#3560

Easy

auto_awesomeMath-driven solution strategy

LeetCode Problem Workspace

Find Minimum Log Transportation Cost

Calculate the minimum cost to transport two logs using a math-driven strategy considering cutting costs and truck limits.

math

Problem Statement

You are given two logs with lengths n and m units and three trucks, each capable of carrying a log of length at most k. You may cut the logs into smaller pieces to fit them into trucks, with a cutting cost defined as the product of the lengths of the two resulting pieces.

Determine the minimum total cost required to cut and transport both logs such that every piece fits into a truck. If the logs already fit within the trucks without cutting, the cost is zero.

Examples

Example 1

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

Output: 5

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

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.

Solution Approach

Check if Cutting is Necessary

If both logs are shorter than or equal to k, no cutting is needed and the minimum cost is zero. This fast check avoids unnecessary calculations for logs already fitting the truck limits.

Evaluate Possible Cuts

For logs exceeding k, consider cutting points that generate pieces under or equal to k. Calculate the cost for each valid split using cost = len1 * len2. Select the cut that yields the minimal cost while ensuring all pieces can fit in trucks.

Compute Total Cost

Sum the minimal cutting costs for all logs that require splitting. Confirm that the resulting pieces fit within the trucks. The final sum is the minimum transportation cost.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time and space complexity depend on the number of valid cut points considered per log. Evaluating all splits is linear in the log length, while the total cost calculation adds constant overhead.

What Interviewers Usually Probe

Look for a direct check if logs fit without cutting.
Evaluate the efficiency of different cut strategies for minimal cost.
Expect reasoning about multiplication cost trade-offs between different cut points.

Common Pitfalls or Variants

Common pitfalls

Assuming each log must be cut even if both are within truck capacity.
Choosing cut points without checking if resulting pieces exceed k.
Forgetting that the cost is the product of split lengths, not the sum.

Follow-up variants

More than two logs with different truck capacities.
Variable cutting costs such as sum or squared length instead of product.
Dynamic truck assignment where each truck can carry multiple small pieces.

FAQ

What is the pattern used to solve Find Minimum Log Transportation Cost?

The solution relies on a math-driven strategy, checking log lengths and computing cutting costs efficiently.

When is the cutting cost zero?

If both logs are shorter than or equal to the truck limit k, no cutting is required, resulting in zero cost.

How do you choose where to cut a log?

Select a cut that produces pieces under k and minimizes the product of the two lengths, which is the cutting cost.

Can the logs be split multiple times?

Yes, multiple splits are allowed as long as all resulting pieces fit within the trucks, summing all cutting costs.

Does the order of logs matter for transportation?

No, as long as all pieces fit individually into the trucks, the order does not affect the minimum cost calculation.

terminal

Solution

Solution 1: Mathematics

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

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class Solution:
    def minCuttingCost(self, n: int, m: int, k: int) -> int:
        x = max(n, m)
        return 0 if x <= k else k * (x - k)

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