi[Óì!à�`yq k*~�á�à�`Z`\Û�� � Û�`k*~�á�à�`Z`\Û�� � ì(`yà�Û*ÞZ[�÷&à�k�ÛRÛ�azqG[\ç§ÚRÛRÛ*ÞZack�ã�`\az]iÛ.az]éÛ�azqG[\ß�k*~�á�à´`y`\Û � � - , l\]Zå_Û*ÞZ[ºhc[�lnì¦k�`�ìk*~�á�à�`Z`\Û � � Þ�l�Yi[õ]Z`iÛDá�[2[�]ÃYZajk�ajÛ�[2å·l�byl�az]ík�a']Zä2[_ã�l�k�k �_l\]Zå:Û*ÞZ[2azà <Zæ!Y�lxh'~Z[2kÓl\à´[ÓÛ*ÞZ~Zk

Page 19: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� 7 �

k�Û�ajhch�`y]Z[\ç�X�~�á�à�`Z`\Û � � Þ�l�kèá�[2[�]íYZajk�ajÛ�[2å `Z]Zhjpé`y]Z[GÛ�azqG[Ólnì(Û�[�àRã�l�k�k�_l\]ZåéajÛ�k:<Zæ!Y�lxh'~Z[ackÓÛ*ÞZ~Zk�� g�çÁX�~�á�à�`Z`\Û!� � Þ�l�kÓá�[2[�]ÁYZack�acÛ�[2å lxÛÓhc[�l�k�ÛÓ`y]Zä2[|l�ì¦Û�[�àGã�l�k�k �Ãl\]ZåÛ*ÞZ~Zk�ajÛ�k2<Zæ!Y�l�hz~Z[Dackfl�Û,hj[�lxk�ÛRÛ�m�`�ç�Ú�ä2ä2`yà�å�az]Zb_Û�`Ó`y~�àÝÛ�ac[Dá�à�[�l\tiaz]Zb¸à�~Zhj[2knß�Û*ÞZ[G~�ã|ã�l�k�k�Ãä2`y]\Û�a']Z~Z[2kÓl\]Zå Û*ÞZ[Ók�az]Zb\hc[|l\]iÛ�kDqõ`�Yi[fì!à�`yq k*~�á�à�`Z`\Û � � Û�`ék*~�á�à�`Z`\Û�� � - , ç_ÚRÛºÛ*ÞZackã�`\az]\ÛGaz]íÛ�azqG[\ß�Û*ÞZ[ <ræ!Y�l�hz~Z[¼`�ìRk*~�á�à�`Z`\Û � � ajkºk�Û�achch�� g·Ü�k�[2[»l\á�`�Y\[�â6çÓX�~�á�à�`y`iÛ� � - ,ÝÞ�l�k�]Z`\ÛRá�[2[�]¼YZack�acÛ�[2å|l�byl�az]Gk�a']Zä2[fã�lxk�k��fl\]Zå¼acÛ�kI<Zæ!Y�lxh'~Z[�ackRÛ�Þy~Zk§k�Û�achch � g�çÚ�ä2ä2`yà�å�az]ZbíÛ�`|`Z~�à�Û�ac[_á�à�[�l\tia']Zb:à�~Zhc[2k�ß�Û*ÞZ[_]Z[2wZÛfk*~�á�à�`Z`\ÛºÛ*Þ�l�Û�Û*ÞZ[_l\]iÛ�kfqG`�Y\[ÓÛ�`éackk*~�á�à´`y`\Û � � çgëRÞZ~Zk�ßZÛ*ÞZ[G~�ã_ã�l�k�kR[�]Zå�k�l\]Zå|Û�ÞZ[�k�az]Zb\hc[Dl\]iÛ�k.qG`�Y\[�Û�`�mRl\à�å�k�k*~�á�à�`Z`\Û�� �l�bZl�az]�ç+��`Z]Zk�[2ór~Z[�]iÛ�hcp\ßOÛ*ÞZ[�à�[Dlià�[flÓhzl\à´b\[f]y~�qGá�[�àR`�ìg~�ã|li]Zå|Û*ÞZ~Zkèlxhjk�`Óå�`�m.]|ã�l�k�k�[2k�çëRÞZ[¼ã�~�à�ã�`\k�[»`�ì�Û*ÞZ[»ã�l\à´Û�acä�~Zh'lià�Û�`yã�`\hc`\b\pí`�ì�Û*ÞZ[Óbyà�liã�Þ|Û�ÞZ[�]íackDÛ�`_[�]Zk*~�à�[ÓÛ*Þ�lxÛ�Û*ÞZ[k�az]Zb\hc[|l\]iÛ�kfqG`�Y\[õì¢à�`Zq k*~�á�à�`Z`\Û$�&,�Û�`|k�~�á�à�`y`iÛ � �Óå�~�à�az]Zbí[�l�ä2Þíå�`�m.]:ã�l�k�kÓl\]ZåÃå�`]Z`\Û�k�Û�`yãGÛ*ÞZ[�å�`�m.]»ã�lxk�k�[�l\à�hcac[�à2ç�&!]¼Û�ÞZ[Rì(`\hchc`�m,a']ZbOßym�[�ã�à�`�Yyacå�[fl k*t�[2Û�ä2ÞÓ`xì}Û�ÞZ[èã�à�`Z`�ìÛ*Þ�lxÛRÛ*ÞZackRack.az]Zå�[2[2åíÛ*ÞZ[�ä�l�k�[\ç

� dj jk����� Ú.k�k*~�qG[gÛ�Þ�l�Û�k�az]Zb\hc[Rl\]iÛ�k�Û*Þ�l�Û�~Zk�[_qR`Zå�[+��`Z~�]\Û�az]Zb�Yyack�acÛ�k*~�á�à´`y`\Û èܦmRacÛ*Þ 4. ���.â,å�~�à�az]Zb ã�l�k�k���ß�m.ÞZ[�à´[ � � � � g�� g�ç�ëRÞZ[GY�l�hz~Z[2k < 2 �DÜ$ �â�ä�li]_Û*ÞZ[�]íá�[ä�l�hcä�~Zhzl�Û�[2å:l�kÝì¦`\hchc`�mRk�ßiì(`yà � �_�

����� ����� �-� ��1 ! � ������� §-�����1�'�������� �-� ��1� ����� -� � 1 ! � � ����� -�� � 1�'�� ����� -� � 1

� ����� -���z1 ! -�� '��z1n�§�#��-�� ��� � ��� -����z1 � � ����� -���� ��� 1(1�'�� ����� -���z1����� ����� Ý-� � 1 ! -�� '��z1n�§�#��-�������� §-�� � ����1 � ����� ����� §-�� � 1(1�'������ �#��� §-� � 1������� §-�� � 1 ! �§�#��-������ � ��� §-� � � �¢1 � ������� Ý-� � 1(1�'�������� µ-�� � 1� ��� ����� -����z1 ! �§�#��-�� ����� -���z1 � � ��� ����� -��� � � 1(1 '�� ��� ����� -����01

� ��� �/� áip�az]Zå�~Zä2Û�ac`y]G`y] Û*ÞZ[Q]Z~�qGá�[�à�`�ì&ã�l�k�k�[2k �èì¦`yà�Û*ÞZ[§[2wrl\qsãZhj[�byà�l\ã�ÞZk�ßnk�[2[�îfg�rnñ�ç

�5qndQ�no=d�j ����� &�ì ��. g�@¼ì(`yà�� � @ � �íß�Û*ÞZ[�]éÛ*ÞZ[få�`�m.]_ã�l�k�k.[�]Zå�kfl�Û,k*~�á�à�`Z`\Û � �l\]ZåéacÛ.ÞZ`\hcå�k.Û�Þ�l�Û

Page 20: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

/�+ � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

� ����� -���z1 !��� �� ��-������ � ��� §-� � 1�'Ó/���1�'�� 'ê7-�� ' �z1$-�� ��� � ��� -����1�'Ó/�� �õ/ �'§7�1�'Ó/�� �Ó/ ��'ê7� '���'¼7

�z� � ! +���z� 4+� ����������$ !³���(�

������� �-�� � 1 ! � � ��� � ��� -� � � � 1�'Ó/�� �D/ � '¼/7�z� 4+� ����������$ !³���(�

� ����� -���z1 ! � � 'ê7/�� �D/ ��'¼77

�z� � ! +�z� 4+� ����������$ !³���(�

� ����� -����z1 ! � /�� �õ/ ��'¼/7�z� 4+� ����������$ !³���(�

� ����� -����z1 ( � ��� � ��� -���� � � 1 �z� g7�� ����� ����� -���z1�� � ��� � ��� -� � � � 1 �z� 4+� ����&�ì ��. g�@ ,ºì(`yà � � @ ���íß�Û*ÞZ[�]éÛ*ÞZ[Ó~�ã ã�lxk�k�[�]Zå�kfl�Û�k*~�á�à�`Z`\Û ��� - , Ü�mRacÛ*Þ

Û*ÞZ[f[2wZä2[�ãZÛ�ac`y]ê`�ìµ~�ã_ã�lxk�kdg�� ,.Û�Þ�l�ÛR[�]Zå�k�l�Û�Û*ÞZ[fb\`Zl�h�Y\[�à´Û�[2wrâ,l\]Zå_acÛèÞZ`\hcå�k�Û*Þ�l�Û

����� ����� �-� � 1 !����� ����� ����� -� �z1-�� '��z1$-�� ��� � ��� -���z1�'Ó/���4/ �z1 's/�� �D/ ��'Ó/� '���'Ó/� '���'¼7

�z� � ! +�z� �+� ������z� �+� � ! �������$ !³���(�

� ��� �#��� -�� �¨1 !�� ������� §-� � 1�'»/�� �õ/ ��' �� ��� ����� -� � � � 1�'Ó/7�z� �+� ������z� � ! ��'¼7�� �������$ !³���(�

����� �#��� §-� � 1 !�� ��'¼7/�� �D/ ��'»/7

�z� � ! +�z� �+� �����������$ !³���(�

����� ����� §-�� � 1 !�� /�� �D/ ��' �/7

�z� �+� ������z� � ! ��'¼7�� �������$ !³���(�

� ��� �#��� -�� �¨1 ( � ����� -���� ��� 1 �z� �+� ���������� ����� �-� � 1�� ������� �-� � ���!1 �z� �+� ������ � ��� � á\péaz]Zå�~Zä2Û�ac`y]:`Z]_Û*ÞZ[Ó]Z~�qDá�[�à�`�ìQÛ�à�lnY\[�à�k�[2å_[2å�b\[2k�ì(`yà�Û*ÞZ[º[2wrliqÓãZhc[Óbyà�l\ã�ÞZknß~Zk�az]Zb¸{}[�qÓqÓlse�ßZk�[2[»îfg@rnñ�ç

ëRÞZ~Zk�ß�Û*ÞZ[�à�[|lià�[ g � g:ã�l�k�k�[2kÓ~�]\Û�ach.léb\`Zl�h�Y\[�à�Û�[2w:ack»à´[�l�ä ÞZ[2å ì(`yàºÛ*ÞZ[¼÷&à�k�ÛÛ�azqG[\ç �gl�ä2Þíå�`�m.]:ã�l�k�kD[�]Zå�kÓl�Ûºk*~�á�à�`Z`\Û � ��ç|ëRÞZ[s÷&]�l�hQ~�ã ã�l�k�kG[�]Zå�kÓl�ÛºÛ*ÞZ[»bi`yl�hY\[�à�Û�[2w�çRÚ�hch4`iÛ*ÞZ[�à.~�ãíã�l�k�k�[2k �é[�]Zå:l�Û�k*~�á�à�`Z`\Û � % �- , ' & � ç &!]êÛ*ÞZ[ºì¦`ihjhc`�mRaz]Zb�ß�m�[f]Z[2[2åÛ�mQ`Gaz]Z[2ó�~�l�hcajÛ�aj[2knç�$�azà�k�Û�ß�l�ä2ä2`Zà�å�az]Zb¸ëRÞZ[2`Zà�[�q , ��ßZacÛ�ÞZ`\hcå�kRì(`yà � ��@ � � Û�Þ�l�Û

Page 21: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� /27

������� Ý-� ��1 ! � � '¼7 �z� � ! +� -�� ��� � ��� -� � 1 's/���1 ' ��'¸7�*�����$ !³���(��¡XZ`\hcYZaz]Zb_Û*ÞZ[fà�[2ä�~�à�k�ac`y]êpyac[2hcå�k

������� §-� ��1 ! � � � � '!� � ��� ' � � ��� �Ó-0/���'Ó/*1�� � '¸- � �D/*1�� ' ��'¼7� � �Ó/ � 'ê7 )

XZ[2Û�Û�a']Zb�@8. � az]êÛ*ÞZack�ì¦`Zà�qG~Zh'lsà�[2k*~ZhcÛ�k�a']

���� � §-� �*1 ! � � � '�� ��� ' � ��� �õ/� � ��� � ��� 'ê7� � �Ó/ � 'ê7 )

ëRÞZack�a'qsãZhjac[2kºÛ*Þ�l�Û;< � � -, 2 �DÜ � � â .7< � � 2 �DÜ � �2â � � � ç�$O`yà�[2w�l\qÓãZhc[\ß�<�* 2 � Ü � �2â . j�r�ßm.ÞZacä2Þ_ack�ä2`y]Zk�ack�Û�[�]iÛRmRacÛ*Þ $�ajbZ~�à�[ , ��ç

XZ[2ä2`y]Zå�ß�ä2`y]Zk�acå�[�àfli] lià�áZacÛ*à�l\à�p_ä2`Z]Zk�Û*l\]iÛ � � � , 9 ð_l\]Zå:l�k�k*~�qõ[DÛ�Þ�l�Û � �

qÓlxw � , � 3h?�� (� � =

��� , & �� � g�ç'qR`iÛ�acä2[�Û*Þ�l�Û2, � �Tì(`yà�`Z~�àgÛ�à�[2[2k�çgëRÞZ[�]�ß

� ! �� � � '�� � ' �� �� � 7 '���� � � ���#����� � �»/� �� � 7 '�� �� � � � � � � ���#����� � � �� � �

! �� 4� � � �l\]ZåÆÛ*ÞZ~Zk�� ��� �( ,ÓÜ ,� h �â � ,ÙÜ�Ú�â2ç &!]LÛ*ÞZ[éì¦`ihjhc`�mRaz]Zb�ß�m�[ l�hck�`Ù~ZÛ�ajhcaco2[ Û*Þ�l�ÛÜ$AQ,�â � � , % , �

� ' � ì¦`Zà:, � � ,% � , &�l\]ZåÁl\à�áZacÛ*à�l\à´p ä2`Z]Zk�Û*l\]iÛ�k:A � �él\]Zå�@ � � Ü(e�â2ç

ëRÞZ[�]�ß

� � � � �� � -(7 � ��� 1 � � 40��� � � � ��� ���#�����,-z�¾1

! ��� �� ���� � � � � /4 � 40��� � � � ���� � � � � ��� /4 � 4

0��� � � � ��� ���#�����,-zÄ}1

! � � � � � ��� 4 /9 ��� � � � ���� � � � � � � ��� /9 ��� � � � ���! � � -�/9 ��� � 80 � 4 1��� � � - /9 ��� 1��

q,`�mfß�hc[2Û � � � , 9 ð á�[:l\]Æl\à�áZacÛ*à�l\à�achcpÙk*qÓlxhjh�ä2`Z]Zk�Û*l\]iÛ�ç -Rk�a']ZbÁÛ�ÞZ[ Û�m�`az]Z[2ó�~�l�hcajÛ�aj[2kÆl\á�`�Y\[\ߺacÛ·ÞZ`\hcå�k Û*Þ�l�Û9< � � - , 2 �DÜ � �2â � � � � , � Ü �� � � â � ì¦`yà�� �

Page 22: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

/*/ � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

� �����-�#��.��� ����*��� ����� �����-�#���������+)���*��� ���*��+�, ���(����� 1 ���*�)+�, ��������� 1�����-�#�� �����)��*��� ����� ���� ������+�������*+�, � ���� )�(���+��������+), �

��������� ��� �������� ������ � 4 �� /� � � � 1��4 /�: 9 0)5 /�2 7 /�7)1 /�1�2 90 0)1 : 4�8)6 :32 4 /�0�:)1 8�9�2 18 8)5 /�1 4�0)6�0 5�432 6 /�4)0�0)1 4�6)832 15 9)0 /�4 0�1�4)5�9 8):�132 0 /�8�4)03/�: 4�4�5)732 19 :)8 /�8 8):3/�8)63/ 5�6)03/�2 : 4)1�9�0)6�0)8 4)8�5�9):32 06 /�1): /�9 73/�4�6�5):3/ :)8�5�/�832 9 0)93/�0)5�6)9�1 0�0)8�5�7)132 8: /�0)5 /�: /�7)7�7)5�6�1)13/ /�8):3/)/�9�0�2 1 6�8)1�8�6)8�8)5�1 5)8�:)8�7�7)532 7

¹���²n��Q/×���µ£���������Ó½�����£���(���z� �Õ42�n�gÐ��£n�����#��)

� ���������.��������*��� ����� �����-�#���������+��*����� ���� �� ���+�������*+�, � ���� )�(���+��������+), ����������.��������*��� ������

4 /�: 4 0)4 /�2 :0 0�1 4 5)9 /�2 78 8�5 4 :)9 /�2 75 9�0 4 /�4)4 /�2 79 :�8 4 /�9)8 432 16 /�1�: 4 43/�4 432 1: /�0�5 4 4�9)9 432 1

¹���²n�� �×����£n��*�����Ó½�����£���(���z� 4ÒZ½�¹���Ô

10

100

1000

10000

100000

1e+06

1e+07

1e+08

1e+09

1e+10

1e+11

0 50 100 150 200 250 300 350 400number of vertices (n)

simulation formula

edge traversals until Node Counting reaches a goal vertex for the first time (from simulation)number of times Node Counting visits vertex g0 until it reaches a goal vertex for the first time (from simulation or formula)

number of times Node Counting visits vertex g0 until it reaches a goal vertex for the first time (from formula)edge traversals until LRTA* reaches a goal vertex for the first time (from simulation)

�Z�)�£n !�R7�7*¡�½�����£n��(��-zÒy�)R×2������.<#1

qÓl�w � , � 3h? � (� � =

� � , & � � l\]Zå Û*ÞZ~Zk|l�hck�` ì(`yà¼k*~0(»ä2ac[�]iÛ�hcpÆh'lià�b\[9,�ç < � � -, 2 � Ü � � â¼ackÛ*ÞZ[�<Zæ!Y�lxh'~Z[_`xì � �»l�ì¦Û�[�à k�az]Zb\hc[|l\]iÛ�k�Û*Þ�l�ÛG~Zk�[ q,`Zå�[ ��`Z~�]\Û�az]Zb:à�[�l�ä2Þ:lêb\`yl�h§Y\[�à�Û�[2wì¦`Zà�Û*ÞZ[�÷&à�k�Û,Û�azqG[f`y]_lõÛ*à�[2[�mRacÛ*Þ � ,.hc[2Y\[2hck�çRë,ÞZajk <Zæ!Y�lxh'~Z[G[2ó�~�l�hck.Û�ÞZ[f]y~�qGá�[�à,`�ìÛ�azqG[2kfÛ�Þ�l�Û�Û*ÞZ[Ók�az]Zb\hc[_l\]iÛ�kèÞ�lnY\[sYyack�acÛ�[2å � �\ß�m.ÞZacä ÞíackGl¼hj`�m�[�à�á�`Z~�]Zå·`y]éÛ*ÞZ[¼]y~�qºæá�[�à,`�ì�[2å�b\[2k.Û�ÞZ[2p|Þ�l�Yi[�Û*à�lnY\[�à�k�[2å|~�]iÛ�ach4lGbi`yl�h�Y\[�à´Û�[2w»ack�à�[�l�ä2ÞZ[2å_ì(`yà�Û*ÞZ[�÷&à´k�ÛRÛ�azqG[\ç��`y]Zk�[2ó�~Z[�]iÛ�hcp\ß�k�az]Zb\hc[Ól\]iÛ�k�Û*Þ�l�ÛR~Zk�[ qR`Zå�[ ��`Z~�]\Û�az]ZbÓÛ�à�lnY\[�à�k�[ÓÜ(l�Û�hc[�l�k�Û*â

