LeetCode #554 — MEDIUM

Brick Wall

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

The Problem

Problem Statement

There is a rectangular brick wall in front of you with n rows of bricks. The i^th row has some number of bricks each of the same height (i.e., one unit) but they can be of different widths. The total width of each row is the same.

Draw a vertical line from the top to the bottom and cross the least bricks. If your line goes through the edge of a brick, then the brick is not considered as crossed. You cannot draw a line just along one of the two vertical edges of the wall, in which case the line will obviously cross no bricks.

Given the 2D array wall that contains the information about the wall, return the minimum number of crossed bricks after drawing such a vertical line.

Example 1:

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

Example 2:

Input: wall = [[1],[1],[1]]
Output: 3

Constraints:

n == wall.length
1 <= n <= 10⁴
1 <= wall[i].length <= 10⁴
1 <= sum(wall[i].length) <= 2 * 10⁴
sum(wall[i]) is the same for each row i.
1 <= wall[i][j] <= 2³¹ - 1

Step 01

Brute Force Baseline

Problem summary: There is a rectangular brick wall in front of you with n rows of bricks. The ith row has some number of bricks each of the same height (i.e., one unit) but they can be of different widths. The total width of each row is the same. Draw a vertical line from the top to the bottom and cross the least bricks. If your line goes through the edge of a brick, then the brick is not considered as crossed. You cannot draw a line just along one of the two vertical edges of the wall, in which case the line will obviously cross no bricks. Given the 2D array wall that contains the information about the wall, return the minimum number of crossed bricks after drawing such a vertical line.

Baseline thinking

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

Pattern signal: Array · Hash Map

Example 1

[[1,2,2,1],[3,1,2],[1,3,2],[2,4],[3,1,2],[1,3,1,1]]

Example 2

[[1],[1],[1]]

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 #554: Brick Wall
class Solution {
    public int leastBricks(List<List<Integer>> wall) {
        Map<Integer, Integer> cnt = new HashMap<>();
        for (var row : wall) {
            int s = 0;
            for (int i = 0; i + 1 < row.size(); ++i) {
                s += row.get(i);
                cnt.merge(s, 1, Integer::sum);
            }
        }
        int mx = 0;
        for (var x : cnt.values()) {
            mx = Math.max(mx, x);
        }
        return wall.size() - mx;
    }
}

// Accepted solution for LeetCode #554: Brick Wall
func leastBricks(wall [][]int) int {
	cnt := map[int]int{}
	for _, row := range wall {
		s := 0
		for _, x := range row[:len(row)-1] {
			s += x
			cnt[s]++
		}
	}
	mx := 0
	for _, x := range cnt {
		mx = max(mx, x)
	}
	return len(wall) - mx
}

# Accepted solution for LeetCode #554: Brick Wall
class Solution:
    def leastBricks(self, wall: List[List[int]]) -> int:
        cnt = Counter()
        for row in wall:
            s = 0
            for x in row[:-1]:
                s += x
                cnt[s] += 1
        return len(wall) - max(cnt.values(), default=0)

// Accepted solution for LeetCode #554: Brick Wall
use std::collections::HashMap;
impl Solution {
    pub fn least_bricks(wall: Vec<Vec<i32>>) -> i32 {
        wall.len() as i32
            - wall
                .into_iter()
                .map(|row| {
                    let mut prefix: Vec<_> = row
                        .into_iter()
                        .scan(0, |sum, x| {
                            *sum += x;
                            Some(*sum)
                        })
                        .collect();
                    prefix.pop();
                    prefix
                })
                .flatten()
                .fold(HashMap::from([(0, 0)]), |mut acc, x| {
                    acc.entry(x).and_modify(|cnt| *cnt += 1).or_insert(1);
                    acc
                })
                .into_iter()
                .map(|(_, v)| v)
                .max()
                .unwrap()
    }
}

// Accepted solution for LeetCode #554: Brick Wall
function leastBricks(wall: number[][]): number {
    const cnt: Map<number, number> = new Map();
    for (const row of wall) {
        let s = 0;
        for (let i = 0; i + 1 < row.length; ++i) {
            s += row[i];
            cnt.set(s, (cnt.get(s) || 0) + 1);
        }
    }
    const mx = Math.max(...cnt.values(), 0);
    return wall.length - mx;
}

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 × 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.

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.