Minimum Cost to Cut a Stick

Minimum Cost to Cut a Stick

What is this problem about?

The Minimum Cost to Cut a Stick problem gives you a stick of length n and a list of cuts you need to make. When you make a cut, the cost is equal to the length of the current piece of stick you are cutting. You can make the cuts in any order. The goal is to find the minimum total cost to complete all the cuts.

Why is this asked in interviews?

This is a classic Minimum Cost to Cut a Stick interview question at Google, Meta, and Microsoft. It is a variation of the "Matrix Chain Multiplication" or "Optimal Binary Search Tree" problem. It tests Dynamic Programming on intervals. It evaluates if a candidate can define a subproblem as "cutting a stick segment between two existing cut points."

Algorithmic pattern used

The Interval Dynamic Programming interview pattern is the standard solution.

1. Add the stick boundaries 0 and n to the cuts array and sort it.
2. Let dp[i][j] be the minimum cost to make all cuts between cuts[i] and cuts[j].
3. The cost to make the first cut at index k (where i < k < j) is: (cuts[j] - cuts[i]) + dp[i][k] + dp[k][j].
4. Iterate through all possible k to find the minimum.
5. Use memoization to solve the overlapping subproblems.

Example explanation

Stick 7, cuts [1, 3, 4, 5].

1. Sorted cuts with boundaries: [0, 1, 3, 4, 5, 7].
2. To cut between 0 and 7 at 3: Cost is 7 (length) + dp(0, 3) + dp(3, 7).
3. The DP systematically explores every possible "first cut" for every possible sub-segment.

Common mistakes candidates make

• Greedy approach: Thinking you should always cut in the middle or at the smallest/largest cut. This doesn't work.
• Not sorting the cuts: The interval DP relies on the cut points being in order to define sub-segments.
• Off-by-one errors: Getting the indices of the cuts array mixed up with the actual coordinates on the stick.

Interview preparation tip

Master Interval DP. If you can solve this and "Burst Balloons," you have a solid grasp of how to split a large problem into two smaller independent segments based on a choice made in the middle.