在 N×N 二进制矩阵中查找仅包含零的最大矩形
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【中文标题】在 N×N 二进制矩阵中查找仅包含零的最大矩形【英文标题】：Find largest rectangle containing only zeros in an N×N binary matrix 【发布时间】：2021-11-03 16:25:46 【问题描述】：给定一个 NxN 二进制矩阵（仅包含 0 或 1），我们如何才能找到包含所有 0 的最大矩形？
例子：
I
0 0 0 0 1 0
0 0 1 0 0 1
II->0 0 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 1 <--IV
0 0 1 0 0 0
IV
对于上面的例子，它是一个 6×6 的二进制矩阵。在这种情况下，返回值将是 Cell 1:(2, 1) 和 Cell 2:(4, 4)。生成的子矩阵可以是正方形或矩形。返回值也可以是全 0 的最大子矩阵的大小，在本例中为 3 × 4。
【问题讨论】：
请考虑将接受的答案更改为 J.F. Sebastian 的答案，该答案现在是正确的并且具有最佳复杂性。 请检查非常相似（我会说重复）的问题：***.com/questions/7770945/…，***.com/a/7353193/684229。解决方案是O(n)。
我正在尝试对任何方向的矩形做同样的事情。见问题：***.com/questions/22604043/…
@TMS 实际上恰恰相反。这些问题与这个问题重复。
【参考方案1】：
这是一个基于 @j_random_hacker 在 cmets 中建议的 "Largest Rectangle in a Histogram" problem 的解决方案：
[算法] 通过迭代来工作 从上到下的行，对于每一行 解决this problem，其中 “柱状图”中的“条形图”包括 所有完整的零向上轨迹 从当前行开始（a 如果列的高度为 1，则列的高度为 0 当前行）。
输入矩阵mat 可以是任意可迭代的，例如文件或网络流。一次只需要一行可用。
#!/usr/bin/env python
from collections import namedtuple
from operator import mul
Info = namedtuple('Info', 'start height')
def max_size(mat, value=0):
"""Find height, width of the largest rectangle containing all `value`'s."""
it = iter(mat)
hist = [(el==value) for el in next(it, [])]
max_size = max_rectangle_size(hist)
for row in it:
hist = [(1+h) if el == value else 0 for h, el in zip(hist, row)]
max_size = max(max_size, max_rectangle_size(hist), key=area)
return max_size
def max_rectangle_size(histogram):
"""Find height, width of the largest rectangle that fits entirely under
the histogram.
"""
stack = []
top = lambda: stack[-1]
max_size = (0, 0) # height, width of the largest rectangle
pos = 0 # current position in the histogram
for pos, height in enumerate(histogram):
start = pos # position where rectangle starts
while True:
if not stack or height > top().height:
stack.append(Info(start, height)) # push
elif stack and height < top().height:
max_size = max(max_size, (top().height, (pos - top().start)),
key=area)
start, _ = stack.pop()
continue
break # height == top().height goes here
pos += 1
for start, height in stack:
max_size = max(max_size, (height, (pos - start)), key=area)
return max_size
def area(size):
return reduce(mul, size)
解决方案是O(N)，其中N 是矩阵中元素的数量。它需要O(ncols) 额外的内存，其中ncols 是矩阵中的列数。
带有测试的最新版本位于https://gist.github.com/776423
【讨论】：
不错的尝试，但是失败了max_size([[0,0,0,0,1,1,1], [0,0,0,0,0,0,0], [0,0,0,1,1,1,1], [0,0,1,1,1,1,1]] + [[1,0,1,1,1,1,1]] * 3)，当左上角有一个 3x3 的正方形时返回 (2, 4)。
基本问题是，仅跟踪（几个）相邻点的最大面积矩形并不总是足够的，就像您在这里所做的那样。我知道唯一正确的 O(N) 算法通过从上到下遍历行来工作，对于解决此问题的每一行：***.com/questions/4311694/…，其中“直方图”中的“条形图”由所有不间断的向上轨迹组成从当前行开始的零（如果在当前行中有 1，则列的高度为 0）。
@j_random_hacker：我已经更新了我的答案以使用基于“直方图”的算法。
这看起来很棒，但是，我正在尝试实际找到最大的矩形（如返回坐标）。该算法将可靠地返回该区域，但是一旦我知道，一个人如何发现一个 3 列 x 2 行矩形的位置，其左上角位于 [3, 5]（例如）？跨度>
从哪里获得边界列信息？ （矩形的左列还是右列？）。我们可以从max_rectangle_size 获取宽度和高度，从for row in it: 迭代中获取底部行，但是我找不到边界列信息。【参考方案2】：
请查看Maximize the rectangular area under Histogram，然后继续阅读下面的解决方案。
Traverse the matrix once and store the following;
For x=1 to N and y=1 to N
F[x][y] = 1 + F[x][y-1] if A[x][y] is 0 , else 0
Then for each row for x=N to 1
We have F[x] -> array with heights of the histograms with base at x.
