《算法设计与分析基础》课程教学课件（PPT讲稿）Chapter 2 Fundamentals of the Analysis of Algorithm Efficiency

Chapter 2 introduction to The Design & Fundamentals of the Analysis Analysis of Algorithms of Algorithm Efficiency 2ND EDITION Anany Levitin PEARSON Addison Wesley Copyright 2007 Pearson Addison-Wesley. All rights reserved

Chapter 2 Fundamentals of the Analysis of Algorithm Efficiency Copyright © 2007 Pearson Addison-Wesley. All rights reserved

Analysis of algorithms 口 Issues correctness time efficiency space efficiency optimality 口 Approaches theoretical analysis ° empirical analysIs Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-1 Analysis of algorithms Issues: • correctness • time efficiency • space efficiency • optimality Approaches: • theoretical analysis • empirical analysis

Theoretical analysis of time efficiency Time efficiency is analyzed by determining the number of repetitions of the basic operation as a function of input size o Basic operation: the operation that contributes the most towards the running time of the algorithm Input size ()c2C() running time execution time Number of times for basic operation basic operation is or cost executed Note: Different basic operations may cost differently Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-2 Theoretical analysis of time efficiency Time efficiency is analyzed by determining the number of repetitions of the basic operation as a function of input size Basic operation: the operation that contributes the most towards the running time of the algorithm T(n) ≈ copC(n) running time execution time for basic operation or cost Number of times basic operation is executed input size Note: Different basic operations may cost differently!

Input size and basic operation examples Problem Input size measure Basic operation Searching for key in a Number of list's items, Kev comparison list of n items l.e. n Multiplication of two Matrix dimensions or Multiplication of two matrices total number of elements numbers Checking primality of n'size= number of digits Di vIsIon a given integer n (in binary representation) Visiting a vertex or ypical graph problem #vertices and/oredges traversing an edge Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-3 Input size and basic operation examples Problem Input size measure Basic operation Searching for key in a list of n items Number of list’s items, i.e. n Key comparison Multiplication of two matrices Matrix dimensions or total number of elements Multiplication of two numbers Checking primality of a given integer n n’size = number of digits (in binary representation) Division Typical graph problem #vertices and/or edges Visiting a vertex or traversing an edge

Empirical analysis of time efficiency o Select a specific(typical) sample of inputs o Use physical unit of time(e.g, milliseconds) or Count actual number of basic operation's executions D Analyze the empirical data Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-4 Empirical analysis of time efficiency Select a specific (typical) sample of inputs Use physical unit of time (e.g., milliseconds) or Count actual number of basic operation’s executions Analyze the empirical data

Best-case, average-case, worst-case For some algorithms, efficiency depends on form of input: o Worst case: Worst(n)- maximum over inputs of size n o Best case: Cbest()- minimum over inputs of size n D Average case: Cavg(n)-"average over inputs of size n Number of times the basic operation will be executed on typical input NOT the average of worst and best case Expected number of basic operations considered as a random variable under some assumption about the probability distribution of all possible inputs. So, avg=expected under uniform distribution. Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2 2-5

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-5 Best-case, average-case, worst-case For some algorithms, efficiency depends on form of input: Worst case: Cworst(n) – maximum over inputs of size n Best case: Cbest(n) – minimum over inputs of size n Average case: Cavg(n) – “average” over inputs of size n • Number of times the basic operation will be executed on typical input • NOT the average of worst and best case • Expected number of basic operations considered as a random variable under some assumption about the probability distribution of all possible inputs. So, avg = expected under uniform distribution

Example: Sequential search ALGORITHM SequentialSearch(A[0.n-1, K) //Searches for a given value in a given array by sequential search //Input: An array A[O n-1 and a search key K //Output: The index of the first element of A that matches K or-1 if there are no matching elements ←0 while i< n and a[i]≠kdo ←-i+1 if i<n return i else return-1 口 Worst case n key comparisons 口 Best case comparisons D Average case (n+1)/2, assuming K is in A Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2 2-6

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-6 Example: Sequential search Worst case Best case Average case n key comparisons 1 comparisons (n+1)/2, assuming K is in A

Types of formulas for basic operation's count 口 Exact formula nn n Formula indicating order of growth with specific multiplicative constant eg,C(m)≈0.5m o Formula indicating order of growth with unknown multiplicative constant C(m)≈cn2 Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2 2-7

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-7 Types of formulas for basic operation’s count Exact formula e.g., C(n) = n(n-1)/2 Formula indicating order of growth with specific multiplicative constant e.g., C(n) ≈ 0.5 n 2 Formula indicating order of growth with unknown multiplicative constant e.g., C(n) ≈ cn2

Order of growth o Most important: Order of growth within a constant multiple as n->00 D Example: How much faster will algorithm run on computer that is twice as fast? How much longer does it take to solve problem of double input size? Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2 2-8

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-8 Order of growth Most important: Order of growth within a constant multiple as n→∞ Example: • How much faster will algorithm run on computer that is twice as fast? • How much longer does it take to solve problem of double input size?

Values of some important functions as n->00 og 272 3.31013.3101021 3.6.10 1026.61026.6-1021041061.310309.31015 10 01031.0.104106109 04131041.31056108102 171051.71061010 1015 106201062010710121018 Table 2.1 Values(some appraximate)of several functions important for analysis of algorithms Copyright 2007 Pearson Addison-Wesley. All rights reserved A Levitin "Intoducion to the Design Analysis of Algorithms, 2nd ed, Ch 2

Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2 nd ed., Ch. 2 2-9 Values of some important functions as n → 