Computer Notes - Data Structures - 30

8/3/2019 Computer Notes - Data Structures - 30

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Class No.30

Data Structures

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2/24

Running Time Analysis

Unionis clearly a constant time operation.

Running time of find(i) is proportional tothe height of the tree containing node i.

This can be proportional to n in the worstcase (but not always)

Goal: Modify unionto ensure that heightsstay small

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3/24

Unionby Size

Maintain sizes (number of nodes) of alltrees, and during union.

Make smaller tree the subtree of the largerone.

Implementation: for each root node i,instead of setting parent[i] to -1, set it

to -k if tree rooted at i has k nodes.

This also called union-by-weight.

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4/24

Unionby Size

union(i,j):

root1 = find(i);root2 = find(j);

if (root1 != root2)

if (parent[root1]


5/24

Unionby Size

Eight elements, initially in different sets.

1 2 3 4 5 6 7 8

-1 -1 -1 -1 -1 -1 -1 -1

1 2 3 4 5 6 7 8



6/24

Unionby Size

Union(4,6)

1 2 3 4 5

6

7 8

-1 -1 -1 -2 -1 4 -1 -1

1 2 3 4 5 6 7 8



7/24

Unionby Size

Union(2,3)

1 2

3

4 5

6

7 8

-1 -2 2 -2 -1 4 -1 -1

1 2 3 4 5 6 7 8



8/24


9/24

Unionby Size

Union(2,4)

12

3

4 5

6

7 8

4 4 2 -5 -1 4 -1 -1

1 2 3 4 5 6 7 8



10/24

Unionby Size

Union(5,4)

12

3

4

56

7 8

4 4 2 -6 4 4 -1 -1

1 2 3 4 5 6 7 8



11/24

Analysis of Unionby Size

If unions are done by weight (size), the depth ofany element is never greater than log2n.



12/24

Analysis of Unionby Size

Intuitive Proof: Initially, every element is at depth zero.

When its depth increases as a result of a

union operation (its in the smaller tree), it isplaced in a tree that becomes at least twiceas large as before (union of two equal sizetrees).

How often can each union be done? -- log2ntimes, because after at most log2nunions,the tree will contain all nelements.



13/24

Unionby Height

Alternative to union-by-size strategy:maintain heights,

During union, make a tree with smaller

height a subtree of the other.

Details are left as an exercise.



14/24

Sprucing up Find

So far we have tried to optimize union.

Can we optimize find?

Yes, using path compression(orcompaction).



15/24

Sprucing up Find

During find(i), as we traverse the path fromito root, update parent entries for all thesenodes to the root.

This reduces the heights of all thesenodes.

Pay now, and reap benefits later!

Subsequent findmay do less work



16/24

Sprucing up Find

Updated code for find

find (i)

{if (parent[i] < 0)

return i;

else

return parent[i] = find(parent[i]);

}



17/24

Path Compression

Find(1)

121420

10

22

7

8 3 6

16

4

2

930

5

13

11

1

31 32

35

13

191817



18/24

Path Compression

Find(1)

121420

10

22

7

8 3 6

16

4

2

930

5

13

11

1

31 32

35

13

191817



19/24

Path Compression

Find(1)

121420

10

22

7

8 3 6

16

4

2

930

5

13

11

1

31 32

35

13

191817



20/24

Path Compression

Find(1)

121420

10

22

7

8 3 6

16

4

2

930

5

13

11

1

31 32

35

13

191817



21/24

Path Compression

Find(1)

121420

10

22

7

8 3 6

16

4

2

930

5

13

11

1

31 32

35

13

191817



22/24

Path Compression

Find(a)

a

b

f

c

d

e



23/24

Path Compression

Find(a)

a b

f

c d e



24/24

Timing with Optimization

Theorem: A sequence of m unionand findoperations, nof which are findoperations, canbe performed on a disjoint-set forest with union

by rank (weight or height) and path compressionin worst case time proportional to (m(n)).

(n)is the inverse Ackermanns function whichgrows extremely slowly. For all practical

puposes,(n) 4. Union-find is essentially proportional to mfor a

sequence of moperations, linear in m.

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Computer Notes - Data Structures - 30