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Maximize Profit From Cutting Rods

AndurilSoftware Engineer · Technical Screen
MediumSoftware EngineerTechnical ScreenCoding & Algorithms

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This problem tests dynamic programming and optimization skills by requiring the calculation of the maximum profit obtainable from cutting a rod into integer-length pieces with given price values.

Problem Statement

You are given a rod of length n meters and an array prices where prices[i] is the price of a rod of length i + 1. Your goal is to determine the maximum total profit you can earn by cutting the rod into smaller pieces of integer lengths and selling them.

Function Signature

def max_rod_profit(n: int, prices: list[int]) -> int:
    pass

Constraints

  • 1 <= n <= 1000
  • len(prices) == n
  • 1 <= prices[i] <= 10^5

Examples

Example 1: Input: n = 4, prices = [1, 5, 8, 9] Output: 10 Explanation: Cut the rod into two pieces of length 2 (5 + 5 = 10).

Example 2: Input: n = 8, prices = [3, 5, 8, 9, 10, 17, 17, 20] Output: 24 Explanation: Cut the rod into lengths 2 and 6 (5 + 17 = 22) or length 2, 2, and 4 (5 + 5 + 9 = 19). The optimal cut is length 2 and 6, but check all combinations: 17 + 7 is not possible, 17 + 5 is 22. Wait, the best is length 2 (5) + length 6 (17) = 22. Actually, looking at the array, length 2 (5) + length 6 (17) = 22 is less than length 8 (20). Max is 24 by cutting 1, 1, 6 (3+3+17).

Edge Cases

  • n = 1: The only option is the first element of prices.
  • Uniform prices: Where it is always better to sell one large rod versus many small ones.

Hints

Hint 1

Define a state `dp[i]` as the maximum profit obtainable for a rod of length `i`.

Hint 2

For each rod length `i`, consider every possible first cut length `j` from 1 to `i`, and observe the recurrence `dp[i] = max(prices[j-1] + dp[i-j])`.

Hint 3

Use a bottom-up approach with an array of size `n + 1` to store the computed maximums, initializing `dp[0] = 0`.

Follow-up questions

  • How would you modify your solution to track the actual lengths of the cuts used to achieve the maximum profit?
  • What is the time and space complexity of your solution, and can you optimize the space complexity further?
  • If you are restricted to at most `k` cuts, how does the state transition change?