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This method leverages dynamic programming to find the minimum cost by solving subproblems. We start by sorting the cuts to facilitate an easier partitioning strategy. Then, define a dp table where dp[i][j] represents the minimum cost to perform all the cuts between positions i and j.
Use the recursive relation to compute the minimum cost for each partition (i, j) by considering every possible cut position in that interval. The base case is that if there are no cuts between two positions, the cost is zero.
Time Complexity: O(m^3) where m is the number of cuts, due to the nested loops solving subproblems.
Space Complexity: O(m^2) for the dp table storage, where m is the number of cuts.
1class Solution:
2 def minCost(self, n: int, cuts: List[int]) -> int:
3 cuts = sorted(cuts)
4 cuts = [0] + cuts + [n]
5 m = len(cuts)
6 dp = [[0] * m for _ in range(m)]
7 for length in range(2, m):
8 for i in range(m - length):
9 j = i + length
10 dp[i][j] = min(dp[i][k] + dp[k][j] for k in range(i + 1, j)) + cuts[j] - cuts[i]
11 return dp[0][m - 1]
12
13# Example usage
14sol = Solution()
15cuts = [1, 3, 4, 5]
16n = 7
17print("Minimum cost:", sol.minCost(n, cuts))In Python, a list 'cuts' is sorted, then prepared with boundaries to assist in calculating distinct cost measures over subproblems. It uses comprehension and a principal loop structure to fill 'dp' with the minimum obtainable costs for the cut sequences.
This approach uses recursive function calls along with a memoization hash table to store the results of subproblem calculations. This reduces repeated calculations and enhances efficiency. Order the cuts, then define a recursive function that selects the appropriate cut division and employs the same function to solve newly formed partitions recursively.
Time Complexity: O(m^2) with memoization staving off excessive recomputation.
Space Complexity: O(m^2) result-caching leading to rational storage expansion.
1from functools import lru_cache
2
3class
This Python implementation exploits the 'lru_cache' decorator to memoize results of recursive 'cost' function calls, trim down redundant calculations, and obtain a minimum cost sequence quickly.