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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 solution for finding the maximum sum in any sublist can be achieved using dynamic programming. The algorithm runs with complexity \(O(n)\).

Step by step solution

01

Identify the Problem

The goal is to find the maximum sum in any contiguous sublist of a given list of \(n\) real values. This is known as the maximum subarray problem.
02

Initiate the Algorithm

Initialize two variables, say max_so_far and max_ending_here, with the first element of the array. These will maintain the maximum sum found so far and the maximum sum till the current index, respectively.
03

Formulate recursive relation

The maximum sum till the current index (max_ending_here) is the maximum of (max_ending_here + current element) and the current element. This gives us the relation:\n\[maxEndingHere[i] = max(maxEndingHere[i - 1] + array[i], array[i])\]
04

Update the Maximum

After calculating max_ending_here for current index, update max_so_far if it is smaller than max_ending_here.
05

Iterate

Iterate through the entire list and perform the above steps. The maximum value at the end of the will be the maximum sum of the contiguous sublist.
06

Analyze the Algorithm

The proposed algorithm runs in linear time, that is, \(O(n)\), where \(n\) is the number of elements in the given list. It only requires a single pass through the elements of the list, and the computation at each step involves constant-time operations.

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Most popular questions from this chapter

Let us consider two sequences of characters \(S_{1}\) and \(S_{2}\). For example, we could have \(S_{1}-\) \(\mathrm{A} \$ \mathrm{CMA}^{*} \mathrm{MN}\) and \(S_{2}=\mathrm{AXMC} 4 \mathrm{ANB}\). Assuming that a subsequence of a sequence can be constructed by deleting any number of characters from any positions, use the dynamic programming approach to create an algorithm that finds the Iongest common subsequence of \(S_{1}\) and \(S_{2}\). This algorithm returns the maximum-length common subsequence of each sequence.

Find an optimization problem in which the principle of optimality does not apply and therefore that the optimal solution cannot be obtained using dynamic programming. Justify your answer.

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