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Chapter 3: Problem 33

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

Expert verified
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

01

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.
02

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.
03

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.

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Most popular questions from this chapter

Write an efficient algorithm that will find an optimal order for multiplying \(n\) matrices \(A_{1} \times A_{2} \times \ldots \times A_{2}\) where the dimension of each matrix is \(1 \times 1\) \(1 \times d, d \times 1,\) or \(d \times d\) for some positive integer \(d .\) Analyze your algorithm, and show the results using order notation.

Write a more detailed version of the dynamic programming algorithm for the Traveling Salesperson Problem (Algorithm 3.11 ).

Show that to fully parenthesize an expression having \(n\) matrices we need \(n-1\) pairs of parentheses.

Generalize the Optimal Binary Search Tree algorithm (Algorithm 3.9 ) to the case where the search key may not be in the tree. That is, you should let \(q_{i}\) where \(i=0,1,2, \ldots, n,\) be the probability that a missing search key can be situated between \(K e y_{i}\) and \(K e y_{i+1}\). Analyze your generalized algorithm, and show the results using order notation.

Find an optimal circuit for the weighted, direct graph represented by the following matrix \(W\). Show the actions step by step. \\[W=\left[\begin{array}{rrrrr}0 & 8 & 13 & 18 & 20 \\\3 & 0 & 7 & 8 & 10 \\\4 & 11 & 0 & 10 & 7 \\ 6 & 6 & 7 & 0 & 11 \\\10 & 6 & 2 & 1 & 0\end{array}\right]\\]
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