mca notes for students

8/8/2019 MCA NOTES FOR STUDENTS

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The UNIVERSITY ofNORTH CAROLINA atCHAPEL HILL

Data Structures For Disjoint

Sets


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Disjoint Sets Some applications involve grouping n

distinct elements into a collection of disjoint

sets Two main operations needed

Which set contains a given element?

Uniting two sets

Two different representations

Linked-List

Tree-based


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Disjoint Sets A disjoint data structure maintains a

collection S = {S1,S

2,....,S

n}

Each set is identified by a representative,which is some member of the set.

Choice of representative is dependent on the

application it could be any member,or theremight be certain criteria that determine which

element to use.


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Required Operations

Make_set(x)

Creates a new set whose only member (and thus

representative) is x.

Union(x,y)

Unions the sets that contain x and y. New

representative needs to be chosen, and the old sets

need to be removed.

Find_set(x)

Returns a pointer to the representative of the set

containing x.


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Running Time Analysis

For both methods, we will analyze the running time by

looking at two parameters:

n number of Make_set operations

m number of Make_set, Union, and Find_set

operations

Maximum value of Union operations = n 1

m n


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Linked List

a b c

/

head

tail


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Operations Make_set and Find_set are easy O(1)

Union Appendxs list onto the end ofy.

Use tail pointer to find the end ofy.

New representative is the representative ofy

Have to update every pointer inxto point to the new

representative (linear in length ofx)


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Union Worst Case n Make_set operations followed by n1Union

operations

ith Union operation updates iobjects, so the totalnumber of objects updated is:

T

otal number of operations is 2n1, and so eachoperation on average requires 5(n) time.

!

!1

1

2)(

n

i

ni


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Weighted Union

Append the shorter list to the larger one.

Union will still take ;(n) time if both sets have;(n) members.

But normally,weighted union reduces the

sequence ofm operations to O(m +n lg n) time.


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Disjoint-set Forests


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Disjoint Set Forests

Make_set

Create a tree with one node

Find_set Follow parent pointers until root is reached (this

path is called thefind path)

Union Make root of one tree point to the root of the

other


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Disjoint-set Forests

Is this any better?

No its possible to have a sequence ofn 1

Union operations which creates a linear chain ofn nodes.

However, there are two heuristics which can be

used in conjunction with this representation to

give a running time of almost O(m)


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UnionB

y Rank Similar to weighted-union

Make the root of the tree with fewer nodes point

to the root of the tree with more nodes For each node, maintain a rankthat is an upper

bound on the height of the node.

In union-by-rank, the root with the smaller rank

is made to point to the other trees root.


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Path Compression

During Find_set operations, make each node on

the find path point to the root.


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Code MAKE_SET

P[x]nx

rank[x]n 0

FIND_SET

ifx { p[x]

then p[x]n FIND_SET(p[x])

return p[x]


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CodeUnion

LINK(FIND_SET(x), FIND_SET(y))

LINKif rank[x] > rank[y]

then p[y] nx

else p[x] n yif rank[x] = rank[y]

then rank[y] n rank[y] +1


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Effects ofH

euristics Separately

Union-by-rank: O(m lg n)

Path Compression: 5(n + f (1+ log2+f/n n))

Together

O(mE(n))

E(n) is a slowly growing function (see 21.4 if

interested)

For most applications, E(n) 4, so running time

is O(m)

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