1547 Minimum Cost to Cut a Stick

🔀 Pick One

Given a wooden stick of length n units. The stick is labelled from 0 to n. For example, a stick of length 6 is labelled as follows:

 ___________________
 |  |  |  |  |  |  |
 0  1  2  3  4  5  6

Given an integer array cuts where cuts[i] denotes a position you should perform a cut at.

You should perform the cuts in order, you can change the order of the cuts as you wish.

The cost of one cut is the length of the stick to be cut, the total cost is the sum of costs of all cuts. When you cut a stick, it will be split into two smaller sticks (i.e. the sum of their lengths is the length of the stick before the cut). Please refer to the first example for a better explanation.

Return the minimum total cost of the cuts.

Example 1:

cuts = [3, 5, 1, 4] (Optimal Ordering)
 ______________________
 |  :     :  :  :     |
 0  1  2  3  4  5  6  7
 __________   _____________
 |  :     |   |  :  :     |
 0  1  2  3   3  4  5  6  7
 __________   _______   _______
 |  :     |   |  :  |   |     |
 0  1  2  3   3  4  5   5  6  7
 ____   _______   _______   _______
 |  |   |     |   |  :  |   |     |
 0  1   1  2  3   3  4  5   5  6  7
 ____   _______   ____   ____   _______
 |  |   |     |   |  |   |  |   |     |
 0  1   1  2  3   3  4   4  5   5  6  7

Input: n = 7, cuts = [1,3,4,5]
Output: 16
Explanation: Using cuts order = [1, 3, 4, 5]
as in the input leads to the following scenario:

cuts = [1, 3, 4, 5]
 ______________________
 |  :     :  :  :     |
 0  1  2  3  4  5  6  7
 ____   ___________________
 |  |   |     :  :  :     |
 0  1   1  2  3  4  5  6  7
 ____   _______   _____________
 |  |   |     |   |  :  :     |
 0  1   1  2  3   3  4  5  6  7
 ____   _______   ____   __________
 |  |   |     |   |  |   |  :     |
 0  1   1  2  3   3  4   4  5  6  7
 ____   _______   ____   ____   _______
 |  |   |     |   |  |   |  |   |     |
 0  1   1  2  3   3  4   4  5   5  6  7

The first cut is done to a rod of length 7 so the cost is 7.
The second cut is done to a rod of length 6 (i.e. the second part of the first cut),
the third is done to a rod of length 4 and the last cut is to a rod of length 3.
The total cost is 7 + 6 + 4 + 3 = 20.
Rearranging the cuts to be [3, 5, 1, 4] for example
will lead to a scenario with total cost = 16
(as shown in the example photo 7 + 4 + 3 + 2 = 16).

Example 2:

Input: n = 9, cuts = [5,6,1,4,2]
Output: 22
Explanation: If you try the given cuts ordering the cost will be 25.
There are much ordering with total cost <= 25, for example,
the order [4, 6, 5, 2, 1] has total cost = 22 which is the minimum possible.

Constraints:

  • 2 <= n <= 106
  • 1 <= cuts.length <= min(n - 1, 100)
  • 1 <= cuts[i] <= n - 1
  • All the integers in cuts array are distinct.
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class Solution:
    def minCost(self, n: int, cuts: List[int]) -> int:
        '''
        Transform the problem to the cost to stick all sticks
        dp[i][j] is the min cost to stick all sticks between A[i] and A[j]
        '''
        A = sorted(cuts + [0, n])
        k = len(A)
        dp = [[0] * k for _ in range(k)]
        for i in range(k-1, -1, -1):
            for j in range(i+2, k):
                dp[i][j] = min(
                    dp[i][m] + dp[m][j] for m in range(i+1, j)
                ) + (A[j] - A[i])
        return dp[0][k - 1]