��- �� -�/

9 ��� 1�� 1[2å�b\[2k�ß4li]Zå Û�ÞZ[»]y~�qGá�[�à�`�ìQ[2å�b\[sÛ*à�lnY\[�à�k*lxhjk�~�]iÛ�achgl¼b\`yl�h�Y\[�à�Û�[2w_ackGà�[�l�ä2ÞZ[2åíì¦`ZàRÛ*ÞZ[

Page 23: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� / �

÷&à�k�ÛRÛ�azqG[�ä�l\]_á�[f[2w�ã�`y]Z[�]iÛ�azl�h�az]·Ü¦Û*ÞZ[�k�ó�~�l\à�[fà�`Z`\Û�`�ì â§Û*ÞZ[f]Z~�qGá�[�à�`�ì�Y\[�à�Û�acä2[2k�[2Y\[�]ì(`yàÓÜ(ãZhzl\]�l\à�â�~�]Zå�a'à´[2ä2Û�[2å·Û�à�[2[2k�çÓëRÞZ[2`yà´[�q ,�,ºì¦`ihjhc`�mRk�ç $O`yà�ä2`yqÓã�lià�ack�`y]�ß�k�a']Zbihj[_l\]iÛ�kÛ*Þ�lxÛÝ~Zk�[f{ �4ë�ÚnmDYZack�acÛ.[�l�ä2Þ¼hc[�lnì�`�ì¾l �Zæ!k*~�á�à�`Z`\Ûg`Z]Zä2[fm.ÞZachc[fÛ*ÞZ[2p_qG`�Y\[�az]¼`y]Z[�å�`�m.]ã�l�k�k�ì¢à�`Zq Û*ÞZ[Gk�Û*l\à�Û�Y\[�à�Û�[2w|Û�` �Zæ!k*~�á�à�`Z`\Û � � Ü�a ìRÛ�ac[2kfl\à�[Óá�à�`Zt�[�]:l�kfá�[�ì¦`yà´[�â2çDë,ÞZ[�]�ßÛ*ÞZ[2pÃYyack�acÛG[�l�ä2Þ:hc[�lnì.`xì,li] �6æ!k*~�á�à�`Z`\Ûº`y]Zä2[»m�ÞZajhc[_Û*ÞZ[2p qG`�Y\[Óaz]:`Z]Z[¸~�ã ã�l�k�kºì!à�`yqÛ*ÞZ[ �Zæ!k*~�á�à�`Z`\Û � �GÛ�`¼Û*ÞZ[Gb\`ylxh4Y\[�à´Û�[2w&ßOì(`yà�l¼Û�`\Û*lxh�`�ì*j � � e � g8. g@, th_[2å�b\[Û*à�l�Yi[�à�k*l�hck�ç

�5qndQ�no=d�j ���� ëRÞZ[|]Z~�qGá�[�às`�ìè[2å�bi[|Û*à�lnY\[�à´k*l�hckÓ~�]\Û�ach�k�az]Zb\hc[ l\]iÛ�kÓÛ�Þ�l�Û»~Zk�[3qR`Zå�[��`Z~�]\Û�az]Zb»à´[�l�ä Þ_lõb\`yl�h�Y\[�à�Û�[2wGì¦`Zà�Û*ÞZ[�÷&à�k�Û�Û�azqG[º`y]¼k�Û*à´`y]Zb\hcpÓä2`y]�]Z[2ä2Û�[2åí~�]Zå�azà�[2ä2Û�[2åbyà�l\ã�ÞZkgä�l\]¼á�[�[2w�ã�`y]Z[�]iÛ�azl�h�az]íÜ�Û*ÞZ[Rk�ó�~�l\à�[�à�`Z`\Ûg`xì â�Û*ÞZ[�]y~�qGá�[�à§`�ì�Y\[�à´Û�acä2[2kQ[2Yi[�]Gì(`yàÜ(ãZhzl\]�lià�âR~�]Zå�azà�[2ä2Û�[2åíÛ*à�[2[2knç

� ��� �/� áipía']Zå�~Zä2Û�aj`Z] `y]íÛ*ÞZ[»]Z~�qGá�[�à `�ìQÛ*à�l�Yi[�à�k�[2åí[2å�b\[2k�ì(`yà Û*ÞZ[Ó[2w�l\qÓãZhc[Óbyà�liã�ÞZk~Zk�az]Zb_{�[�qÓqÓl eÓl\]Zå ë,ÞZ[2`yà�[�q ,�,ißyk�[2[Óîfg�rnñ�ç

ë�`fä2`y]i÷&à�qIÛ*ÞZack�Û*ÞZ[2`yà�[2Û�ajä�lxh&à�[2k*~ZhcÛ�ßim�[.ã�[�à$ì(`yà�qõ[2å¸l�k�azqG~Zhzl�Û�ac`y]¼k�Û�~Zå�pD`xì}k�a']Zbihj[l\]iÛ�k�Û*Þ�lxÛè~Zk�[sqR`Zå�[ ��`y~�]iÛ�az]Zb_`y]í`Z~�à�[2wrl\qsãZhj[sbyà�l\ã�ÞZk�ß�k�[2[»ë4liáZhj[sg�çfòÃ[Ók�Û�`yã�ã�[2åÛ*ÞZ[�k�a'qG~Zhzl�Û�ac`y]êm.ÞZ[�]�� à´[�l�ä ÞZ[2å_[2acbyÞiÛ.á�[2ä�l\~Zk�[fÛ�ÞZ[D]Z~�qDá�[�à�`�ì�[2å�b\[�Û*à�lnY\[�à´k*l�hckRl\]ZåÛ*ÞZ~Zk�Û*ÞZ[Gk�azqG~Zhzl�Û�ac`y]ÃÛ�azqG[Óá�[2ä�l\qG[sh'lià�b\[\çGëRÞZ[Ók�azqG~Zhzl�Û�ac`y]íä2`y]i÷&à�qG[2å:`y~�à,ì(`yà�qG~Zhzl�kì(`yàGÞZ`�môÛ*ÞZ[|]Z~�qDá�[�àG`�ì�Y\[�à�Û�acä2[2k ,�ߧÛ*ÞZ[|]Z~�qDá�[�àG`�ìfã�l�k�k�[2k¼~�]\Û�ach.léb\`Zl�h�Y\[�à�Û�[2w ackà�[�lxä ÞZ[2åsì¦`ZàgÛ�ÞZ[R÷&à�k�ÛQÛ�a'qõ[\ßrli]Zå»Û*ÞZ[;<Zæ!Y�l�hz~Z[�`�ì � � m.ÞZ[�]_lºb\`Zl�h�Y\[�à�Û�[2wsajkRà�[�lxä ÞZ[2å¼ì(`yàÛ*ÞZ[õ÷&à�k�ÛfÛ�a'qõ[Óå�[�ã�[�]ZåÁ`y] �íç»ëRÞZ[Ók�a'qG~Zhzl�Û�ac`y] l�hck�`íã�à�`�Yyacå�[2å·~ZkºmRacÛ*Þ:Û�ÞZ[»]y~�qGá�[�à`�ì�[2å�b\[ÓÛ*à�l�Yi[�à�k*l�hck�~�]\Û�ach�l¼b\`yl�h�Y\[�à´Û�[2w|ackDà´[�l�ä ÞZ[2åéì(`yà�Û*ÞZ[G÷&à´k�Û�Û�azqG[\ß�ì(`yà�m.ÞZacä2Þím�[å�`»]Z`iÛ.Þ�lnY\[flºì(`yà�qG~ZhzlGaz]|ä2hc`\k�[2å_ì(`yà�q_ßZ`y]Zhcp|lGhc`�m�[�àRá�`y~�]Zåéaz]_ì(`yà�q `�ì�Û*ÞZ[W<Zæ!Y�lxh'~Z[`�ì � �fm.ÞZ[�]:l¼bi`yl�h�Y\[�à´Û�[2w|ackDà´[�l�ä ÞZ[2åéì(`yà,Û*ÞZ[G÷&à�k�Û�Û�azqG[\ç�,R`�mIÛ*ÞZ[Ó]Z~�qGá�[�à�`�ì�[2å�b\[Û*à�l�Yi[�à�k*l�hckRl\]Zå|acÛ�k,hj`�m�[�àRá�`y~�]Zå à�[2hzl�Û�[fack.k�ÞZ`�m�]¼a'] $�acby~�à�[ ,�,\çµö.ack*~�l�h�a']Zk�ã�[2ä2Û�ac`y]í`xìÛ*ÞZ[,byà�l\ã�ÞZk§k*~Zb\b\[2k�Û�k�Û*Þ�l�Û�Û*ÞZ[Rhc`�m�[�à�á�`y~�]Zå¼ackQl\ã�ã�à´`�wZazqÓl�Û�[2hcp»lxÛgl�ä2`y]Zk�Û*l\]iÛ�å�ack�Û*l\]Zä2[ì!à�`yqÆÛ�ÞZ[g]y~�qGá�[�à�`�ìy[2å�bi[gÛ*à�lnY\[�à´k*l�hck�a'] Û*ÞZ[ghc`\b�ãZhj`iÛ�ß�k�~Zb\b\[2k�Û�az]ZbRÛ�Þ�l�Û�Û*ÞZ[�hc`�m�[�à�á�`y~�]Zå~�]Zå�[�à�[2k�Û�azqÓl�Û�[2k�Û*ÞZ[G]y~�qGá�[�à�`�ì¾[2å�bi[DÛ�à�lnY\[�à�k*l�hck.áip|à�`y~ZbZÞZhjpílÓä2`Z]Zk�Û*l\]iÛ�ì!l�ä2Û�`yà2ç $O`yàä2`yqsã�l\à�ack�`y]�ßgë�l\áZhc[ jíl\]Zå $�ajbZ~�à�[ ,0,»ä2`Z]\Û*lxa'] à�[2k*~ZhcÛ�kõì¦`yàº{ ��ë�Únm|`y]ÃÛ*ÞZ[¼[2wrl\qsãZhj[byà�l\ã�ÞZkDÜ�a ì�Û�ac[2kèl\à´[fá�à�`yt�[�]_l�k�á�[�ì(`yà�[�â2ç

ëRÞZ[2`Zà�[�q ,�,sä�l\]Ã[�l�k�achcp á�[»[2wZÛ�[�]Zå�[2åÁÛ�`ébyà�`y~�ãZkº`�ì/@·l\]iÛ�k_ܨì¦`Zàfl\]ipíä2`y]Zk�Û*l\]iÛ@�â2ç%&$]:Û*ÞZackGä�l�k�[\ß�m�[»à�[�ãZhcacä�l�Û�[_`Z~�à�[2w�l\qÓãZhc[»bZà�l\ã�Þ-@ÃÛ�azqG[2k�ß�k�[2[.$�acby~�à�[re Ü(à�acbyÞiÛ*â2çò:[ºaz]\Û*à´`yå�~Zä2[GlÓ]Z[2mBk�Û*l\à�ÛRYi[�à�Û�[2w�ßym,ajÛ�Þ·Ü(~�]Zå�azà�[2ä2Û�[2å�â�[2å�bi[2kèá�[2Û�m�[2[�]êajÛ�l\]Zå l�hch�`\hcåk�Û�l\à�ÛfYi[�à�Û�acä2[2k�ç·ë,ÞZ[¸]Z[2môbi`yl�hgYi[�à�Û�[2w:ack|Ü(l\]iprâº`y]Z[¼`�ì.Û�ÞZ[»`\hcåÁb\`Zl�hgY\[�à´Û�acä2[2k�ç'&¦ì @l\]iÛ�k�Û*Þ�lxÛf~Zk�[ q,`Zå�[ ��`y~�]iÛ�az]Zb l\à´[Ók�Û*l\à´Û�[2åía'] Û*ÞZ[»]Z[2m k�Û*l\à�Û�Y\[�à´Û�[2w&ß�[�l�ä2Þ:l\]iÛ�ä�l\]qG`�Y\[sÛ�` lêå�a #�[�à�[�]iÛG`y]Z[¼`�ìRÛ*ÞZ[»`ihjå k�Û*lià�Û�Y\[�à�Û�acä2[2kÓl\]Zå Û*ÞZ[�]Ã[2wrÞZazáZacÛÓÛ*ÞZ[_á�[�Þ�lnYZaj`Zàå�[2k�ä�à´a'á�[2å l\á�`�Y\[.az]sÛ*ÞZ[�ä2`y]iÛ�[2wZÛg`�ì4k�a']Zbihj[fli]\Û�knç¾XZaz]Zä2[�Û*ÞZ[�]Z~�qDá�[�à�`�ì�Y\[�à�Û�ajä2[2k�`�ì�Û*ÞZ[]Z[2mI[2w�l\qÓãZhc[Gbyà�l\ã�Þêack�`y]Zhcp l¼ä2`y]Zk�Û�l\]\ÛÝì¢l�ä2Û�`yàRhzl\à´b\[�àRÛ�Þ�l\]_Û*ÞZ[Ó]Z~�qGá�[�à�`�ìgY\[�à´Û�acä2[2k`�ì�Û*ÞZ[�`\hcåé[2wrl\qsãZhj[ºbyà�l\ã�Þ�ß�Û*ÞZ[�ì¦`ihjhc`�mRaz]Zb¼Û*ÞZ[2`yà�[�q ì(`\hchc`�mRk�ç

�:��o=�ng&g��o<CD��b�� ëRÞZ[RÛ�a'qõ[QacÛ�Û*lit�[2kglRbiajYi[�]Ó]y~�qGá�[�à�`�ì&l\]iÛ�k�Û*Þ�lxÛ¾~Zk�[_q,`yå�[ ��`y~�]iÛ�az]Zb

Page 24: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

/ � � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

Eulerian Graph Converted Graph

�Z�)�£n !�R7�/2¡&×6�*�§®n��g��£n��$ (����ÓÑ� !��®n�f��������(�4Ð����¶*�$ !�����Û�`Óà�[�l�ä2Þ_lºb\`yl�h�Y\[�à�Û�[2wGì¦`ZàgÛ�ÞZ[�÷&à�k�Û�Û�azqG[f`Z]Ók�Û�à�`y]Zb\hcp¼ä2`y]�]Z[2ä2Û�[2å|~�]Zå�azà�[2ä2Û�[2åíbyà�liã�ÞZkä�l\]Gá�[�[2w�ã�`y]Z[�]iÛ�azl�h�az]_Ü�Û*ÞZ[�k�ó�~�l\à�[Qà´`y`\Û�`�ì â}Û*ÞZ[Q]Z~�qGá�[�à�`�ì�Y\[�à´Û�acä2[2k�[2Y\[�] ì¦`yà�Ü(ãZhzl\]�l\à�â~�]Zå�azà�[2ä2Û�[2åíÛ*à�[2[2k�ç

XZaz]Zä2[|Û*ÞZ[_Û�azqG[_acÛÓÛ�l\t�[2kÓlébyà�`y~�ãÃ`�ìfl\]\Û�kfaz]ÁÛ*ÞZ[_m�`yà´k�Ûfä�l�k�[_Û�`:à�[�l�ä2Þ·líbi`yl�hY\[�à�Û�[2wõì¦`yàµÛ*ÞZ[,÷&à�k�Û�Û�azqG[�ajk�l�hc`�mQ[�à�á�`y~�]Zå_`y]sÛ*ÞZ[.ä2`�Y\[�àµÛ�azqG[�az]¼Û*ÞZ[�mQ`Zà�k�Û�ä�l�k�[\ßZÛ*ÞZ[ì¦`ihjhc`�mRaz]Zb¼ä2`yà�`\hchzl\à�p¼ì(`\hchc`�m,k�ç

�:�no=��g gi�o<CD� ��� ëRÞZ[�ä2`�Yi[�à�Û�azqG[�`�ì&l,b\acY\[�]G]Z~�qDá�[�à�`�ì&l\]iÛ�k�Û*Þ�l�Û4~Zk�[_q,`yå�[���`Z~�]\Û�az]Zb`y]_k�Û*à�`y]Zb\hcp_ä2`y]�]Z[2ä2Û�[2å ~�]Zå�azà�[2ä2Û�[2å bZà�l\ã�ÞZk�ä�l\]|á�[D[2w�ã�`Z]Z[�]\Û�azl�h�az]ÁÜ�Û*ÞZ[�k�ó�~�l\à�[Gà�`y`iÛ`�ì â§Û*ÞZ[G]y~�qGá�[�àÝ`�ì¾Yi[�à�Û�acä2[2kR[2Y\[�]¸ì¦`yàºÜ(ãZhzl\]�l\à�â�~�]Zå�azà�[2ä2Û�[2åíÛ*à�[2[2knç

���� / ����� � � � � �� �� �� �0����� � �� ��� � ����� � � � �� � XZ`�ì!l\à2ß�m�[RÞ�lnY\[gk*ÞZ`�m.]ºÛ*Þ�lxÛ4Û*ÞZ[ä2`�Y\[�à�Û�azqG[�`�ì¾li]\Û�k§Û*Þ�l�Û�~Zk�[ q,`Zå�[ ��`Z~�]\Û�az]ZbGackR]Z`\Ûgby~�lià�l\]iÛ�[2[2å»Û�`Dá�[�ã�`ihjp�]Z`yqGazl�h�az]Û*ÞZ[R]y~�qGá�[�à�`�ì�Y\[�à�Û�acä2[2k�ì(`yà�[2acÛ*ÞZ[�à�å�azà�[2ä2Û�[2åÓ`Zà4~�]Zå�azà�[2ä2Û�[2åsbyà�l\ã�ÞZk�ß�az]Zä2hz~Zå�az]Zb Ü(ãZhzl\]�l\à�â~�]Zå�azà�[2ä2Û�[2å_Û*à�[2[2knç4òÃ[Ýlià�[.l�hck�`fl\áZhc[�Û�`�å�[2k�ä�à�azá�[èl byà�l\ã�ÞGã�à�`yã�[�à�Û�pGÛ*Þ�l�Û�by~�l\à�l\]\Û�[2[2kglã�`\hcp�]Z`yqGazl�h�ä2`�Y\[�à§Û�azqG[º`�ìgk�az]Zb\hc[Dl\]iÛ�kRÛ�Þ�l�Û.~Zk�[ qR`Zå�[ ��`Z~�]\Û�az]Zb�ç§Ú �g~Zhc[�à�azl\]ébyà�l\ã�Þack»líå�azà�[2ä2Û�[2åÁbyà�l\ã�Þ�ß�[�l�ä2Þ `�ì.m.ÞZ`ik�[»Yi[�à�Û�acä2[2k»Þ�lnY\[_l\] [2ór~�lxhÝ]Z~�qGá�[�àG`xì.az]Zä2`yqGaz]Zbl\]Zå|`Z~ZÛ�b\`\az]ZbÓå�azà�[2ä2Û�[2åí[2å�b\[2knç+��`Z]Zk�acå�[�à.l\]_lià�áZacÛ*à�l\à�p¼k�Û*à´`y]Zb\hcp_ä2`y]�]Z[2ä2Û�[2å �g~Zhc[�à�azl\]byà�l\ã�Þsm.ÞZ`\k�[è]Z~�qGá�[�à§`�ì4[2å�b\[2k�ack.l�Û�qG`\k�Û�ã�`\hcp�]Z`yqGazl�h�az]¼Û*ÞZ[�]Z~�qDá�[�à§`�ì4acÛ�k�Y\[�à�Û�acä2[2kܨì(`yà�[2w�l\qÓãZhc[\ß�á�[2ä�l\~Zk�[f]Z`GÛ�mQ`º[2å�b\[2k�ä2`y]�]Z[2ä2ÛRÛ*ÞZ[�k*l\qõ[RY\[�à�Û�acä2[2k*â6ßrli]Zå|l�byà�liã�ÞÓÛ*Þ�lxÛack»å�[�à�acY\[2åLì!à�`yq Û�ÞZ[ �g~Zhc[�à�azl\]Lbyà�l\ã�ÞLá\pÁà�[�ãZhzl�ä2az]ZbÁ[�l�ä2ÞÁ`�ì�Û*ÞZ[_å�azà�[2ä2Û�[2åÆ[2å�b\[2k¼`�ìÛ*ÞZ[ �§~Zhj[�à´a'li]íbyà�l\ã�ÞémRacÛ*ÞéÛ�m�`»å�azà�[2ä2Û�[2å:[2å�b\[2kfÛ�Þ�l�Ûèlià�[Gä2`y]�]Z[2ä2Û�[2å:mRacÛ*ÞÃl Ü(~�]Zacó�~Z[�âaz]\Û�[�à�qG[2å�azl�Û�[_Y\[�à�Û�[2w&ç.$�acby~�à�[ ,�g_k�ÞZ`�m,kDl\]í[2w�l\qÓãZhc[\ç -Rk�a']Zb:à�[2k�~ZhjÛ�kfì!à�`yqFî�,�gnñ�ß�m�[ä�l\]Ák�ÞZ`�m Û*Þ�lxÛÓÛ*ÞZ[|ã�[�à�ì¦`yà�qÓl\]Zä2[_`�ì�k�az]Zb\hc[:l\]\Û�kDÛ�Þ�l�Û»~Zk�[ q,`yå�[ ��`y~�]iÛ�az]Zb:`y] Û*ÞZ[ä2`y]iY\[�à�Û�[2åGbyà�liã�Þfack�by~�lià�l\]iÛ�[2[2åfÛ�`fá�[Rã�`\hcp�]Z`yqGazl�h�az]GacÛ�kg]Z~�qGá�[�à�`�ì�acÛ�k�Y\[�à�Û�acä2[2k�ç�ëRÞZackackgk�`fá�[2ä�l\~Zk�[RÛ*ÞZ[Rá�[�Þ�l�YZac`yà�`xì&k�az]Zb\hc[�l\]\Û�k4Û*Þ�lxÛg~Zk�[_qR`Zå�[ ��`y~�]iÛ�az]Zb�`y]ºÛ*ÞZ[Rä2`Z]\Y\[�à´Û�[2åbyà�l\ã�Þêack.Û*ÞZ[ºk*l\qG[flxk.Û*ÞZ[Gá�[�Þ�lnYZaj`ZàR`�ì�k�az]Zb\hc[Ól\]iÛ�kRÛ*Þ�l�Û�~Zk�[ �µå�b\[ ��`Z~�]\Û�az]Zbíî�,�gnñ�`y]Û*ÞZ[�`yà�acb\az]�l�h�byà�l\ã�Þ�ßZlGà�[�l�h æ!Û�azqG[�k�[�l\à�ä2Þ»qõ[2Û*ÞZ`yå¼Û*Þ�l�Û�l�hcmRl�pZkQqsl\t�[2kgÛ�ÞZ[.l\]iÛQÛ�à�lnY\[�à�k�[l\]¼[2å�b\[�Û*Þ�lxÛ.Þ�l�kRá�[2[�]êÛ*à�lnY\[�à�k�[2å¼Û*ÞZ[�hc[�l�k�Û�]y~�qGá�[�à,`�ì4Û�a'qõ[2k�ß�l\]Zå_Û*ÞZ[fã�[�à$ì(`yà�qÓl\]Zä2[

