Dynamic Programming Questions
The Rod Cutting problem is a classic optimization problem in which we are given a rod of length n and a price list for different lengths of the rod. The goal is to determine the maximum revenue that can be obtained by cutting the rod into smaller pieces and selling them.
Dynamic Programming can be used to solve the Rod Cutting problem efficiently. The approach involves breaking down the problem into smaller subproblems and solving them in a bottom-up manner.
To solve the problem using Dynamic Programming, we can create an array dp[] of size n+1 to store the maximum revenue for each length of the rod. We initialize dp[0] as 0 since there is no revenue for a rod of length 0.
Then, for each length i from 1 to n, we iterate through all possible cuts j from 1 to i and calculate the maximum revenue by considering two cases:
1. If we make a cut at length j, the revenue will be the price of the rod of length j plus the maximum revenue obtained from the remaining length (i-j). This can be represented as dp[i] = max(dp[i], price[j] + dp[i-j]).
2. If we do not make a cut at length j, the revenue will be the maximum revenue obtained so far for length i. This can be represented as dp[i] = max(dp[i], dp[i-j]).
Finally, the maximum revenue for the rod of length n will be stored in dp[n].
By using this approach, we can solve the Rod Cutting problem in O(n^2) time complexity, where n is the length of the rod.