Arrays aren t going to work. What can we do? Use pointers Copy a large section of a heap, with a single pointer assignment
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1 CS5-008S-0 Leftist Heaps 0-0: Leftist Heaps Operations: Add an element Remove smallest element Merge two heaps together 0-: Leftist Heaps Operations: Add an element Remove smallest element Merge two heaps together n is the # of elements in the merged heap Using standard heaps, time for each operation? 0-: Leftist Heaps Operations: Add an element Θ(lg n) Remove smallest element Θ(lg n) Merge two heaps together Θ(n) n is the # of elements in the merged heap Using standard heaps, Time for each operation 0-: Leftist Heaps Mergeable heaps Merge two heaps together Total time to be less Θ(n) n is the # of elements in the combined heap Arrays aren t going to work. What can we do? 0-: Leftist Heaps Mergeable heaps Merge two heaps together Total time to be less Θ(n) n is the # of elements in the combined heap Arrays aren t going to work. What can we do? Use pointers Copy a large section of a heap, with a single pointer assignment
2 CS5-008S-0 Leftist Heaps 0-5: Null Path Length Given a node in a binary tree, the Null Path Length (NPL) of the node is: Shortest path from the node to an element that has fewer than children NPL(null) defined to be - 0-: NPL Examples Calculating NPL for each node in the tree 0-: NPL Examples Calculating NPL for each node in the tree 0-8: NPL Examples
3 CS5-008S-0 Leftist Heaps Calculating NPL for each node in the tree 0-9: NPL Examples Calculating NPL for each node in the tree 0-0: NPL Examples Calculating NPL for each node in the tree 0-: NPL Examples Calculating NPL for each node in the tree 0-: Leftist Heap
4 CS5-008S-0 Leftist Heaps Binary Tree Pointers, not array Heap property For every node in the tree: NPL(left child) NPL(right child) 0-: Leftist Heap Examples : Leftist Heap Examples : Leftist Heaps Leftist heaps are unbalanced Unlike standard heaps, very balanced Left subtrees tend to be deep Right subtrees tend to be shallow 0-: Leftist Heaps Unbalanced is a plus, not a weakness Allows us to merge heaps quickly 0-: Merging Heaps To merge two leftist heaps H, H
5 CS5-008S-0 Leftist Heaps 5 If either heap is empty, return the other If Root(H ) Root(H ) else H.right = Merge(H.right, H ) return H H.right = Merge(H.right, H ) return H 0-8: Merging Heaps Examples : Merging Heaps Examples : Merging Heaps Examples : Merging Heaps Examples
6 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples : Merging Heaps Examples : Merging Heaps Examples
7 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples : Merging Heaps Examples
8 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples : Merging Heaps Examples
9 CS5-008S-0 Leftist Heaps : Merging Heaps Examples Heap is no longer leftist! What can we do to keep the heaps leftist? 0-: Merging Heaps Examples Merge might make heap non-leftist How can we make a heap leftist after a merge? 0-: Merging Heaps Examples Merge might make heap non-leftist How can we make a heap leftist after a merge? Swap children of the root 0-5: Merging Heaps To merge two leftist heaps H, H
10 CS5-008S-0 Leftist Heaps 0 If either heap is empty, return the other If Root(H ) < Root(H ) else H.right = Merge(H.right, H ) RH = H H.right = Merge(H.right, H ) RH = H If (RH.left().NPL() < RH.right().NPL()) RH.swapchildren() RH.setNPL(MIN(RH.left().NPL(), RH.right().NPL())+) return RH 0-: Merging Heaps Examples : Merging Heaps Examples : Merging Heaps Examples
11 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples
12 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples
13 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples
14 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples : Merging Heaps Examples
15 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples
16 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples
17 CS5-008S-0 Leftist Heaps : Merging Heaps Examples : Merging Heaps Examples
18 CS5-008S-0 Leftist Heaps : Code class LHeap { private LHeapNode left; private LHeapNode right; private int data; private int NPL; LHeap(LHeapNode newleft, LHeapNode newright; int newdata, int newnpl) { left = newleft; right = newright; data = newdata; NPL = newnlp; } LHeap(int newdata) { left = null; right = null; data = newdata; NPL = 0; }... } 0-55: Code LHeap Merge(LHeap H, LHeap H) { LHeap retheap; if (H == null) return H; if (H == null) return H; if (H.data() < H.data()) { H.setRight(Merge(H.Right(),H)); retheap = H; } else { H.setLeft(Merge(H.Right(),H)); retheap = H; } : Code (Continued) LHeap Merge(LHeap H, LHeap H) { LHeap retheap; }... if ((retheap.left == null) ((retheap.right!= null) && (retheap.left.npl() < retheap.right.npl()))) retheap.swapchildren(); if ((retheap.left() == null) retheap.right() == null) retheap.setnpl(0); else retheap.setnpl( + MIN(retHeap.left().NPL(), retheap.right().npl())); return retheap;
19 CS5-008S-0 Leftist Heaps 9 0-5: Time to Merge Given two heaps H and H, how long does a merge take? 0-58: Time to Merge Given two heaps H and H, how long does a merge take? Sum of the lengths of the rightmost branches in the two heaps When is the rightmost branch longest in a leftist heap of n elements? 0-59: Time to Merge Given two heaps H and H, how long does a merge take? Sum of the lengths of the rightmost branches in the two heaps When is the rightmost branch longest in a leftist heap of n elements? In a balanced tree! 0-0: Time to Merge Worst case time to merge occurs when merging two balanced trees In a balanced tree of n elements, what is the length of the rightmost branch of the tree? 0-: Time to Merge Worst case time to merge occurs when merging two balanced trees In a balanced tree of n elements, what is the length of the rightmost branch of the tree? Θ(lg n) 0-: Time to Merge Worst case time to merge occurs when merging two balanced trees In a balanced tree of n elements, what is the length of the rightmost branch of the tree? Θ(lg n) Time to merge two heaps, that have n elements between them, is O(lg n) 0-: Inserting We can merge two heaps quickly How can we insert an element quickly? 0-: Inserting We can merge two heaps quickly How can we insert an element quickly? Use Merge to insert
20 CS5-008S-0 Leftist Heaps 0 Merge heap with a heap containing a single element Time: 0(lg n) 0-5: Remove Min How can we remove the minimum element? 0-: Remove Min How can we remove the minimum element? Remove the root Merge the children Time: 0(lg n)
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