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Chapter 2 Fundamentals of the Analysis of Algorithm Efficiency Copyright © 2007 Pearson Addison-Wesley. All rights reserved. Analysis of algorithms Issues: • • • • correctness time efficiency space efficiency optimality Approaches: • theoretical analysis • empirical analysis Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-1 Analysis of Algorithms The term "analysis of algorithms" usually means an investigation of the efficiency of an algorithm with respect to two resources, namely execution time and memory space. Why? Time efficiency: How fast the algorithm runs? Space efficiency: How much memory space the algorithm requires? We will concentrate on analyzing for time efficiency. 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 most towards the running time of the algorithm input size T(n) ≈ copC(n) running time execution time for basic operation Copyright © 2007 Pearson Addison-Wesley. All rights reserved. Number of times basic operation is executed A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-3 Measuring an Input's Size Observation: Almost all algorithms run longer on larger inputs Example: Sorting The time efficiency of an algorithm is usually calculated as a function of a parameter n indicating the input size of the algorithm. For some algorithms, n is obvious. Example: Sorting, Searching Sometimes, there is a choice. Example: Calculating the product of two n x n matrices 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 Division (in binary representation) Typical graph problem #vertices and/or edges Copyright © 2007 Pearson Addison-Wesley. All rights reserved. Visiting a vertex or traversing an edge A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-5 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 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-6 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: Average case: Cavg(n) – “average” over inputs of size n Cbest(n) – minimum 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 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-7 Best-case, Average-Case, Worst-Case There are many algorithms the running time of which can be different for the same list size n, depending on the condition of the input. Example: Sorting Example: Sequential Search (see next slide). Worst case: no matches, or the first match is the last element of the list. Cworst(n) = n Best case: the first match is the first element of the list. Cbest(n) = 1 Best-case, Average-Case, Worst-Case Average case Calculation assumptions: Probability of successful search is equal to p Probability of the first match occurring in the ith position of the list is the same for every i. Best-case, Average-Case, Worst-Case 1 2 3 ... n Cavg (n) p( ) (1 p)n n p n(n 1) (1 p)n n 2 p(n 1) (1 p)n 2 • • p = 0 (search must be unsuccessful)? p = 1 (search must be successful)? Example: Sequential search Worst case Best case Average case (p48) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-11 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 n2 Formula indicating order of growth with unknown multiplicative constant e.g., C(n) ≈ cn2 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-12 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? (p45) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-13 Values of some important functions as n Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-14 Asymptotic order of growth A way of comparing functions that ignores constant factors and small input sizes O(g(n)): class of functions f(n) that grow no faster than g(n) Θ(g(n)): class of functions f(n) that grow at same rate as g(n) Ω(g(n)): class of functions f(n) that grow at least as fast as g(n) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-15 Establishing order of growth using the definition Definition (p53): f(n) is in O(g(n)) if order of growth of f(n) ≤ order of growth of g(n) (within constant multiple), i.e., there exist positive constant c and non-negative integer n0 such that f(n) ≤ c g(n) for every n ≥ n0 Examples: 10n is O(n2) 5n+20 is O(n) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-16 Big-oh Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-17 Establishing order of growth using the definition Definition : f(n) is in Ω(g(n)) if order of growth of f(n) ≥ order of growth of g(n) (within constant multiple), i.e., there exist positive constant c and non-negative integer n0 such that f(n) ≥ c g(n) for every n ≥ n0 Examples: 10n2 is O(n2) 5n2+20 is O(n) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-18 Big-omega Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-19 Establishing order of growth using the definition Definition: f(n) is in Θ(g(n)) if there exist positive constant c1 and c2 and non-negative integer n0 such that c2 g(n) ≤ f(n) ≤ c1 g(n) for every n ≥ n0 Examples: 10n is Θ(n) 5n3+20 is Θ(n3) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-20 Big-theta Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-21 Some properties of asymptotic order of growth f(n) O(f(n)) f(n) O(g(n)) iff g(n) (f(n)) If f (n) O(g (n)) and g(n) O(h(n)) , then f(n) O(h(n)) Note similarity with a ≤ b If f1(n) O(g1(n)) and f2(n) O(g2(n)) , then f1(n) + f2(n) O(max{g1(n), g2(n)}) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-22 Establishing order of growth using limits 0 order of growth of T(n) < order of growth of g(n) c > 0 order of growth of T(n) = order of growth of g(n) lim T(n)/g(n) = n→∞ ∞ order of growth of T(n) > order of growth of g(n) Examples: • 10n vs. n2 • n(n+1)/2 vs. n2 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-23 L’Hôpital’s rule and Stirling’s formula L’Hôpital’s rule: If limn f(n) = limn g(n) = and the derivatives f´, g´ exist, then lim n f(n) g(n) = lim n f ´(n) g ´(n) Example: log n vs. n Example: 2n vs. n! Stirling’s formula: n! (2n)1/2 (n/e)n Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-24 Orders of growth of some important functions All logarithmic functions loga n belong to the same class (log n) no matter what the logarithm’s base a > 1 is