Use O(N) algorithm to find the largest area of rectangle in this histogram = H[x]
From all areas computed, report the largest.
时间复杂度为 O(N*N) = O(N²)（对于 NxN 二进制矩阵）
例子：
Initial array F[x][y] array
0 0 0 0 1 0 1 1 1 1 0 1
0 0 1 0 0 1 2 2 0 2 1 0
0 0 0 0 0 0 3 3 1 3 2 1
1 0 0 0 0 0 0 4 2 4 3 2
0 0 0 0 0 1 1 5 3 5 4 0
0 0 1 0 0 0 2 6 0 6 5 1
For x = N to 1
H[6] = 2 6 0 6 5 1 -> 10 (5*2)
H[5] = 1 5 3 5 4 0 -> 12 (3*4)
H[4] = 0 4 2 4 3 2 -> 10 (2*5)
H[3] = 3 3 1 3 2 1 -> 6 (3*2)
H[2] = 2 2 0 2 1 0 -> 4 (2*2)
H[1] = 1 1 1 1 0 1 -> 4 (1*4)
The largest area is thus H[5] = 12
【讨论】：
很好的例子解释 你确定这是 O(N*N) 吗？整个矩阵有两次传递，但我的印象是这是 O(N)。 很好的解释.. :) 我希望你也能解释“最大化直方图下的矩形区域”.. :D 为了更清楚。解决方案是 O(N*N)，其中 N 是行/列中的项目数，因为问题表明输入的大小为 NxN。如果 N 是输入中的项目总数，那么它是 O(N)【参考方案3】：这是一个Python3的解决方案，除了最大矩形的面积之外，还返回位置：
#!/usr/bin/env python3
import numpy
s = '''0 0 0 0 1 0
0 0 1 0 0 1
0 0 0 0 0 0
1 0 0 0 0 0
0 0 0 0 0 1
0 0 1 0 0 0'''
nrows = 6
ncols = 6
skip = 1
area_max = (0, [])
a = numpy.fromstring(s, dtype=int, sep=' ').reshape(nrows, ncols)
w = numpy.zeros(dtype=int, shape=a.shape)
h = numpy.zeros(dtype=int, shape=a.shape)
for r in range(nrows):
for c in range(ncols):
if a[r][c] == skip:
continue
if r == 0:
h[r][c] = 1
else:
h[r][c] = h[r-1][c]+1
if c == 0:
w[r][c] = 1
else:
w[r][c] = w[r][c-1]+1
minw = w[r][c]
for dh in range(h[r][c]):
minw = min(minw, w[r-dh][c])
area = (dh+1)*minw
if area > area_max[0]:
area_max = (area, [(r-dh, c-minw+1, r, c)])
print('area', area_max[0])
for t in area_max[1]:
print('Cell 1:(, ) and Cell 2:(, )'.format(*t))
输出：
area 12
Cell 1:(2, 1) and Cell 2:(4, 4)
【讨论】：
效果很好！我从中制作了一个 Fortran 版本并编译它以在 Python 中使用，因为像这样在 Python 中遍历一个大数组非常慢。【参考方案4】：这是 J.F. Sebastians 方法翻译成 C#：
private Vector2 MaxRectSize(int[] histogram)
Vector2 maxSize = Vector2.zero;
int maxArea = 0;
Stack<Vector2> stack = new Stack<Vector2>();
int x = 0;
for (x = 0; x < histogram.Length; x++)
int start = x;
int height = histogram[x];
while (true)
if (stack.Count == 0 || height > stack.Peek().y)
stack.Push(new Vector2(start, height));
else if(height < stack.Peek().y)
int tempArea = (int)(stack.Peek().y * (x - stack.Peek().x));
if(tempArea > maxArea)
maxSize = new Vector2(stack.Peek().y, (x - stack.Peek().x));
maxArea = tempArea;
Vector2 popped = stack.Pop();
start = (int)popped.x;
continue;
break;
foreach (Vector2 data in stack)
int tempArea = (int)(data.y * (x - data.x));
if(tempArea > maxArea)
maxSize = new Vector2(data.y, (x - data.x));
maxArea = tempArea;
return maxSize;
public Vector2 GetMaximumFreeSpace()
// STEP 1:
// build a seed histogram using the first row of grid points
// example: [true, true, false, true] = [1,1,0,1]
int[] hist = new int[gridSizeY];
for (int y = 0; y < gridSizeY; y++)
if(!invalidPoints[0, y])
hist[y] = 1;
// STEP 2:
// get a starting max area from the seed histogram we created above.
