1. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 1 Analysis of algorithms b Issues: correctnesscorrectness time efficiencytime efficiency space efficiencyspace efficiency optimalityoptimality b Approaches: theoretical analysistheoretical analysis empirical analysisempirical analysis

2. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 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 b Basic operation: the operation that contributes most towards the running time of the algorithm T(n) ≈ c op C(n) T(n) ≈ c op C(n) running time execution time for basic operation Number of times basic operation is executed input size

3. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 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

4. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 4 Empirical analysis of time efficiency b Select a specific (typical) sample of inputs b Use physical unit of time (e.g., milliseconds) or or Count actual number of basic operation’s executions Count actual number of basic operation’s executions b Analyze the empirical data b We mostly do theoretical analysis (may do empirical in assignment)

5. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 5 Best-case, average-case, worst-case For some algs C(n) is independent of the input set. For others, C(n) depends on which input set (of size n) is used. Example on next slide (Search). Consider three possibilities: b Worst case: C worst (n) – maximum over inputs of size n b Best case: C best (n) – minimum over inputs of size n b Average case: C avg (n) – “average” over inputs of size n Number of times the basic operation will be executed on typical inputNumber of times the basic operation will be executed on typical input NOT the average of worst and best caseNOT 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 of size nExpected number of basic operations considered as a random variable under some assumption about the probability distribution of all possible inputs of size n Consider all possible input sets of size n, average C(n) for all setsConsider all possible input sets of size n, average C(n) for all sets b Some algs are same for all three (eg all case performance)

6. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 6 Example: Sequential search b Worst case b Best case b Average case: depends on assumputions about input (eg proportion of found vs not-found keys)

7. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 7 Example: Find maximum b Worst case b Best case b Average case: depends on assumputions about input (eg proportion of found vs not-found keys) b All case

8. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 8 Types of formulas for basic operation’s count b Formula for C(n) can have different levels of detail b Exact formula e.g., C(n) = n(n-1)/2 e.g., C(n) = n(n-1)/2 b Formula indicating order of growth with specific multiplicative constant e.g., C(n) ≈ 0.5 n 2 e.g., C(n) ≈ 0.5 n 2 b Formula indicating order of growth with unknown multiplicative constant e.g., C(n) ≈ cn 2 e.g., C(n) ≈ cn 2 b Which depends on problem, analysis technique, need

9. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 9 Order of growth b Most important: Order of growth within a constant multiple as n→∞ b Examples: How much faster will algorithm run on computer that is twice as fast? What say you?How much faster will algorithm run on computer that is twice as fast? What say you? –Time = … How much longer does it take to solve problem of double input size? What say you?How much longer does it take to solve problem of double input size? What say you? –Time =

10. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 10 Values of some important functions as n   Focus: asymptotic order of growth: Main concern: which function describes behavior. Less concerned with constants

11. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 11 Asymptotic order of growth Critical factor for problem size n: - IS NOT the exact number of basic ops executed for given n - IS how number of basic ops grows as n increases - IS how number of basic ops grows as n increasesConsequently: - Focus on how performance grows as n   - Ignore constant factors, constants, and small input sizes - Ignore constant factors, constants, and small input sizes - Example: We consider 100n^2 +1000 better than n+1 - Example: We consider 100n^2 +1000 better than n+1 Call this: Asymptotic Order of Growth

12. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 12 Order of growth: Sets of functions Approach: classify functions (ie of alg performance) based on their growth rates and divide into sets Example: Θ(n^2) is set of functions whose growth rate is n^2 These are all in Θ(n^2): - f(n) = 100n^2 + 1000 - f(n) = n^2 + 1 - f(n) = 0.001n^2 + 1000000

13. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 13 Order of growth: Upper, tight, lower bounds More formally: - Θ(g(n)): class of functions f(n) that grow at same rate as g(n) - Θ(g(n)): class of functions f(n) that grow at same rate as g(n) Upper, tight, and lower bounds on performance: b O(g(n)): class of functions f(n) that grow no faster than g(n) [ie f ’s speed is same as or faster than g, f bounded above by g][ie f ’s speed is same as or faster than g, f bounded above by g] b Θ(g(n)): class of functions f(n) that grow at same rate as g(n) b Ω(g(n)): class of functions f(n) that grow at least as fast as g(n) [ie f ’s speed is same as or slower than g, f bounded below by g][ie f ’s speed is same as or slower than g, f bounded below by g]

14. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 14 t(n)  O(g(n)) iff t(n) n 0 t(n) = 10n 3 in O(n 3 ) and in O(n 5 )

15. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 15 t(n)  Ω(g(n)) iff t(n) >=cg(n) for n > n 0 t(n) = 10n 3 in Ω(n 2 ) and in Ω(n 3 )

16. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 16 t(n)  Θ(g(n)) iff t(n)  O(g(n)) and  Ω(g(n)) t(n) = 10n 3 in Θ(n 3 ) but NOT in Ω(n 2 ) or O(n 4 )

17. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 17 Terminology Informally: f(n) is O(g) means f(n)  O(g) Example: 10n is O(n 2 ) Use similar terms for Big Omega and Big Theta

18. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 18 Informal Definitions: Big O, Ω, Θ Informal Definition: f(n) is in O(g(n)) if order of growth of f(n) ≤ order of growth of g(n) (within constant multiple). Informal Definition: f(n) is in Ω(g(n)) if order of growth of f(n) ≥ order of growth of g(n) (within constant multiple). Informal Definition: f(n) is in Θ(g(n)) if order of growth of f(n) = order of growth of g(n) (within constant multiples).

19. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 19 Formal Definitions: Big O, Ω, Θ Definition: f(n)  O(g(n)) iff there exist positive constant c and non-negative integer n 0 such that f(n) ≤ c g(n) for every n ≥ n 0 f(n) ≤ c g(n) for every n ≥ n 0 Definition: f(n)  Ω(g(n)) iff there exist positive constant c and non-negative integer n 0 such that f(n) ≥ c g(n) for every n ≥ n 0 f(n) ≥ c g(n) for every n ≥ n 0 Definition: f(n)  Θ(g(n)) iff there exist positive constants c 1 and c 2 and non-negative integer n 0 such that c 1 g(n) ≤ f(n) ≤ c 2 g(n) for every n ≥ n 0 c 1 g(n) ≤ f(n) ≤ c 2 g(n) for every n ≥ n 0

20. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 20 Using the Definition: Big O Informal Definition: f(n) is in O(g(n)) if order of growth of f(n) ≤ order of growth of g(n) (within constant multiple), Definition: f(n)  O(g(n)) iff there exist positive constant c and non-negative integer n 0 such that f(n) ≤ c g(n) for every n ≥ n 0 f(n) ≤ c g(n) for every n ≥ n 0Examples: b 10n is O(n 2 ) [Can choose c and n 0. Solve for 2 different c’s][Can choose c and n 0. Solve for 2 different c’s] b 5n + 20 is O(n) [Solve for c and n 0 ]

21. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 21 Using the Definition: Big Omega Informal Definition: f(n) is in  (g(n)) iff order of growth of f(n) ≥ order of growth of g(n) (within constant multiple), Definition: f(n)   (g(n)) iff there exist positive constant c and non-negative integer n 0 such that f(n) ≥ c g(n) for every n ≥ n 0 f(n) ≥ c g(n) for every n ≥ n 0Examples: b 10n 2 is  (n) b 5n + 20 is  (n)

22. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 22 Using the Definition: Theta Informal Definition: f(n) is in  (g(n)) if order of growth of f(n) = order of growth of g(n) (within constant multiple), Definition: f(n)  Θ(g(n)) iff there exist positive constants c 1 and c 2 and non-negative integer n 0 such that c 1 g(n) ≤ f(n) ≤ c 2 g(n) for every n ≥ n 0 c 1 g(n) ≤ f(n) ≤ c 2 g(n) for every n ≥ n 0Examples: b 10n 2 is  (n 2 ) b 5n + 20 is  (n)

23. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 23 Some properties of asymptotic order of growth b f(n)  O(f(n)) b f(n)  O(g(n)) iff g(n)  (f(n)) b If f (n)  O(g (n)) and g(n)  O(h(n)), then f(n)  O(h(n)) Note similarity with a ≤ b b If f 1 (n)  O(g 1 (n)) and f 2 (n)  O(g 2 (n)), then f 1 (n) + f 2 (n)  O(max{g 1 (n), g 2 (n)}) f 1 (n) + f 2 (n)  O(max{g 1 (n), g 2 (n)})

24. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 24 Establishing order of growth using limits lim T(n)/g(n) = 0T(n)g(n) 0 order of growth of T(n) < order of growth of g(n) c > 0T(n)g(n) c > 0 order of growth of T(n) = order of growth of g(n) ∞T(n)g(n) ∞ order of growth of T(n) > order of growth of g(n) Examples: 10n vs. n 2 n(n+1)/2 vs. n 2 n→∞

25. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 25 L’Hôpital’s rule and Stirling’s formula L’Hôpital’s rule: If lim n  f(n) = lim n  g(n) =  and the derivatives f´, g´ exist, then the derivatives f´, g´ exist, then Stirling’s formula: n!  (2  n) 1/2 (n/e) n f(n)f(n)g(n)g(n)f(n)f(n)g(n)g(n)lim n  = f ´(n) g ´(n) lim n  Example: log n vs. n Example: 2 n vs. n!

26. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 26 Orders of growth of some important functions b All logarithmic functions log a nbelong to the same class  (log n)no matter what the logarithm’s base a > 1 is b All logarithmic functions log a n belong to the same class  (log n) no matter what the logarithm’s base a > 1 is b All polynomials of the same degree k belong to the same class: a k n k + a k-1 n k-1 + … + a 0   (n k ) b Exponential functions a n have different orders of growth for different a’s b order log n 0) 0) < order a n < order n! < order n n

27. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 27 Basic asymptotic efficiency classes 1constant Best case log n logarithmic Divide ignore part nlinear Examine each n log n n-log-n or linearithmic Divide use all parts n2n2n2n2quadratic Nested loops n3n3n3n3cubic 2n2n2n2nexponential All subsets n!n!n!n!factorial All permutations

28. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 28 Time efficiency of nonrecursive algorithms General Plan for Analysis b Decide on parameter n indicating input size b Identify algorithm’s basic operation b Determine worst, average, and best cases for input of size n b Set up a sum for the number of times the basic operation is executed b Simplify the sum using standard formulas and rules (see Appendix A)

29. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 29 Useful summation formulas and rules  l  i  u 1 = 1+1+ ⋯ +1 = u - l + 1 In particular,  l  i  u 1 = n - 1 + 1 = n   (n) In particular,  l  i  u 1 = n - 1 + 1 = n   (n)  1  i  n i = 1+2+ ⋯ +n = n(n+1)/2  n 2 /2   (n 2 )  1  i  n i 2 = 1 2 +2 2 + ⋯ +n 2 = n(n+1)(2n+1)/6  n 3 /3   (n 3 )  0  i  n a i = 1 + a + ⋯ + a n = (a n+1 - 1)/(a - 1) for any a  1 In particular,  0  i  n 2 i = 2 0 + 2 1 + ⋯ + 2 n = 2 n+1 - 1   (2 n ) In particular,  0  i  n 2 i = 2 0 + 2 1 + ⋯ + 2 n = 2 n+1 - 1   (2 n )  (a i ± b i ) =  a i ±  b i  ca i = c  a i  l  i  u a i =  l  i  m a i +  m+1  i  u a i

30. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 30 Example 1: Maximum element

31. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 31 Example 2: Element uniqueness problem

32. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 32 Example 3: Matrix multiplication

33. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 33 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] A[j,k]  A[j,k] - A[i,k]  A[j,i] / A[i,i] Find the efficiency class and a constant factor improvement.

34. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 34 Example 5: Counting binary digits It cannot be investigated the way the previous examples are.

35. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 35 Plan for Analysis of Recursive Algorithms b Decide on a parameter indicating an input’s size. b Identify the algorithm’s basic operation. b 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.) b Set up a recurrence relation with an appropriate initial condition expressing the number of times the basic op. is executed. b Solve the recurrence (or, at the very least, establish its solution’s order of growth) by backward substitutions or another method.

36. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 36 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 F(0) = 1Size: Basic operation: Recurrence relation:

37. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 37 Solving the recurrence for M(n) M(n) = M(n-1) + 1, M(0) = 0

38. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 38 Example 2: The Tower of Hanoi Puzzle Recurrence for number of moves:

39. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 39 Solving recurrence for number of moves M(n) = 2M(n-1) + 1, M(1) = 1

40. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 40 Tree of calls for the Tower of Hanoi Puzzle

41. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 41 Example 3: Counting #bits

42. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 42 Fibonacci numbers 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 2 nd order linear homogeneous recurrence with constant coefficients: aX(n) + bX(n-1) + cX(n-2) = 0 aX(n) + bX(n-1) + cX(n-2) = 0

43. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 43 Solving aX(n) + bX(n-1) + cX(n-2) = 0 b Set up the characteristic equation (quadratic) ar 2 + br + c = 0 ar 2 + br + c = 0 b Solve to obtain roots r 1 and r 2 b General solution to the recurrence if r 1 and r 2 are two distinct real roots: X(n) = α r 1 n + β r 2 n if r 1 = r 2 = r are two equal real roots: X(n) = α r n + β nr n b Particular solution can be found by using initial conditions

44. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 44 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: Characteristic equation: Roots of the characteristic equation: General solution to the recurrence: Particular solution for F(0) =0, F(1)=1:

45. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 3rd ed., Ch. 2 ©2012 Pearson Education, Inc. Upper Saddle River, NJ. All Rights Reserved. 45 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) F(n) F(n+1) 0 1 1 1 =n for n≥1, assuming an efficient way of computing matrix powers.