Question

Use the dynamic programming approach to write an algorithm to find the maximum sum in any contiguous sublist of a given list of \(n\) real values. Analyze your algorithm, and show the results using order notation.

Short answer

The algorithm will iterate through the list, maintaining a running sum of the elements and using that sum to update a variable holding the maximum sum. In every iteration, the running sum (current_max) will be the maximum between the current number and the sum of current_max and the current number. If current_max is greater than global_max, global_max will be updated. At the end, global_max will be the maximum sum of any sublist. The algorithm operates at a time complexity of \(O(n)\), where \(n\) is the number of elements in the list.

Step-by-step solution

Understanding Dynamic Programming and Problem

Dynamic programming is a method for solving problems by breaking them down into simpler subproblems and using the fact that the optimal solution to the overall problem depends upon the optimal solution to its subproblems. In this exercise, a list of real numbers is given, the task is to find the maximum sum obtained by adding the numbers in a contiguous sublist.

Applying Kadane's Algorithm

Kadane's algorithm can be used to find out the maximum sum of a contiguous sublist. Here is how it can be implemented: Initialize two variables, say \(global\_max\) and \(current\_max\), to the first element of the list. For each element in the list from the second one, update \(current\_max\) to be the maximum between the current number and the sum of \(current\_max\) and the current number. Then, if \(current\_max\) is greater than \(global\_max\), update \(global\_max\). By the end of the list, \(global\_max\) will hold the maximum sum of any contiguous sublist.

Analyzing the Algorithm

Since every element in the list is processed only once in this algorithm, the time complexity for this algorithm can be given in terms of order notation or Big-O notation as \(O(n)\), where \(n\) is the number of elements in the list. This means that the time taken by the algorithm will increase linearly with the increase in the size of the list.