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Find Minimum Log Transportation Cost - Solution & Explanation

EasyMath6 min read
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Problem Statement

You are given integers n, m, and k.

There are two logs of lengths n and m units, which need to be transported in three trucks where each truck can carry one log with length at most k units.

You may cut the logs into smaller pieces, where the cost of cutting a log of length x into logs of length len1 and len2 is cost = len1 * len2 such that len1 + len2 = x.

Return the minimum total cost to distribute the logs onto the trucks. If the logs don't need to be cut, the total cost is 0.

 

Example 1:

Input: n = 6, m = 5, k = 5

Output: 5

Explanation:

Cut the log with length 6 into logs with length 1 and 5, at a cost equal to 1 * 5 == 5. Now the three logs of length 1, 5, and 5 can fit in one truck each.

Example 2:

Input: n = 4, m = 4, k = 6

Output: 0

Explanation:

The two logs can fit in the trucks already, hence we don't need to cut the logs.

 

Constraints:

  • 2 <= k <= 105
  • 1 <= n, m <= 2 * k
  • The input is generated such that it is always possible to transport the logs.

Approach Overview

Problem Overview: You are given constraints around transporting logs and need to compute the minimum total transportation cost. The key task is determining how to group logs efficiently so the total cost of all transport operations is minimized.

Approach 1: Direct Simulation (O(n) time, O(1) space)

A straightforward idea is to simulate the transportation process step by step. You repeatedly move a batch of logs, update the remaining count, and accumulate the cost for each trip. This works because you explicitly model each transportation operation until all logs are delivered. The drawback is unnecessary iteration when the number of logs is large, since the same calculation can be derived mathematically instead of simulating each step.

Approach 2: Mathematics / Closed-Form Formula (O(1) time, O(1) space)

The optimal solution comes from observing that the number of transport operations follows a predictable pattern. Instead of simulating trips, compute how many batches are required using ceiling division. If each operation can move a fixed number of logs, the number of trips becomes ceil(totalLogs / capacity). The final transportation cost is then calculated directly using this value and the cost rule defined in the problem.

This approach relies purely on arithmetic operations. No iteration over logs is required. The main insight is recognizing that batching determines the number of transport operations, and that batching can be calculated with integer math. Problems like this frequently appear in math-focused interview questions where deriving the correct formula eliminates simulation.

In practice, you compute the number of trips with (n + k - 1) / k style integer math and multiply it by the cost per operation. This technique is common in problems involving batching, grouping, or resource allocation and appears across many greedy and mathematical optimizations.

Recommended for interviews: Start by explaining the simulation idea to demonstrate understanding of the process. Then derive the mathematical shortcut using ceiling division. Interviewers expect the constant-time formula because it shows you can recognize patterns and remove unnecessary iteration.

Solution

If the lengths of both logs do not exceed the truck's maximum load k, then no cutting is needed, and we simply return 0.

Otherwise, it means that only one log has a length greater than k, and we need to cut it into two pieces. Let the longer log have length x, then the cutting cost is k times (x - k).

The time complexity is O(1), and the space complexity is O(1).

Code

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Python

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C++

Go

TypeScript

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Detailed Complexity Analysis

ApproachTimeSpaceWhen to Use
Direct SimulationO(n)O(1)Useful for understanding the transportation process step by step
Mathematical Formula (Ceiling Division)O(1)O(1)Best choice when trips can be computed directly using arithmetic batching

Video Solution

Find Minimum Log Transportation Cost | Weekly Contest 451 | Java | Developer Coder • Developer Coder • 691 views views

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Frequently Asked Questions

Is Find Minimum Log Transportation Cost easy or hard?
Find Minimum Log Transportation Cost is classified as an Easy problem. The main challenge is recognizing that the transportation process can be simplified into a mathematical formula instead of being simulated step by step.
Find Minimum Log Transportation Cost Python/Java solution
The implementation is very small in both Python and Java because the logic reduces to a mathematical formula. Compute the number of required batches using ceiling division and multiply by the transportation cost. The same logic works identically in C++, Go, TypeScript, and other languages.
How to solve Find Minimum Log Transportation Cost in O(1)?
Derive a formula for the number of transport operations instead of simulating them. Use ceiling division to compute how many batches of logs are needed, then apply the cost rule defined in the problem. Since only arithmetic operations are performed, the runtime stays constant.
What is the best approach for Find Minimum Log Transportation Cost?
The best approach uses a mathematical formula instead of simulation. Compute the number of required transport operations using ceiling division and calculate the total cost directly. This reduces the solution to O(1) time and O(1) space with only a few arithmetic operations.
Is Find Minimum Log Transportation Cost asked at Google/Amazon/Meta?
Math-based optimization problems like this frequently appear in coding interviews at large tech companies. While the exact problem may vary, the pattern of replacing simulation with a mathematical formula is commonly tested at companies such as Amazon, Google, and Meta.
What data structure is used in Find Minimum Log Transportation Cost?
No specialized data structure is required. The solution relies purely on mathematical reasoning and integer arithmetic such as ceiling division, making it a typical math-focused interview problem.
What is the time complexity of Find Minimum Log Transportation Cost?
The optimal solution runs in O(1) time because it relies on a constant number of arithmetic calculations. No loops or additional data structures are required, so the space complexity is also O(1).

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