Page 25: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� / �

`�ìgk�a']Zbihj[¼l\]\Û�k.Û*Þ�lxÛè~Zk�[ �µå�b\[ ��`y~�]iÛ�az]Zb_`y]_k�Û�à�`y]Zb\hcp|ä2`Z]�]Z[2ä2Û�[2å �g~Zhc[�à�azl\]íbyà�l\ã�ÞZk�ackã�`\hcpr]Z`yqõa'lxhyaz]�Û�ÞZ[Qã�à�`Zå�~Zä2Û4`xìrÛ*ÞZ[g]Z~�qGá�[�à�`�ì�ajÛ�k4[2å�b\[2k�l\]ZåfÛ*ÞZ[g]Z~�qGá�[�à�`�ì�ajÛ�k4Y\[�à´Û�acä2[2kî�,0gnñ�ç &(Û_ack_ä�~�à�à�[�]iÛ�hcp ~�]�ty]Z`�m.]Lm.ÞZ[2Û�ÞZ[�à¼byà�acå�k|l\]ZåÆ`\Û*ÞZ[�à¼byà�l\ã�ÞÁÛ�`Zã�`\hc`\b\ac[2kéÛ*Þ�l�Ûà�[�ã�à´[2k�[�]\Û�à�[�l�hcack�Û�acäGÛ�[�à�à�l�az]ZkRby~�l\à�li]\Û�[2[fli]\Û�k,Û*Þ�l�Û.~Zk�[ qR`Zå�[ �Ý`y~�]\Û�a']Zb¼lÓã�`\hcp�]Z`yqGazl�hä2`�Y\[�à�Û�a'qõ[\ç

� ��������� � � � ���� ��� ��� � � ���� ��� �� ��0� �������� � � ���XZ`Ãì¢lià2ß�m�[|Þ�l�Yi[|k*ÞZ`�m.] Û*Þ�l�Û¼Û*ÞZ[|ä2`�Y\[�àGÛ�a'qõ[ `xìfk�az]Zb\hc[ li]\Û�k¼Û*Þ�l�Û¼~Zk�[ qR`Zå�[

��`Z~�]\Û�az]ZbÓä�li]_á�[f[2w�ã�`Z]Z[�]\Û�azl�h�a'] Ü�Û*ÞZ[�k�ó�~�l\à�[fà�`Z`\Û�`�ì â§Û*ÞZ[f]Z~�qGá�[�à�`�ì�Y\[�à�Û�ajä2[2k,[2Y\[�]ì(`yàRÜ(ãZhzl\]�l\à�âg~�]Zå�azà�[2ä2Û�[2å¼Û*à�[2[2k�ç�ò ÞZachc[�mQ[Rà�[�lxhjaco2[2å_Û*Þ�lxÛ�Û*ÞZ[Rä2`�Y\[�à�Û�azqG[R`xì}k�a']Zbihj[�l\]iÛ�kÛ*Þ�lxÛ.~Zk�[ q,`Zå�[ ��`y~�]iÛ�az]Zb¼ajk�[2w�ã�`y]Z[�]iÛ�azl�h�`y]_å�azà�[2ä2Û�[2åébyà�l\ã�ÞZknß�m�[fÛ*à�ac[2åéÛ�`»ã�à�`�Y\[�ì(`yàléhc`y]ZbíÛ�azqG[¼Û*Þ�l�ÛGacÛDmRl�kGã�`\hcp�]Z`yqGazl�hR`Z]·~�]Zå�azà�[2ä2Û�[2åÁbZà�l\ã�ÞZk�çíëRÞ�l�ÛºmQ[¼[2Y\[�]iÛ*~�l�hchcpã�à�`�Y\[2åsÛ*ÞZ[R`yã�ã�`\k�acÛ�[RmRl�k�k*~�à�ã�à�ack�az]ZbDÛ�`f~ZkQá�[2ä�li~Zk�[.Û�ÞZ[RY�l�hz~Z[�æ�~�ã�å�l�Û�[fà�~Zhc[�`�ì}{ ��ë�ÚFmack�k�azqGach'liàRÛ�`sÛ*Þ�l�Û�`�ì'qR`Zå�[ ��`Z~�]\Û�az]Zb»á�~ZÛ,by~�l\à�l\]iÛ�[2[2kRk�a']Zbihj[Gl\]iÛ�k.lGä2`�Y\[�à§Û�azqG[ºÛ*Þ�l�ÛackÓã�`\hcp�]Z`yqGazl�hRaz]ÃÛ*ÞZ[_]y~�qGá�[�àº`�ìRY\[�à´Û�acä2[2kDaz]Zk�Û�[�l�å·`xì.[2wrã�`y]Z[�]iÛ�azl�h(ç ëRÞZ[�à´[¸l�hck�`í[2wiæack�ÛG`\Û*ÞZ[�àGà�[�lxhæ!Û�a'qõ[»k�[�lià�ä Þ qG[2Û*ÞZ`Zå�kGm.ÞZ`\k�[¼ä2`�Y\[�à�Û�a'qõ[»`y]Ãå�azà�[2ä2Û�[2åÁ`yàº~�]Zå�azà�[2ä2Û�[2åbyà�l\ã�ÞZk_ack ã�`ihjp�]Z`yqGazl�hGaz]ÆÛ*ÞZ[:]Z~�qGá�[�à_`xìDY\[�à�Û�ajä2[2kÁÜ(ã�[�à�Þ�l\ãZk|mRacÛ*Þ Û*ÞZ[·à�[2k�Û�à�acä2Û�ac`y]Û*Þ�lxÛDÛ�ÞZ[|byà�liã�ÞZkõÞ�lnY\[|]Z`Ã[2å�b\[2kÓÛ*Þ�l�ÛGhc[�lnY\[|Û�ÞZ[_ä�~�à�à�[�]iÛÓY\[�à�Û�[2w ~�]Zä Þ�l\]Zbi[2å�â2çÙë�lnæáZhc[ h_k*ÞZ`�mRk�k�`yqõ[D`xìQÛ*ÞZ[2k�[Óà�[�lxhæ!Û�a'qõ[Ók�[�l\à�ä2ÞíqG[2Û*ÞZ`Zå�k�ç�ò·lxby]Z[�à2ß�{�az]Zå�[�]Zá�l\~�q l\]Zåe�à�~Zä2t\k�Û�[2az]��#k.öglxh'~Z[�æ -�ã�å�lxÛ�[ �R~Zhc[ , îkg�pnñQÜ�k*ÞZ`yà´Û���ò·l�by]Z[�à�� kQöglxh'~Z[�æ -�ã�å�lxÛ�[ �R~Zhc[�â�l\]ZåëRÞ�à�~�]��#k�ögl�hz~Z[�æ -.ã�å�l�Û�[��R~Zhc[Óîfg\ð�ñ}à�[2k�[�qGáZhc[XqR`Zå�[ �Ý`y~�]\Û�a']ZbG[2Y\[�]¼qG`yà´[Rä2hc`\k�[2hcp»Û�Þ�l\]{ ��ë�ÚFmfà�[2k�[�qGáZhc[2kdq,`yå�[��`y~�]iÛ�az]Zbfk�a']Zä2[�Û�ÞZ[2a'à§Y�l�hz~Z[�æ�~�ã�å�l�Û�[fà�~Zhj[2k�ä2`y]iÛ*l�az]sÛ*ÞZ[�Û�[�à�q, Z<�Ü$ �â H´~Zk�Û.hcazt�[rq,`yå�[ ��`Z~�]\Û�az]Zb�ç�ëRÞZ[2azà�ä2`�Y\[�à�Û�a'qõ[2k�mQ[�à´[Dl\]�l�hcpZo2[2å·áip_Û*ÞZ[2azà�l\~iæÛ*ÞZ`Zà�k.l\]Zåék*ÞZ`�m.]_Û�`¼á�[Óã�`\hcp�]Z`yqGazl�h�az]|Û�ÞZ[D]Z~�qDá�[�àR`xìgY\[�à�Û�acä2[2k,ì¦`yàÝk�az]Zb\hc[»l\]iÛ�k�ç§ò:[Þ�lnY\[�k*ÞZ`�m.]ºÛ*Þ�l�Û�Û�ÞZ[Rä2`�Y\[�à�Û�azqG[R`�ìEqR`Zå�[ ��`Z~�]\Û�az]Zb�ä�l\]sá�[R[2w�ã�`y]Z[�]iÛ�azl�hOa']õÛ*ÞZ[.]Z~�qºæá�[�à�`�ì�Y\[�à�Û�acä2[2k�ì(`yà�k�az]Zb\hc[»l\]iÛ�k�ç��Ý`y]Zk�[2ó�~Z[�]\Û�hcp\ß�Û*ÞZ[2k�[Ók*qsl�hchgqG`yå�a ÷�ä�l�Û�aj`Z]Zkf`�ì_qR`Zå�[��`Z~�]\Û�az]Zb»lià�[f]Z[2ä2[2k�k*l\à´p_Û�`Gby~�l\à�l\]iÛ�[2[flÓã�`\hcpr]Z`ZqGazl�h�ä2`�Y\[�à�Û�azqG[\ç

¹���²��� ����#Â\�$ !�!����½������#�0¹�����R×2���� !�¢�è �������2���

Ø����#£��$��#�®x�n�*�(�§½�£������¿¸%¨¹�Ö�Å #4¹�����­�����§�����Ð��¶��$ 4¹�����gÑ�£��� !�����(���$&% "1' ) +�, - $&% "1'04 Õ�2���gÐ��£������#��)

Ø����#£��$��#�®\�����(�g½�£n�����¿_%z¹�Ö�����­�����§�����Ð��¶��! �¹�����gÑ�£��� !�����(���$&% "('*) +�,�- $&% " $/.0.1% " 2 %�'�'54 ÒZ½�¹��¾Ô

� $&% "('� $&% " $/.0.1% " 2)%�'�' ������� $&% "1' ) +�, - $&% "1'04 ¿º�*)����$ �� ��Ø����#£��$��#�®x�n�*�(�§½�£�����#/ ���$&% "('*) +�� d�� % ,�- $&% "1'021, - $&% " $/.0.1% " 2 %�'�'�'04 ¹��n (£�� � ��ØO���#£��!��#�®\�����(�g½}£�����#/ ���

� ¹����.��£������ !�������#��������§ !�����#�0��������(���� !�¢�»��������2�ÓØ��$ !�(��°��0�����g¿º���#�\¡G¿f�������*�§£��(�,����������*������$ !��²\������£��(�.������­f�����#������� !�����#������Q���(���� !�¢�Ó�Q������2�DØO�$ !�(��°���������¿º���#�G�#��� / ��� �����G�.�6�#Âx�$ !�$�������gØO�$ !�(��°���������¿º���#���#��� / � � ¡&¿º�� !�$�'�$ ��(������� !�����#�0��������(���� !�(�f��������2���n���(�! (�#²\���º�#��� / ��� ¡&¹}���������$ 4 !�����#�0������g�(���� !�(�f��������2�,��4��¶��! !�����f���Òy½�¹���Ô2¡

Page 26: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

/ � � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

ò:[»]Z`�m lxÛ�Û�[�qÓãZÛ�Û�`éb\acY\[_l\]íaz]\Û�~ZajÛ�aj`Z]:m.ÞipíÛ*ÞZ[¼ä2`�Yi[�à�Û�azqG[»`xì,li]\Û�k�Û�Þ�l�ÛD~Zk�[k�`yqG[Rà�[�lxhæ!Û�a'qõ[Rk�[�l\à�ä2ÞGqG[2Û*ÞZ`Zå�k�ä�l\]Gá�[�[2w�ã�`y]Z[�]iÛ�azl�h�az]GÛ�ÞZ[Ý]Z~�qGá�[�à�`�ì�Y\[�à�Û�acä2[2k�m.ÞZachc[Û*ÞZ[Gä2`�Y\[�àRÛ�a'qõ[f`�ìRl\]\Û�kRÛ*Þ�l�Û�~Zk�[G`\Û*ÞZ[�à�à�[�l�h æ!Û�azqG[Gk�[�l\à´ä ÞéqG[2Û*ÞZ`Zå�k�ajk�bZ~�l\à�l\]iÛ�[2[2å|Û�`á�[íã�`\hcp�]Z`yqGazl�h(çÁò:[¼å�`:Û*ÞZackÓaz] ì(`yà�q `�ì�l|bZ~Zajå�[2hcaz]Z[íì(`yàGÞZ`�m Û�` å�[2k�ajbZ]Áà�[�l�h æ!Û�azqG[k�[�l\à�ä2Þ¼qG[2Û*ÞZ`Zå�kgÛ*Þ�l�Û�à�[2k*~ZhcÛ�az]»lGã�`ihjp�]Z`yqGazl�h�ä2`�Y\[�àµÛ�azqG[\ç�$�`yàµk�azqÓãZhcajä2acÛ�p\ßOmQ[,å�ajk�ä�~Zk�k`y]ZhcpÁÛ*ÞZ[_ä�l�k�[_m�ÞZ[�à�[_k�az]Zb\hc[ l\]iÛ�kÓÞ�lnY\[¼Û�`·à�[�l�ä2Þ líb\`yl�h§Y\[�à�Û�[2wÃì(`yàºÛ*ÞZ[¼÷&à�k�ÛGÛ�azqG[\çÚ.k�l\à�bZ~Z[2å|[�l\à�hcac[�à2ß�Û*ÞZ[ºÛ�azqG[GacÛ.Û*lit�[2kRk�az]Zb\hc[Ól\]iÛ�k.az]éÛ*ÞZ[ºmQ`Zà�k�Û�ä�l�k�[GÛ�`»à´[�l�ä ÞélÓbi`yl�hY\[�à�Û�[2wsì(`yà�Û*ÞZ[ ÷&à�k�ÛRÛ�azqG[�ack�Û*ÞZ[fk�l\qG[fl�kRÛ�ÞZ[�ä2`�Yi[�à�Û�azqG[f`xì¾k�az]Zb\hc[Gl\]\Û�kRaz]_Û*ÞZ[ºmQ`Zà�k�Ûä�l�k�[\çfë,ÞZ[õà´[2k*~ZhcÛ�k�ä�l\]_Û�ÞZ[�]|[�lxk�achjp:á�[Dbi[�]Z[�à�l�hcaco2[2å·Û�`¼byà�`Z~�ãZk.`�ìRl\]iÛ�k�ß�m.ÞZacä2Þím�[få�`[2hck�[2m.ÞZ[�à�[\ç

�Wq�df��o=dj � �[� ëRÞZ[fÛ�azqG[ºacÛ.Û*lit�[2k.lsk�az]Zb\hc[õli]\ÛRÛ�`»à�[�l�ä2Þ|lsb\`yl�h�Y\[�à´Û�[2w_`y]_k�Û*à�`y]Zb\hcpä2`y]�]Z[2ä2Û�[2å:Ü�å�azà�[2ä2Û�[2å¼`yà�~�]Zå�azà�[2ä2Û�[2å�ß�ãZhzl\]�lià�`yà�]Z`y]iæ�ãZhzl\]�lià�â§byà�l\ã�ÞZk�ì(`yà�Û�ÞZ[g÷&à´k�ÛgÛ�azqG[ackèl�Û�qG`\k�ÛfÜ$U�Ü$,�â ,�â U¾ÜE,&â � ")R�� ����Ü$ nâga ìQl�hch�`xì¾Û*ÞZ[�ì(`\hchc`�mRaz]Zb»ä2`y]Zå�acÛ�ac`y]Zk�ÞZ`\hcå�ç��`y]Zå�acÛ�ac`y] ,��RÚRÛ�[2Y\[�à´p¸ã�`\az]\Û�az]_Û�azqG[\ß�Û�ÞZ[:<Zæ!Y�lxh'~Z[2k:<�Ü$ �â,l\à�[�az]\Û�[2b\[�à�k�ç��`y]Zå�acÛ�ac`y] g��RÚRÛ�[2Y\[�à�p�ã�`\az]iÛ�a'] Û�azqG[\ß�<4ÜE �#!â�ack�]Z`\Û4å�[2ä�à´[�l�k�[2å�ß�m.ÞZ[�à�[ �#�ack�Û*ÞZ[�ä�~�à�à´[�]\Û

Y\[�à´Û�[2w»`xì¾Û*ÞZ[fli]\Û�ç��`y]Zå�acÛ�ac`y] j��RÚRÛ�[2Y\[�à´p¸ã�`\az]\Û�az]_Û�azqG[\ß<�Ü$ �#!â§ack�a']Zä�à´[�l�k�[2å áip_l�ÛRhc[�l�k�ÛR`Z]Z[�aì <�Ü$ �#!â��

qGaz]�%�R��:<4Ü$ =<>�>\Ü$ # ?aAyâ�â2ßZm.ÞZ[�à�[: �#�ack�Û*ÞZ[�ä�~�à�à�[�]iÛ�Y\[�à�Û�[2wÓ`xì¾Û*ÞZ[fli]\Û�ç��`y]Zå�acÛ�ac`y] h�RÚRÛ�[2Y\[�à�píã�`\az]iÛfaz]íÛ�azqG[\ß�acÛfÞZ`\hcå�kºÛ*Þ�l�Û /1<�Ü$ �â C<�Ü$ �<�>�>\Ü$ �?aAyâ�â=/ �7U�Ü$,�â

