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Heap

1. 1. 12/08/2006 CMSC 131 Fall 2006 Rance Cleaveland ©2006 Univeristy of Maryland Lecture 41: Heapsort Last time: 1. Analyzing selection sort 2. Trees Today: 1. Project #8 assigned! 2. Heaps 3. Heapsort
2. 2. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 1 Recall: Binary Trees Data structures consisting of Nodes (where data it stored) Edges Every node has one parent except root, which has none Every node has ≤ two children Nodes with no children = leaves Nodes with children = internal Nodes can be divided into levels based on distance from root The height of a tree is the longest path from root A binary tree is complete if: Every level except the last is full In the last level, every leaf is as far to the left as possible A binary tree is perfect if every level (including the last) is full ‘a’ ‘y’ ‘J’ ‘3’‘c’
3. 3. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 2 The Tree Quiz 1. Let T be a perfect tree of height n a. How many leaves does T have? 2n b. How many internal nodes does T have? 2n - 1 c. How many total nodes does T have? 2n+1 - 1 2. Let T be a complete binary tree of height n a. How many leaves does T have (give range)? 2n-1 to 2n, inclusive b. How many total nodes may T have (give range)? 2n to 2n+1 - 1, inclusive 3. Let T be a complete binary tree with n nodes. What is its height? Floor (rounding down) of log2n ‘a’ ‘y’ ‘J’ ‘3’‘c’ ‘z’ ‘:’
4. 4. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 3 Heaps Assume data stored in tree nodes is ordered by ≤ A heap is: A complete binary tree such that: Every parent’s data is ≥ its child(ren)’s data Fact: greatest value is stored at root! 17 14 11 1015
5. 5. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 4 Heapsort Idea: Keep unsorted elements in a heap Keep sorted elements in array, filling in from right (largest) to left (smallest) Maintain following property: Every unsorted element ≤ every sorted element (cf. selection sort) To increase sorted part: Remove root (largest element) from heap and put into correct position in array Restore remaining heap elements into a heap Questions How do you “restore” a heap? How do you create a heap in the first place? 17 14 11 1015 Unsorted elements 242219 Sorted elements
6. 6. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 5 Restoring a Heap After removing root of a heap … … how do we rearrange the remainder into a heap? One approach: Move bottom-most, right- most leaf to root Repeatedly swap node with greater of children until heap property restored 17 14 11 1015
7. 7. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 6 Example 11 14 1015 14 11 1015 15 14 1011 15 11 1014
8. 8. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 7 What Is Complexity of Restoring a Heap? Each swap operation takes O(1) Swap always results in level of “out-of-place” node increasing How many times can level increase, in terms of n = # of nodes in tree? log2n! So complexity of restoration operation is O(log2n)
9. 9. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 8 Creating a Heap To create a heap, we repeatedly insert elements from original array into heap How to insert a new node into a heap? Put new node into heap as bottom-most, right- most leaf Swap node with parent until heap property restored
10. 10. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 9 Example: Inserting 13 17 14 11 1015 13 17 11 1315 1014
11. 11. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 10 Complexity of Insertion Each swap is O(1) Swap decreases level of new node by one So complexity of insertion in terms of n = # of nodes in heap is: O(log2n)
12. 12. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 11 Complexity of Heap Creation Recall heap-creation procedure (a is array to be sorted) start with empty heap h for i = 0, …, a.length insert a[i] into h Complexity analysis Insertion called for each element in a (so, n times) Each insertion costs O(log2n) So complexity of heap creation is O(n log2n)
13. 13. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 12 Complexity of Heapsort Here is heapsort pseudo-code for sorting array a Create heap h from a for j = a.length-1, …, 0 do remove root from h and assign to a[j] restore heap Complexity analysis Heap creation: O(n log2n) Each loop iteration: O(log2n) Number of iterations: n Total: O(n log2n)!
14. 14. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 13 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 0 8 16 24 32 40 48 56 64 72 80 Selection sort Heapsort Comparing Sorts # of elements in array runningtime
15. 15. CMSC 131 Fall 2006 Rance Cleaveland ©2006 University of Maryland 14 How To Implement Heaps? The heap can also be stored in a, along with sorted portion Root is stored in a[0] Level 1 is stored in a[1], a[2] Level 2 is stored in a[3] – a[6] etc. Using this scheme: Children of node stored in a[i] are a[2i+1], a[2i+2] Parent of node stored in a[i] is a[(i-1) / 2] 17 14 11 1015 242219 2422191114101517 heap sorted