Intro to Computer Algorithms Lecture 6

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Intro to Computer Algorithms Lecture 6

• Phillip G. Bradford
• Computer Science
• University of Alabama

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Announcements

• Colloquia on Monday 22-Sept
• David M. Mount presents
• Approximation Algorithms for Clustering
• At 1100 in 108 Houser

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Lecture Outline

• Homework Assignment
• Review last lecture
• More on Recurrences
• Exhaustive Search
• Divide and Conquer

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Reading

• Up through Chapter 3
• Start Chapter 4.

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Exhaustive Search

• Brute-Force String Matching
• Elements in string n
• Elements in search string m
• Closest Point Problems
• Traveling Salesperson
• Knapsack Problem

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More Recurrence Examples

• T(n) 2T(n/2) 1
• Solution?
• How about
• T(n) 2T(n/2) n
• Assume n is a power of 2.

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Divide and Conquer

• Divide instance in to parts
• Solve problem on the parts
• Combine the solutions

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The Master Theorem

• T(n) aT(n/b) f(n)
• M. Theorem. Suppose f(n) is Q(nd) and d gt 0. Then
  T(n) is
• Q(nd) if a lt bd
• Q(nd log n) if a bd
• Q(nlogba) if a gt bd

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Examples

• Recall, T(n) 2T(n/2) 1
• Here, f(n) 1 and a b 2 and d 0.
• So, a gt bd
• Giving T(n) is Q(nlogba) Q(n)