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Recurrences

Recurrences. recurrence. 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:. R ecurrence M ethods. Substitution Method, Recursion-tree method, Master method.

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Recurrences

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  1. Recurrences

  2. recurrence • 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:

  3. Recurrence Methods • Substitution Method, • Recursion-tree method, • Master method

  4. The substitution method • 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.

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

  6. The recursion-tree method • 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.

  7. 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

  8. guess

  9. The master method

  10. Proof of the master theorem (no need)

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