ì(`yàRl�hch4]Z`Z]iæ!b\`yl�h�Yi[�à�Û�acä2[2k *��Ll\]ZåkA!*�3GÜ$ �â6ß�m.ÞZ[�à�[:U�Ü$,�â � ,�ackèlõì¢~�]Zä2Û�aj`Z]_`�ìÛ*ÞZ[f]Z~�qGá�[�à,`�ì�Y\[�à�Û�acä2[2k2,�ç

� � ��� � ëRÞZ[Óã�à�`Z`�ì¾ack�az]|Û�Þ�à�[2[fk�Û�[�ãZk�çQ{}[2Û #gå�[�]Z`\Û�[ºÛ*ÞZ[fä�~�à�à�[�]\Û�Y\[�à�Û�[2w»`�ì�Û*ÞZ[Gl\]\Ûnßl\]Zå9 �#�- ,fÛ�ÞZ[DYi[�à�Û�[2w_`�ì�Û*ÞZ[Ól\]\Û�lnì(Û�[�à,ajÛfÞ�lxk�Û*à�lnY\[�à�k�[2åéÛ*ÞZ[Gä ÞZ`ik�[�]é[2å�b\[\çD{}[2Û:<�#*Ü$ nâå�[�]Z`\Û�[�Û*ÞZ[;<Zæ!Y�l�hz~Z[f`�ì�Y\[�à�Û�[2w8 �azqÓqõ[2å�a'lxÛ�[2hcp|á�[�ì(`yà�[�Û*ÞZ[fl\]iÛQ~�ã�å�lxÛ�[2kRÛ*ÞZ[;<Zæ!Y�l�hz~Z[f`�ì �#�ß�li]Zåk<�#�-,�ÜE �â§Û*ÞZ[fk�l\qG[:<Zæ!Y�l�hz~Z[fazqÓqG[2å�azl�Û�[2hjpílnì(Û�[�à§Û*ÞZ[D~�ã�å�l�Û�[\ç,\ç $�azà�k�Ûnß4m�[_ã�à�`�Y\[_á\p:az]Zå�~Zä2Û�ac`y] `Z]:Û*ÞZ[_]Z~�qDá�[�àG`�ìRÛ�azqG[_k�Û�[�ãZk���Û�Þ�l�Û�ß�l�Ûf[2Yi[�à�pã�`\az]\Û_az]LÛ�azqG[\ßRacÛ_ÞZ`\hcå�k_Û*Þ�l�Û�<4Ü$ nâ � U�Ü$,�â ���&Ü$ nâºì(`yàÓl�hch�Y\[�à�Û�acä2[2k� \ç &(Û|ÞZ`\hcå�kl�ÛGÛ�azqG[¼k�Û�[�ã o2[�à´` k�a']Zä2[_Û�ÞZ[�]C<4Ü$ nâ . �êì(`yàflxhjh�Y\[�à´Û�acä2[2k \ç:X�~�ã�ã�`\k�[|Û�Þ�l�ÛfÛ*ÞZ[az]Zå�~Zä2Û�ac`y]ÆÞiprã�`\Û*ÞZ[2k�ack»ÞZ`\hcå�k_l�ÛGÛ�azqG[_k�Û�[�ã���çÁëRÞZ[|`y]ZhcpC<Zæ!Y�l�hz~Z[|Û*Þ�l�ÛGä2Þ�l\]Zb\[2kl�ÛGÛ�azqG[¼k�Û�[�ã���ack <4ÜE �#!â�ì(`yàºÛ*ÞZ[_]Z`y]iæ!b\`ylxhgY\[�à�Û�[2w #*çíëRÞZ[�]�ß�acÛÓÞZ`\hcå�kGì(`yà:A�� .l\à´bQqGaz] % R�� % "1' ����Ü$ =<>�>\Ü$ # ?aAyâ�âµÛ*Þ�l�Û

� ������-����01 ����� �#��-�����*-�� � � ��� 1(1�'��Z- � 1��� � -������-�� � � ��� 1(1�'��Z- � 1���Z- � 1 ���-�����*-�� � � ��� 1(1�'��Z- � 1! �Z- � 1$-(7 '����-������-�� � � ��� 1(1(1! �Z- � 1 ���-�����1 �

Page 27: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� / �

k�az]Zä2[k �<�>�>\Ü$ �#)?BA���âG4. #�ç'$�~�à�Û*ÞZ[�à�qõ`yà�[\ß�ì(`yàGl�hch�Y\[�à�Û�acä2[2k 4. #�ß�acÛ»ÞZ`\hcå�k¼Û*Þ�l�Û<�#�- ,�Ü$ nâI. <�#�Ü$ �â � U�Ü$,�â ���&ÜE �â2ç��`y]Zk�[2ór~Z[�]iÛ�hcp\ß�Û*ÞZ[Gaz]Zå�~Zä2Û�ac`y] Þ\p�ã�`iÛ*ÞZ[2k�ackDÞZ`\hcå�kl�byl�az]_l�ÛRÛ�a'qõ[�k�Û�[�ã�� ,\ç

g�ç_qR[2wZÛ�ß\m�[.ã�à´`�Yi[.á\pGaz]Zå�~Zä2Û�ac`y]¼`y]õÛ*ÞZ[è]Z~�qGá�[�àµ`�ì�Û�azqG[.k�Û�[�ãZk ��Û*Þ�l�Û§Û*ÞZ[.ã�`iÛ�[�]\Û�a'lxhÜ$U�Ü$,�â ,�â � " R � <4Ü$ nâ C<4Ü$ #!âRaz]Zä�à�[�l�k�[2kfá\píl�Û.hc[�l�k�Û�`y]Z[GmRacÛ*Þé[�l�ä ÞéqG`�Y\[�qG[�]iÛ�ç��`y]Zk�acå�[�àRl\]¼l\à�áZacÛ*à�l\à�pGÛ�azqG[�k�Û�[�ã��*ç§ëRÞZ[.`Z]Zhjp_~iæ!Y�lxh'~Z[ºÛ*Þ�l�Û§ä Þ�l\]Zbi[2kQack2<�Ü$ �#!âµì(`yàÛ*ÞZ[f]Z`y]iæ!b\`ylxh�Y\[�à�Û�[2w! # ç�ò:[ºå�ack�Û�az]Zby~Zack*ÞéÛ*Þ�à�[2[�ä�l�k�[2k��Ü¢l\â%&�ì: �#�- ,94. #»Ü ,�âfli]ZåZ<�#*Ü$ #!â � <_#*Ü$ �#�-,*â_Ü�giâ2ß�Û*ÞZ[�]C<�#�- ,�Ü$ #�- ,*â8. <�#*ÜE �#�- ,�â

l�ä2ä2`yà�å�az]ZbÓÛ�`¸Ü ,�â6ç <_#�- ,�Ü$ �#(â � , 9<�#*Ü$ #!â�l�ä2ä2`yà�å�az]ZbGÛ�`|Ü�g\â§l\]Zå �Ý`y]Zå�acÛ�ac`y] j�ç< # Ü$ # â < # Ü$ #�- , â�� IU�Ü$,�â�l�ä2ä2`Zà�å�az]Zb_Û�` ��`y]Zå�acÛ�ac`y] j�ç �Ý`y]Zk�[2ó�~Z[�]\Û�hcp\ß�Û*ÞZ[ã�`\Û�[�]\Û�azl�h�a']Zä�à´[�l�k�[2k�á\p

-(- �Z- � 1�'¼7�1 � �� �� � ����� -���1#��� � �#� -�� ����� 1(1�s-(- �Z- � 1�'ê7´1 � �� �� � � -���1��� � -�� � 1(1! -(- �Z- � 1 '_7�1�� � �#� -�� � 1���� ����� -�� ����� 1(1��¼-(- �Z- � 1�'¸7´1�� � -�� � 1���� � -�� � 1(1( -(- �Z- � 1 '_7�1$-(7�'&� � -�� � 1(1#�$� ����� -�� ����� 1(1� �Z- � 1�� � -�� � 1! -(- �Z- � 1 '_7�1$-(7�'&� � -�� � 1(1#�$� � -�� ����� 1(1�� �r- � 1�� � -�� � 1! - �Z- � 1 '_7�1�'ê-�� �¢-�����1������$-��������!1(1( - �Z- � 1 '_7�1�'ê-�� �Z- � 1(1! 7")

Ü(á�â%&�ì: �#�- ,94. #»Ü ,�âfli]ZåZ<�#*Ü$ #!â � <_#*Ü$ �#�-,*â_Ü�giâ2ß�Û*ÞZ[�]C<�#�- ,�Ü$ #�- ,*â8. <�#*ÜE �#�- ,�âl�ä2ä2`yà�å�az]ZbÓÛ�`¸Ü ,�â6ç <_#*Ü$ �#�-,�â � <�#*Ü$ #¦â ,�l�ä2ä2`yà�å�az]ZbGÛ�`|Ü�g\â§l\]Zå �Ý`y]Zå�acÛ�ac`y] ,\ç<�#�- ,�Ü$ #!â � <_#*Ü$ �#!âRlxä2ä2`yà�å�az]Zb_Û�` ��`Z]Zå�ajÛ�aj`Z] g�ç ��`y]Zk�[2ór~Z[�]iÛ�hcp\ß�Û�ÞZ[Dã�`iÛ�[�]\Û�a'lxhaz]Zä�à�[�l�k�[2k�á\p

-(- �Z- � 1�'¸7´1 � �� �� ��������-���1�����������-��������!1(1��Ó-(- �Z- � 1�'ê7�1 � �� �� ���¢-���1������(-�����1(1! -(- �Z- � 1�'_7�1�� ����� -�� � 1���� ����� -�� ����� 1(1��¼-(- �Z- � 1 '¼7´1�� � -�� � 1��� � -�� � 1(1! -(- �Z- � 1�'_7�1���������-����01������¢-��������!1(1�Ó-(- �r- � 1�'ê7´1����¢-����01#�����$-����z1(1( -(- �Z- � 1�'_7�1�� ����� -�� � 1��Ó-�� � -�� � 1#�»7�1��¼-(- �Z- � 1�'¼7�1�� � -�� � 1���� � -�� � 1(1! - �Z- � 1�'_7�1$-�� ����� -�� � 1���� � -�� � 1(1 '_7( 7")

Ü�ä�â%&�ì+ �#�- , . �#�ßZÛ*ÞZ[�]�<�#�-,�ÜE �#(â � , F<_#*Ü$ �#(â�l�ä2ä2`yà�å�az]ZbÓÛ�` ��`Z]Zå�ajÛ�aj`Z] jDli]Zå_Û*ÞZ[ã�`\Û�[�]\Û�azl�h�a']Zä�à´[�l�k�[2k�á\p

-(- �Z- � 1�'¼7�1 � �� �� ��������-���1#����� �#��-��������!1(1�s-(- �Z- � 1�'ê7´1 � �� �� ���¢-���1��� �¢-�����1(1

Page 28: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

/ � � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

! -(- �Z- � 1�'¼7´1���������-�� �z1��� ������-��������!1(1��»-(- �r- � 1�'¼7�1����$-����z1������(-�����1(1! -(- �Z- � 1�'¼7´1�� ����� -�� � 1��� ����� -�� � 1(1#�Ó-(- �Z- � 1�'¼7�1�� � -�� � 1��&� � -�� � 1(1! �Z- � 1$-�� ����� -�� � 1���� � -�� � 1(1( �Z- � 1$-(7�'&� � -�� � 1��$� � -�� � 1(1! �Z- � 1( 7%)

j�ç $�az]�l�hchcp\ß�m�[Dã�à´`�Yi[fÛ*ÞZ[ÓqÓl�az]êÛ*ÞZ[2`yà�[�q_ç�ëRÞZ[Gã�`\Û�[�]\Û�azl�hRÜ�Ü$U�Ü$,�â ,�â � ")R�� <4Ü$ nâ <�Ü$ # âºackÓaz]ZacÛ�azl�hchjpLo2[�à�`�ß�az]Zä�à�[�lxk�[2k»m,ajÛ�Þ:[2Y\[�à´p qG`�Y\[�qõ[�]\ÛGá\p·lxÛDhc[�l�k�ÛÓ`y]Z[\ßgl\]Zåack_á�`y~�]Zå�[2åLì!à�`yqïl\á�`�Y\[|á\p Ü$U�Ü$,�â� ,�â � ")R�� U�Ü$,�â ���&Ü$ nâ2çÁëRÞZ[ qsl�az]ÁÛ*ÞZ[2`yà´[�qì(`\hchc`�m,k�ç

��[�l�h æ!Û�azqG[�k�[�lià�ä ÞsqG[2Û*ÞZ`Zå�kgÛ*Þ�l�Û§k*l�Û�ack�ì(p ��`y]Zå�acÛ�ac`y]Zk,\ßOg�ßyl\]Zå j�`xì�ë,ÞZ[2`yà�[�q ,Zhä2`�Y\[�à�Ü�k�Û*à�`Z]Zb\hcpDä2`Z]�]Z[2ä2Û�[2å�â�byà�liã�ÞZkÝà´[�ã�[�l�Û�[2å�hcp¼k�az]Zä2[fÛ*ÞZ[�ã�à�`Z`�ì4az]_XZ[2ä2Û�ac`y] jGl\ã�ãZhcac[2kÛ�`¼Û*ÞZ[�q_ç�&$]íÛ*ÞZ[�ì(`\hchc`�mRaz]Zb�ß�m�[Gk*ÞZ`�m ÞZ`�mIëRÞZ[2`yà�[�q ,Zh»ä�l\]íá�[õ~Zk�[2å:Û�`_ã�à�`�Y\[fÛ*Þ�lxÛÛ*ÞZ[fä2`�Y\[�àÝÛ�azqG[2k.`xìgk�az]Zb\hc[õli]\Û�k,Û*Þ�l�Û.~Zk�[G{ ��ë�ÚnmyßOò·l�by]Z[�à�� kRögl�hz~Z[�æ -.ã�å�l�Û�[ �,~Zhj[\ß�`ZàëRÞ�à�~�]��#ksögl�hz~Z[�æ -.ã�å�l�Û�[ �R~Zhc[ l\à´[|ã�`\hcp�]Z`yqGazl�hRaz]Z,�çÁëRÞZack¼ajksk�`·á�[2ä�li~Zk�[|l�hchRÛ*Þ�à�[2[ä2`y]Zå�acÛ�ac`y]Zk_`�ì�Û*ÞZ[_Û*ÞZ[2`yà´[�qïÞZ`\hcåÁì(`yàGÛ*ÞZ[2k�[_Û*Þ�à�[2[|à�[�l�h æ!Û�azqG[ík�[�lià�ä ÞÁqõ[2Û*ÞZ`yå�kõì¦`yàslì¢~�]Zä2Û�aj`Z]8U�Ü$,�â�Û*Þ�lxÛ4ackgã�`ihjp�]Z`yqGazl�h�az]8,�ç�ëRÞZ[�]�ß�Ü$U�Ü$,�â %,�â(U�Ü$,�â � ")R�� ����Ü$ �â��BÜEU¾ÜE,&â ,�â(U�Ü$,�â � � � ,� +�� . ÜEU¾ÜE,&â .,�â U¾ÜE,&ânÜ$,�Ü$, ,�â 9 g\â2ß�l�ä2ä2`yà�å�az]Zb�Û�`�`y~�à�l�k�k*~�qsãZÛ�ac`y]ºÛ*Þ�l�Û�Û*ÞZ[byà�l\ã�Þºackgk�Û*à�`y]Zb\hcpºä2`y]�]Z[2ä2Û�[2å�ç���`Z]Zk�[2ór~Z[�]iÛ�hcp\ß\Û�ÞZ[Rä2`�Y\[�à�Û�a'qõ[RackQã�`ihjp�]Z`yqGazl�hOaz]8,�ç *�]Û*ÞZ[�`\Û*ÞZ[�à,Þ�l\]Zå�ß�l�hch�Û*Þ�à�[2[�ä2`y]Zå�acÛ�ac`y]Zk�`�ì�Û*ÞZ[�Û*ÞZ[2`yà�[�qôÞZ`ihjå¼ì(`yà_q,`Zå�[���`y~�]iÛ�az]ZbÓ`y]Zhcpì¦`Zà4l§ì!~�]Zä2Û�ac`y]WU�Ü$,�â�Û�Þ�l�Û�ä�l\]ºá�[�[2wrã�`y]Z[�]iÛ�azl�h�az]:,�ç�$�`yà�[2w�l\qÓãZhc[\ßQU¾ÜE,&â � g � & � � + & ��ì¦`ZàÛ*ÞZ[fbZà�l\ã�Þ¼Û�`yã�`\hc`\b\pÓaz] $�ajbZ~�à�[ oGk�a']Zä2[W<4Ü � , â 9<4Ü � � â . g � & � � + & �.l�ì¦Û�[�à§Û�[�à�qGaz]�l�Û�ac`y]�ç

� � � �"! � �Ý`y]Zå�acÛ�ac`y] g,`�ì&ëRÞZ[2`yà�[�q ,ZhèÞZ`\hcå�kgl�k�k*ÞZ`�m.]GáipfëRÞZ[2`yà�[�q g�ç���`y]Zå�acÛ�ac`y]Zk ,l\]Zå jíÞZ`\hcå á�[2ä�l\~Zk�[\ß�l�ä2ä2`yà�å�az]Zb:Û�`íÛ*ÞZ[¼Y�lxh'~Z[�æ�~�ã�å�l�Û�[:à�~Zhc[_`�ìf{ ��ë�Únmyß < #�- , Ü$ # â8., ºqGaz]% R��:<_#*Ü$ �<�>�>\Ü$ �#)?BAZâ�â2ß6m.ÞZ[�à�[+<�#*ÜE �â�å�[�]Z`\Û�[2k4Û*ÞZ[ <Zæ!Y�l�hz~Z[Q`xìrY\[�à´Û�[2w §azqÓqG[2å�azl�Û�[2hjpá�[�ì(`yà�[�Û*ÞZ[�l\]\Û�~�ã�å�lxÛ�[2k�Û*ÞZ[ <ræ!Y�l�hz~Z[�`�ìn #}li]Zå <�#�-,�ÜE �â�Û*ÞZ[gk*liqG[ <ræ!Y�l�hz~Z[RazqÓqG[2å�azl�Û�[2hjplnì(Û�[�àRÛ�ÞZ[D~�ã�å�lxÛ�[\ç.ëRÞ�lxÛ ��`y]Zå�acÛ�ac`y] h_ÞZ`\hcå�k�ì¦`yà U¾ÜE,&â . ,fmRl�k�ã�à�`�Y\[2å|az] îfg�enñ�çQëRÞZackà�[2k*~ZhcÛ�ä�l\]_á�[ºb\[�]Z[�à�l�hcaco2[2åíÛ�`»qG~ZhcÛ�azãZhj[Ól\]iÛ�k�ç

� � ����/ �� ��� ��� � � �����0 � �� � ��`Z]Zå�ajÛ�aj`Z]Zk ,\ßUg li]Zå j|`xìèëRÞZ[2`yà�[�q ,�h:ÞZ`\hcåÙá�[�æä�l\~Zk�[\ß|lxä2ä2`yà�å�az]Zb Û�` ò·l�bZ]Z[�à��#kÁögl�hz~Z[�æ -�ã�å�l�Û�[ �R~Zhc[\ß9< #�- , Ü$ # â7. , < # ÜE # â a ì<�#*Ü$ #!â��ÙqGaz]% R��:<�#*Ü$ =<>�>\Ü$ # ?aAyâ�â6ç¾ë,Þ�l�Û���`y]Zå�acÛ�ac`y]shÓÞZ`\hcå�k�ì(`yà U¾ÜE,&â�. ,�mRl�kRã�à�`�Y\[2åaz]:îfg�p�ñ�ç�ëRÞZackèà�[2k�~ZhjÛ,ä�l\]|á�[fb\[�]Z[�à�lxhjaco2[2åíÛ�`ÓqG~ZhcÛ�azãZhj[Ól\]iÛ�k�ç

