Leftist Heaps Text Leftist Heap Building a Leftist Heap Read Weiss, §6.6 Leftist Heap Definition of null path length Definition of leftist heap Building a Leftist Heap Sequence of inserts Re-heapification if leftist heap property is violated

Motivation A binary heap provides O(log n) inserts and O(log n) deletes but suffers from O(n log n) merges A leftist heap offers O(log n) inserts and O(log n) deletes and O(log n) merges Note, however, leftist heap inserts and deletes are more expensive than Binary Heap inserts and deletes

Definition A 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 Length 4 8 19 12 27 20 15 25 43 node 4 8 19 12 15 25 27 20 43 npl 1 0 1 1 0 0 0 0 0

Example: Null Path Length 4 8 19 1 What is the npl of the right child of 8? 12 1 27 20 What are the npl’s of the children of 20 and the right child of 27? 15 25 43 node 4 8 19 12 15 25 27 20 43 npl 1 0 1 1 0 0 0 0 0

Example: Null Path Length 4 node 4 violates leftist heap property  fix by swapping children 8 19 1 12 1 27 20 15 25 43 node 4 8 19 12 15 25 27 20 43 npl 1 0 1 1 0 0 0 0 0

Leftist Heap 4 19 20 27 43 1 25 12 15 8 1 node 4 8 19 12 15 25 27 20 43 npl 1 0 1 1 0 0 0 0 0

Merging Leftist Heaps Consider two leftist heaps … 4 6 19 8 8 7 27 20 12 14 15 25 43 Task: merge them into a single leftist heap

Merging Leftist Heaps First, instantiate a Stack 4 6 19 8 8 7 27 20 12 14 15 25 43 First, instantiate a Stack

Merging Leftist Heaps Compare root nodes x y merge(x,y) 4 6 19 8 8 7 27 20 12 14 15 25 43 Compare root nodes merge(x,y)

Merging Leftist Heaps Remember smaller value x y 4 4 6 19 8 8 7 27 20 12 14 15 25 43 Remember smaller value

Merging Leftist Heaps y 4 4 x 6 19 8 8 7 27 20 12 14 15 25 43 Repeat the process with the right child of the smaller value

Merging Leftist Heaps Remember smaller value y x 6 4 4 6 19 8 8 7 27 20 12 14 15 25 43 Remember smaller value

Merging Leftist Heaps 6 4 4 x 6 y 19 8 8 7 27 20 12 14 15 25 43 Repeat the process with the right child of the smaller value

Merging Leftist Heaps Remember smaller value x y 7 6 4 4 6 19 8 8 7 27 20 12 14 15 25 43 Remember smaller value

Merging Leftist Heaps 7 6 4 4 x 6 19 8 8 7 y 27 20 12 14 null 15 25 43 Repeat the process with the right child of the smaller value

Merging Leftist Heaps 7 6 4 4 x 6 19 8 8 7 27 20 12 14 15 25 43 8 Because one of the arguments is null, return the other argument

Merging Leftist Heaps Make 8 the right child of 7 x Refers to node 8 7 6 4 4 6 x 19 Refers to node 8 8 7 27 20 14 8 12 43 8 15 25 Make 8 the right child of 7

Merging Leftist Heaps Make 7 leftist (by swapping children) 6 4 4 6 19 Refers to node 8 8 7 8 27 20 14 12 43 8 15 25 Make 7 leftist (by swapping children)

Merging Leftist Heaps Return node 7 Refers to node 8 6 4 7 4 6 19 8 7 27 20 14 12 43 15 25 7 Return node 7

Make 7 the right child of 6 (which it already is) Merging Leftist Heaps 6 4 4 6 19 Refers to node 8 8 7 8 27 20 14 12 43 15 25 7 Make 7 the right child of 6 (which it already is)

Merging Leftist Heaps Make 6 leftist (it already is) Refers to node 8 4 4 6 19 Refers to node 8 8 7 8 27 20 14 12 43 15 25 7 Make 6 leftist (it already is)

Merging Leftist Heaps Return node 6 Refers to node 8 4 6 4 6 19 8 7 8 27 20 14 12 43 15 25 6 Return node 6

Merging Leftist Heaps Make 6 the right child of 4 4 6 4 19 6 27 20 8 7 14 8 43 12 15 25 6 Make 6 the right child of 4

Merging Leftist Heaps Make 4 leftist (it already is) 4 6 4 19 6 27 20 8 7 14 8 43 12 15 25 6 Make 4 leftist (it already is)

Final Leftist Heap Verify that the tree is heap 4 19 6 27 20 8 7 14 8 43 12 15 25 Verify that the tree is heap Verify that the heap is leftist 4 Return node 4

Analysis Height 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 Deletes To insert a node into a leftist heap, merge the leftist heap with the node After deleting root X from a leftist heap, merge its left and right subheaps In summary, there is only one operation, a merge.

