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Average-Case Analysis of Algorithms and Data Structures L’Analyse En Moyenne Des Algorithmes Et Des Structures De Donn Ees
(page i) Average-Case Analysis of Algorithms and Data Structures L'analyse en moyenne des algorithmes et des structures de donn¶ees Je®rey Scott Vitter 1 and Philippe Flajolet 2 Abstract. This report is a contributed chapter to the Handbook of Theoretical Computer Science (North-Holland, 1990). Its aim is to describe the main mathematical methods and applications in the average-case analysis of algorithms and data structures. It comprises two parts: First, we present basic combinatorial enumerations based on symbolic methods and asymptotic methods with emphasis on complex analysis techniques (such as singularity analysis, saddle point, Mellin transforms). Next, we show how to apply these general methods to the analysis of sorting, searching, tree data structures, hashing, and dynamic algorithms. The emphasis is on algorithms for which exact \analytic models" can be derived. R¶esum¶e. Ce rapport est un chapitre qui para^³t dans le Handbook of Theoretical Com- puter Science (North-Holland, 1990). Son but est de d¶ecrire les principales m¶ethodes et applications de l'analyse de complexit¶e en moyenne des algorithmes. Il comprend deux parties. Tout d'abord, nous donnons une pr¶esentation des m¶ethodes de d¶enombrements combinatoires qui repose sur l'utilisation de m¶ethodes symboliques, ainsi que des tech- niques asymptotiques fond¶ees sur l'analyse complexe (analyse de singularit¶es, m¶ethode du col, transformation de Mellin). Ensuite, nous d¶ecrivons l'application de ces m¶ethodes g¶enerales a l'analyse du tri, de la recherche, de la manipulation d'arbres, du hachage et des algorithmes dynamiques. -
Design and Analysis of Algorithms Credits and Contact Hours: 3 Credits
Course number and name: CS 07340: Design and Analysis of Algorithms Credits and contact hours: 3 credits. / 3 contact hours Faculty Coordinator: Andrea Lobo Text book, title, author, and year: The Design and Analysis of Algorithms, Anany Levitin, 2012. Specific course information Catalog description: In this course, students will learn to design and analyze efficient algorithms for sorting, searching, graphs, sets, matrices, and other applications. Students will also learn to recognize and prove NP- Completeness. Prerequisites: CS07210 Foundations of Computer Science and CS04222 Data Structures and Algorithms Type of Course: ☒ Required ☐ Elective ☐ Selected Elective Specific goals for the course 1. algorithm complexity. Students have analyzed the worst-case runtime complexity of algorithms including the quantification of resources required for computation of basic problems. o ABET (j) An ability to apply mathematical foundations, algorithmic principles, and computer science theory in the modeling and design of computer-based systems in a way that demonstrates comprehension of the tradeoffs involved in design choices 2. algorithm design. Students have applied multiple algorithm design strategies. o ABET (j) An ability to apply mathematical foundations, algorithmic principles, and computer science theory in the modeling and design of computer-based systems in a way that demonstrates comprehension of the tradeoffs involved in design choices 3. classic algorithms. Students have demonstrated understanding of algorithms for several well-known computer science problems o ABET (j) An ability to apply mathematical foundations, algorithmic principles, and computer science theory in the modeling and design of computer-based systems in a way that demonstrates comprehension of the tradeoffs involved in design choices and are able to implement these algorithms. -
Introduction to the Design and Analysis of Algorithms
This page intentionally left blank Vice President and Editorial Director, ECS Marcia Horton Editor-in-Chief Michael Hirsch Acquisitions Editor Matt Goldstein Editorial Assistant Chelsea Bell Vice President, Marketing Patrice Jones Marketing Manager Yezan Alayan Senior Marketing Coordinator Kathryn Ferranti Marketing Assistant Emma Snider Vice President, Production Vince O’Brien Managing Editor Jeff Holcomb Production Project Manager Kayla Smith-Tarbox Senior Operations Supervisor Alan Fischer Manufacturing Buyer Lisa McDowell Art Director Anthony Gemmellaro Text Designer Sandra Rigney Cover Designer Anthony Gemmellaro Cover Illustration Jennifer Kohnke Media Editor Daniel Sandin Full-Service Project Management Windfall Software Composition Windfall Software, using ZzTEX Printer/Binder Courier Westford Cover Printer Courier Westford Text Font Times Ten Copyright © 2012, 2007, 2003 Pearson Education, Inc., publishing as Addison-Wesley. All rights reserved. Printed in the United States of America. This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to 201-236-3290. This is the eBook of the printed book and may not include any media, Website access codes or print supplements that may come packaged with the bound book. Many of the designations by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. -