���� ���� ������� � � � �����0 � � �� ��`y]Zå�acÛ�ac`y]Zk ,\ßXgÁl\]Zå j `�ìõëRÞZ[2`Zà�[�q ,�hÙÞZ`ihjå á�[�æä�l\~Zk�[\ß�l�ä2ä2`yà�å�az]Zb_Û�`_ëRÞ�à�~�]��#k�öglxh'~Z[�æ -�ã�å�lxÛ�[ �,~Zhj[\ß <_#�- ,6Ü$ #!â2. qÓl�w&Ü , <�#*Ü$ #!â�? ,�qGaz]% R��:<_#*Ü$ �<�>�>\Ü$ �#)?BAZâ�â�â6ç4ëRÞ�l�Û���`Z]Zå�ajÛ�aj`Z] hÓÞZ`\hcå�k�ì(`yà U¾ÜE,&â�.D,émRl�kQã�à�`�Y\[2å_az]íîfgiðnñ�çëRÞZack.Y�l�hz~Z[Ga']Zä�à´[�l�k�[2k�á\p_lºì!l�ä2Û�`yà§`�ìA@sì¦`Zà @él\]iÛ�k�ç

Page 29: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� / �

start

�Z�)�£n !�R7 � ¡�¹��$ ( !���#�

2 4 6 8 10 12 14 16 18 200

500

1000

1500

2000

2500

3000

3500

Number of Ants

Tim

e S

teps

to F

irst C

over

age

Wagner’s Rule LRTA* Node Counting Thrun’s Rule

2 4 6 8 10 12 14 16 18 203000

3200

3400

3600

3800

4000

4200

4400

4600

4800

5000

Number of Ants

Tot

al N

umbe

r of

Mov

es to

Firs

t Cov

erag

e

Wagner’s Rule LRTA* Node Counting Thrun’s Rule

�Z�)�£� !�R7 � ¡&Ð��¶��$ 4¹������������¹�*�(���yÕ�£��µ²x�$ ���� ��¶������z� [�Z�# !�(�4Ð��¶��$ !�*)��

� � ��� �� �// � ���������� �����/� �0��� �� � � ���

&$]fÛ�ÞZajk�k�[2ä2Û�ac`y]�ß�m�[Ýà´[�ã�`yà�Û�[2w�ã�[�à�azqG[�]iÛ*l�h�à�[2k�~ZhjÛ�k�ì¦`yà�Û*ÞZ[§ä2`�Yi[�à�Û�azqG[2k�`xì�Y�l�ä�~�~�qºæä2hc[�l\]Zaz]Zb·l\]iÛ�kºÛ*Þ�l�Ûf~Zk�[¼Y�l\à´aj`Z~Zkõà´[�l�h æ!Û�azqG[»k�[�l\à�ä2Þ:qG[2Û*ÞZ`Zå�k�ß�az]Zä2h'~Zå�az]ZbÁ{ ��ë�Únm|l\]Zåq,`yå�[ ��`y~�]iÛ�az]Zb�ߧa']Lã�l\à�ÛÓ`xì l\] ` (»ä2[|á�~Zachjå�az]ZbLÛ*Þ�l�ÛGä2`y]iÛ*l�az]Zk»Û�Þ�à�[2[_`�(»ä2[2k»li]ZåÙlk*qsl�hch�mRl�acÛ�az]Zb»l\à�[�l�ß�k�[2[$�acby~�à�[ ,0j¸Ü¦Û*ÞZ[fà�[2k*~ZhcÛ�k,ì¦`Zàgå�a/#�[�à�[�]iÛ.Û�[�à�à�l�az]ZkRm�[�à�[�k�azqGachzl\à�â6çò:[íazqÓã�`\k�[2å lÁà�[2by~Zhzl\àõì¦`Z~�à$æ!ä2`y]�]Z[2ä2Û�[2å bZà�acå `�Y\[�àõÛ*ÞZ[íÛ�[�à�à�l�az]�ß�à�[2k*~ZhcÛ�az]ZbÙaz] h ���j�� ä2[2hchck�çôëRÞZ[ l\]\Û�k»ä2`y~ZhcåIqG`�Y\[|Û�`Á[�l�ä ÞL`�ìGÛ*ÞZ[éì¦`Z~�à_]Z[2ajbZÞyá�`yà�az]ZbÆä2[2hjhckí`�ìGÛ*ÞZ[2azàä�~�à�à´[�]\Û§ä2[2hjh�ã�à�`�YZacå�[2å»Û*Þ�lxÛ�Û*ÞZ[Rå�[2k�Û�az]�l�Û�aj`Z]Óä2[2hch�mRl�kgÛ*à�l�Yi[�à�k*l\áZhc[fÜ�m.ÞZacÛ�[�â2ç ��[2hchck�m�[�à�[~�]iÛ*à�lnY\[�à�k�l\áZhc[fa ìgÛ*ÞZ[2p|ä2`Z]\Û*lxa']Z[2åé[2acÛ*ÞZ[�à�mRl�hchjk»Ü¢áZh'lxä tZâ�`yàÝì¢~�à�]ZajÛ�~�à�[|Ü�byà�[2p�â2ç§Ú.hchgl\]iÛ�kk�Û�l\à�Û�[2åÁaz]ÁÛ�ÞZ[|k*l\qõ[_ä2[2hjhÓÜ(qÓlià�t�[2å L�k�Û*l\à´Û(NyâGl\]ZåÙ~Zk�[2åLÛ*ÞZ[|k*liqG[|à�[�l�h æ!Û�azqG[ék�[�l\à�ä2ÞqG[2Û�ÞZ`yå�ß�á�à�[�l\tiaz]ZbÓÛ�ac[2k.à�li]Zå�`yqGhcp\çgëRÞZ[�l\]iÛ�k�[2wy[2ä�~ZÛ�[2å_Û�ÞZ[fà�[�l�h æ!Û�azqG[�k�[�l\à´ä Þ¼qG[2Û*ÞZ`Zåaz]Zå�[�ã�[�]Zå�[�]iÛ�hcpÆ[2Y\[�]Ãaì�k�[2Y\[�à�l�hRl\]\Û�kDk�Þ�l\à�[2åÁl|ä2[2hch(ç �gl�ä2ÞÁl\]iÛÓqG`�Y\[2å:`y]Zä2[_å�~�à�az]Zb[�l�ä2Þ¼Û�azqG[ºk�Û�[�ã¼az]|lGbiajYi[�]_k�[2ó�~Z[�]\Û�a'lxh�`yà�å�[�à2ç �� ¿f������(g®\�$ (�z� !��������°�®\�$ (����$���(�4³����$ !�4�����4�����(���Q�¶*�����*�(­����(�� !����£����­�� ��#�R !��������Á� !���$ �²�£��������µ����f�(R����§�#��� Q !����£n��(����°2���$®��������*�������g¶��� (��������R��&�������(��(�����2£���²\�$ g������¶*�����z� ¾�����g"n !�(����¶��! !��)��g�����$ !���*�(����³�����f�������2£��µ²x�$ ���������(���#���(�(��������� !�������#���#��)R !�£�)����­�������(�(������¡

Page 30: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� + � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

&!]í`Z~�à,÷&à�k�ÛfÛ�Þ�à�[2[G[2wrã�[�à�azqG[�]\Û�k�ß�mQ[Gk�Û�~Zå�aj[2å Û*ÞZ[Óä2`�Y\[�à�Û�a'qõ[2kf`�ì�Û*ÞZ[Óà�[�l�h æ!Û�azqG[k�[�l\à�ä2Þ¼qG[2Û*ÞZ`Zå�k�ì¦`Zà�`y]Z[�æ!Û�azqG[�Y�l�ä�~�~�q ä2hc[�l\]Zaz]Zb_l\]Zå»Û�ÞZ[2a'àÝYyack�acÛ�ì!à�[2ó�~Z[�]Zä2aj[2kÝì¦`yà§ä2`y]iæÛ�az]y~Z`Z~ZkDY�l�ä�~�~�q ä2hc[�l\]Zaz]Zb�ç &$]:`Z~�à�÷&à�k�Ûº[2wrã�[�à�azqG[�]iÛ�ß�mQ[¼qG[�l�k*~�à�[2å Û*ÞZ[»]Z~�qGá�[�àº`�ìÛ�azqG[�k�Û�[�ãZk�~�]\Û�ach�Û*ÞZ[fl\]iÛ�k�ä2`�Y\[�à�[2å¼Û*ÞZ[�Û�[�à�à�l�az]Gì(`yà§Û*ÞZ[�÷&à�k�Û�Û�azqG[»Ü¦ä2`�Yi[�àgÛ�a'qõ[�â2ß�l�Yi[�à$æl�b\[2å¼`�Y\[�àYg�ß �����ºà�~�]Zk�ç�ëRÞZackRackRazqÓã�`yà�Û*l\]iÛ§ì¦`Zà�`y]Z[�æ!Û�azqG[�Y�l�ä�~�~�q ä2hc[�l\]Zaz]Zb¼k�az]Zä2[�[�l�ä Þä2[2hchÝÞ�lxkfÛ�`|á�[»Y�l�ä�~�~�qG[2åÁl�Û�hc[�l�k�Ûf`Z]Zä2[\ç $�acby~�à´[ ,�h·Ü�hc[�ì(Û*â�k*ÞZ`�mRk�Û�ÞZ[¸à�[2k�~ZhjÛ�k�ç»ëRÞZ[Û*à�[�]ZåímRl�kRÛ*ÞZ[ºk*l\qG[ ì¦`Zà.l�hch�à�[�l�h æ!Û�azqG[Gk�[�l\à´ä ÞéqG[2Û*ÞZ`Zå�k�ç.ëRÞZ[ºä2`�Y\[�àRÛ�a'qõ[2k.azqÓã�à�`�Y\[2ål�kgqG`yà´[Ýli]\Û�k�m�[�à�[Rl�å�å�[2å¼l�hcÛ*ÞZ`y~ZbyÞGÛ�ÞZ[.à�l�Û�[§`�ì&azqÓã�à´`�Yi[�qG[�]\Û�å�[2ä�à´[�l�k�[2å�ç�&$]Gb\[�]Z[�à�l�h(ßl�hch�ä2`�Y\[�à§Û�azqG[2kRm�[�à�[�k�azqGachzl\à2ç�ë�`Óå�a/#}[�à´[�]\Û�azl�Û�[fá�[2Û�Û�[�à,á�[2Û�m�[2[�]¼Û*ÞZ[fà�[�l�h æ!Û�azqG[ºk�[�l\à�ä2ÞqG[2Û*ÞZ`Zå�k�ß�$�acby~�à�[ ,�h·Ü¢à�acbyÞ\Û�â�k*ÞZ`�mRkfÛ*ÞZ[sÛ�`\Û*l�h�]Z~�qDá�[�à�`�ì.qG`�Y\[2kfqsl�å�[»áip lxhjhgli]\Û�knßÛ*Þ�l�Û�ack�ß�Û*ÞZ[Ó]y~�qGá�[�à�`�ìQl\]iÛ�k.Û�a'qõ[2k�Û*ÞZ[Gä2`�Yi[�àRÛ�azqG[\ç&¢Ûfl\ã�ã�[�l\à´k.Û*Þ�lxÛ.Û*ÞZ[Gå�a/#�[�à�[�]Zä2[az]_Û*ÞZ[�Û�`iÛ*l�h&]Z~�qGá�[�à�`xì¾qG`�Y\[2k�`�ì4Û*ÞZ[�å�a/#�[�à�[�]\Û�à�[�l�h æ!Û�azqG[fk�[�l\à�ä2Þ»qG[2Û�ÞZ`yå�kRmRl�k�az]Zå�[�æã�[�]Zå�[�]iÛG`�ìQÛ*ÞZ[¼]y~�qGá�[�à `�ìÝli]\Û�knçÓëRÞ�à�~�]��#k�ögl�hz~Z[�æ -�ã�å�l�Û�[ �R~Zhc[»mRl�kfá�[2k�Û�ß�ì(`\hchc`�mQ[2åá\p|{ �4ë�ÚFm»li]Zå q,`yå�[ ��`y~�]iÛ�az]Zb:Ü�Û*Þ�l�ÛRm�[�à�[Gl�hzqG`\k�Û,a']Zå�ack�Û�a']ZbZ~Zajk�Þ�l\áZhz[�â2ß�l\]Zåê÷&]�l�hchjpì¦`ihjhc`�m�[2å áip_ò·l�by]Z[�à �#kRögl�hz~Z[�æ -�ã�å�l�Û�[ �R~Zhc[\ç�ëRÞZ[fä2`�Y\[�à�Û�a'qõ[f`�ì�k�az]Zb\hc[Ól\]\Û�kRmRl�kRá�[�æÛ�mQ[2[�] j�ßfg�����l\]Zårj�ß r����,mRacÛ*ÞÓl k�Û*l\]Zå�l\à´åDå�[2YZazl�Û�ac`y]¼á�[2Û�m�[2[�]Óð�����li]Zåro�����ç�ëRÞZ~Zk�ß\Û*ÞZ[å�a/#}[�à´[�]Zä2[faz]_ã�[�à$ì(`yà�qsl\]Zä2[.mRl�k§å�`yqGaz]�l�Û�[2å|áipÓÛ*ÞZ[�k�Û*li]Zå�l\à�å¼å�[2Yyazl�Û�ac`y]�ç�ëRÞZack�[2wrã�[�à$æazqG[�]\Û�l�h¾à´[2k*~ZhcÛ�ajk�az]\Û�[�à�[2k�Û�az]Zb|á�[2ä�li~Zk�[G`y~�à�Û*ÞZ[2`yà´[2Û�acä�l�h4m�`yà´k�Û�æ!ä�l�k�[fà´[2k*~ZhcÛ�k.k�~Zb\b\[2k�Û�[2åÛ*Þ�l�Û�Û*ÞZ[�ä2`�Y\[�à�Û�a'qõ[�`�ì1q,`yå�[ ��`y~�]iÛ�az]ZbõqõajbZÞ\Û.á�[DqG~Zä2ÞÓhzl\à�bi[�à�Û*Þ�l\]¼Û*ÞZ[�ä2`�Y\[�à§Û�azqG[`�ì.{ ��ë�ÚnmyçÓòÃ[»l�hck�`_[2w�ã�[�à�azqG[�]iÛ�[2å mRacÛ*Þ:à�l\]Zå�`Zq mRl�hzt\kfá�~ZÛºÛ*ÞZ[2azà�ä2`�Y\[�à�Û�azqG[smÝlxkqG~Zä Þ¼hzl\à�bi[�àgÛ�Þ�l\]¼Û*ÞZ`\k�[�`�ì4Û�ÞZ[fà�[�l�h æ!Û�azqG[fk�[�l\à�ä2Þ»qG[2Û�ÞZ`yå�k.liá�`�Y\[\ç�$�`yà§[2w�l\qÓãZhc[\ß�Û*ÞZ[ä2`�Y\[�àRÛ�a'qõ[D`xì¾k�az]Zb\hc[»li]\Û�k�Û*Þ�l�Û�~Zk�[Óà�l\]Zå�`yqTmRl�hzt\k�mRl�k.l\ã�ã�à�`�wZazqÓl�Û�[2hcp hWe�ß/�����GmRacÛ*ÞlGk�Û*li]Zå�l\à�å_å�[2YZa'lxÛ�ac`y]_`�ìgl\á�`y~ZÛdg���ß/������ç

&!]é`y~�à,k�[2ä2`y]Zåí[2w�ã�[�à�azqG[�]iÛ�ß�m�[DqG[�lxk*~�à�[2åíÛ*ÞZ[�ì!à�[2ó�~Z[�]Zä2pímRacÛ*Þ_m.ÞZacä2ÞíÛ*ÞZ[Gl\]\Û�kYyack�acÛ�[2åí[�l�ä2Þ_ä2[2hch�`�ì�Û*ÞZ[�Û�[�à�à�lxa']êm.ÞZ[�]_Û�ÞZ[2p»ä2`�Y\[�à�[2åéÛ*ÞZ[�Û�[�à�à�l�az]|à�[�ã�[�l�Û�[2å�hcp\çRëRÞZack.ackazqÓã�`yà´Û*l\]iÛ�ì(`yà�ä2`y]iÛ�az]Z~Z`y~Zk�Y�l�ä�~�~�q ä2hc[�l\]Zaz]ZbDá�[2ä�l\~Zk�[�`y]Z[Qã�à´`yá�l\áZhcp�mRl\]\Û�k�Û�`�Y�l�ä�~�~�q[�l�ä2Þ_ä2[2hch4[2ó�~�l�hchcpí`�ì¦Û�[�]¼a']éÛ*ÞZ[�hc`y]Zb_à�~�] Ü(l�hcÛ*ÞZ`y~ZbyÞê`y]Z[fä2`Z~Zhjå:l\à´by~Z[�Û*Þ�l�Û.qõ`\k�ÛRå�azà�Ûl�ä2ä�~�qG~Zhzl�Û�[2kGm.ÞZ[�à�[_ã�[2`yãZhc[émÝlxh'tíl\]Zå Û*ÞZ~ZkGÛ*Þ�l�Ûf`Z]Z[ÓmRl\]\Û�kfÛ�`éY�l�ä�~�~�q ä2[2hchck»ä2hc`\k�[Û�`í`yáZk�Û*lxä2hj[2kshj[2k�kÓ`�ì(Û�[�]�â2çéò:[_k�Û�`yã�ã�[2åÁÛ*ÞZ[_l\]\Û�kõl�ì¦Û�[�àng�ß ���0��ß �����¼Û�a'qõ[_k�Û�[�ãZk�ç:ëRÞZ[Yyack�acÛsì¢à´[2ór~Z[�]Zä2ac[2k_å�acå ]Z`\Ûsä Þ�l\]Zbi[|k�acby]Za ÷�ä�l\]iÛ�hcpÙl�kÓÛ*ÞZ[í]y~�qGá�[�àG`�ìfl\]\Û�kÓaz]Zä�à�[�l�k�[2å�ß]Z`:qÓl�Û�Û�[�à�m�ÞZajä2Þ à�[�l�h æ!Û�azqG[_k�[�l\à´ä Þ:qõ[2Û*ÞZ`yå�kºm�[�à�[»~Zk�[2å�çéò:[ÓÛ*ÞZ[�à´[�ì¦`yà´[»à�[�ã�`Zà�ÛfÛ*ÞZ[Yyack�acÛºì!à�[2ór~Z[�]Zä2ac[2kGì(`yà�k�az]Zb\hc[_l\]iÛ�k�`y]Zhcp\ç.$�ajbZ~�à�[ ,=r¼k*ÞZ`�mRk�Û*ÞZ[»à�[2k�~ZhjÛ�k�ç.�fl\à�t�[�à�ä2[2hchckaz]ÁÛ*ÞZ[¼÷�by~�à´[_mQ[�à´[_Yyack�acÛ�[2åLhj[2k�kÓì!à�[2ó�~Z[�]\Û�hcp\çÁÚ�kÓÛ*ÞZ[¼÷�bZ~�à�[_k*ÞZ`�mRk�ß�k�`yqG[_à�[�l�h æ!Û�azqG[k�[�l\à�ä2Þ_qG[2Û*ÞZ`Zå�k�Yyack�acÛ�[2åéÛ*ÞZ[�ä2[2hchckèqG`Zà�[è~�]Za ì(`yà�qGhcp¼Û*Þ�l\]¼`\Û�ÞZ[�à�k�ç�$�`yà§[2w�l\qÓãZhc[\ß(q,`yå�[��`y~�]iÛ�az]ZbÓYZajk�ajÛ�[2å_Û*ÞZ[.ä2[2hchck�qD~Zä2Þ¼qG`yà�[�~�]Za ì¦`yà�qGhcp_Û*Þ�l\]¼{ �4ë�ÚnmyçgëRÞZackRackRaz]iÛ�[�à�[2k�Û�az]Zbá�[2ä�l\~Zk�[QÛ�ÞZ[gÛ�`\Û�l�hy]Z~�qDá�[�à�`xì�qG`�Y\[�qõ[�]\Û�k�`�ì&{ ��ë�Únm.l\]Zå q,`Zå�[+��`y~�]iÛ�az]Zb�mÝlxk�k�azqGachzl\àaz]Ó`y~�à¾÷&à�k�Û§[2wrã�[�à�azqG[�]\Ûnç�*�]Z[èqõ[�l�k*~�à�[�ì(`yàµÛ*ÞZ[.~�]Za ì¦`Zà�qGacÛ�pD`xì}Û�ÞZ[RYyack�acÛ§ì!à�[2ór~Z[�]Zä2ac[2kRackÛ*ÞZ[2azàR[�]iÛ*à�`Zã\p\ç�ëRÞZ[fhzl\à´b\[�àRÛ�ÞZ[�[�]\Û*à´`yã\p � "�� Ü$ nâ�hc`\b � � Ü$ �â2ß�Û*ÞZ[fä2hc`\k�[�à,Û�`»~�]Za ì(`yà�qÛ*ÞZ[�Yyack�acÛ�ì!à�[2ó�~Z[�]Zä2aj[2k � Ü$ �â2ç§ëRÞZ[.[�]iÛ*à�`Zã\pG`�ì4~�]Za ì¦`Zà�q Yyack�acÛ�ì!à�[2ó�~Z[�]Zä2aj[2k,ajk_e �ko�p�j���l\]ZåÛ*ÞZ[�[�]iÛ*à�`yãZac[2k�`�ì4Û*ÞZ[�YZack�acÛ�ì!à�[2ór~Z[�]Zä2ac[2k�`�ì4Û*ÞZ[�à�[�l�h æ!Û�azqG[�k�[�l\à�ä2Þ»qG[2Û�ÞZ`yå�kRm�[�à�[���q,`yå�[��`y~�]iÛ�az]Zb¸Ü�e �fo�p�g�eiâ2ß2ò·l�bZ]Z[�à��#k�ögl�hz~Z[�æ -.ã�å�l�Û�[�R~Zhc[õÜ�e �fo�o�o�eiâ2ß�ëRÞ�à�~�]�� k�ögl�hz~Z[�æ -.ã�å�l�Û�[�R~Zhc[ Ü�e �ko�o�o�g\â6ßrli]Zå { ��ë�Únm Ü e �fo�o�g=o\â2ç�Ú�]Z`\Û*ÞZ[�à�qG[�l�k*~�à´[fì(`yà�Û�ÞZ[õ~�]Za ì(`yà�qGacÛ�p_`�ìgÛ*ÞZ[Yyack�acÛ ì¢à�[2ó�~Z[�]Zä2ac[2kGajkºÛ*ÞZ[Gå�a/#}[�à´[�]Zä2[¸á�[2Û�m�[2[�]íÛ�ÞZ[õqõ[�l\]éÛ�azqG[2kDá�[2Û�m�[2[�]íYZack�acÛ�kfÛ�`»Û*ÞZ[