// using the example from above, this value would be [1, 1], as the only valid area is a single point.
// another example for [0,0,0,1,0,0] would be [1, 3], because the largest area of contiguous free space is 3.
// Note that at this step, the heigh fo the found rectangle will always be 1 because we are operating on
// a single row of data.
Vector2 maxSize = MaxRectSize(hist);
int maxArea = (int)(maxSize.x * maxSize.y);
// STEP 3:
// build histograms for each additional row, re-testing for new possible max rectangluar areas
for (int x = 1; x < gridSizeX; x++)
// build a new histogram for this row. the values of this row are
// 0 if the current grid point is occupied; otherwise, it is 1 + the value
// of the previously found historgram value for the previous position.
// What this does is effectly keep track of the height of continous avilable spaces.
// EXAMPLE:
// Given the following grid data (where 1 means occupied, and 0 means free; for clairty):
// INPUT: OUTPUT:
// 1.) [0,0,1,0] = [1,1,0,1]
// 2.) [0,0,1,0] = [2,2,0,2]
// 3.) [1,1,0,1] = [0,0,1,0]
//
// As such, you'll notice position 1,0 (row 1, column 0) is 2, because this is the height of contiguous
// free space.
for (int y = 0; y < gridSizeY; y++)
if(!invalidPoints[x, y])
hist[y] = 1 + hist[y];
else
hist[y] = 0;
// find the maximum size of the current histogram. If it happens to be larger
// that the currently recorded max size, then it is the new max size.
Vector2 maxSizeTemp = MaxRectSize(hist);
int tempArea = (int)(maxSizeTemp.x * maxSizeTemp.y);
if (tempArea > maxArea)
maxSize = maxSizeTemp;
maxArea = tempArea;
// at this point, we know the max size
return maxSize;
有几点需要注意：
-
此版本适用于 Unity API。您可以通过将 Vector2 的实例替换为 KeyValuePair 来轻松地使其更通用。 Vector2 仅用于存储两个值的便捷方式。
invalidPoints[] 是一个 bool 数组，其中 true 表示网格点“正在使用”，false 表示未使用。
【讨论】：
【参考方案5】：空间复杂度 O(columns) [也可以修改为 O(rows)] 和时间复杂度 O(rows*columns) 的解决方案
public int maximalRectangle(char[][] matrix)
int m = matrix.length;
if (m == 0)
return 0;
int n = matrix[0].length;
int maxArea = 0;
int[] aux = new int[n];
for (int i = 0; i < n; i++)
aux[i] = 0;
for (int i = 0; i < m; i++)
for (int j = 0; j < n; j++)
aux[j] = matrix[i][j] - '0' + aux[j];
maxArea = Math.max(maxArea, maxAreaHist(aux));
return maxArea;
public int maxAreaHist(int[] heights)
int n = heights.length;
Stack<Integer> stack = new Stack<Integer>();
stack.push(0);
int maxRect = heights[0];
int top = 0;
int leftSideArea = 0;
int rightSideArea = heights[0];
for (int i = 1; i < n; i++)
if (stack.isEmpty() || heights[i] >= heights[stack.peek()])
stack.push(i);
else
while (!stack.isEmpty() && heights[stack.peek()] > heights[i])
top = stack.pop();
rightSideArea = heights[top] * (i - top);
leftSideArea = 0;
if (!stack.isEmpty())
leftSideArea = heights[top] * (top - stack.peek() - 1);
else
leftSideArea = heights[top] * top;
maxRect = Math.max(maxRect, leftSideArea + rightSideArea);
stack.push(i);
while (!stack.isEmpty())
top = stack.pop();
rightSideArea = heights[top] * (n - top);
leftSideArea = 0;
if (!stack.isEmpty())
leftSideArea = heights[top] * (top - stack.peek() - 1);
else
leftSideArea = heights[top] * top;
maxRect = Math.max(maxRect, leftSideArea + rightSideArea);
return maxRect;
但是当我在 LeetCode 上尝试这个时，我得到了 Time Limite exceeded excpetion。有没有更简单的解决方案？
【讨论】：
简单易懂..谢谢！【参考方案6】：我提出了一个 O(nxn) 方法。
首先，您可以列出所有最大的空矩形。空意味着它只覆盖 0。最大的空矩形是这样的，它不能在不覆盖（至少）一个 1 的情况下在一个方向上扩展。