All polynomials of the same degree k belong to the same class: aknk + ak-1nk-1 + … + a0 (nk) Exponential functions an have different orders of growth for different a’s order log n < order n (>0) < order an < order n! < order nn Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-25 Basic asymptotic efficiency classes 1 constant log n logarithmic n linear n log n n-log-n n2 quadratic n3 cubic 2n exponential n! factorial Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-26 Time efficiency of nonrecursive algorithms General Plan for Analysis Decide on parameter n indicating input size Identify algorithm’s basic operation Determine worst, average, and best cases for input of size n Set up a sum for the number of times the basic operation is executed Simplify the sum using standard formulas and rules (see Appendix A) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-27 Useful summation formulas and rules liu1 = 1+1+…+1 = u - l + 1 In particular, liu1 = n - 1 + 1 = n (n) 1in i = 1+2+…+n = n(n+1)/2 n2/2 (n2) 1in i2 = 12+22+…+n2 = n(n+1)(2n+1)/6 n3/3 (n3) 0in ai = 1 + a +…+ an = (an+1 - 1)/(a - 1) for any a 1 In particular, 0in 2i = 20 + 21 +…+ 2n = 2n+1 - 1 (2n ) (ai ± bi ) = ai ± bi m+1iuai Copyright © 2007 Pearson Addison-Wesley. All rights reserved. cai = cai liuai = limai + A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-28 Example 1: Maximum element Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-29 What indicates the algorithm’s input size? What is the algorithm’s basic operation? No need to consider worst, average and best cases for this example. Why? Calculate C(n) n 1 C (n) 1 (n 1) 1 1 n 1 i 1 C ( n ) ( n ) Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-30 Example 2: Element uniqueness problem Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-31 What indicates the algorithm’s input size? What is the algorithm’s basic operation? The number of times its basic operation is executed depends not only on its input size. Consider Cworst(n) n 2 n 1 n2 i 0 j i 1 i 0 C (n) 1 (n 1) (i 1) 1 n2 n2 n2 i 0 i 0 i 0 (n 1 i ) (n 1) i Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-32 (n 2)( n 1) (n 1)1 2 i 0 n2 (n 2)( n 1) (n 1) 2 (n 2) (n 1) (n 1) 2 2 (n 1)n 1 2 n ( n 2 ) 2 2 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-33 Example 3: Matrix multiplication Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-34 What indicates the algorithm’s input size? What is the algorithm’s basic operation? The number of times its basic operation is executed depends only on its input size. Determine C(n) n 1 n 1 n 1 n 1 n 1 n 1 i 0 j 0 k 0 i 0 j 0 i 0 C (n) 1 n n 2 n3 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-35 Example 4: Gaussian elimination Algorithm GaussianElimination(A[0..n-1,0..n]) //Implements Gaussian elimination of an n-by-(n+1) matrix A for i 0 to n - 2 do for j i + 1 to n - 1 do for k i to n do A[j,k] A[j,k] - A[i,k] A[j,i] / A[i,i] Find the efficiency class and a constant factor improvement. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-36 Example 5: Counting binary digits It cannot be investigated the way the previous examples are. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-37 Plan for Analysis of Recursive Algorithms Decide on a parameter indicating an input’s size. Identify the algorithm’s basic operation. Check whether the number of times the basic op. is executed may vary on different inputs of the same size. (If it may, the worst, average, and best cases must be investigated separately.) Set up a recurrence relation with an appropriate initial condition expressing the number of times the basic op. is executed. Solve the recurrence (or, at the very least, establish its solution’s order of growth) by backward substitutions or another method. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-38 Example 1: Recursive evaluation of n! Definition: n ! = 1 2 … (n-1) n for n ≥ 1 and 0! = 1 Recursive definition of n!: F(n) = F(n-1) n for n ≥ 1 and F(0) = 1 Size: Basic operation: Recurrence relation: Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-39 Solving the recurrence for M(n) M(n) = M(n-1) + 1, M(0) = 0 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-40 Example 2: The Tower of Hanoi Puzzle 1 3 2 Recurrence for number of moves: Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-41 Solving recurrence for number of moves M(n) = 2M(n-1) + 1, M(1) = 1 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-42 Tree of calls for the Tower of Hanoi Puzzle n n-1 n-1 n-2 2 1 ... 1 n-2 n-2 ... ... 2 1 n-2 1 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. 2 1 2 1 1 A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 1 2-43 Example 3: Counting #bits Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-44 Fibonacci numbers Textbook p78 – p83 The Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, … The Fibonacci recurrence: F(n) = F(n-1) + F(n-2) F(0) = 0 F(1) = 1 General 2nd order linear homogeneous recurrence with constant coefficients: aX(n) + bX(n-1) + cX(n-2) = 0 Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-45 Solving aX(n) + bX(n-1) + cX(n-2) = 0 Set up the characteristic equation (quadratic) ar2 + br + c = 0 Solve to obtain roots r1 and r2 General solution to the recurrence if r1 and r2 are two distinct real roots: X(n) = αr1n + βr2n if r1 = r2 = r are two equal real roots: X(n) = αrn + βnr n Particular solution can be found by using initial conditions Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-46 Application to the Fibonacci numbers F(n) = F(n-1) + F(n-2) or F(n) - F(n-1) - F(n-2) = 0 Characteristic equation: Roots of the characteristic equation: General solution to the recurrence: Particular solution for F(0) =0, F(1)=1: Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-47 Computing Fibonacci numbers 1. Definition-based recursive algorithm 2. Nonrecursive definition-based algorithm 3. Explicit formula algorithm 4. Logarithmic algorithm based on formula: F(n-1) F(n) 0 1 n = 1 1 F(n) F(n+1) for n≥1, assuming an efficient way of computing matrix powers. Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 2 2-48