02 - Maximum Contiguous Subarray

02 - Maximum Contiguous Subarray - Part I: Divide and...

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Unformatted text preview: Part I: Divide and Conquer Lecture 2: Maximum Contiguous Subarray Problem Lecture 2: Maximum Contiguous Subarray Problem Part I: Divide and Conquer Introduction to Part I Divide and conquer (D&C) is an important algorithm design paradigm. It works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. In Part I, we will illustrate Divide-and-Conquer using several examples: Maximum contiguous subarray Polynomial Multiplication Partition and Selection Lecture 2: Maximum Contiguous Subarray Problem Part I: Divide and Conquer Objective and Outline Objective : Discuss first example of divide-and-conquer Reference : Chapter 8 in Programming Pearls, (2nd ed) by Jon Bentley Outline: Problem definition A brute force algorithm A data-reuse algorithm A divide-and-conquer algorithm Analysis of the divide-and-conquer algorithm Summary Lecture 2: Maximum Contiguous Subarray Problem Part I: Divide and Conquer Maximum Contiguous Subarray Problem ACME CORP 1 – PROFIT HISTORY Year 1 2 3 4 5 6 7 8 9 Profit M$-3 2 1-4 5 2-1 3-1 Between years 1 and 9: ACME earned- 3 + 2 + 1- 4 + 5 + 2- 1 + 3- 1 = 4 M$ Between years 2 and 6: ACME earned 2 + 1- 4 + 5 + 2 = 6 M$ Betweeen years 5 and 8: ACME earned 5 + 2- 1 + 3 = 9 M$ Problem: Find the span of years in which ACME earned the most Answer: Year 5-8 , 9 M$ 1 A Company that Makes Everything Lecture 2: Maximum Contiguous Subarray Problem Part I: Divide and Conquer Formal Definition Input : An array of reals A [1 . . . N ] The value of sub array A [ i . . . j ] is V ( i , j ) = j X x = i A ( x ) Definition (Maximum Contiguous Subarray Problem) Find i ≤ j such that V ( i...
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02 - Maximum Contiguous Subarray - Part I: Divide and...

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