1. 1Computer Sciences Department

3. Objectives Recurrences.  Substitution Method,  Recursion-tree method,  Master method

4. Recurrence 4Computer Sciences Department

5.  A recurrence is an equation or inequality that describes a function in terms of its value on smaller inputs.  Example:  MERGE-SORT procedure could be described by the recurrence: 5 recurrence Computer Sciences Department

6.  Substitution Method,  Recursion-tree method,  Master method 6 Recurrence M ethods Computer Sciences Department

7.  The substitution method for solving recurrences entails two steps:  1. Guess the form of the solution.  2. Use mathematical induction to find the constants and show that the solution works.  The substitution method can be used to establish either upper or lower bounds on a recurrence. 7 The substitution method Computer Sciences Department

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9. As an example, let us determine an upper bound on the recurrence 9 lgn/2=lgn-lg2 Computer Sciences Department

10.  A recursion tree is best used to generate a good guess, which is then verified by the substitution method.  Using recursion trees to generate good guesses. 10 The recursion-tree method Computer Sciences Department

11. 11 Solution Computer Sciences Department

12. 12 Solution

13. Computer Sciences Department13 Solution

14. Computer Sciences Department14 Solution

15. Computer Sciences Department15 Solution

16. 16 For example We start by focusing on finding an upper bound for the solution. create a recursion tree for the recurrence we assume that n is an exact power of 4 Computer Sciences Department

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19. 19 guess Computer Sciences Department

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21. 21 The master method Computer Sciences Department

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27. Proof of the master theorem (no need) 27Computer Sciences Department