LeetCode #3218 — MEDIUM

Minimum Cost for Cutting Cake I

Move from brute-force thinking to an efficient approach using array strategy.

Solve on LeetCode

The Problem

Problem Statement

There is an m x n cake that needs to be cut into 1 x 1 pieces.

You are given integers m, n, and two arrays:

horizontalCut of size m - 1, where horizontalCut[i] represents the cost to cut along the horizontal line i.
verticalCut of size n - 1, where verticalCut[j] represents the cost to cut along the vertical line j.

In one operation, you can choose any piece of cake that is not yet a 1 x 1 square and perform one of the following cuts:

Cut along a horizontal line i at a cost of horizontalCut[i].
Cut along a vertical line j at a cost of verticalCut[j].

After the cut, the piece of cake is divided into two distinct pieces.

The cost of a cut depends only on the initial cost of the line and does not change.

Return the minimum total cost to cut the entire cake into 1 x 1 pieces.

Example 1:

Input: m = 3, n = 2, horizontalCut = [1,3], verticalCut = [5]

Output: 13

Explanation:

Perform a cut on the vertical line 0 with cost 5, current total cost is 5.
Perform a cut on the horizontal line 0 on 3 x 1 subgrid with cost 1.
Perform a cut on the horizontal line 0 on 3 x 1 subgrid with cost 1.
Perform a cut on the horizontal line 1 on 2 x 1 subgrid with cost 3.
Perform a cut on the horizontal line 1 on 2 x 1 subgrid with cost 3.

The total cost is 5 + 1 + 1 + 3 + 3 = 13.

Example 2:

Input: m = 2, n = 2, horizontalCut = [7], verticalCut = [4]

Output: 15

Explanation:

Perform a cut on the horizontal line 0 with cost 7.
Perform a cut on the vertical line 0 on 1 x 2 subgrid with cost 4.
Perform a cut on the vertical line 0 on 1 x 2 subgrid with cost 4.

The total cost is 7 + 4 + 4 = 15.

Constraints:

1 <= m, n <= 20
horizontalCut.length == m - 1
verticalCut.length == n - 1
1 <= horizontalCut[i], verticalCut[i] <= 10³

Roadmap

Brute Force Baseline
Core Insight
Algorithm Walkthrough
Edge Cases
Full Annotated Code
Interactive Study Demo
Complexity Analysis

Step 01

Brute Force Baseline

Problem summary: There is an m x n cake that needs to be cut into 1 x 1 pieces. You are given integers m, n, and two arrays: horizontalCut of size m - 1, where horizontalCut[i] represents the cost to cut along the horizontal line i. verticalCut of size n - 1, where verticalCut[j] represents the cost to cut along the vertical line j. In one operation, you can choose any piece of cake that is not yet a 1 x 1 square and perform one of the following cuts: Cut along a horizontal line i at a cost of horizontalCut[i]. Cut along a vertical line j at a cost of verticalCut[j]. After the cut, the piece of cake is divided into two distinct pieces. The cost of a cut depends only on the initial cost of the line and does not change. Return the minimum total cost to cut the entire cake into 1 x 1 pieces.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Two Pointers · Dynamic Programming · Greedy

Example 1

3
2
[1,3]
[5]

Example 2

2
2
[7]
[4]

Related Problems

Minimum Cost for Cutting Cake II (minimum-cost-for-cutting-cake-ii)

Step 02

Core Insight

What unlocks the optimal approach

The intended solution uses Dynamic Programming.
Let <code>dp[sx][sy][tx][ty]</code> denote the minimum cost to cut the rectangle into <code>1 x 1</code> pieces.
Iterate on the row or column on which you will perform the next cut, after the cut, the current rectangle will be decomposed into two sub-rectangles.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #3218: Minimum Cost for Cutting Cake I
class Solution {
    public int minimumCost(int m, int n, int[] horizontalCut, int[] verticalCut) {
        Arrays.sort(horizontalCut);
        Arrays.sort(verticalCut);
        int ans = 0;
        int i = m - 2, j = n - 2;
        int h = 1, v = 1;
        while (i >= 0 || j >= 0) {
            if (j < 0 || (i >= 0 && horizontalCut[i] > verticalCut[j])) {
                ans += horizontalCut[i--] * v;
                ++h;
            } else {
                ans += verticalCut[j--] * h;
                ++v;
            }
        }
        return ans;
    }
}

