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Algorithms Analysis & Methods | Lecture Summary

Explore recursion tree, substitution, & iteration methods in this computer algorithms lecture. Learn about The Master Theorem & examples of application. Understand how to solve equations using The Master Method.

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Algorithms Analysis & Methods | Lecture Summary

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  1. CS 575Design and Analysis ofComputer AlgorithmsProfessor Michal CutlerLecture 5September 13, 2005

  2. This class • Recursion tree method • Substitution method • Iteration • The master theorem • Application examples of Master theorem

  3. Recursion trees • The tree is expanded one depth at a time. • In the result tree: • Each “recursive call” is represented by a node containing the “non recursive” number of operations done during the call (t(size)) • Each base case is represented by a leaf containing the count for the base case • The number of operations is summed for each depth, and then over the depths

  4. T(n)=2T(n/2)+n2 n2 T(n/2) T(n/2)

  5. T(n)=2T(n/2)+n2 n2 (n/2)2 (n/2)2 T(n/4) T(n/4) T(n/4) T(n/4)

  6. The recursion tree for T(n) n2 n2 0 1 (n/2)2 (n/2)2 (1/2) n2 2 (n/4)2 (n/4)2 (n/4)2 (n/4)2 (1/4) n2 . . . . . . lgn

  7. Recursion trees

  8. Recursion trees

  9. The substitution method • We guess the form of the solution (from experience mostly). • We then verify the solution and solve for constants. • Not possible to guess correctly in every case. • T(n) = 4T(n/2) + n • We guess O(n3) and show T(n) £cn3 for all n³n0

  10. Example (cont.) • We assume that T(k) £ ck3 for k < n and prove T(n) £ cn3 by induction. T(n)=4T(n/2) + n £ 4c(n/2)3 + n = = (c/2)n3 + n = cn3 - ((c/2)n3 - n) £cn3 provided (c/2)n3 - n ³0 • So (c/2)n3 ³ n and c ³ 2/n2 for all n³1 • c³2 and n³1

  11. Initial Condition • Assume that T(1)=3 • 3=T(1)c13=c . So cmax{3, 2}=3.

  12. A tighter bound • Improve bound to T(n)=O(n2) • We assume that T(k) £ ck2 for k < n and prove T(n) £ cn2. T(n)=4T(n/2) + n£4c(n/2)2 + n = = cn2 + n. • Bad guess! Try T(n)£c1n2-c2n

  13. Subtract a lower term • We assume that T(k) £ c1k2-c2k for k < n and prove T(n) £ c1n2-c2n. T(n)=4T(n/2) + n£4(c1(n/2)2 - c2n/2)+ n = (c1n2 - c2n) - (c2n - n) £(c1n2 - c2n) only if c2n- n³0 and c2³1 for n³1 • Since T(1)=3 £ c112 - c21 we pick c1=4 and c2=1

  14. The iteration method • We expand the recursion, then get a summation, then compute the summation using algebraic methods.

  15. The iteration method • We have to know rules for summations to use this method. • We often use this method to guess the solution for the substitution method.

  16. Example • T(n) = n + 4T(n/2) = n + 4(n/2 + 4T(n/4))= = n + 2n + 16(n/4 + 4T(n/8))= = n + 2n + 4n + 64(n/8 + 4T(n/16))= = n + 2n + 22n + 23n + …+2k-1n + 4kT(n/2k) • Let n=2k and k=lgn • We know 4lg n=nlg4 =n2 (because algb = blga)

  17. Example (cont)

  18. The Master method • Used to solve equations of the form T(n) = aT(n/b) + f(n) where a ³1, b > 1, and f(n) is an asymptotically nonnegative function. Note: Version more general than one in text

  19. Master theorem • More general version than book

  20. Master theorem • A different version of previous slide

  21. The recursion tree (integer a) f(n) f(n) af(n/b) f(n/b) f(n/b) f(n/b) a2f(n/b2) f(n/b2) f(n/b2) f(n/b2) k Q(1) Q(1) Q(1) Q(1) Q(1) Q(1) Q(1) Q(1) Q(1) ak Q(1) n=bk, k=logbn, T(n/bk )=T(1)= Q (1)

  22. The Master method • Let n=bk and k=logbn. • The number of leaves in the recursion tree is alogbn = nlogba.

  23. Master method examples • T(n) = 2T(n/2) + Q(n) • a = b =2 • f(n)= Q(n) • nlog22 = n, • Q(n)/n = Q(1) = Q(lg0n) • Case 2 and T(n)= Q(nlgn)

  24. Master method examples • T(n) = 7T(n/2) + Q(n2). • a =7, b = 2 • nlogba = nlog27 • Q(n2)/nlog27 = Q(n2-lg 7) = Q(n-0.8) • Case 1 and T(n) = Q(nlog27)

  25. Master method examples • T(n) = T(n/2) + Q(1) Since nlog21 = n0 = 1, Q(1)/1 = Q(log0n), we are dealing with Case 2 and T(n) = Q(log n)

  26. Master method examples • T(n) = 4T(n/2) + n3 Since n3/nlg4 = n, we are dealing with Case 3. • Is af(n/b) <= cf(n) for some c<1 and all sufficiently large n? yes 4(n/2)3 =n3/2 so c=1/2 • So T(n) = Q(n3) • The master method does not solve all recurrences of the given form.

  27. Master method fails • T(n) = 4T(n/2) + n2/lgn (n2/lgn)/n2 = 1/lgn and no case applies

  28. Master method fails • The solution is T(n) = Q(n2lglgn) and can be verified by the substitution method.

  29. The Three Cases

  30. Case 1: nlogba is polynomially larger than f(n). • f(n)= O(nlogba-e), or • f(n)/nlogba = O(n-e) , or • nlogba/ f(n) = W(ne) for some constant e > 0 • We bound the sum within the O-notation by factoring the terms, the remaining sum is an increasing geometric series

  31. g(n) for case 1

  32. g(n) for Case 1

  33. g(n) for Case 1

  34. Case 1: nlogba is polynomially larger than f(n). • Total weight is (constant *number leaves).

  35. Case 2: nlogba and f(n) are within a polylogarithmic factor. • f(n)/ nlogba = Q(lgkn) for some constant k³0. • T(n) = Q(nlogba) + Q(nlogba lgk+1n)= = Q(nlogba lgk+1n)

  36. g(n) for Case 2

  37. g(n) for Case 2

  38. g(n) for Case 2

  39. g(n) for Case 2

  40. Case 3:nlogba is polynomially smaller than f(n). • f(n)/ nlogba = W (ne) for some constant e > 0. • Total weight is (constant times weight of root). • T(n) = Q(nlogba) + Q(f(n))= Q(f(n))

  41. Case 3:nlogba is polynomially smaller than f(n). • We will show that 1. g(n)= W(f(n)), and 2. g(n)=O(f(n)) and therefore g(n)= Q(f(n))

  42. Case 3:nlogba is polynomially smaller than f(n). 1. We know that a³1 and that f(n)³0 for all large enough n • So all terms of g(n) are nonnegative for large enough n. • The first term of g(n) is f(n), so g(n)³f(n) for all large enough n and g(n)=W(f(n))

  43. Case 3

  44. Case 3

  45. Next class • Quicksort

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