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Objectives

• Recurrences.

Substitution Method,

Recursion-tree method,

Master method

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Recurrence

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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:

recurrence

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Substitution Method,

Recursion-tree method,

Master method

Recurrence Methods

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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.

The substitution method

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As an example, let us determine an upper bound on the recurrence

lgn/2=lgn-lg2

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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.

The recursion-tree method

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Solution

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Solution

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Solution

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Solution

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Solution

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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

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The master method

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and

and

Conditions of using the master method

1

2

3

5

4

and

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?

Try

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TRY (LAB)

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Proof of the master theorem (no need)