LeetCode #2943 — MEDIUM

Maximize Area of Square Hole in Grid

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

Solve on LeetCode
The Problem

Problem Statement

You are given the two integers, n and m and two integer arrays, hBars and vBars. The grid has n + 2 horizontal and m + 2 vertical bars, creating 1 x 1 unit cells. The bars are indexed starting from 1.

You can remove some of the bars in hBars from horizontal bars and some of the bars in vBars from vertical bars. Note that other bars are fixed and cannot be removed.

Return an integer denoting the maximum area of a square-shaped hole in the grid, after removing some bars (possibly none).

Example 1:

Input: n = 2, m = 1, hBars = [2,3], vBars = [2]

Output: 4

Explanation:

The left image shows the initial grid formed by the bars. The horizontal bars are [1,2,3,4], and the vertical bars are [1,2,3].

One way to get the maximum square-shaped hole is by removing horizontal bar 2 and vertical bar 2.

Example 2:

Input: n = 1, m = 1, hBars = [2], vBars = [2]

Output: 4

Explanation:

To get the maximum square-shaped hole, we remove horizontal bar 2 and vertical bar 2.

Example 3:

Input: n = 2, m = 3, hBars = [2,3], vBars = [2,4]

Output: 4

Explanation:

One way to get the maximum square-shaped hole is by removing horizontal bar 3, and vertical bar 4.

Constraints:

  • 1 <= n <= 109
  • 1 <= m <= 109
  • 1 <= hBars.length <= 100
  • 2 <= hBars[i] <= n + 1
  • 1 <= vBars.length <= 100
  • 2 <= vBars[i] <= m + 1
  • All values in hBars are distinct.
  • All values in vBars are distinct.

Roadmap

  1. Brute Force Baseline
  2. Core Insight
  3. Algorithm Walkthrough
  4. Edge Cases
  5. Full Annotated Code
  6. Interactive Study Demo
  7. Complexity Analysis
Step 01

Brute Force Baseline

Problem summary: You are given the two integers, n and m and two integer arrays, hBars and vBars. The grid has n + 2 horizontal and m + 2 vertical bars, creating 1 x 1 unit cells. The bars are indexed starting from 1. You can remove some of the bars in hBars from horizontal bars and some of the bars in vBars from vertical bars. Note that other bars are fixed and cannot be removed. Return an integer denoting the maximum area of a square-shaped hole in the grid, after removing some bars (possibly none).

Baseline thinking

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

Pattern signal: Array

Example 1

2
1
[2,3]
[2]

Example 2

1
1
[2]
[2]

Example 3

2
3
[2,3]
[2,4]

Related Problems

  • Maximal Square (maximal-square)
  • Maximum Square Area by Removing Fences From a Field (maximum-square-area-by-removing-fences-from-a-field)
Step 02

Core Insight

What unlocks the optimal approach

  • Sort <code>hBars</code> and <code>vBars</code> and consider them separately.
  • Compute the longest sequence of consecutive integer values in each array, denoted as <code>[hx, hy]</code> and <code>[vx, vy]</code>, respectively.
  • The maximum square length we can get is <code>min(hy - hx + 2, vy - vx + 2)</code>.
  • Square the maximum square length to get the area.
Interview move: turn each hint into an invariant you can check after every iteration/recursion step.
Step 03

Algorithm Walkthrough

Iteration Checklist

  1. Define state (indices, window, stack, map, DP cell, or recursion frame).
  2. Apply one transition step and update the invariant.
  3. Record answer candidate when condition is met.
  4. 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 #2943: Maximize Area of Square Hole in Grid
class Solution {
    public int maximizeSquareHoleArea(int n, int m, int[] hBars, int[] vBars) {
        int x = Math.min(f(hBars), f(vBars));
        return x * x;
    }

    private int f(int[] nums) {
        Arrays.sort(nums);
        int ans = 1, cnt = 1;
        for (int i = 1; i < nums.length; ++i) {
            if (nums[i] == nums[i - 1] + 1) {
                ans = Math.max(ans, ++cnt);
            } else {
                cnt = 1;
            }
        }
        return ans + 1;
    }
}
Step 06

Interactive Study Demo

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

Press Step or Run All to begin.
Step 07

Complexity Analysis

Time
O(n × log n)
Space
O(n)

Approach Breakdown

BRUTE FORCE
O(n²) time
O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED
O(n) time
O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.
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.