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Design and Analysis of Algorithms: Maximum Subarray Problem and Matrix Multiplication

Learn about the divide-and-conquer algorithm for the maximum subarray problem and matrix multiplication with detailed examples. Explore the time complexity and Strassen's algorithm.

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Design and Analysis of Algorithms: Maximum Subarray Problem and Matrix Multiplication

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  1. Design and Analysis of AlgorithmsMaximum-subarray problem and matrix multiplication (with more examples) HaidongXue Summer 2012, at GSU

  2. 1 -4 3 2 Target array : What is a maximum subarray? 1 1 All the sub arrays: -4 -4 3 3 The subarray with the largest sum 2 2 1 -4 -3 -4 3 -1 3 2 What is the brute-force T(n)? Max! 5 1 -4 3 0 -4 3 2 1 O() 1 -4 3 2 2

  3. Maximum-subarray • Can we do it in a divide-and-conquer manner? • Yes, we can

  4. Divide target array into 2 arrays 1 -4 3 2 Target array : We then have 3 types of subarrays: 1 1 All the sub arrays: -4 -4 The problem can be then solved by: The ones belong to the left array 3 3 1. Find the max in left sub arrays The ones belong to the right array 2 2 1 -4 -3 2. Find the max in right sub arrays The ones crossing the mid point -4 3 -1 3. Find the max in crossing sub arrays 3 2 5 1 -4 3 0 -4 3 2 1 4. Choose the largest one from those 3 as the final result 1 -4 3 2 2

  5. How to find the max subarray crossing the midpoint? 1 -4 3 2 Find the left part -4 Sum=-4 1 -4 Sum=-3 largest Find the right part 3 Sum=3 3 2 Sum=5 largest The largest crossing subarray is :

  6. The whole algorithm 1 -4 3 2 FindMaxSub ( ) 1 -4 1. Find the max in left sub arrays FindMaxSub ( ) 3 2 2. Find the max in right sub arrays FindMaxSub ( ) 3. Find the max in crossing sub arrays 3 2 1 -4 Scan once, and scan once 4. Choose the largest one from those 3 as the final result

  7. Maximum-subarrayproblem – divide-and-conquer algorithm • Input: array A[i, …, j] • Ouput: sum of maximum-subarray, start point of maximum-subarray, end point of maximum-subarray • FindMaxSubarray: • if(j<=i) return (A[i], i, j); • mid = floor(i+j); • (sumCross, startCross, endCross) = FindMaxCrossingSubarray(A, i, j, mid); • (sumLeft, startLeft, endLeft) = FindMaxSubarray(A, i, mid); • (sumRight, startRight, endRight) = FindMaxSubarray(A, mid+1, j); • Return the largest one from those 3 (n) Time complexity?

  8. Maximum-subarray problem – divide-and-conquer algorithm FindMaxCrossingSubarray(A, i, j, mid) • Scan A[i, mid] once, find the largest A[left, mid] • Scan A[mid+1, j] once, find the largest A[mid+1, right] • Return (sum of A[left, mid] and A[mid+1, right], left, right) Let’s try it in Java

  9. Matrix multiplication • How to multiply two matrices? • If each matrix is nxn, for each , we need • There are so = A B C 1 0 0 -1 0 -1 2 3 3 4 9 10

  10. Matrix multiplication • How to multiply two matrices? • If each matrix is nxn, for each , we need • There are so = A B C 1 0 0 -1 0 -1 2 3 3 4 9 10

  11. Matrix multiplicationdivide-and-conquer algorithm • C=AB • Recurrence equation? • () 1 0 1 5 0 0 1 5 2 3 2 3 2 3 2 3 1 6 1 0 1 0 0 0 7 3 2 3 0 3 2 2

  12. Matrix multiplicationdivide-and-conquer algorithm • () • What is the time complexity? • From Master method we know it is ()

  13. Matrix multiplicationStrassen’s algorithm • () • () • =O()O()

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