LeetCode #939 — MEDIUM

Minimum Area Rectangle

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

The Problem

Problem Statement

You are given an array of points in the X-Y plane points where points[i] = [x_i, y_i].

Return the minimum area of a rectangle formed from these points, with sides parallel to the X and Y axes. If there is not any such rectangle, return 0.

Example 1:

Input: points = [[1,1],[1,3],[3,1],[3,3],[2,2]]
Output: 4

Example 2:

Input: points = [[1,1],[1,3],[3,1],[3,3],[4,1],[4,3]]
Output: 2

Constraints:

1 <= points.length <= 500
points[i].length == 2
0 <= x_i, y_i <= 4 * 10⁴
All the given points are unique.

Step 01

Brute Force Baseline

Problem summary: You are given an array of points in the X-Y plane points where points[i] = [xi, yi]. Return the minimum area of a rectangle formed from these points, with sides parallel to the X and Y axes. If there is not any such rectangle, return 0.

Baseline thinking

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

Pattern signal: Array · Hash Map · Math

Example 1

[[1,1],[1,3],[3,1],[3,3],[2,2]]

Example 2

[[1,1],[1,3],[3,1],[3,3],[4,1],[4,3]]

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

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 #939: Minimum Area Rectangle
class Solution {
    public int minAreaRect(int[][] points) {
        TreeMap<Integer, List<Integer>> d = new TreeMap<>();
        for (var p : points) {
            int x = p[0], y = p[1];
            d.computeIfAbsent(x, k -> new ArrayList<>()).add(y);
        }
        Map<Integer, Integer> pos = new HashMap<>();
        int ans = 1 << 30;
        for (var e : d.entrySet()) {
            int x = e.getKey();
            var ys = e.getValue();
            Collections.sort(ys);
            int n = ys.size();
            for (int i = 0; i < n; ++i) {
                int y1 = ys.get(i);
                for (int j = i + 1; j < n; ++j) {
                    int y2 = ys.get(j);
                    int p = y1 * 40001 + y2;
                    if (pos.containsKey(p)) {
                        ans = Math.min(ans, (x - pos.get(p)) * (y2 - y1));
                    }
                    pos.put(p, x);
                }
            }
        }
        return ans == 1 << 30 ? 0 : ans;
    }
}

// Accepted solution for LeetCode #939: Minimum Area Rectangle
func minAreaRect(points [][]int) int {
	d := map[int][]int{}
	xs := []int{}
	for _, p := range points {
		x, y := p[0], p[1]
		d[x] = append(d[x], y)
	}
	for x := range d {
		xs = append(xs, x)
	}
	sort.Ints(xs)
	type pair struct{ x, y int }
	pos := map[pair]int{}
	ans := 1 << 30
	for _, x := range xs {
		ys := d[x]
		sort.Ints(ys)
		for i, y1 := range ys {
			for _, y2 := range ys[i+1:] {
				p := pair{y1, y2}
				if x1, ok := pos[p]; ok {
					ans = min(ans, (x-x1)*(y2-y1))
				}
				pos[p] = x
			}
		}
	}
	if ans == 1<<30 {
		return 0
	}
	return ans
}

# Accepted solution for LeetCode #939: Minimum Area Rectangle
class Solution:
    def minAreaRect(self, points: List[List[int]]) -> int:
        d = defaultdict(list)
        for x, y in points:
            d[x].append(y)
        pos = {}
        ans = inf
        for x in sorted(d):
            ys = d[x]
            ys.sort()
            n = len(ys)
            for i, y1 in enumerate(ys):
                for y2 in ys[i + 1 :]:
                    if (y1, y2) in pos:
                        ans = min(ans, (x - pos[(y1, y2)]) * (y2 - y1))
                    pos[(y1, y2)] = x
        return 0 if ans == inf else ans

// Accepted solution for LeetCode #939: Minimum Area Rectangle
struct Solution;
use std::collections::HashSet;
use std::i32;

impl Solution {
    fn min_area_rect(points: Vec<Vec<i32>>) -> i32 {
        let n = points.len();
        let mut hs: HashSet<(i32, i32)> = HashSet::new();
        for i in 0..n {
            hs.insert((points[i][0], points[i][1]));
        }
        let mut min = i32::MAX;
        for i in 0..n - 1 {
            for j in i + 1..n {
                let x1 = points[i][0];
                let y1 = points[i][1];
                let x2 = points[j][0];
                let y2 = points[j][1];
                if x2 != x1 && y2 != y1 && hs.contains(&(x1, y2)) && hs.contains(&(x2, y1)) {
                    min = min.min((x2 - x1).abs() * (y2 - y1).abs())
                }
            }
        }
        if min == i32::MAX {
            0
        } else {
            min
        }
    }
}

#[test]
fn test() {
    let points = vec_vec_i32![[1, 1], [1, 3], [3, 1], [3, 3], [2, 2]];
    let res = 4;
    assert_eq!(Solution::min_area_rect(points), res);
    let points = vec_vec_i32![[1, 1], [1, 3], [3, 1], [3, 3], [4, 1], [4, 3]];
    let res = 2;
    assert_eq!(Solution::min_area_rect(points), res);
}

// Accepted solution for LeetCode #939: Minimum Area Rectangle
// Auto-generated TypeScript example from java.
function exampleSolution(): void {
}
// Reference (java):
// // Accepted solution for LeetCode #939: Minimum Area Rectangle
// class Solution {
//     public int minAreaRect(int[][] points) {
//         TreeMap<Integer, List<Integer>> d = new TreeMap<>();
//         for (var p : points) {
//             int x = p[0], y = p[1];
//             d.computeIfAbsent(x, k -> new ArrayList<>()).add(y);
//         }
//         Map<Integer, Integer> pos = new HashMap<>();
//         int ans = 1 << 30;
//         for (var e : d.entrySet()) {
//             int x = e.getKey();
//             var ys = e.getValue();
//             Collections.sort(ys);
//             int n = ys.size();
//             for (int i = 0; i < n; ++i) {
//                 int y1 = ys.get(i);
//                 for (int j = i + 1; j < n; ++j) {
//                     int y2 = ys.get(j);
//                     int p = y1 * 40001 + y2;
//                     if (pos.containsKey(p)) {
//                         ans = Math.min(ans, (x - pos.get(p)) * (y2 - y1));
//                     }
//                     pos.put(p, x);
//                 }
//             }
//         }
//         return ans == 1 << 30 ? 0 : 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(n)

Space

O(1)

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.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.