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Sets

Lecture-17Kiran Ijaz

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A Bit-Vector Implementation of Sets

A Bit-Vector(Boolean Array) can be used if all the

sets in the domain are subsets of a small universal

set, whose elements are the integers 1,.Nfor

some fixed N.

A Set is represented by a bit-vector in which the ith

bit is true ifiis an element of the set.

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A Bit-Vector Implementation of Sets

MEMBER, INSERT and DELETE operations can be

performed in constant time by directly addressing

the appropriate bit.

UNION, INTERSECTION and DIFFERENCE can

be performed in time proportional to the size of the

universal set.

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A Bit-Vector Implementation of Sets

Bit-Vector implementation can be used when the

universal set is a finite set other than a set of

consecutive integers.

A mapping would be required to translate the set

members to the integers 1,.N

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Linked-List Implementation of Sets

The Items of the linked-list are the members of the

set.

Linked-list uses space proportional to the size of theset represented, not the universal set.

Linked-List can represent the sets where the

universal set is infinite.

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Intersection in Unsorted List

An element is in the intersection of lists L1 and L2 if

and only if it is on both lists.

In unsorted lists, we must match each element ofL1with each element on L2.

The process will take O(n2) steps on lists of length

n.

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Intersection in Sorted List

To Match an element e on one list L1 with the

elements of another list L2, look down L

2until;

Either find e, that is the match has been found.

Or, find an element greater than e, which indicates

the match does not exist.

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Intersection in Sorted List

Ifd is the element on L1 that immediately precedes e.

And the first element found on L2 is f, such that d

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Assign in Sorted List

A =B

Copy all elements in B to A.

Cannot be implemented by pointing the header cell

ofA to the header cell ofB.

Subsequent, changes in B will result in unexpected

changes in A.

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Union in Sorted List

C=A B Attach all the elements from eitherA or B list to the C list, in their

proper, sorted order.

Compare the elements ofA with B.

If the elements are equal add once to C.

If the elements are unequal, add the elements from the smallerelements list until a larger element is found.

If one list exhausts, append the elements of the other list as it is.

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Difference in Sorted List

C=A B

Do not add an element to the C list when equal elements

are found.

Add the current A list element to the C list when it is

smaller than the current B list element; since the former

cannot be found on the B list.

IfB exhausts then append all the remaining elements of

A.

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OtherOperations

MIN: Return the first element on the list.

FIND: Search through the list and return when the target

element is found.

DELETE: Same as FIND but dispose of the target

element.

INSERTION: Find out the position of the element to be

inserted in order, and then change the pointers

appropriately.

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Dictionaries Collection of pairs.

(key, element)

Pairs have different keys.

Operations.

get(theKey)

put(theKey, theElement)

remove(theKey)

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Application

Collection of student records in this class.

(key, element) = (student name, linear list of

assignment and exam scores) All keys are distinct.

Get the element whose key is John Adams.

Update the element whose key isD

ianaRoss.

put() implemented as update when there isalready a pair with the given key.

remove() followed by put().

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Dictionary With

Duplicates

Keys are not required to be distinct.

Word dictionary. Pairs are of the form (word, meaning).

May have two or more entries for the same word.

(bolt, a threaded pin)

(bolt, a crash of thunder)

(bolt, to shoot forth suddenly)

(bolt, a gulp)

(bolt, a standard roll of cloth)

etc.

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Represent As A Linear

List

L = (e0, e1, e2, e3, , en-1)

Each ei is a pair(key, element).

5-pair dictionary D = (a, b, c, d, e).

a = (aKey, aElement), b = (bKey, bElement),

etc.

Array or linked representation.

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Array Representationa b c d e

get(theKey)

O(size) time

put(theKey, theElement)

O(size) time to verify duplicate, O(1) to add at rightend.

remove(theKey)

O(size) time.

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Sorted ArrayA B C D E

elements are in ascending order of key.

get(theKey)O(log size) time

put(theKey, theElement)

O(log size) time to verify duplicate, O(size) to add.

remove(theKey)

O(size) time.

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Unsorted Chain

get(theKey)

O(size) time

put(theKey, theElement)

O(size) time to verify duplicate, O(1) to add at left end.

remove(theKey)

O(size) time.

a b c d e

nullfirstNode

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Sorted Chain

Elements are in ascending order of Key.

get(theKey)

O(size) time

put(theKey, theElement)

O(size) time to verify duplicate, O(1) to put at properplace.

A B C D E

nullfirstNode

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Sorted Chain

Elements are in ascending order of Key.

A B C D E

nullfirstNode

remove(theKey)

O(size) time.