Page 31: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� � 7

���K^���� �'$�V����.$������C�[�=B

ZB����Q!� � �.������;��S'TOVW��C��������� ^���!C���� ���A������ �\SUTWVWZ�������=�[��Z�)�£n !�.7 � ¡�Ø������ �n !��À�£��$���$������z� 4½��$®\���*�(���fÐ��¶��$ !�*)��

2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

1400

Number of Ants

The

Gap

of M

ean

Tim

e be

twee

n V

isits

Thrun’s Rule LRTA* Wagner’s Rule Node Counting

�Z�)�£� !�§7 � ¡ ���#Â\�! !�$��������yÒy�� !)����(�O�����.×2�����#����(�& ������.¹������²\���0³����$�RØ������(�O�z� �½��!®\�����(����Ð��¶��$ !�*)��

ä2[2hchckÓmRacÛ*ÞÃÛ*ÞZ[Óhzl\à�bi[2k�ÛDl\]Zå k*qÓlxhjhc[2k�ÛGqG[�l\]ÃÛ�azqG[»á�[2Û�m�[2[�]ÃYyack�acÛ�knç|ëRÞZ[¼k*qÓl�hchc[�àºÛ*ÞZackY�l�hz~Z[\ßgÛ*ÞZ[sä2hj`ik�[�àºÛ�` ~�]Za ì(`yà�q Û*ÞZ[¼Yyack�acÛGì!à�[2ó�~Z[�]Zä2aj[2knç $�acby~�à�[ ,�ðêk*ÞZ`�mRkfÛ*ÞZ[_à�[2k*~ZhcÛ�knßÛ*Þ�lxÛgm�[�à�[Rk�azqGachzl\à§Û�`ºÛ*ÞZ`\k�[�ì(`yà�Û*ÞZ[,[�]\Û*à´`yãZac[2k�ç�*�]ZhcpÓÛ*ÞZ[�ã�`\k�ajÛ�aj`Z]ZkQ`xì�ë,Þ�à�~�]��#kgöglxh'~Z[�æ-�ã�å�l�Û�[ �R~Zhc[Dli]Zå { ��ë�ÚnmGmQ[�à´[�k�mRacÛ�ä2ÞZ[2å�ç

Page 32: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� / � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

2 4 6 8 10 12 14 16 18 200

500

1000

1500

2000

2500

3000

3500

Number of Ants

Wei

ghte

d A

vera

ge o

f Sta

ndar

d D

evia

tion

of T

ime

betw

een

Vis

its

Node Counting Wagner’s Rule LRTA* Thrun’s Rule

�Z�)�£n !�R7 � ¡�×2�(�����n�� !������¶���*�����D���������¹�����µ²x����³����$��Ø������(���z� �½��$®x�����(���fÐ��¶��! !��)��

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.23500

3600

3700

3800

3900

4000

4100

Probability of Being Kicked

Tot

al N

umbe

r of

Mov

es to

Firs

t Cov

erag

e

Wagner’s Rule LRTA* Node Counting Thrun’s Rule

�Z�)�£n !�R7 � ¡O¹�*�(���rÕ�£��µ²\�! ���� ��¶������z� �Z�# !�(�¾Ð��¶*�$ !��)*���#�������������(��¿º�$ !������(�2���&!]Æ`Z~�à¼Û*ÞZazà�å [2w�ã�[�à�azqG[�]iÛ�ß�m�[ qõ[�l�k*~�à�[2å m�ÞZ[2Û*ÞZ[�à¼Û*ÞZ[:Û�azqG[2k|á�[2Û�mQ[2[�] Yyack�acÛ�k

Û�`íÛ*ÞZ[_ä2[2hchck»m�[�à�[_k*ã�à´[�l�åÁ`y~ZÛG[2Y\[�]Zhcp\çÙëRÞZackÓackÓazqÓã�`yà�Û*l\]iÛºì¦`ZàGä2`y]iÛ�az]y~Z`y~ZksY�l�ä�~�~�qä2hc[�l\]Zaz]Zb á�[2ä�l\~Zk�[ÓacÛºk�[2[�qGkfá�[2Û�Û�[�à�a ì,l¼ä2[2hch�b\[2Û�k�Yyack�acÛ�[2å:[2Yi[�à�p , ���sÛ�azqG[Gk�Û�[�ãZk�Û*Þ�l\]a ìèacÛGb\[2Û�kGYZajk�ajÛ�[2åÁ[2Y\[�à�p , �êÛ�azqG[¼k�Û�[�ãZkºì¦`Zà eêÛ�azqG[2kÓaz]Ál|à´`�m l\]Zå Û*ÞZ[�]Ál�byl�az]Ã`y]Zhcplnì(Û�[�àFe�, �_Û�azqG[_k�Û�[�ãZk�ç *�]Z[|qG[�l�k�~�à�[»`xì,ÞZ`�m [2Y\[�]Zhcp·Û*ÞZ[¼Û�azqG[2kÓá�[2Û�m�[2[�]ÃYyack�acÛ�k»lià�[k*ã�à�[�l�åí`y~ZÛ�ack�Û*ÞZ[DlnY\[�à�l�b\[f`xìgÛ*ÞZ[fk�Û�l\]Zå�l\à�åéå�[2Yyazl�Û�ac`y]Zkº`�ìgÛ*ÞZ[ºÛ�azqG[2kfá�[2Û�m�[2[�]éYyack�acÛ�k`�Y\[�à�l�hch�ä2[2hchjknß�m�[2acbyÞiÛ�[2å_mRacÛ*Þ¼Û*ÞZ[2azà�YZack�acÛRì!à�[2ó�~Z[�]Zä2aj[2knçQëRÞZ[�ä2hc`\k�[�à�Û�`Óo2[�à�`GÛ*ÞZackRY�l�hz~Z[\ßÛ*ÞZ[Dqõ`yà�[f[2Yi[�]Zhjp_k�ã�à�[�l�å|`Z~ZÛ.Û*ÞZ[�Û�a'qõ[2kfá�[2Û�m�[2[�]_YZajk�ajÛ�k�ç�ò:[fk�Û�`yã�ã�[2åéÛ*ÞZ[Dli]\Û�k�lnì¦Û�[�àg�ß ���0��ß ������Û�a'qõ[.k�Û�[�ãZk�ç�$�ajbZ~�à�[ ,0ofk�ÞZ`�m,kgÛ*ÞZ[�à�[2k*~ZhcÛ�k�ç§ëRÞZ[.Û�a'qõ[2kÝá�[2Û�mQ[2[�]sYZajk�ajÛ�kRm�[�à�[Y\[�à�p|~�]Z[2Y\[�]Zhcpék*ã�à�[�l�åé`y~ZÛ�ì(`yà_q,`yå�[ ��`y~�]iÛ�az]Zb¸l\]ZåíqG`yà�[�[2Yi[�]Zhjpék*ã�à�[�l�åé`y~ZÛ�ì(`yàRl�hch`\Û*ÞZ[�àÓà´[�l�h æ!Û�azqG[ík�[�l\à�ä2ÞÁqG[2Û*ÞZ`Zå�k�ߧmRacÛ*ÞÙl�hzqG`ik�Û»]Z`:å�a/#�[�à�[�]Zä2[:l\qG`y]ZbíÛ*ÞZ[�qêç ëRÞZackackDaz]iÛ�[�à�[2k�Û�a']Zbíá�[2ä�l\~Zk�[ q,`yå�[.��`y~�]iÛ�az]Zb Þ�l�å:Yi[�à�p:~�]Zaì(`yà�q Yyack�acÛºì!à�[2ór~Z[�]Zä2ac[2kGaz]:`Z~�àk�[2ä2`y]Zåé[2wrã�[�à�azqG[�]\Ûnç

&!]íÛ�ÞZ[õ]Z[2wZÛ�Û*Þ�à�[2[G[2w�ã�[�à�azqG[�]iÛ�k�ß�m�[Dk�Û*~Zå�ac[2å·Û*ÞZ[Gä2`�Y\[�à,Û�azqG[2kf`xìgÛ*ÞZ[Óà�[�l�h æ!Û�azqG[k�[�l\à�ä2Þ qG[2Û*ÞZ`Zå�kfì(`yà `y]Z[�æ!Û�azqG[¼Y�l�ä�~�~�q ä2hc[�l\]Zaz]Zb·~�]Zå�[�àºY�l\à�ac`y~ZkGì!l�achz~�à�[»ä2`Z]Zå�ajÛ�aj`Z]Zk�çÜ�ò:[|l�hck�` ã�[�à$ì(`yà�qG[2åÁ[2w�ã�[�à´a'qõ[�]\Û�k¼m.ÞZ[�à�[_m�[_k�Û�~Zå�aj[2åLÛ*ÞZ[_YZajk�ajÛõì¢à�[2ó�~Z[�]Zä2ac[2k_`�ì.Û*ÞZ[à�[�l�h æ!Û�azqG[_k�[�lià�ä Þ qG[2Û*ÞZ`Zå�kºì¦`Zàfä2`Z]\Û�az]Z~Z`y~ZkGY�l�ä�~�~�q ä2hj[�li]Za']Zb á�~ZÛÓÛ*ÞZ[¼[)#}[2ä2ÛG`�ìRÛ*ÞZ[

Page 33: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� �B�

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1400

500

600

700

800

900

1000

Probability of Recovery

Tim

e S

teps

to F

irst C

over

age

Wagner’s Rule LRTA* Node Counting Thrun’s Rule

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.13400

3500

3600

3700

3800

3900

4000

Probability of Recovery

Tot

al N

umbe

r of

Mov

es to

Firs

t Cov

erag

e

Wagner’s Rule LRTA* Node Counting Thrun’s Rule

�Z�)�£� !�G7 � ¡»Ð}�¶*�$ R¹�����������»¹���(����Õ�£��µ²x�$ .��4 ��¶������z� �Z�# !�(�.Ð��¶��$ !�*)��,�#�g�����(�§ ����#�c£����������-c !�����¶*�$ �®� !�²���²��#�#���­f¶��� (����!1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1400

500

600

700

800

900

1000

Probability of Fail

Tim

e S

teps

to F

irst C

over

age

Wagner’s Rule LRTA* Node Counting Thrun’s Rule

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.13400

3500

3600

3700

3800

3900

4000

Probability of Fail

Tot

al N

umbe

r of

Mov

es to

Firs

t Cov

erag

e

Wagner’s Rule LRTA* Node Counting Thrun’s Rule

�Z�)�£� !�f/�+�¡»Ð}�¶*�$ R¹�����������»¹���(����Õ�£��µ²x�$ .��4 ��¶������z� �Z�# !�(�.Ð��¶��$ !�*)��,�#�g�����(�§ ����#�c£����������-c�'���#�#£� !�µ®� !�²���²��#�#�c�0­º¶��� (����!1

ì!l�achz~�à�[Gä2`y]Zå�acÛ�ac`y]Zkf`y]êÛ*ÞZ[fYZack�acÛ�ì¢à´[2ór~Z[�]Zä2ac[2k�mRl�kRk*qsl�hch¦ç â &!]é`y~�àÝì(`y~�à�Û*Þ¼[2w�ã�[�à�azqG[�]iÛ�ßm�[fqG[�l�k�~�à�[2å_Û*ÞZ[fà�`yá�~Zk�Û*]Z[2k�k�`�ì�Û*ÞZ[fà�[�l�h æ!Û�azqG[�k�[�lià�ä Þ_qõ[2Û*ÞZ`yå�k,m.ÞZ[�]¼Û*ÞZ[fl\]iÛ�k�m�[�à�[qG`�Y\[2åÆlnmÝlnpíì!à�`yq Û*ÞZ[2azà¼ä�~�à�à�[�]iÛ»ä2[2hch�mRacÛ*ÞZ`y~ZÛ_à�[�l�hcaco2az]ZbÁÛ*ÞZack�çBë,ÞZajk_ack_azqÓã�`yà�Û*l\]iÛá�[2ä�l\~Zk�[»ã�[2`yãZhc[Óä�l\]é[�l�k�achcp à�~�]éaz]iÛ�`_k*qÓl�hch�Y�l�ä�~�~�qDæ!ä2hj[�li]Za']Zb l\]\Û�k.l\]Zå:l�ä2ä2acå�[�]\Û�l�hchjpã�~Zk*ÞõÛ*ÞZ[�q Û�`�lRå�a/#�[�à�[�]iÛ�hj`Zä�l�Û�ac`y]�ç �f~�à�az]Zb�[�l�ä ÞºÛ�azqG[�k�Û�[�ã�ß\mRacÛ*ÞGl,b\acY\[�]Óã�à�`yá�liáZajhcacÛ�p[2w�l�ä2Û�hcp_`y]Z[fl\]iÛRmRl�kRqG`�Y\[2å·Ü�`iÛ*ÞZ[�à�mRack�[f]Z`»li]\ÛRmRl�kRqG`�Y\[2å�â6ç &¦ìQl\]_l\]iÛRmRl�kRqG`�Y\[2å�ß[2w�l�ä2Û�hcp|`Z]Z[Dl\]iÛèli]Zå acÛ�k�]Z[2mIä2[2hch�m�[�à�[fä2ÞZ`\k�[�]émRacÛ*Þí~�]Za ì¦`Zà�q ã�à´`yá�l\áZachcajÛ�p\ß�mRacÛ*ÞéÛ*ÞZ[à�[2k�Û*à�acä2Û�ac`y]ºÛ*Þ�l�Û�Û*ÞZ[Rl\]iÛ�mÝlxk¾qG`�Y\[2åGá\pGl�Û�qG`\k�Û�Û�mQ`,ä2[2hjhck�ç $�ajbZ~�à�[,�p�k*ÞZ`�mRk�Û*ÞZ[�Û�`\Û*l�h]Z~�qGá�[�à,`�ìµqõ`�Yi[2kÝ~�]iÛ�ach�[2ajbZÞ\Û�l\]iÛ�k�ä2`�Yi[�à�[2å_Û*ÞZ[fÛ�[�à�à�l�az]sì(`yà�Û*ÞZ[ ÷&à�k�ÛRÛ�azqG[\ßOl�Yi[�à�l�b\[2å`�Y\[�à�g�ß/������à�~�]Zk�çµÚ.hch&à´[�l�h æ!Û�azqG[.k�[�l\à�ä2ÞÓqG[2Û*ÞZ`Zå�kgä2`y]iÛ�az]Z~Z[2å»Û�`ºä2`�Y\[�à�Û*ÞZ[�Û�[�à�à�l�az]�ßZl\]ZåÛ*ÞZ[f]Z~�qGá�[�à§`�ì¾qõ`�Yi[�qG[�]\Û�kgaz]Zä�à�[�l�k�[2åébyà�l�ä2[�ì!~Zhchcp¸l�k�Û*ÞZ[fã�à´`yá�l\áZachcajÛ�p¼`�ì¾á�[2a']Zb_qG`�Y\[2åaz]Zä�à�[�l�k�[2å�çgëRÞZ[R`yà´å�[�à�`�ì&Û*ÞZ[Rà�[�lxhæ!Û�a'qõ[.k�[�lià�ä ÞGqG[2Û�ÞZ`yå�kgà�[�qsl�az]Z[2å¸~�]�l #�[2ä2Û�[2åÓáipGÛ*ÞZackì!l�achz~�à�[fä2`y]Zå�acÛ�ac`y]�ç

&$]¼`y~�à§÷�ì(Û*ÞÓ[2w�ã�[�à´a'qõ[�]\Û�ßOm�[èqõ[�l�k*~�à�[2å_Û*ÞZ[fà´`yá�~Zk�Û*]Z[2k�k,`�ì4Û*ÞZ[fà�[�lxhæ!Û�a'qõ[�k�[�l\à�ä2ÞqG[2Û�ÞZ`yå�k§m.ÞZ[�]GÛ*ÞZ[Rl\]iÛ�k�ì!l�achj[2å�ç§ëRÞZackgackgazqÓã�`yà�Û*l\]iÛgá�[2ä�li~Zk�[.à�`yá�`\Û�k�ä�l\]GqÓlxhì!~�]Zä2Û�ac`y]�ç

