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85. 最大矩形

题目描述

给定一个仅包含 0 和 1 、大小为 rows x cols 的二维二进制矩阵，找出只包含 1 的最大矩形，并返回其面积。

 

示例 1：

输入：matrix = [["1","0","1","0","0"],["1","0","1","1","1"],["1","1","1","1","1"],["1","0","0","1","0"]]
输出：6
解释：最大矩形如上图所示。

示例 2：

输入：matrix = [["0"]]
输出：0

示例 3：

输入：matrix = [["1"]]
输出：1

 

提示：

  • rows == matrix.length
  • cols == matrix[0].length
  • 1 <= rows, cols <= 200
  • matrix[i][j] 为 '0' 或 '1'

解法

方法一：单调栈

我们把每一行视为柱状图的底部，对每一行求柱状图的最大面积即可。

时间复杂度 \(O(m \times n)\)，其中 \(m\) 表示 \(matrix\) 的行数，\(n\) 表示 \(matrix\) 的列数。

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class Solution:
    def maximalRectangle(self, matrix: List[List[str]]) -> int:
        heights = [0] * len(matrix[0])
        ans = 0
        for row in matrix:
            for j, v in enumerate(row):
                if v == "1":
                    heights[j] += 1
                else:
                    heights[j] = 0
            ans = max(ans, self.largestRectangleArea(heights))
        return ans

    def largestRectangleArea(self, heights: List[int]) -> int:
        n = len(heights)
        stk = []
        left = [-1] * n
        right = [n] * n
        for i, h in enumerate(heights):
            while stk and heights[stk[-1]] >= h:
                stk.pop()
            if stk:
                left[i] = stk[-1]
            stk.append(i)
        stk = []
        for i in range(n - 1, -1, -1):
            h = heights[i]
            while stk and heights[stk[-1]] >= h:
                stk.pop()
            if stk:
                right[i] = stk[-1]
            stk.append(i)
        return max(h * (right[i] - left[i] - 1) for i, h in enumerate(heights))

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class Solution {
    public int maximalRectangle(char[][] matrix) {
        int n = matrix[0].length;
        int[] heights = new int[n];
        int ans = 0;
        for (var row : matrix) {
            for (int j = 0; j < n; ++j) {
                if (row[j] == '1') {
                    heights[j] += 1;
                } else {
                    heights[j] = 0;
                }
            }
            ans = Math.max(ans, largestRectangleArea(heights));
        }
        return ans;
    }

    private int largestRectangleArea(int[] heights) {
        int res = 0, n = heights.length;
        Deque<Integer> stk = new ArrayDeque<>();
        int[] left = new int[n];
        int[] right = new int[n];
        Arrays.fill(right, n);
        for (int i = 0; i < n; ++i) {
            while (!stk.isEmpty() && heights[stk.peek()] >= heights[i]) {
                right[stk.pop()] = i;
            }
            left[i] = stk.isEmpty() ? -1 : stk.peek();
            stk.push(i);
        }
        for (int i = 0; i < n; ++i) {
            res = Math.max(res, heights[i] * (right[i] - left[i] - 1));
        }
        return res;
    }
}

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class Solution {
public:
    int maximalRectangle(vector<vector<char>>& matrix) {
        int n = matrix[0].size();
        vector<int> heights(n);
        int ans = 0;
        for (auto& row : matrix) {
            for (int j = 0; j < n; ++j) {
                if (row[j] == '1')
                    ++heights[j];
                else
                    heights[j] = 0;
            }
            ans = max(ans, largestRectangleArea(heights));
        }
        return ans;
    }

    int largestRectangleArea(vector<int>& heights) {
        int res = 0, n = heights.size();
        stack<int> stk;
        vector<int> left(n, -1);
        vector<int> right(n, n);
        for (int i = 0; i < n; ++i) {
            while (!stk.empty() && heights[stk.top()] >= heights[i]) {
                right[stk.top()] = i;
                stk.pop();
            }
            if (!stk.empty()) left[i] = stk.top();
            stk.push(i);
        }
        for (int i = 0; i < n; ++i)
            res = max(res, heights[i] * (right[i] - left[i] - 1));
        return res;
    }
};

