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

Recurrence 4Computer Sciences Department

 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

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

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

 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

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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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21 The master method Computer Sciences Department

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