Page 34: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

�M� � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.23500

3600

3700

3800

3900

4000

4100

Probability of Marks Being Erased

Tot

al N

umbe

r of

Mov

es to

Firs

t Cov

erag

e

Wagner’s Rule Node Counting LRTA* Thrun’s Rule

�Z�)�£n !�Q/67*¡O¹�*�(���yÕ�£���²\�$ ���� ��¶������z� �Z�# !�(�4Ð��¶��$ !�*)����#�� ��� (�2�#��)���¿f�$ !���� !���(����f~�à�az]ZbÓ[�l�ä2Þ¼Û�azqG[�k�Û�[�ã�ßZ[�l�ä2Þsì¢~�]Zä2Û�ac`y]�lxh4l\]iÛ�ì!l�achj[2åémRacÛ*Þ_lºb\acY\[�]sì!l�ach'~�à´[fã�à�`yá�l\áZachcacÛ�pl\]Zå [�l�ä Þ ì¢l�achc[2åÁl\]\Ûºä�l\]:à�[2ä2`�Y\[�à´[2å:mRacÛ*Þ l_b\acY\[�] à�[2ä2`�Y\[�à�píã�à�`yá�l\áZachcacÛ�p\ç $�ajbZ~�à�[ ,0ek*ÞZ`�mRkGÛ*ÞZ[¼ä2`�Yi[�àfÛ�a'qõ[ Ü�hc[�ì(Û*âGl\]ZåÁÛ�`\Û�l�hg]y~�qGá�[�àG`�ìèqG`�Y\[2k_Ü(à�acbyÞiÛ*âf~�]iÛ�achR[2acbyÞiÛÓl\]\Û�kä2`�Y\[�à�[2åÙÛ�ÞZ[_Û�[�à�à�l�az] ì¦`yàºÛ*ÞZ[ê÷&à�k�ÛÓÛ�a'qõ[\ßRl�Yi[�à�l�b\[2å `�Y\[�à g�ß �0���íà�~�]Zk�çÁò:[ít�[�ãZÛÓÛ*ÞZ[ì¢lxajhz~�à�[fã�à´`yá�l\áZachcajÛ�p¼ä2`y]Zk�Û*li]\Û�l�Û+��ç/��,.l\]Zå¼Y�lià�ac[2å_Û*ÞZ[.à�[2ä2`�Y\[�à�p¼ã�à�`yá�l\áZachcacÛ�psì!à�`yq ��ç ��,Û�`%��ç�, �_Ü�azqÓãZhcpZa']ZbéÛ*Þ�l�ÛRacÛ�Û�`Z`yt¼lGì!l�achc[2å:l\]\Û�l\]|lnY\[�à�lxb\[�`�ì , �GÛ�` , �0�GÛ�azqG[fk�Û�[�ãZk.Û�`à�[2ä2`�Y\[�à�â2ç $�acby~�à�[ g��Gk*ÞZ`�mRkRÛ�ÞZ[Dà�[2k*~ZhcÛ�k�`�ìglÓk�a'qõajhzl\à�[2w�ã�[�à´a'qõ[�]\Û�m.ÞZ[�à�[ºmQ[Gt�[�ãZÛRÛ*ÞZ[à�[2ä2`�Y\[�à�pÓã�à�`yá�liáZajhcacÛ�p»ä2`Z]Zk�Û*l\]iÛQl�Û���ç , ��l\]Zå¼Y�l\à�ac[2å_Û*ÞZ[,ì!l�ach'~�à´[fã�à�`yá�l\áZachcacÛ�psì!à�`yq ��ç ���Û�`���ç�, ��ç�Ú.hch�à�[�lxhæ!Û�a'qõ[Rk�[�l\à�ä2ÞGqG[2Û*ÞZ`Zå�k�ä2`y]iÛ�az]y~Z[2åÓÛ�`.ä2`�Y\[�à�Û*ÞZ[�Û�[�à�à�lxa']Gl�k�hc`y]Zbflxk4Û*ÞZ[ì¢lxajhz~�à�[_ã�à�`yá�liáZajhcacÛ�p·mRl�kf]Z`éhzl\à�b\[�à�Û*Þ�li]íÛ*ÞZ[»à�[2ä2`�Y\[�à´p ã�à´`yá�l\áZachcajÛ�p\ç_ëRÞZ[»]Z~�qGá�[�àº`�ìqG`�Y\[�qG[�]iÛ�kgà�[�qÓlxa']Z[2å¼à�`y~ZbyÞZhcpGÛ*ÞZ[Rk*liqG[Rl\]ZåÓÛ*ÞZ[Rä2`�Y\[�à�Û�azqG[Raz]Zä�à�[�lxk�[2å»bZà�l�ä2[�ì!~Zhjhcp»lxkÛ*ÞZ[�ì!l�achz~�à�[fã�à�`yá�l\áZachcacÛ�pÓaz]Zä�à�[�l�k�[2å¼`yà�Û�ÞZ[.à�[2ä2`�Y\[�à�pGã�à�`yá�l\áZachcacÛ�pså�[2ä�à�[�l�k�[2å�çgë,ÞZ[.`yà´å�[�à`�ì�Û*ÞZ[fà�[�l�h æ!Û�azqG[ºk�[�l\à�ä2Þ_qG[2Û*ÞZ`Zå�kèà´[�qÓl�az]Z[2å ~�]�l #�[2ä2Û�[2å|áip¼Û*ÞZackRì!l�achz~�à�[fä2`y]Zå�acÛ�ac`y]�ç

&!]í`Z~�à�k�acwZÛ*Þ:l\]Zå:hzl�k�Ûf[2w�ã�[�à�azqG[�]iÛ�ß�m�[õqõ[�l�k*~�à�[2å:Û�ÞZ[»à�`yá�~Zk�Û�]Z[2k�kf`�ì�Û*ÞZ[»à�[�lxhæÛ�azqG[fk�[�l\à�ä2Þ»qG[2Û�ÞZ`yå�kRm�ÞZ[�]_Û*ÞZ[fqÓlià�t\az]Zb\k�m�[�à�[�[�à�lxk�[2å�çgëRÞZackRack�a'qsã�`yà�Û�l\]\ÛRá�[2ä�l\~Zk�[ã�Þ\pZk�acä�l�h�qÓl\à�tiaz]Zb\k�ä�l\]_b\[2Û,å�[2k�Û*à�`�p\[2å�ç �f~�à�az]Zb¼[�l�ä2Þ_Û�azqG[ºk�Û�[�ã�ß�mRacÛ*ÞélÓbiajYi[�]|ã�à�`yáiæl\áZachcajÛ�pí[2w�l�ä2Û�hcp|`y]Z[GqÓl\à�tiaz]Zb»mRl�k�[�à�l�k�[2å Ü�`\Û�ÞZ[�à�mRack�[Ó]Z`¸qsl\à�tia']Zb¼mRl�k�[�à�l�k�[2å�â2ç &�ìRlqÓl\à�tiaz]ZbGmÝlxk�[�à�l�k�[2å�ßZ[2w�l�ä2Û�hcpÓ`y]Z[Rä2[2hch�mRl�k�ä2ÞZ`\k�[�]smRacÛ*Þ¼~�]Za ì¦`yà�q ã�à´`yá�l\áZachcajÛ�p»li]Zå»acÛ�k<Zæ!Y�l�hz~Z[ÓmRl�k.k�[2Û.Û�`¼o2[�à�`:Ü(á�[2ä�li~Zk�[Ól\]\Û�k.ä�l\]�]Z`iÛ.å�ack�Û�az]Zby~Zack*ÞÃá�[2Û�m�[2[�] lsä2[2hch�mRacÛ*ÞZ`y~ZÛl_qÓl\à�tiaz]Zb|l\]Zå:l¼ä2[2hch�m.ÞZ`\k�[ÓqÓl\à�t\az]Zb_mRl�k�å�[2k�Û*à�`�p\[2å�â2ç $�acby~�à�[ g�,Gk�ÞZ`�m,k.Û*ÞZ[GÛ�`iÛ*l�h]y~�qGá�[�àR`�ìgqG`�Y\[2k.~�]iÛ�ach�[2acbyÞiÛèli]\Û�k�ä2`�Y\[�à�[2åéÛ*ÞZ[�Û�[�à�à�l�az]¼ì(`yà§Û*ÞZ[�÷&à�k�ÛRÛ�azqG[\ß�lnY\[�à�lxb\[2å`�Y\[�à g�ß �0���íà�~�]Zk�çÁÚ�hch.à�[�l�h æ!Û�azqG[ék�[�l\à�ä2ÞÙqG[2Û�ÞZ`yå�kÓä2`Z]\Û�az]Z~Z[2åÙÛ�`Ãä2`�Yi[�àfÛ�ÞZ[|Û�[�à�à�l�az]�ßl\]Zå¼Û*ÞZ[è]Z~�qGá�[�à�`xì�qõ`�Yi[�qG[�]\Û�kgaz]Zä�à�[�l�k�[2å_bZà�l�ä2[�ì!~Zhjhcp_l�kgÛ�ÞZ[èã�à�`Zá�l\áZachjacÛ�p¼mRacÛ*ÞÓm�ÞZajä2ÞqÓl\à�tiaz]Zb\kfm�[�à�[G[�à�lxk�[2åía']Zä�à´[�l�k�[2å�ç»ë,ÞZ[Ó`yà�å�[�à�`�ì�Û*ÞZ[»à�[�lxhæ!Û�a'qõ[Ók�[�l\à�ä2Þ:qG[2Û�ÞZ`yå�kfà�[�æqÓl�az]Z[2å ~�]�l�#}[2ä2Û�[2å áip»Û*ÞZack,ì¢lxajhz~�à�[ºä2`y]Zå�acÛ�ac`y]�ç

D�� �:����p�g�� =�����ò:[_k�Û*~Zå�ac[2åÆà�[�l�h æ!Û�azqG[|k�[�l\à�ä2ÞÁqG[2Û*ÞZ`Zå�kÓÛ*Þ�l�ÛÓl�hchc`�m<l\]iÛ�kGÛ�`íä2`�Y\[�àGÛ�[�à�à�l�az]�ßQlxk

à�[2ó�~Za'à´[2å_ì(`yà�`y]Z[�æ!Û�a'qõ[�`yà�ä2`y]iÛ�az]Z~Z`y~ZkRY�lxä�~�~�q ä2hc[�l\]Zaz]Zb�ß�hzlnm.]»qõ`�m,a']ZbOßrqõa']Z[�k�mQ[2[�ãiæ

Page 35: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� � �

az]Zb�ß�`yà�k*~�à´Y\[2achjhzl\]Zä2[\ç%&$]:ã�l\à�Û�acä�~Zhzl\à2ß�m�[Gk�Û*~Zå�ac[2åÁ{ �4ë�Únmêli]Zå q,`Zå�[ ��`Z~�]\Û�az]Zb�ß�Û�m�`à�[�lxhæ!Û�a'qõ[_k�[�l\à�ä2Þ·qõ[2Û*ÞZ`yå�kGÛ*Þ�lxÛDl\à�[_l�hcb\`yà´ajÛ�Þ�qGacä�l�hchjpÁk�a'qõajhzl\à6ç q,`yå�[%��`y~�]iÛ�az]Zb�ß�ì(`yà[2w�l\qÓãZhc[\ß�l�hcmRl�pZkfqG`�Y\[2k�Û*ÞZ[_l\]iÛ�Û�`|l|]Z[2acbyÞZá�`Zà�az]Zb hc`Zä�l�Û�ac`y]ÃÛ*Þ�l�ÛfÞ�l�kGá�[2[�]ÃYyack�acÛ�[2åÛ*ÞZ[shj[�lxk�ÛD]Z~�qGá�[�àº`�ìQÛ�a'qõ[2k�ç%*�~�à�[2w�ã�[�à�azqG[�]iÛ*l�hRà�[2k*~ZhcÛ�kº`y]:à�[2bZ~Zh'lià�ì(`y~�à$æ!ä2`y]�]Z[2ä2Û�[2åbyà´ajå�ksa']Zå�acä�l�Û�[2å Û�Þ�l�ÛGÛ*ÞZ[2p à´[2k*~ZhcÛÓaz]Ák�azqGachzl\àsä2`�Yi[�àGÛ�azqG[2k�çÁëRÞZ[�à�[�ì(`yà�[\ßµajÛ¼ä2`y~ZhcåÙá�[k*ã�[2ä�~Zhzl�Û�[2å Û*Þ�l�Û�l\]\Û�k�Û*Þ�l�Û�~Zk�[ qR`Zå�[ ��`y~�]iÛ�az]Zb_by~�l\à�l\]\Û�[2[õlsk�azqGachzl\àºä2`�Y\[�àRÛ�a'qõ[õlxkl\]iÛ�k�Û*Þ�lxÛf~Zk�[»{ ��ë�Únmyçsò:[Ók*ÞZ`�m�[2åíÛ*Þ�l�Û�Û*ÞZ[Óä2`�Y\[�à�Û�a'qõ[Ó`�ìÝli]\Û�k�Û�Þ�l�Ûf~Zk�[»{ ��ë�ÚFmack�by~�l\à�l\]iÛ�[2[2åéÛ�`»á�[Óã�`\hcpr]Z`yqõa'lxh4az]éÛ*ÞZ[D]Z~�qGá�[�à,`�ì�hj`Zä�l�Û�ac`y]Zk�`xì¾Û*ÞZ[ºÛ�[�à�à�lxa']�çRë,ÞZ[�]�ßm�[�k*ÞZ`�mQ[2å�ßZk*~�à�ã�à�ack�az]Zb\hcp\ß�Û*Þ�l�Û�Û*ÞZ[�ä2`�Yi[�àgÛ�a'qõ[�`�ì¾l\]iÛ�k�Û*Þ�l�ÛR~Zk�[Fq,`yå�[��Ý`y~�]\Û�a']Zbsä�l\]á�[_[2wrã�`y]Z[�]\Û�a'lxhQaz] Ü�Û�ÞZ[»k�ó�~�l\à�[_à�`Z`\Ûf`xì â�Û*ÞZ[_]y~�qGá�[�àõ`�ìRhj`Zä�l�Û�ac`y]Zks[2Y\[�]:az] Û�[�à�à�l�az]Û*Þ�lxÛ�ä2`yà�à�[2k*ã�`y]Zå�kGÛ�`·Ü(ãZhzl\]�l\à�âf~�]Zå�azà�[2ä2Û�[2åÁÛ*à´[2[2k�ç»ëRÞZ~Zk�ß�m.ÞZachj[|li]\Û�k�Û�Þ�l�Ûf~Zk�[ qR`Zå�[��`Z~�]\Û�az]ZbG[2w�ã�[�à�azqG[�]iÛ*l�hchcp¸ã�[�à$ì(`yà�qG[2å¼m�[2hjhOaz]Ók�`ZqG[.Û�[�à�à�l�az]ZkgÛ�Þ�l�Ûgä2`yà�à�[2k*ã�`Z]Zå»Û�`fãZhzlnæ]�l\à�~�]Zå�azà�[2ä2Û�[2å:byà�l\ã�ÞZknß�Û*ÞZackDã�à�`yã�[�à�Û�p|l�hc`y]Z[Óackf]Z`\Û�k*~0(¼ä2aj[�]iÛ�Û�`_[2w�ãZhzl�az]ím.Þip|Û*ÞZ[2pã�[�à$ì¦`Zà�qG[2å|m�[2hch�k�az]Zä2[ÓÛ*ÞZ[2azàRä2`�Y\[�à,Û�azqG[º`y]|ãZhzl\]�l\à�~�]Zå�azà�[2ä2Û�[2å:bZà�l\ã�ÞZk�ajk�]Z`\Û�l�hcmRl�pZkb\`Z`Zå�çÙëRÞZack¸qõ[�l\]ZkÓÛ*Þ�l�Ûs`y]Z[|ack»]Z`\Û¼l�k�k*~�à�[2åLÛ*Þ�l�ÛÓl\]iÛ�kÓÛ*Þ�lxÛ»~Zk�[ q,`yå�[ ��`y~�]iÛ�az]Zb[2w�ÞZazáZajÛ¼l_b\`Z`yåÃä2`�Yi[�à�Û�azqG[Óaz] Û�[�à�à�l�az]Zkfm.ÞZ`ik�[ÓÛ�`yã�`\hc`\b\píå�a/#�[�à�kºì¢à�`Zq Û*ÞZ[ÓÛ�`yã�`\hc`\b\p`�ì�Û*ÞZ[_Û�[�à�à�lxa']Zk¼~Zk�[2å ì¦`ZàfÛ�[2k�Û�az]Zb�ç *�] Û*ÞZ[_`\Û*ÞZ[�àGÞ�l\]Zå�ß�l\]iÛ�kGÛ*Þ�l�ÛÓ~Zk�[|{ ��ë�ÚFm:l�h æmRlnpyk§by~�l\à�li]\Û�[2[�lºb\`Z`yå¼ä2`�Y\[�à�Û�azqG[\ßO]Z`õqsl�Û�Û�[�àµm.ÞZ[2Û*ÞZ[�à�Û*ÞZ[�Û�[�à�à�l�az]ZkQä2`Zà�à�[2k*ã�`y]Zå_Û�`å�azà�[2ä2Û�[2åÁbZà�l\ã�ÞZkG`yà�m�ÞZ[2Û*ÞZ[�àºÛ*ÞZ[2p:ä2`yà�à´[2k*ã�`y]Zå Û�`:ãZhzl\]�l\àºbyà�liã�ÞZk�ç|ëRÞZ[_ä2`�Y\[�à Û�azqG[by~�lià�l\]iÛ�[2[�`�ìgl\]\Û�kRÛ*Þ�l�Û.~Zk�[D{ ��ë�ÚnmGå�`y[2k�]Z`\ÛRk�[2[�qTÛ�`Óä2`ZqG[Dl�Û,Û*ÞZ[�ä2`\k�ÛR`xìQlGm�`yà�k�[lnY\[�à�lxb\[�æ!ä�l�k�[ºä2`�Yi[�àRÛ�azqG[Gaz]:ã�à�lxä2Û�acä2[Ók�az]Zä2[»l\]iÛ�k�Û*Þ�l�Û�~Zk�[Ó{ �4ë�Únm»k�[2[�q Û�`_ã�[�à�ì¦`yà�ql�k�m�[2hchgl�kfl\]iÛ�k�Û*Þ�lxÛè~Zk�[sqR`Zå�[ ��`y~�]iÛ�az]Zb_`y]íà�[2by~Zhzl\à,ì(`y~�à$æ!ä2`Z]�]Z[2ä2Û�[2å:byà�acå�k�ç *�Û�ÞZ[�àl�å�Y�l\]iÛ*l�b\[2kR`xì¾{ ��ë�ÚnmÓ`�Y\[�àYqR`Zå�[ ��`Z~�]\Û�az]Zb»Þ�lnY\[fá�[2[�]_k�Û�~Zå�aj[2åéaz]:î�,�hxñ�ç

