Solution README

Dynamic programming becomes a valuable tool for improving code efficiency and tackling advanced challenges in programming and algorithms. In this particular case, the goal is to maximize profits given a rod of length 'n'. The task is to determine the best possible cuts to make, where each resulting segment of the rod has a value that we aim to maximize using the corresponding algorithm.

To solve this problem, we can imagine cutting the rod in any possible way, with each possible cut representing a potential solution. Our objective is to find the optimal way to maximize the total value based on the price of each segment. Dynamic programming is applied here by following four key steps that are crucial for conceptualizing and solving the problem:

Characterize the structure of an optimal solution.
Recursively define the value of an optimal solution.
Compute the value of an optimal solution using dynamic programming.
Construct an optimal solution from computed information.
When deciding to make the first cut, we may encounter several scenarios where additional decisions are required, such as making further cuts or leaving the rod as is. Storing the optimal solutions of these subproblems and combining them can provide a globally optimal solution to the original problem. As defined in step one, the structure involves starting with the initial rod and making specific cuts to increase its value, or choosing not to cut it at all. It is essential to compare all initial cutting possibilities—from making a single cut to making 'n' cuts (or none)—using a recursive function that calculates the values and returns only the largest one. This function calculates the price of the left sub-rod (if divided) plus the value returned by the recursive call of the right sub-rod, thus fulfilling step two of defining an optimal value recursively.

Step three, which involves computing the optimal value using dynamic programming, introduces memoization. Here, we take the recursive algorithm and add an array where, before calculating a new value, we first check if it already exists in the array. If it does, we use that value; otherwise, we calculate it normally in the recursive algorithm and store it in the array for future use, thereby saving the need to perform multiple calculations. In this approach, we trade time for space, as storing the computed values in the array for future use requires more memory and, consequently, more computer resources. However, this significantly reduces execution times.

By storing the optimal value results for each subproblem, we eventually find a globally optimal solution to the original problem. Furthermore, to calculate not only the value but also the cuts or decisions made for each subproblem, we store the cut positions in another array. This array is used to print the cuts made, thereby fulfilling the fourth step in efficiently designing an algorithm using dynamic programming.