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Chapter 2: Problem 42

Use the divide-and-conquer approach to write a recursive algorithm that finds the maximum sum in any contiguous sublist of a given list of \(n\) real values. Analyze your algorithm, and show the results in order notation.

Short Answer

Expert verified
The algorithm designed for finding the maximum sum in any contiguous sublist of a given list of \(n\) real values using divide-and-conquer approach has a time complexity of \(O(n \log n)\) in the worst case scenario.

Step by step solution

01

Problem analysis and Algorithm design

Divide and conquer strategy divides the problem into subproblems until they can be solved independently. So, divide the array into two halves. Note that the maximum contiguous sublist may lie entirely in the left half, entirely in the right half, or it may cross the boundary. First, solve for these three conditions individually. The recursive function has an array and a value \(l\) and \(r\) as its argument, which indicates the current array range.
02

Base case

In base case scenario, if \(l = r\), return (l, r, array[l], array[l]) which in order indicates the starting index, ending index, maximum subarray sum and total array sum.
03

Recursive case

In recursive case scenario, let \(m = middle\) and recursively calculate the value for \(left__[l, m]\) and \(right__[m+1, r]\). Calculate \(cross__[l, r]\) by finding maximum suffix sum in the left half and maximum prefix sum in the right half. Return the maximum of \(left__, right__\) and \(cross__\).
04

Algorithm Analysis

The resulting approach scans the entire array only three times, so the algorithm is linear (\(O(n)\)). But the recursive nature of the algorithm also includes a logarithmic factor making it \(O(n \log n)\) in worst-case scenario.

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