可以在www.ulg.ac.be/telecom/rectangles 找到一篇介绍 O(nxn) 算法来创建此类列表的论文以及源代码（未优化）。不需要存储列表，每次算法找到一个矩形时调用一个回调函数就足够了，并且只存储最大的一个（或者如果需要，可以选择另一个标准）。
请注意，存在一个证据（参见论文），即最大空矩形的数量受图像像素数的限制（在本例中为 nxn）。
因此，选择最优矩形可以在O(nxn)内完成，整体方法也是O(nxn)。
在实践中，这种方法速度很快，用于实时视频流分析。
【讨论】：
【参考方案7】：这里是jfs的解决方案的一个版本，它也传递了最大矩形的位置：
from collections import namedtuple
from operator import mul
Info = namedtuple('Info', 'start height')
def max_rect(mat, value=0):
"""returns (height, width, left_column, bottom_row) of the largest rectangle
containing all `value`'s.
Example:
[[0, 0, 0, 0, 0, 0, 0, 0, 3, 2],
[0, 4, 0, 2, 4, 0, 0, 1, 0, 0],
[1, 0, 1, 0, 0, 0, 3, 0, 0, 4],
[0, 0, 0, 0, 4, 2, 0, 0, 0, 0],
[0, 0, 0, 2, 0, 0, 0, 0, 0, 0],
[4, 3, 0, 0, 1, 2, 0, 0, 0, 0],
[3, 0, 0, 0, 2, 0, 0, 0, 0, 4],
[0, 0, 0, 1, 0, 3, 2, 4, 3, 2],
[0, 3, 0, 0, 0, 2, 0, 1, 0, 0]]
gives: (3, 4, 6, 5)
"""
it = iter(mat)
hist = [(el==value) for el in next(it, [])]
max_rect = max_rectangle_size(hist) + (0,)
for irow,row in enumerate(it):
hist = [(1+h) if el == value else 0 for h, el in zip(hist, row)]
max_rect = max(max_rect, max_rectangle_size(hist) + (irow+1,), key=area)
# irow+1, because we already used one row for initializing max_rect
return max_rect
def max_rectangle_size(histogram):
stack = []
top = lambda: stack[-1]
max_size = (0, 0, 0) # height, width and start position of the largest rectangle
pos = 0 # current position in the histogram
for pos, height in enumerate(histogram):
start = pos # position where rectangle starts
while True:
if not stack or height > top().height:
stack.append(Info(start, height)) # push
elif stack and height < top().height:
max_size = max(max_size, (top().height, (pos - top().start), top().start), key=area)
start, _ = stack.pop()
continue
break # height == top().height goes here
pos += 1
for start, height in stack:
max_size = max(max_size, (height, (pos - start), start), key=area)
return max_size
def area(size):
return size[0] * size[1]
【讨论】：
【参考方案8】：为了完整起见，这里是输出矩形坐标的 C# 版本。 它基于 dmarra 的回答，但没有任何其他依赖项。 只有 bool GetPixel(int x, int y) 函数，当像素设置在坐标 x,y 时返回 true。
public struct INTRECT
public int Left, Right, Top, Bottom;
public INTRECT(int aLeft, int aTop, int aRight, int aBottom)
Left = aLeft;
Top = aTop;
Right = aRight;
Bottom = aBottom;
public int Width get return (Right - Left + 1);
public int Height get return (Bottom - Top + 1);
public bool IsEmpty get return Left == 0 && Right == 0 && Top == 0 && Bottom == 0;
public static bool operator ==(INTRECT lhs, INTRECT rhs)
return lhs.Left == rhs.Left && lhs.Top == rhs.Top && lhs.Right == rhs.Right && lhs.Bottom == rhs.Bottom;
public static bool operator !=(INTRECT lhs, INTRECT rhs)
return !(lhs == rhs);
public override bool Equals(Object obj)
return obj is INTRECT && this == (INTRECT)obj;
public bool Equals(INTRECT obj)
return this == obj;
public override int GetHashCode()
return Left.GetHashCode() ^ Right.GetHashCode() ^ Top.GetHashCode() ^ Bottom.GetHashCode();
public INTRECT GetMaximumFreeRectangle()
int XEnd = 0;
int YStart = 0;
int MaxRectTop = 0;
INTRECT MaxRect = new INTRECT();
// STEP 1:
// build a seed histogram using the first row of grid points
// example: [true, true, false, true] = [1,1,0,1]
int[] hist = new int[Height];
for (int y = 0; y < Height; y++)
if (!GetPixel(0, y))
hist[y] = 1;
// STEP 2:
// get a starting max area from the seed histogram we created above.