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func maximalRectangle(matrix [][]byte) int {
    n := len(matrix[0])
    heights := make([]int, n)
    ans := 0
    for _, row := range matrix {
        for j, v := range row {
            if v == '1' {
                heights[j]++
            } else {
                heights[j] = 0
            }
        }
        ans = max(ans, largestRectangleArea(heights))
    }
    return ans
}

func largestRectangleArea(heights []int) int {
    res, n := 0, len(heights)
    var stk []int
    left, right := make([]int, n), make([]int, n)
    for i := range right {
        right[i] = n
    }
    for i, h := range heights {
        for len(stk) > 0 && heights[stk[len(stk)-1]] >= h {
            right[stk[len(stk)-1]] = i
            stk = stk[:len(stk)-1]
        }
        if len(stk) > 0 {
            left[i] = stk[len(stk)-1]
        } else {
            left[i] = -1
        }
        stk = append(stk, i)
    }
    for i, h := range heights {
        res = max(res, h*(right[i]-left[i]-1))
    }
    return res
}

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impl Solution {
    #[allow(dead_code)]
    pub fn maximal_rectangle(matrix: Vec<Vec<char>>) -> i32 {
        let n = matrix[0].len();
        let mut heights = vec![0; n];
        let mut ret = -1;

        for row in &matrix {
            Self::array_builder(row, &mut heights);
            ret = std::cmp::max(ret, Self::largest_rectangle_area(heights.clone()));
        }

        ret
    }

    /// Helper function, build the heights array according to the input
    #[allow(dead_code)]
    fn array_builder(input: &Vec<char>, heights: &mut Vec<i32>) {
        for (i, &c) in input.iter().enumerate() {
            heights[i] += match c {
                '1' => 1,
                '0' => {
                    heights[i] = 0;
                    0
                }
                _ => panic!("This is impossible"),
            };
        }
    }

    /// Helper function, see: https://leetcode.com/problems/largest-rectangle-in-histogram/ for details
    #[allow(dead_code)]
    fn largest_rectangle_area(heights: Vec<i32>) -> i32 {
        let n = heights.len();
        let mut left = vec![-1; n];
        let mut right = vec![-1; n];
        let mut stack: Vec<(usize, i32)> = Vec::new();
        let mut ret = -1;

        // Build left vector
        for (i, h) in heights.iter().enumerate() {
            while !stack.is_empty() && stack.last().unwrap().1 >= *h {
                stack.pop();
            }
            if stack.is_empty() {
                left[i] = -1;
            } else {
                left[i] = stack.last().unwrap().0 as i32;
            }
            stack.push((i, *h));
        }

        stack.clear();

        // Build right vector
        for (i, h) in heights.iter().enumerate().rev() {
            while !stack.is_empty() && stack.last().unwrap().1 >= *h {
                stack.pop();
            }
            if stack.is_empty() {
                right[i] = n as i32;
            } else {
                right[i] = stack.last().unwrap().0 as i32;
            }
            stack.push((i, *h));
        }

        // Calculate the max area
        for (i, h) in heights.iter().enumerate() {
            ret = std::cmp::max(ret, (right[i] - left[i] - 1) * *h);
        }

        ret
    }
}

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using System;
using System.Collections.Generic;
using System.Linq;

public class Solution {
    private int MaximalRectangleHistagram(int[] height) {
        var stack = new Stack<int>();
        var result = 0;
        var i = 0;
        while (i < height.Length || stack.Any())
        {
            if (!stack.Any() || (i < height.Length && height[stack.Peek()] < height[i]))
            {
                stack.Push(i);
                ++i;
            }
            else
            {
                var previousIndex = stack.Pop();
                var area = height[previousIndex] * (stack.Any() ? (i - stack.Peek() - 1) : i);
                result = Math.Max(result, area);
            }
        }

        return result;
    }

    public int MaximalRectangle(char[][] matrix) {
        var lenI = matrix.Length;
        var lenJ = lenI == 0 ? 0 : matrix[0].Length;
        var height = new int[lenJ];
        var result = 0;
        for (var i = 0; i < lenI; ++i)
        {
            for (var j = 0; j < lenJ; ++j)
            {
                if (matrix[i][j] == '1')
                {
                    ++height[j];
                }
                else
                {
                    height[j] = 0;
                }
            }
            result = Math.Max(result, MaximalRectangleHistagram(height));
        }
        return result;
    }
}

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