leftist skew heaps

Leftist HeapsText Read Weiss, 23.1 (Skew Heaps)Leftist HeapDefinition of null path length Definition of leftist heapBuilding a Leftist HeapSequence of inserts Re-heapification if leftist heap property is violated

MotivationA binary heap provides O(log n) inserts and O(log n) deletes but suffers from O(n log n) mergesA leftist heap offers O(log n) inserts and O(log n) deletes and O(log n) mergesNote, however, leftist heap inserts and deletes are more expensive than Binary Heap inserts and deletes

DefinitionA Leftist (min)Heap is a binary tree that satisfies the follow-ing conditions. If X is a node and L and R are its left and right children, then: X.value L.value X.value R.value null path length of L null path length of R

where the null path length of a node is the shortest distance between from that node to a descendant with 0 or 1 child.If a node is null, its null path length is 1.

Example: Null Path Length1920274325121584node4 8 19 12 15 25 27 20 43npl1 0 1 1 0 0 0 0 0

Example: Null Path Length1920274325121584node4 8 19 12 15 25 27 20 43npl1 0 1 1 0 0 0 0 001100000What are the npls of the children of 20 and the right child of 27?What is the npl of the right child of 8?

Example: Null Path Length1920274325121584node4 8 19 12 15 25 27 20 43npl1 0 1 1 0 0 0 0 0node 4 violates leftist heap property fix by swapping children01100000

Leftist Heap4node4 8 19 12 15 25 27 20 43npl1 0 1 1 0 0 0 0 0

Merging Leftist Heaps425121581920274367814Consider two leftist heaps Task: merge them into a single leftist heap

Merging Leftist Heaps425121581920274367814 First, instantiate a Stack

Merging Leftist Heaps425121581920274367814 Compare root nodesmerge(x,y)

xy

Merging Leftist Heaps425121581920274367814Remember smaller value

4xy

Merging Leftist Heaps425121581920274367814Repeat the process with the right child of the smaller value

4xy


6 4xy


7 6 4xy


7 6 4xynull

Merging Leftist Heaps425121581920274367814Because one of the arguments is null, return the other argument

7 6 4x8

Merging Leftist Heaps4251215192027436814Make 8 the right child of 7

7 6 4x887Refers to node 8

Merging Leftist Heaps4251215192027436814Make 7 leftist (by swapping children)

7 6 4887Refers to node 8

Merging Leftist Heaps4251215192027436814Return node 7

6 4787Refers to node 8

Merging Leftist Heaps4251215192027436814Make 7 the right child of 6 (which it already is)


Merging Leftist Heaps4251215192027436814Make 6 leftist (it already is)


Merging Leftist Heaps4251215192027436814Return node 6

4687Refers to node 8

Merging Leftist Heaps4251215192027436814Make 6 the right child of 4

4687

Merging Leftist Heaps4251215192027436814Make 4 leftist (it already is)

4687

Final Leftist Heap4251215192027436814Return node 4

487 Verify that the tree is heap Verify that the heap is leftist

AnalysisHeight of a leftist heap O(log n)Maximum number of values stored in Stack 2 * O(log n) O(log n)Total cost of merge O(log n)

Inserts and DeletesTo insert a node into a leftist heap, merge the leftist heap with the nodeAfter deleting root X from a leftist heap, merge its left and right subheapsIn summary, there is only one operation, a merge.

Skew HeapsText Read Weiss, 6.7Skew HeapNo need for null path length Definition of skew heapBuilding a Skew HeapSequence of inserts Swap children at every merge step

MotivationSimplify leftist heap by not maintaining null path lengthsswapping children at every merge step

DefinitionA Skew (min)Heap is a binary tree that satisfies the follow-ing conditions. If X is a node and L and R are its left and right children, then: X.value L.value X.value R.value

A Skew (max)Heap is a binary tree that satisfies the follow-ing conditions. If X is a node and L and R are its left and right children, then: X.value L.value X.value R.value

Merging Skew Heaps425121581920274367814Consider two skew heaps Task: merge them into a single skew heap

Merging Skew Heaps425121581920274367814 First, instantiate a Stack

Merging Skew Heaps425121581920274367814 Compare root nodesmerge(x,y)

xy

Merging Skew Heaps425121581920274367814Remember smaller value

4xy

Merging Skew Heaps425121581920274367814Repeat the process with the right child of the smaller value

4xy


6 4xy


7 6 4xy


7 6 4xynull

Merging Skew Heaps425121581920274367814Because one of the arguments is null, return the other argument

7 6 4x8

Merging Skew Heaps4251215192027436814Make 8 the right child of 7

7 6 4x887Refers to node 8

Merging Skew Heaps4251215192027436814Swap children of node 7

7 6 4887Refers to node 8

Merging Skew Heaps4251215192027436814Return node 7


Merging Skew Heaps4251215192027436814Make 7 the right child of 6 (which it already is)


Merging Skew Heaps4251215192027436814Swap children of node 6


Merging Skew Heaps419202743Return node 6

46Refers to node 8251215681487

Merging Skew HeapsMake 6 the right child of 4

46419202743251215681487

Merging Skew HeapsSwap children of node 4

46419202743251215681487

Final Skew HeapReturn node 4

4 Verify that the tree is heap Verify that the heap is skew419202743251215681487

AnalysisHeight of a skew heap O(log n)Maximum number of values stored in Stack 2 * O(log n) O(log n)Total cost of merge O(log n)

Inserts and DeletesTo insert a node into a skew heap, merge the leftist heap with the nodeAfter deleting root X from a skew heap, merge its left and right subheapsIn summary, there is only one operation, a merge.

leftist skew heaps

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