Skew Heaps Text Skew Heap Building a Skew Heap Read Weiss, §6.7 No need for null path length Definition of skew heap Building a Skew Heap Sequence of inserts Fix heap if leftist heap property is violated

Motivation Simplify leftist heap by not maintaining null path lengths swapping children at every step

Definition A 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 Heaps Consider two skew heaps … 4 6 19 8 8 7 27 20 12 14 15 25 43 Task: merge them into a single skew heap

Merging Skew Heaps First, instantiate a Stack 4 6 19 8 8 7 27 20 12 14 15 25 43 First, instantiate a Stack

Merging Skew Heaps Compare root nodes x y merge(x,y) 4 6 19 8 8 7 27 20 12 14 15 25 43 Compare root nodes merge(x,y)

Merging Skew Heaps Remember smaller value x y 4 4 6 19 8 8 7 27 20 12 14 15 25 43 Remember smaller value

Merging Skew Heaps y 4 4 x 6 19 8 8 7 27 20 12 14 15 25 43 Repeat the process with the right child of the smaller value

Merging Skew Heaps Remember smaller value y x 6 4 4 6 19 8 8 7 27 20 12 14 15 25 43 Remember smaller value

Merging Skew Heaps 6 4 4 x 6 y 19 8 8 7 27 20 12 14 15 25 43 Repeat the process with the right child of the smaller value

Merging Skew Heaps Remember smaller value x y 7 6 4 4 6 19 8 8 7 27 20 12 14 15 25 43 Remember smaller value

Merging Skew Heaps 7 6 4 4 x 6 19 8 8 7 y 27 20 12 14 null 15 25 43 Repeat the process with the right child of the smaller value

Merging Skew Heaps 7 6 4 4 x 6 19 8 8 7 27 20 12 14 15 25 43 8 Because one of the arguments is null, return the other argument

Merging Skew Heaps Make 8 the right child of 7 x Refers to node 8 7 6 4 4 6 x 19 Refers to node 8 8 7 27 20 14 8 12 43 8 15 25 Make 8 the right child of 7

Merging Skew Heaps Swap children of node 7 Refers to node 8 7 6 4 8 4 19 Refers to node 8 8 7 8 27 20 14 12 43 8 15 25 Swap children of node 7

Merging Skew Heaps Return node 7 Refers to node 8 6 4 7 4 6 19 8 7 8 27 20 14 12 43 15 25 7 Return node 7

Make 7 the right child of 6 (which it already is) Merging Skew Heaps 6 4 4 6 19 Refers to node 8 8 7 8 27 20 14 12 43 15 25 7 Make 7 the right child of 6 (which it already is)

Merging Skew Heaps Swap children of node 6 Refers to node 8 6 4 7 4 6 19 Refers to node 8 7 8 8 27 20 14 12 43 15 25 7 Swap children of node 6

Merging Skew Heaps Return node 6 Refers to node 8 4 6 4 6 19 7 8 8 27 20 14 12 43 15 25 6 Return node 6

Merging Skew Heaps Make 6 the right child of 4 4 6 4 19 6 7 8 27 20 8 14 43 12 15 25 6 Make 6 the right child of 4

Merging Skew Heaps Swap children of node 4 4 6 4 6 19 7 8 27 20 8 14 43 12 15 25 6 Swap children of node 4

Final Skew Heap Verify that the tree is heap 4 6 19 7 8 27 20 8 14 43 12 15 25 Verify that the tree is heap Verify that the heap is skew 4 Return node 4

Analysis Height 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 Deletes To insert a node into a leftist heap, merge the leftist heap with the node After deleting root X from a leftist heap, merge its left and right subheaps In summary, there is only one operation, a merge.