Algorithms and Computational Complexity: an Overview
Algorithms and Computational Complexity: an Overview Winter 2011 Larry Ruzzo Thanks to Paul Beame, James Lee, Kevin Wayne for some slides 1 goals Design of Algorithms – a taste design methods common or important types of problems analysis of algorithms - efficiency 2 goals Complexity & intractability – a taste solving problems in principle is not enough algorithms must be efficient some problems have no efficient solution NP-complete problems important & useful class of problems whose solutions (seemingly) cannot be found efficiently 3 complexity example Cryptography (e.g. RSA, SSL in browsers) Secret: p,q prime, say 512 bits each Public: n which equals p x q, 1024 bits In principle there is an algorithm that given n will find p and q:" try all 2512 possible p’s, but an astronomical number In practice no fast algorithm known for this problem (on non-quantum computers) security of RSA depends on this fact (and research in “quantum computing” is strongly driven by the possibility of changing this) 4 algorithms versus machines Moore’s Law and the exponential improvements in hardware... Ex: sparse linear equations over 25 years 10 orders of magnitude improvement! 5 algorithms or hardware? 107 25 years G.E. / CDC 3600 progress solving CDC 6600 G.E. = Gaussian Elimination 106 sparse linear CDC 7600 Cray 1 systems 105 Cray 2 Hardware " 104 alone: 4 orders Seconds Cray 3 (Est.) 103 of magnitude 102 101 Source: Sandia, via M. Schultz! 100 1960 1970 1980 1990 6 2000 algorithms or hardware? 107 25 years G.E. / CDC 3600 CDC 6600 G.E. = Gaussian Elimination progress solving SOR = Successive OverRelaxation 106 CG = Conjugate Gradient sparse linear CDC 7600 Cray 1 systems 105 Cray 2 Hardware " 104 alone: 4 orders Seconds Cray 3 (Est.) 103 of magnitude Sparse G.E. -
From Analysis of Algorithms to Analytic Combinatorics a Journey with Philippe Flajolet
From Analysis of Algorithms to Analytic Combinatorics a journey with Philippe Flajolet Robert Sedgewick Princeton University This talk is dedicated to the memory of Philippe Flajolet Philippe Flajolet 1948-2011 From Analysis of Algorithms to Analytic Combinatorics 47. PF. Elements of a general theory of combinatorial structures. 1985. 69. PF. Mathematical methods in the analysis of algorithms and data structures. 1988. 70. PF L'analyse d'algorithmes ou le risque calculé. 1986. 72. PF. Random tree models in the analysis of algorithms. 1988. 88. PF and Michèle Soria. Gaussian limiting distributions for the number of components in combinatorial structures. 1990. 91. Jeffrey Scott Vitter and PF. Average-case analysis of algorithms and data structures. 1990. 95. PF and Michèle Soria. The cycle construction.1991. 97. PF. Analytic analysis of algorithms. 1992. 99. PF. Introduction à l'analyse d'algorithmes. 1992. 112. PF and Michèle Soria. General combinatorial schemas: Gaussian limit distributions and exponential tails. 1993. 130. RS and PF. An Introduction to the Analysis of Algorithms. 1996. 131. RS and PF. Introduction à l'analyse des algorithmes. 1996. 171. PF. Singular combinatorics. 2002. 175. Philippe Chassaing and PF. Hachage, arbres, chemins & graphes. 2003. 189. PF, Éric Fusy, Xavier Gourdon, Daniel Panario, and Nicolas Pouyanne. A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics. 2006. 192. PF. Analytic combinatorics—a calculus of discrete structures. 2007. 201. PF and RS. Analytic Combinatorics. 2009. Analysis of Algorithms Pioneering research by Knuth put the study of the performance of computer programs on a scientific basis. “As soon as an Analytic Engine exists, it will necessarily guide the future course of the science. -
A Survey and Analysis of Algorithms for the Detection of Termination in a Distributed System
