HEAP EXAMPLES-2 14 May-16 May 2014

Ps-Gözde ALP

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• Remember…

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Leftist

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Example 1: Consider the heap below is it leftist?

Leftist

4

7

12 16

21 8

4

11 19

17 46 25

Example 2: Consider the heaps below are they leftist?

0 0 0 0 0

0 0 1 1

1 1

Leftist- Recursive Merging

5

7

12 16

21 8

4

11 19

17 46 25

Example 3: Recursively merge the two leftist heaps showing each immediate

step.

Compare

6

7

12 16

21 8

4

11 19

17 46 25

Compare

7

7

12 16

21 8

4

11

19 17 46

25

Compare

8

7

12 16

21 8

4

11

19

17 46

25

Leftist - Recursive Merging 9

Example 4: Recursively merge the two leftist heaps showing each immediate

step.

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11

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Leftist – Nonrecursive Merging 13

Example 5: Non recursively merge the two leftist heaps showing each

immediate step.

Remember Non recursive Merging Algorithm

Requires 2 passes on the heap:

First pass: Create a new tree by merging

the right paths of both heaps in sorted order

Second pass: Perform child swaps at

nodes that violate the leftist heap property.

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3 8 6 7 18 SORT 3 6 7 8 18

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Heaps

First pass

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First pass

Second pass

Binomial Trees

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Useful properties of order k binomial tree Bk.

• Number of nodes = 2𝑘.

• Height = k.

• Degree of root = k.

• Deleting root yields binomial trees B k-1 , … , B0.

The binomial trees B0 through B4. Node depths in B4 are shown. below

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Binomial Heaps

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Example 6: What is the Node number, tree number, height and the binary

equivalent of the following tree.

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Example 7: Merge the following binomial trees and show the result.

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24

+

Example 8: Merge the following binomial heaps and show the result step by

step.

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Example 9: Delete the node with minimum key in binomial heap H and show

the result.

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H ← Union(H’, H)

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Example 10: If x=11 insert x into the following heap and show the result.

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Example 11: How many binomial trees are there in a binomial heap with n

element?

a. At least

b. At most

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Example 11: How many binomial trees are there in a binomial heap with n

element?

a. At least

b. At most

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Example 12: How many binomial trees are there in a binomial heap with n

element?

If n is equal to 4378

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Example 12: How many binomial trees are there in a binomial heap with n

element?

If n is equal to 4378

4378=4096 + 256 + 16 + 8 + 2

4378=212+ 28 + 24 + 23 + 21

4378= 𝐵12 + 𝐵8 + 𝐵4 + 𝐵3 + 𝐵1

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Example 13: How many leaves does a binomial tree with n elements have?

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Example 14: Where and how many nodes would you search to find the max

key in a binomial tree Bi confirming to the min-heap order property?

References

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https://courses.cs.washington.edu/courses/cse326/00wi/handouts/lecture7/sld0

18.htm

http://www.cs.princeton.edu/~wayne/cs423/lectures/heaps-4up.pdf

T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, “Introduction to Algorithms,”

2nd edition, MIT Press, 2003