// using the example from above, this value would be [1, 1], as the only valid area is a single point.
// another example for [0,0,0,1,0,0] would be [1, 3], because the largest area of contiguous free space is 3.
// Note that at this step, the heigh fo the found rectangle will always be 1 because we are operating on
// a single row of data.
Tuple<int, int> maxSize = MaxRectSize(hist, out YStart);
int maxArea = (int)(maxSize.Item1 * maxSize.Item2);
MaxRectTop = YStart;
// STEP 3:
// build histograms for each additional row, re-testing for new possible max rectangluar areas
for (int x = 1; x < Width; x++)
// build a new histogram for this row. the values of this row are
// 0 if the current grid point is occupied; otherwise, it is 1 + the value
// of the previously found historgram value for the previous position.
// What this does is effectly keep track of the height of continous avilable spaces.
// EXAMPLE:
// Given the following grid data (where 1 means occupied, and 0 means free; for clairty):
// INPUT: OUTPUT:
// 1.) [0,0,1,0] = [1,1,0,1]
// 2.) [0,0,1,0] = [2,2,0,2]
// 3.) [1,1,0,1] = [0,0,1,0]
//
// As such, you'll notice position 1,0 (row 1, column 0) is 2, because this is the height of contiguous
// free space.
for (int y = 0; y < Height; y++)
if (!GetPixel(x, y))
hist[y]++;
else
hist[y] = 0;
// find the maximum size of the current histogram. If it happens to be larger
// that the currently recorded max size, then it is the new max size.
Tuple<int, int> maxSizeTemp = MaxRectSize(hist, out YStart);
int tempArea = (int)(maxSizeTemp.Item1 * maxSizeTemp.Item2);
if (tempArea > maxArea)
maxSize = maxSizeTemp;
maxArea = tempArea;
MaxRectTop = YStart;
XEnd = x;
MaxRect.Left = XEnd - maxSize.Item1 + 1;
MaxRect.Top = MaxRectTop;
MaxRect.Right = XEnd;
MaxRect.Bottom = MaxRectTop + maxSize.Item2 - 1;
// at this point, we know the max size
return MaxRect;
private Tuple<int, int> MaxRectSize(int[] histogram, out int YStart)
Tuple<int, int> maxSize = new Tuple<int, int>(0, 0);
int maxArea = 0;
Stack<Tuple<int, int>> stack = new Stack<Tuple<int, int>>();
int x = 0;
YStart = 0;
for (x = 0; x < histogram.Length; x++)
int start = x;
int height = histogram[x];
while (true)
if (stack.Count == 0 || height > stack.Peek().Item2)
stack.Push(new Tuple<int, int>(start, height));
else if (height < stack.Peek().Item2)
int tempArea = (int)(stack.Peek().Item2 * (x - stack.Peek().Item1));
if (tempArea > maxArea)
YStart = stack.Peek().Item1;
maxSize = new Tuple<int, int>(stack.Peek().Item2, (x - stack.Peek().Item1));
maxArea = tempArea;
Tuple<int, int> popped = stack.Pop();
start = (int)popped.Item1;
continue;
break;
foreach (Tuple<int, int> data in stack)
int tempArea = (int)(data.Item2 * (x - data.Item1));
if (tempArea > maxArea)
YStart = data.Item1;
maxSize = new Tuple<int, int>(data.Item2, (x - data.Item1));
maxArea = tempArea;
return maxSize;
【讨论】：
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