Lecture 4 Divide-and-Conquer

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Presentation transcript:

Lecture 4 Divide-and-Conquer

What’s Self-Reducibility? A problem can be reduced to several “same” problems with smaller inputs.

Divide and Conquer Divide the problem into subproblems. Conquer the subproblems by solving them recursively. Combine the solutions to subproblems into the solution for original problem.

Tree Structure problem subproblem subproblem subsubproblem

Insertion Sort

key

Merge Sort

Procedure divide conquer combine

Heapsort 16 14 10 3 8 7 9 2 4 1

Heapsort

Heapsort

Quicksort divide conquer combine is simplified

Selection with O(n) Comparisons divide conquer divide conquer

Remark on Divide and Conquer Key Point:

Merge Sort Quick Sort Selection Analysis requires to solve a recurrence!

Maximum Subarray

Problem

Closest pair of points

Matrix Multiplication

n/2 Step 1 n/2

Step 2 Step 3

Step 4

Master Theorem