Rochester Institute of Technology RIT Scholar Works Theses 1991 A Survey and analysis of algorithms for the detection of termination in a distributed system Lois Rixner Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Rixner, Lois, "A Survey and analysis of algorithms for the detection of termination in a distributed system" (1991). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Rochester Institute of Technology Computer Science Department A Survey and Analysis of Algorithms for the Detection of Termination rna Distributed System by Lois R. Romer A thesis, submitted to The Faculty of the Computer Science Department, in partial fulfillment of the requirements for the degree of Master of Science in Computer Science. Approved by: Dr. Andrew Kitchen Dr. James Heliotis "22 Jit~ 1/ Dr. Peter Anderson Date Title of Thesis: A Survey and Analysis of Algorithms for the Detection of Termination in a Distributed System I, Lois R. Rixner hereby deny permission to the Wallace Memorial Library, of RITj to reproduce my thesis in whole or in part. Date: ~ oj'~, /f7t:J/ Abstract - This paper looks at algorithms for the detection of termination in a distributed system and analyzes them for effectiveness and efficiency. A survey is done of the pub lished algorithms for distributed termination and each is evaluated. Both centralized dis tributed systems and fully distributed systems are reviewed. -
Algorithm Analysis
Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 Analysis of Algorithms Input Algorithm Output © 2015 Goodrich and Tamassia Analysis of Algorithms 1 Scalability q Scientists often have to deal with differences in scale, from the microscopically small to the astronomically large. q Computer scientists must also deal with scale, but they deal with it primarily in terms of data volume rather than physical object size. q Scalability refers to the ability of a system to gracefully accommodate growing sizes of inputs or amounts of workload. © 2015 Goodrich and Tamassia Analysis of Algorithms 2 Application: Job Interviews q High technology companies tend to ask questions about algorithms and data structures during job interviews. q Algorithms questions can be short but often require critical thinking, creative insights, and subject knowledge. n All the “Applications” exercises in Chapter 1 of the Goodrich- Tamassia textbook are taken from reports of actual job interview questions. xkcd “Labyrinth Puzzle.” http://xkcd.com/246/ Used with permission under Creative Commons 2.5 License. © 2015 Goodrich and Tamassia Analysis of Algorithms 3 Algorithms and Data Structures q An algorithm is a step-by-step procedure for performing some task in a finite amount of time. n Typically, an algorithm takes input data and produces an output based upon it. Input Algorithm Output q A data structure is a systematic way of organizing and accessing data. © 2015 Goodrich and Tamassia Analysis of Algorithms 4 Running Time q Most algorithms transform best case input objects into output average case worst case objects. -
The History of Algorithmic Complexity
City University of New York (CUNY) CUNY Academic Works Publications and Research Borough of Manhattan Community College 2016 The history of Algorithmic complexity Audrey A. Nasar CUNY Borough of Manhattan Community College How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/bm_pubs/72 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] The Mathematics Enthusiast Volume 13 Article 4 Number 3 Number 3 8-2016 The history of Algorithmic complexity Audrey A. Nasar Follow this and additional works at: http://scholarworks.umt.edu/tme Part of the Mathematics Commons Recommended Citation Nasar, Audrey A. (2016) "The history of Algorithmic complexity," The Mathematics Enthusiast: Vol. 13: No. 3, Article 4. Available at: http://scholarworks.umt.edu/tme/vol13/iss3/4 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME, vol. 13, no.3, p.217 The history of Algorithmic complexity Audrey A. Nasar1 Borough of Manhattan Community College at the City University of New York ABSTRACT: This paper provides a historical account of the development of algorithmic complexity in a form that is suitable to instructors of mathematics at the high school or undergraduate level. The study of algorithmic complexity, despite being deeply rooted in mathematics, is usually restricted to the computer science curriculum. -
Algorithm Efficiency Measuring the Efficiency of Algorithms the Topic Of
Algorithm Efficiency Measuring the efficiency of algorithms The topic of algorithms is a topic that is central to computer science. Measuring an algorithm’s efficiency is important because your choice of an algorithm for a given application often has a great impact. Word processors, ATMs, video games and life support systems all depend on efficient algorithms. Consider two searching algorithms. What does it mean to compare the algorithms and conclude that one is better than the other? The analysis of algorithms is the area of computer science that provides tools for contrasting the efficiency of different methods of solution. Notice the use of the term methods of solution rather than programs; it is important to emphasize that the analysis concerns itself primarily with significant differences in efficiency – differences that you can usually obtain only through superior methods of solution and rarely through clever tricks in coding. Although the efficient use of both time and space is important, inexpensive memory has reduced the significance of space efficiency. Thus, we will focus primarily on time efficiency. How do you compare the time efficiency of two algorithms that solve the same problem? One possible approach is to implement the two algorithms in C++ and run the programs. There are three difficulties with this approach: 1. How are the algorithms coded? Does one algorithm run faster than another because of better programming? We should not compare implementations rather than the algorithms. Implementations are sensitive to factors such as programming style that cloud the issue. 2. What computer should you use? The only fair way would be to use the same computer for both programs. -
