Heapsort Running Time Build-Heap(A

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Heapsort Heap of n elements has height (show) - sorts in place – worst case running time All operations run in time proportional to the height - special data structure heap – complete binary tree Extract-Max, Insert, Extract-Min, Delete (completely filled except the lowest level) - Useful for priority queues Heap property : value of the node is at most - Discrete event simulators value of the parent A[Parent(i)] >= A[i] Algorithms for Heapify how to manipulate heaps Height of the node – number of edges on the longest path Buildheap how to build a heap from leaf to that node Heapsort how to sort using heap Idea: Implicit data structure – heap is represented by an Extract_max, array two attributes Insert maintain heap property is length(A) – length of the array heapsize(A) – number of elements of the heap Heap-Extract-Max(A) Remove A[1] How to get back and forth in the heap A[1] <- A[n] N <- n –1 Heapify(A,1,n) Different from the book Heapify(A,i,n) % assumes that L and R subtrees are heaps Build-Heap(A,n) % makes i’s subtree a heap if 2i · n and A[2i] > A[i] % build heap from unsorted array then largest <- 2i For i = n downto 1 else largest <- i do Heapify(A,i,n) if 2i+1 · n and A[2i+1] > A[largest] then largest <= 2i + 1 Simple bound on running time ≠ if largest i In fact more tight bound then exchange A[I] <-> A[largest] Heapify(A,largest,n) (idea of amortized analysis – more in chapter 18) Running time 1 Decision Tree Model Heapsort(A,n) Build-Heap(A,n) How to make sorting more efficient ? for i Å n downto 2 do exchange A[1] ÅÆ A[i] Can we do better then ? how to beat it ? Heapify(A,1,i-1) • establish the lower bound using the decision tree model Until now – sorting using comparisons only We for sure have to do at least comparisons to Variations – heap can be min (max) heap See all the input – can we provide tighter lower bound ? - heap can be k-ary tree instead of binary tree Decision Tree model – abstraction of comparison sorts Data Structures for dynamic sets – need to support -different algorithms follow different path in the decision tree efficiently basic operations (insert, delete, minimum, search) - there are at least n! leaves e.g. priority queues - decision tree for sorting n-elements has height - any sorting algorithm using comparisons takes at least on the order of n lg n operations 2.

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Advanced Data Structures
Advanced Data Structures PETER BRASS City College of New York CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521880374 © Peter Brass 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 978-0-511-43685-7 eBook (EBL) ISBN-13 978-0-521-88037-4 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page xi 1 Elementary Structures 1 1.1 Stack 1 1.2 Queue 8 1.3 Double-Ended Queue 16 1.4 Dynamical Allocation of Nodes 16 1.5 Shadow Copies of Array-Based Structures 18 2 Search Trees 23 2.1 Two Models of Search Trees 23 2.2 General Properties and Transformations 26 2.3 Height of a Search Tree 29 2.4 Basic Find, Insert, and Delete 31 2.5ReturningfromLeaftoRoot35 2.6 Dealing with Nonunique Keys 37 2.7 Queries for the Keys in an Interval 38 2.8 Building Optimal Search Trees 40 2.9 Converting Trees into Lists 47 2.10
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