// Accepted solution for LeetCode #3218: Minimum Cost for Cutting Cake I
func minimumCost(m int, n int, horizontalCut []int, verticalCut []int) (ans int) {
	sort.Sort(sort.Reverse(sort.IntSlice(horizontalCut)))
	sort.Sort(sort.Reverse(sort.IntSlice(verticalCut)))
	i, j := 0, 0
	h, v := 1, 1
	for i < m-1 || j < n-1 {
		if j == n-1 || (i < m-1 && horizontalCut[i] > verticalCut[j]) {
			ans += horizontalCut[i] * v
			h++
			i++
		} else {
			ans += verticalCut[j] * h
			v++
			j++
		}
	}
	return
}

# Accepted solution for LeetCode #3218: Minimum Cost for Cutting Cake I
class Solution:
    def minimumCost(
        self, m: int, n: int, horizontalCut: List[int], verticalCut: List[int]
    ) -> int:
        horizontalCut.sort(reverse=True)
        verticalCut.sort(reverse=True)
        ans = i = j = 0
        h = v = 1
        while i < m - 1 or j < n - 1:
            if j == n - 1 or (i < m - 1 and horizontalCut[i] > verticalCut[j]):
                ans += horizontalCut[i] * v
                h, i = h + 1, i + 1
            else:
                ans += verticalCut[j] * h
                v, j = v + 1, j + 1
        return ans

// Accepted solution for LeetCode #3218: Minimum Cost for Cutting Cake I
impl Solution {
    pub fn minimum_cost(m: i32, n: i32, mut horizontal_cut: Vec<i32>, mut vertical_cut: Vec<i32>) -> i32 {
        horizontal_cut.sort();
        vertical_cut.sort();
        let (mut i, mut j) = ((m - 2) as isize, (n - 2) as isize);
        let (mut h, mut v) = (1, 1);
        let mut ans = 0;

        while i >= 0 || j >= 0 {
            if j < 0 || (i >= 0 && horizontal_cut[i as usize] > vertical_cut[j as usize]) {
                ans += horizontal_cut[i as usize] * v;
                i -= 1;
                h += 1;
            } else {
                ans += vertical_cut[j as usize] * h;
                j -= 1;
                v += 1;
            }
        }

        ans
    }
}

// Accepted solution for LeetCode #3218: Minimum Cost for Cutting Cake I
function minimumCost(m: number, n: number, horizontalCut: number[], verticalCut: number[]): number {
    horizontalCut.sort((a, b) => b - a);
    verticalCut.sort((a, b) => b - a);
    let ans = 0;
    let [i, j] = [0, 0];
    let [h, v] = [1, 1];
    while (i < m - 1 || j < n - 1) {
        if (j === n - 1 || (i < m - 1 && horizontalCut[i] > verticalCut[j])) {
            ans += horizontalCut[i] * v;
            h++;
            i++;
        } else {
            ans += verticalCut[j] * h;
            v++;
            j++;
        }
    }
    return ans;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(m × log m + n × log n)

Space

O(log m + log n)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair of elements. The outer loop picks one element, the inner loop scans the rest. For n elements that is n × (n−1)/2 comparisons = O(n²). No extra memory — just two loop variables.

TWO POINTERS

O(n) time

O(1) space

Each pointer traverses the array at most once. With two pointers moving inward (or both moving right), the total number of steps is bounded by n. Each comparison is O(1), giving O(n) overall. No auxiliary data structures are needed — just two index variables.

Shortcut: Two converging pointers on sorted data → O(n) time, O(1) space.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Moving both pointers on every comparison

Wrong move: Advancing both pointers shrinks the search space too aggressively and skips candidates.

Usually fails on: A valid pair can be skipped when only one side should move.

Fix: Move exactly one pointer per decision branch based on invariant.

State misses one required dimension

Wrong move: An incomplete state merges distinct subproblems and caches incorrect answers.

Usually fails on: Correctness breaks on cases that differ only in hidden state.

Fix: Define state so each unique subproblem maps to one DP cell.

Using greedy without proof

Wrong move: Locally optimal choices may fail globally.

Usually fails on: Counterexamples appear on crafted input orderings.

Fix: Verify with exchange argument or monotonic objective before committing.