Problem Link - Cut Ribbon
Problem Statement:
Given a ribbon of length N units. It needs to be cut into parts such that the length of each part is even and each part is atmost L units. How many ways are there to cut the ribbon? Two ways are different if and only if
- Number of cuts are different (OR)
- Number of cuts are same and there is some i such that the i-th part from the left is of different lengths in each of the two ways
If there is no way to cut it such that all parts are even, print 0.
Approach:
The key idea of this solution is to use dynamic programming to count the number of ways to cut the ribbon. We maintain an array DP where DP[i] represents the number of ways to cut a ribbon of length i.
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Initialization: Set DP[0] to 1, as there is one way to cut a ribbon of length 0 (doing nothing).
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Dynamic Programming Transition:
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For each length i from 1 to n, initialize DP[i] to 0.
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For possible last cut lengths starting from i-1 down to 1 (stepping by 2 to ensure even lengths), check if the remaining length (i - j + 1) is valid (less than or equal to L). If valid, update DP[i] by adding DP[j-1] while applying MOD to prevent overflow.
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This dynamic programming approach efficiently counts valid configurations by breaking the problem into manageable parts and leveraging previously computed results to avoid redundant calculations. Each length is evaluated based on its possible last cuts, ensuring we only consider valid configurations under the constraints.
Time Complexity:
- O(n^2) in the worst case, as we examine each possible length and check for valid cuts.
Space Complexity:
- O(n) for the DP array.