*�~�àÓà´[2k*~ZhcÛ�kGå�[�qG`y]Zk�Û*à�l�Û�[¼Û*Þ�l�ÛGÛ*ÞZ[2`yà�[2Û�ajä�lxhÝà´[2k*~ZhcÛ�k»l\à�[¼azqÓã�`Zà�Û*l\]iÛÓá�[2ä�l\~Zk�[éacÛack�~�]Zhjazt�[2hcpíÛ*Þ�l�Û�`y]Z[ºmRachch�÷&]ZåémQ`Zà�k�Û�æ!ä�l�k�[.[2w�l\qÓãZhc[fbZà�l\ã�ÞZk.áip¸à�l\]Zå�`yqGhcp_b\[�]Z[�à�lxÛ�az]Zbbyà�l\ã�ÞZkfl\]Zå:[2w�ã�[�à�azqG[�]iÛ*l�hchcp:å�[2Û�[�à�qGaz]Za']Zb:ÞZ`�mIhc`y]Zb|acÛºÛ*l\t�[2k�l\]iÛ�k�Û*Þ�l�Ûf~Zk�[Ólêb\acY\[�]à�[�lxhæ!Û�a'qõ[»k�[�lià�ä ÞÃqG[2Û*ÞZ`Zå·Û�`éä2`�Y\[�à�Û*ÞZ[�q_ç%*�~�àGà�[2k*~ZhcÛ�kÓl�hck�`éå�[�qG`y]Zk�Û*à�l�Û�[GÛ*Þ�l�ÛGl\ãiæãZhcacä�l�Û�ac`y]Zk�`�ì4à�[�l�h æ!Û�azqG[�k�[�lià�ä Þ¼qG[2Û*ÞZ`Zå�kRl\à�[�azqÓã�`yà´Û*l\]iÛÝá�[2ä�l\~Zk�[�Û*ÞZ[2psk*~Zb\b\[2k�Û§m.ÞZacä Þtiaz]Zå�k.`�ì�Û�[�à�à�l�az]Zk.l\à�[�azqÓã�`yà�Û*l\]iÛRaz]|ã�à�l�ä2Û�acä2[Dl\]Zå_Û�Þy~ZkRk�ÞZ`y~Zhcå á�[Gl\]�l�hcpZo2[2å|Û*ÞZ[2`yà´[2Û�æacä�l�hchcp\ç�$O`yà�[2w�l\qÓãZhc[\ß�`y~�à�m�`yà�k�Û´æ!ä�l�k�[�[2w�l\qÓãZhc[Óbyà�liã�ÞZkèqGacbyÞiÛf]Z`\Û�`yä2ä�~�à�az]íã�à�l�ä2Û�ajä2[l\]Zå Û*ÞZ~ZkDl\]iÛ�kºÛ*Þ�l�ÛD~Zk�[¸à�[�lxhæ!Û�a'qõ[»k�[�lià�ä ÞÃqG[2Û*ÞZ`Zå�kõqõajbZÞ\ÛGá�[_l\áZhc[_Û�`|by~�lià�l\]iÛ�[2[Ólá�[2Û�Û�[�à�ä2`�Y\[�à,Û�azqG[Ga']íã�à�lxä2Û�acä2[DÛ�Þ�l\]éajk�k�~Zb\b\[2k�Û�[2åíá\pílÓm�`yà´k�ÛRä�l�k�[Ól\]�lxhjpZk�ack�`�Y\[�à�l�hchÛ�[�à�à�l�az]GÛ�`yã�`\hc`\b\ac[2k�ç��Ý`y]Zk�[2ó�~Z[�]\Û�hcp\ßZ`y~�à§à�[2k*~ZhcÛ�kQà�l�ack�[èl�Y�l\à�ac[2Û�pÓ`�ì�ór~Z[2k�Û�aj`Z]ZkÝli]ZåÓÛ*ÞZ~Zkk*~Zbib\[2k�Û�ì!~ZÛ*~�à�[Óm�`yà�têÛ*Þ�l�Û ì¦`\hchc`�mRk�ì!à�`yq Û�ÞZ[Óå�ack�ä�à�[�ã�l\]Zä2p:`xìÝÛ�ÞZ[ÓÛ*ÞZ[2`yà�[2Û�ajä�lxh�mQ`Zà�k�Û�æä�l�k�[|à�[2k*~ZhcÛ�k_l\]ZåÙÛ*ÞZ[_[2w�ã�[�à´a'qõ[�]\Û*lxhèlnY\[�à�lxb\[�æ!ä�l�k�[_à�[2k*~ZhcÛ�k¼`y] byà�acå�k�ç $�`ZàG[2wrliqÓãZhc[\ßacÛÓm�`y~ZhcåÙá�[_az]iÛ�[�à�[2k�Û�a']Zb Û�`:ã�[�à�ì¦`yà�q l Û*ÞZ[2`yà�[2Û�acä�l�hRlnY\[�à�lxb\[�æ!ä�l�k�[_l\]�l�hcpZk�ack Ü�k*lnp\ß�ì(`yàà�li]Zå�`yq Û�ac[�æ�á�à�[�l\tia']ZbZâ2ç���Yi[�]Óaz]sÛ*ÞZ[.ä2`Z]\Û�[2wZÛg`xì¾l�Û*ÞZ[2`Zà�[2Û�acä�l�h�m�`yà�k�Û�æ!ä�lxk�[Rl\]�l�hcpyk�ajknß�acÛm�`y~Zhcå_á�[�az]\Û�[�à�[2k�Û�az]ZbGÛ�`facå�[�]iÛ�a ì¦p_à�[�l�hcack�Û�acäfã�à�`Zã�[�à�Û�ac[2k�`�ì4bZà�l\ã�ÞZkgÛ�Þ�l�Ûgby~�lià�l\]iÛ�[2[.l\]iÛ�kÛ*Þ�lxÛ¾~Zk�[_q,`yå�[���`Z~�]\Û�az]Zbfl�ä2`�Y\[�à�Û�azqG[�Û*Þ�l�Û�ackgã�`\hcp�]Z`yqGazl�h�az]GÛ*ÞZ[R]Z~�qDá�[�à�`�ì�Y\[�à�Û�ajä2[2knçJºà�acå�k�ßiì¦`yàµ[2wrliqÓãZhc[\ßrlià�[è~�]Zå�azà�[2ä2Û�[2å_byà�l\ã�ÞZk§m.ÞZ`\k�[�Y\[�à�Û�acä2[2kRÞ�lnY\[.lºá�`y~�]Zå�[2å_å�[2byà´[2[\ßl\]Zåí]Z`Óm�`yà�k�Û´æ!ä�l�k�[�à�[2k*~ZhcÛ�k.l\à´[fty]Z`�m.]sì(`yà�byà�l\ã�ÞZkRmRacÛ*Þ¼Û*ÞZ[2k�[fã�à�`yã�[�à´Û�ac[2k�ç

Page 36: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� � � ������� � ��©����*�§�2�n���2� �����c�����»���j�������2�� §�!�����¼���c�������������r�������0���6���è��� ���� ��

��pS� �n���!g�d�� ��djGd����ëRÞZ[�&!]iÛ�[2hchcacb\[�]\Û ��[2ä2ack�ac`y]iæ :l\tia']Zb�Jfà´`y~�ã»lxÛ-J [2`yà�b\azlGë�[2ä Þ¼ackRk*~�ã�ã�`yà�Û�[2å|áip»l\]

q.X�$ �Rlià�[2[�à�Ú�mÝlià�å|Û�`_XZY\[�]é^�`y[�]Zacb ~�]Zå�[�à�ä2`Z]\Û*à�l�ä2Û & &!Xiæ e�e=p�hWp�g�o�ç.ëRÞZ[ºYyac[2mRkfl\]Zåä2`y]Zä2hz~Zk�ac`y]Zkfä2`Z]\Û*lxa']Z[2åíaz]éÛ*ÞZack�å�`yä�~�qõ[�]\Û�l\à�[ºÛ*ÞZ`\k�[º`�ìgÛ*ÞZ[Dli~ZÛ*ÞZ`yà�k.li]Zå k�ÞZ`y~Zhcå ]Z`iÛá�[�az]iÛ�[�à�ã�à�[2Û�[2å|lxkÝà´[�ã�à�[2k�[�]iÛ�az]ZbÓÛ*ÞZ[�`�(»ä2azl�h�ã�`\hcajä2ac[2k�ßO[2acÛ*ÞZ[�à�[2w�ã�à�[2k�k�[2å_`ZàgazqÓãZhcac[2å�ß�`�ìÛ*ÞZ[fk�ã�`y]Zk�`Zà�az]ZbÓ`yà�byli]Zajo�lxÛ�ac`y]Zk.l\]Zåíl�b\[�]Zä2ac[2k.`Zà�Û*ÞZ[�-ºçX�ç�b\`�Y\[�à�]�qõ[�]\Û�ç

c8d d�o=d��pSd��7 � ��¡2�4�6��$ 4����� �§¡�Ñ�� !�n��Z¡�%c�n�'� !���*����������#����������f��������®� !������²�­�������³�� (������\®����� !��#�#�#��).�����(��¡

����� ���§���!�c�$�2�����2�z�������� ��� ±n7 � +n- � 1$� � � � 5 � +*+n±�7 � � /2¡� / � ¹�¡ZÄ&����(�¼�����D½µ¡r�� (�2�#�y¡��4¶����2�#��)G������®����(���Q� ����§®����±O²�£��§�$Âx�������¶��.�(�� !���(��)*­º�z� � !���������¶*�����¶��)*�������Z¡�%c� �!���¨���!���2�z�c�2���������2�������0�����$�&�2�ݪ&���!�2�¨�j�´���2�� ������¨�2�g�2�z�c�2� ±(®���)���� � � � 5 � � � ±i7 � � � ¡

� ��� �§¡*Ä��� !�(n±�×�¡2ÄO !�*�n��� ��±�������×�¡ × �#��)��y¡�ÒZ���� (���#��)g�(�������£����#��)§ !�����#�0���������­������§���®� !�)� !�����§�#��)n¡���$�z� �&���j���n�!���z���c��#�2�����!� ± � � -(7´1$� � 7$5�7 � � ±�7 � � � ¡

� ��� Ä�¡*Ä��������������Ö§¡*Ñ��!Â����! �¡ ������n���#��)Q³}����Q�#����*�§®�����(���#���z� !�Q�������è���Z���$£� (��(������(���� !�(���#��²\�!�#��$���®�������¡è%'� «��0���!�!� 2�c� � �¾�(�R� �����!���z���!���2�z�c�2���������2�������������!���2���µ�!�z� �����c���\�!���z���c��#�2�����!�µ«���2�����c� ��2�� §©x�$��� 2�2��c�2� ±�®��*)���� � /�5 � 7*±\/�+�+�+�¡

� � � Ä�¡�Ä��������±yѧ¡\ÒZ2�! (�#������±y������Ö§¡xÑ��$Â����$ �¡4�é !�²�£��(���������'���(���*�������D�(�$����������Ó�����¢��������(��¡¾%'�«&�����$�!� 2�c�2�����(����������2�z�c�2�����y���2�������0�����$���2�����!�z� �����c�����!���z�´�j��#�2�����$� ±�®���)���� � 7 � 5 � 7 � ±�7 � � � ¡

� ��� ½�¡ZÄ� !2����������G�g¡Q�Z�­����y¡;�����(��±r�¢������®y±������G�£��g���������� !��z�R�: !�²\*�µ�#��¶��������_��������R�(���� �(­2�(�(����¡�� �2���$�����\�(���������!�c�z� �¦�,�!���z���'�\��2�����z�2�!��©\�����c���z� ±�®���)*��� � � � 5 � � � ±�7 � � � ¡

� � � ¹�¡�%¨��������n¡ ª��!������y�c�§��©x�$�2�0�$�����2� �\�$�2�!���c� �����n�z�2���2�g�2�2�����2�����c� ¡ ���#£�³��$ 4�4���������§�� ��£n²��#�������$ !��±y7 � � � ¡

� ��� ¹�¡�%z���n����G�����Ó½�¡ ��� (�¦¡� ��¶��#��)G�(�� !)����g�(���� !�¢�y¡D%'� «&�����!�$� 2�c�2���g�(�R����,�!���z���$���2�¨�j�2�n��� � �2�c������2�������0�����$�4�2�����$�z� �����c�����!���z���c��#�2�����!� ±�®���)*����/�+ � 5�/67�+n±�7 � � 7*¡

� � � ÒO¡ ���*�$�#²��#�#��)�±� G¡yÒi��(�(�����Z±������D�§¡y �2� !��¡Q½��!�#�n�z� !�������!���§����� (���#��)n���·��£n !¶*��­�¡�� �2���$�������(����$�z� �&���j���n�!���z���c��#�2�����!��ª������$�2�0�$� ± � � / � � 5�/ � � ±y7 � � � ¡

�7�+ � ×�¡ ��2�$���)n±��g¡�ÄO�#£���±�¹�¡2%z���n����n±�������½�¡ ��� (�(±*���6��(� !��¡ «��0���$�!� 2�j�2�����(������ ��� �µ� ������.�2� ���¦���$��2������ ���j����©\�!�2���!� ¡Z�4���¾%��� !���(��±Z7 � � � ¡��4¶����#���²�����*���4���¾%�¹����(���n�������½��$®\� !�4¿_×�� � � �!7�+n¡

�7�7 � ×�¡ ��2�!�n�)G�����Ó½�¡ ѧ¡�×�����������¡»½������#�0��������(���� !�(�¼�#�Ó����n�0�����(�$ !�§�#����(���j�G�n*�����#����¡¸%'� «��0���!�$� 2�c�2�����(�g����µ�!���z���!���2�z�c�2����� � �2�c�������2�������������!���2�����$�z� �&���j�����!���z���c��#�2�����$� ±�®��*)����g7 � � +�5�7 � � � ±7 � � � ¡

�7�/ � ×�¡ ��2�$���)R������½µ¡ ѧ¡\×�����������¡&�O�*�(­����������� !���(���(��²x���n���z� � !�����#�0������§�(���� !�(�f���)�� (������Q��¡4%'�«&�����$�!� 2�c�2�����(����������2�z�c�2�����y���2�������0�����$���2�����!�z� �����c�����!���z�´�j��#�2�����$� ±�®���)�����/ � � 5�/ � � ±�7 � � � ¡

�7 ��� ×�¡ ��2�$���)�������½�¡ ѧ¡\×�����������¡¾×6��¶��#��)� !�²\�������¶��)��������G®� !�²��������³�������#�n��������®\*�(�µ£������$ (��(���#���0­�£����#��)� !�����#������Q�§���!£n (��(����R�(���� !�¢�y¡g%'� «&�����$�!� 2�c�2���4�(�§����§�!���z���!���2�z�c�2�n���r���2�������������!���2����$�z� �&���j���n�!���z���c��#�2�����!��«O��2�����c�2�µ©x�����¨����� ±*®��*)����g7 � � 5�7 � � ±�7 � � � ¡

�7 ��� ×�¡ ��2�$���)������RÄ�¡�×2¬�­2���������2�z¡�Ø����#£��$�z£n®x�n�*�(�� (£n������z� � !�����#�0��������(���� !�(�Z¡r%'� «&�����!�$� 2�c�2���&�(��� ������2�z�c�2�����Z���2�������������!���2�����$�z� �&���j���n�!���z���c��#�2�����!� ±�®��*)���� � 7 � 5 � / � ±�7 � � � ¡

�7 � � ½�¡ ��� (�(¡ ½������#�0����������$£� (��(������(���� !�(�Z� �Z�# !�(�� !����£���(��¡�%'� «��0���$�!� 2�j�2�����(����������2�z�c�2���������2�������������!��2�����$�z� �&���j���n�!���z���c��#�2�����!� ±�®��*)�����7 �B� 5�7 � � ±�7 � � � ¡

�7 ��� ½�¡ ��� (�(¡�½������#�0����������$£n (��(����R�(���� !�¢�y¡ ���$�z� �����c�����!���z���c��#�2�����!� ± � /6-0/�� � 1$� 7 � � 5�/67*7�±�7 � � +�¡�7 � � Õ§¡�Õ��#��(�(��y¡ «��0���´������©\��� ���c� � �è��� ���� ��4�c�����!�¨� �����c���n�!���¨���c�� �6�����!� ¡� ���Ñ� !��³��0Ö��#�#�z±�7 � � 7�¡�7 ����� ¡������� (�z¡�� �����$����z�c�������!���z���c��#�2������©\�!�2���!��©\�z���2�z���2�c�������2�����2���y�n�z����«��0���´����í©\��� ���c�2� ¡������6��(��n�

¿f�������­�±�7 � � � ¡�7 � � �§¡ ���# !¬������$�_�����Ó¿|¡�× ��­2���$ �¡D�ã��n�#"����¼�(��#£������|�(����¶��$ !�*)��.�����s�(���� !�(�s�#�»��°�®��� !���s�����

£�����°�®��� !���G�(�$ ( !���#����£����#��)R�#���2�# !�����g������� !��z¡�%c� «&�����!�$� 2�c�2���4�(�§������!���z���$���2�z�c�2���������2�������������!�

Page 37: # Z© - Computer Science at RPIszymansk/papers/amai.00.pdf · 2007-09-05 · [lk p_Û`»ã à `\bZàl\q_ß l\]Zåéä ÞZ[l\ãêÛ`»á ~Zachcå ç ëRÞZackèqÓlit[2kRacÛRì([lkazáZhc[ÓÛ`så

� �������#���Z©����*�§�2�n���2� �����c��� ��¼���c�������2�� g�!�����¼���c�������������������������2����������� �� � �

�2�,ª&���$�2�z�c���4�2�� �����¨�2�g�2�z�c�2� ±�®��*)����¾/67*7 � 56/67*7 � ±�7 � � +n¡�#/�+ � ½µ¡*½�£��(�(�!�#�z¡ZÖ4�������� !���#������������ !���z�#�¶���������¶��)*���������������� (�2�$ !���z� &���²n�#��� !�²\*�(��¡\%'� «��0���!�!� 2�c� � �

�(��� �����!���¨���!���2�z�c�2�����\���2�������������!���2��ª����!�2�z�c�����2�� ����n�z�2�g�2�¨�j�2� ±�®��*)���� � � � � 5 � � � � ±�7 � � � ¡�#/27 � ½µ¡ ½}£��(�(�$�#�z± �§¡2¹�����$�z±������R�§¡* ����(�*��­��(×����¡�×2�$�����#��)Q2�n�£n ��� !���#����z� &���²��#��� !�²x��O����¶��)��������Z¡

%'� «&�����$�!� 2�c�2���4�(������§�!���z���$���2�z�c�2���������2�������������!�§�2��ª����!�2�z�c�����2�� +���n�z�2�g�2�¨�j�2� ±�®���)����g/ � � /´5/ � � � ±y7 � � � ¡

�#/*/ � ×x¡2½�£��(�(�$�#�y�����R��¡2¿f�$�z����i¡ ¥ �g�����ª��#�����������c�2���g©\�z� 2�c�����c� �r�c�§�j�¨� �ª��2�z�c�2������j�¨� ¡2 f%¨¹ �� !���(��±7 � � 7�¡

�#/ ��� ½µ¡6× ���� (®\�������RÄ�¡ ¿º�$²�²y¡�×���µ£�����(���������������£����(���R��2���$�����r�¢�����§������y�� !���#�\�z��#��³��#��)Ý�#�.�����(��¡%'� «&�����!�$� 2�c�2�����(��������!���z���$���2�z�c�2�����y���2�������������!���2��©x�c�§�2��2�z�c�2�R�(��� 2�$�y�¨� ��� ���!�������c�2� ±�®���)����7 � � 5�/�+ � ±y7 � � � ¡

�#/ ��� ½µ¡�×�£��(�(��f�������§¡�Ä��� !�(n¡ ª����c�����2���!���§�������\�$�2�!���c� � � �����!���z���� 2�����¨�j��� ¡\ f%z¹��� !���(��±y7 � � � ¡�#/ � � Ä�¡&×6¬�­2�Q�������2�������|×�¡ ��2�$���)n¡ ¹�������*�§®����°����­|�����2�n�f���£n�����#��)¼��¼£����2�# !�����(���|)� !��®n����¡

¹����¢����������� !�$®\� !��±�Ð���g®n£��(�$ .×2�$��$���������!®��� !�(���$����±�½��$���(�(�$��*�$ ����­2�(���¢������G%c���(�����£��(��±�¹Z !�­-zÕ���³���� (��1$±\7 � � � ¡

�#/ ��� ×x¡�¹}�� (£��Z¡����R�$��!������°�®n�� !�������s�#�f !�$�#���z� !�������$��������� (�n�#��)n¡g¹����(���n�������½��$®x� !�gÐ� #��¦Ð�× � � /´�7�+2/2±\×2�¢��2��&��&Ð�*�§®n£��(�$ µ×6�$��!�����*±�Ð��� (����)���� ��$�#��� #����¶��$ !����0­�± ����(�(��²�£n !)��¼- ���$�����(­��¶����n���1$±7 � � /6¡

�#/ � � ×x¡�¹}�� (£��Z¡4¹����� !���������°�®��� !�������Ó�#������� (���#��)f������� !��z¡¾%'� �§¡\¿Ó�n��(�Q�������§¡y×2��z)*��±\���2��(� !��±� �2�� ��!�������(�µ�!���z���c��#�2���������2���¨�0��� ��¾���n�0���#���\� � �����2�� �� 2�$�y�z� ��� �}�2�y�����2�!����� ±2®��*)���� � / � 5 ��� � ¡ØO����Õ�*�(�� !������½��$�#�������i±&7 � � /6¡

�#/ ��� %¢¡�¿f��)����$ �±2 G¡*Òi�#�����$�2²���£��.±��������g¡�ÄO (£��(���(�(�$�#�y¡�Å����z�#�#����)� !��®��R�(���� !�(���#��)�²�­§�4�(���$�#�#�0� (��!���(���¶��$ !�(��°�®� !2�����(��¡4%c�f×�¡ ��2�$�n�)�±\�§¡�ÄO�#£���±y¹µ¡�%z���n����n±y������½�¡ ��� (�¦±n���6��(� !��± «&�����!�$� 2�c�2�����(�����������������2� ���(���$�õ�2� �&� �r�c����©\�!�2���!� ±n®���)*����7�/*/´5�7�/ � ±�7 � � � ¡���¶����#���²���������4���¾%}¹����¢���n������½��!®\� !��¿_×�� � � �$7�+n¡

�#/ � � %¢¡�¿f��)����! �±\ G¡\Òi�#���n�$�2²���£���±������f�§¡�ÄO (£��¢���(�(�$�#�y¡����R�$��$�����­f�(���� !�¢���#��).�R�n­������§��R)� !��®��º²�­�§�(���!�#�#��� (��$���(����¶*�$ !�(��°�®n !2�����(��¡ ���������#���(� ���2������§�2�z�c���&�2�� ���!�¨� �����c���2�!���z���c��#�2�����$� ±´/ � � /67*7$5/�/ � ±Z7 � � � ¡