CSC 8301 Lecture 9 CSC 8301- Design and Analysis of Algorithms 8301Transform and Conquer representation change: Heapsort Reading: Chapter 6, Section 6.4 Exercises: 6.4: 1, 2, 3a, 6, 10 Next time: Space-time tradeoffs (Chapter 7) Lecture 9 Transform and Conquer Solve problem by transforming into: a more convenient instance of the same problem (instance implification) (instance implification) presorting Gaussian elimination a different representation of the same instance (representation change) (representation change) balanced search trees heaps and heapsort polynomial evaluation by Horner s rule Fast Fourier Transform a different problem altogether (problem reduction) (problem reduction) reductions to graph problems linear programming CSC 8301 Design and Analysis of Algorithms - Lecture 9 2 Heapsort Solve instance of problem by transforming into another simpler/easier instance of the same problem Definition: A heap is a binary tree with the following conditions: it is essentially complete: The key at each node is keys at its children CSC 8301 Design and Analysis of Algorithms - Lecture 9 3 1 CSC 8301 Lecture 9 Definition implies: Given n, there exists a unique binary tree with n nodes that is essentially complete, with h= lg n The root has the largest key The <a href="/keyword/subtree-rooted/" >subtree rooted</a> at any node of a heap is also a heap CSC 8301 Design and Analysis of Algorithms - Lecture 9 4 Heapsort Algorithm: 1. Build heap Remove root exchange with last (rightmost) leaf Fix up heap (excluding last leaf) 2. 3. Repeat 2, 3 until heap contains just one node. CSC 8301 Design and Analysis of Algorithms - Lecture 9 5 Heap construction Insert elements in the order given breadth-first in a binary breadthtree Starting with the last (rightmost) parental node, fix the heap rooted at it, if it does not satisfy the heap condition: 1. exchange it with its largest child 2. fix the <a href="/keyword/subtree-rooted/" >subtree rooted</a> at it (now in the child s position) Example: 1 2 4 9 3 5 CSC 8301 Design and Analysis of Algorithms - Lecture 9 6 2 CSC 8301 Lecture 9 Representation Use an array to store breadth-first traversal of heap tree: breadthExample: 9 5 1 4 2 3 1 2 3 4 5 6 9 5 3 1 4 2 Left child of node j is at 2j 2j Right child of node j is at 2j+1 2j Parent of node j is at j /2 Parental nodes are represented in the first n /2 locations CSC 8301 Design and Analysis of Algorithms - Lecture 9 7 Analysis of Heapsort See algorithm HeapBottomUp in section 6.4 Fix heap with problem at level h: 2h comparisons For <a href="/keyword/subtree-rooted/" >subtree rooted</a> at level i it does 2(h-i) comparisons 2(h Total for heap construction phase: 2(h-i) 2 i=0 h-1 i = 2 ( n lg (n + 1)) = (n) # nodes at level i CSC 8301 Design and Analysis of Algorithms - Lecture 9 8 Analysis of Heapsort (continued) Recall algorithm: (n) 1. Build heap 2. Remove root exchange with last (rightmost) leaf (log n) 3. Fix up heap (excluding last leaf) Repeat 2, 3 until heap contains just one node. n 1 times Total: (n) + (log n) = (log n) Note: this is the worst case. Average case also (log n). CSC 8301 Design and Analysis of Algorithms - Lecture 9 9 3 CSC 8301 Lecture 9 Priority queues A priority queue is the ADT of an ordered set with the operations: find element with highest priority delete element with highest priority insert element with assigned priority Heaps are very good for implementing priority queues CSC 8301 Design and Analysis of Algorithms - Lecture 9 10 Insertion of a new element Insert element at last position in heap. Compare with its parent and if it violates heap condition exchange them Continue comparing the new element with nodes up the tree until the heap condition is satisfied Example: Efficiency: CSC 8301 Design and Analysis of Algorithms - Lecture 9 11 Bottom-up vs. Top-down heap construction BottomTopTop down: Heaps can be constructed by successively inserting elements into an (initially) empty heap Bottom-up: Put everything in and then fix it BottomWhich one is better? CSC 8301 Design and Analysis of Algorithms - Lecture 9 12 4