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Recurrence Relations

This guide delves into the methods for analyzing the performance of recursive algorithms, focusing on worst-case and average-case analysis. It covers the selection of input sets that are expandable and independent of algorithm behavior, employing static behavior analysis where necessary. The discussion includes binary search algorithm (BinS) performance, exploring recursive behavior and recurrence relations. Learn to determine the amount of work required for a range of inputs and compute mean values for average-case scenarios.

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Recurrence Relations

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  1. Recurrence Relations Analyzing the performance of recursive algorithms

  2. Worst Case Analysis • Select a Hypothetical Set of Inputs • Should be expandable to any size • Make Algorithm run as long as possible • Must not depend on Algorithm Behavior • May use static behavior analysis • Inputs may not change dynamically

  3. Average Case Analysis • Determine the range of inputs over which the average will be taken • Select a probability distribution • Characterize Amount of work required for each input • Compute the Mean Value

  4. Binary Search BinS(X:Item;First,Last:Integer):Integer var Middle:Integer; begin if First <= Last then Middle = (First + Last)/2; if L[Middle] = X then return Middle; if L[Middle] < X then return BinS(X,Middle+1,Last); if L[Middle] > X then return BinS(X,First,Middle-1); else return -1; endif; end;

  5. Analysis of BinS: Basis • If List Size = 1 then First = Last • If List Size = 0 then First > Last if First <= Last then ... else return -1; • Then W(0) = 0

  6. Analysis of BinS: Recurrsion I • If comparison fails, find size of new list • If list size is odd: • First=1,Last=5,Middle=(1+5)/2=3 • If list size is even: • First=1,Last=6,Middle=(1+6)/2=3

  7. Analysis of BinS: Recurrsion II • Exercise: Verify that these examples are characteristic of all lists of odd or even size • Worst case: Item X not in list, larger than any element in list. • If list is of size n, new list will be of size:

  8. Analysis of BinS: Recurrence Is W(n) ÎQ(n)?

  9. Solving the Recurrence We need to eliminate W from the RHS.

  10. Continuing the Exercise ...

  11. Ah Ha! A General Formula!

  12. Ah Ha! A General Formula! But, if k is big enough ...

  13. Ah Ha! A General Formula! But, if k is big enough ... Then the argument of W will be zero, and ...

  14. Ah Ha! A General Formula! But, if k is big enough ... Then the argument of W will be zero, and ... Since W(0) = 0 ...

  15. Ah Ha! A General Formula! But, if k is big enough ... Then the argument of W will be zero, and ... Since W(0) = 0 ... The W on the RHS ...

  16. Ah Ha! A General Formula! But, if k is big enough ... Then the argument of W will be zero, and ... Since W(0) = 0 ... The W on the RHS ... Will Disappear!

  17. When Will W disappear? When:

  18. When Will W Disappear? When: Take the Log:

  19. When Will W Disappear? When: Take the Log: Take the Floor:

  20. Finally ...

  21. Some Other Recurrences

  22. Recurrences with T(1)=1

  23. The Challenger

  24. The Master Theorem I

  25. The Master Theorem II

  26. The Master Theorem III (for sufficiently large n)

  27. Homework • Page 72, 4-1 • Page 73, 4-4

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