Bioinformatics Algorithms Techniques and Appli
BIOINFORMATICS ALGORITHMS Techniques and Applications Edited by Ion I. Mandoiu˘ and Alexander Zelikovsky A JOHN WILEY & SONS, INC., PUBLICATION BIOINFORMATICS ALGORITHMS BIOINFORMATICS ALGORITHMS Techniques and Applications Edited by Ion I. Mandoiu˘ and Alexander Zelikovsky A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2008 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201)-748-6011, fax (201)-748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. -
A Theoretician's Guide to the Experimental Analysis of Algorithms
A Theoretician's Guide to the Experimental Analysis of Algorithms David S. Johnson AT&T Labs { Research http://www.research.att.com/ dsj/ ∼ November 25, 2001 Abstract This paper presents an informal discussion of issues that arise when one attempts to an- alyze algorithms experimentally. It is based on lessons learned by the author over the course of more than a decade of experimentation, survey paper writing, refereeing, and lively dis- cussions with other experimentalists. Although written from the perspective of a theoretical computer scientist, it is intended to be of use to researchers from all fields who want to study algorithms experimentally. It has two goals: first, to provide a useful guide to new experimen- talists about how such work can best be performed and written up, and second, to challenge current researchers to think about whether their own work might be improved from a scientific point of view. With the latter purpose in mind, the author hopes that at least a few of his recommendations will be considered controversial. 1 Introduction It is common these days to distinguish between three different approaches to analyzing al- gorithms: worst-case analysis, average-case analysis, and experimental analysis. The above ordering of the approaches is the one that most theoretical computer scientists would choose, with experimental analysis treated almost as an afterthought. Indeed, until recently exper- imental analysis of algorithms has been almost invisible in the theoretical computer science literature, although experimental work dominates algorithmic research in most other areas of computer science and related fields such as operations research. The past emphasis among theoreticians on the more rigorous and theoretical modes of analysis is of course to be expected. -
Design and Analysis of Algorithms Application of Greedy Algorithms
Design and Analysis of Algorithms Application of Greedy Algorithms 1 Huffman Coding 2 Shortest Path Problem Dijkstra’s Algorithm 3 Minimal Spanning Tree Kruskal’s Algorithm Prim’s Algorithm ©Yu Chen 1 / 83 Applications of Greedy Algorithms Huffman coding Proof of Huffman algorithm Single source shortest path algorithm Dijkstra algorithm and correctness proof Minimal spanning trees Prim’s algorithm Kruskal’s algorithm ©Yu Chen 2 / 83 1 Huffman Coding 2 Shortest Path Problem Dijkstra’s Algorithm 3 Minimal Spanning Tree Kruskal’s Algorithm Prim’s Algorithm ©Yu Chen 3 / 83 Motivation of Coding Let Γ beP an alphabet of n symbols, the frequency of symbol i is fi. n Clearly, i=1 fi = 1. In practice, to facilitate transmission in the digital world (improve efficiency and robustness), we use coding to encode symbols (characters in a file) into codewords, then transmit. encoding transmit decoding Γ C C Γ What is a good coding scheme? efficiency: efficient encoding/decoding & low bandwidth other nice properties about robustness, e.g., error-correcting ©Yu Chen 4 / 83 Fix-length coding seems neat. Why bother to introduce variable-length coding? If each symbol appears with same frequency, then fix-length coding is good. If (Γ;F ) is non-uniform, we can use less bits to represent more frequent symbols ) more economical coding ) low bandwidth Fix-Length vs. Variable-Length Coding Two approaches of encoding a source (Γ;F ) Fix-length: the length of each codeword is a fixed number Variable-length: the length of each codeword could be different ©Yu Chen 5 / 83 If each symbol appears with same frequency, then fix-length coding is good.