我有一个N * N矩阵（N = 2到10000）的数字，范围从0到1000。 如何找到由相同数字组成的最大（矩形）子矩阵？
示例：
1 2 3 4 5
-- -- -- -- --
1 | 10 9 9 9 80
2 | 5 9 9 9 10
3 | 85 86 54 45 45
4 | 15 21 5 1 0
5 | 5 6 88 11 10
输出应该是子矩阵的区域，后面是其左上角元素的基于1的坐标。例如，它是(6, 2, 1)，因为有6个9位于第2列第1行。
答案 0 :(得分：0)
这是一项正在进行中的工作
我想到了这个问题，我想我可能有O(w*h)算法。
这个想法是这样的：
(i,j)，从j开始计算(i,j)列中具有相同值的最大单元格数。将此值存储为heights[i][j]。height > heights[i][j]的所有子矩阵。因为子高度> gt; heights[i][j]无法在此单元格上继续(i,j,heights[i][j])定义的子矩阵，其中j是我们可以拟合高度子矩阵的最近坐标：heights[i][j] 棘手的部分是在内循环中。我使用类似于max子窗口算法的东西来确保每个单元格平均O(1)。
我会尝试制定证明，但同时这里是代码。
#include <algorithm>
#include <iterator>
#include <iostream>
#include <ostream>
#include <vector>
typedef std::vector<int> row_t;
typedef std::vector<row_t> matrix_t;
std::size_t height(matrix_t const& M) { return M.size(); }
std::size_t width (matrix_t const& M) { return M.size() ? M[0].size() : 0u; }
std::ostream& operator<<(std::ostream& out, matrix_t const& M) {
for(unsigned i=0; i<height(M); ++i) {
std::copy(begin(M[i]), end(M[i]),
std::ostream_iterator<int>(out, ", "));
out << std::endl;
}
return out;
}
struct sub_matrix_t {
int i, j, h, w;
sub_matrix_t(): i(0),j(0),h(0),w(1) {}
sub_matrix_t(int i_,int j_,int h_,int w_):i(i_),j(j_),h(h_),w(w_) {}
bool operator<(sub_matrix_t const& rhs) const { return (w*h)<(rhs.w*rhs.h); }
};
// Pop all sub_matrix from the vector keeping only those with an height
// inferior to the passed height.
// Compute the max sub matrix while removing sub matrix with height > h
void pop_sub_m(std::vector<sub_matrix_t>& subs,
int i, int j, int h, sub_matrix_t& max_m) {
sub_matrix_t sub_m(i, j, h, 1);
while(subs.size() && subs.back().h >= h) {
sub_m = subs.back();
subs.pop_back();
sub_m.w = j-sub_m.j;
max_m = std::max(max_m, sub_m);
}
// Now sub_m.{i,j} is updated to the farest coordinates where there is a
// submatrix with heights >= h
// If we don't cut the current height (because we changed value) update
// the current max submatrix
if(h > 0) {
sub_m.h = h;
sub_m.w = j-sub_m.j+1;
max_m = std::max(max_m, sub_m);
subs.push_back(sub_m);
}
}
void push_sub_m(std::vector<sub_matrix_t>& subs,
int i, int j, int h, sub_matrix_t& max_m) {
if(subs.empty() || subs.back().h < h)
subs.emplace_back(i, j, h, 1);
}
void solve(matrix_t const& M, sub_matrix_t& max_m) {
// Initialize answer suitable for an empty matrix
max_m = sub_matrix_t();
if(height(M) == 0 || width(M) == 0) return;
// 1) Compute the heights of columns of the same values
matrix_t heights(height(M), row_t(width(M), 1));
for(unsigned i=height(M)-1; i>0; --i)
for(unsigned j=0; j<width(M); ++j)
if(M[i-1][j]==M[i][j])
heights[i-1][j] = heights[i][j]+1;
// 2) Run through all columns heights to compute local sub matrices
std::vector<sub_matrix_t> subs;
for(int i=height(M)-1; i>=0; --i) {
push_sub_m(subs, i, 0, heights[i][0], max_m);
for(unsigned j=1; j<width(M); ++j) {
bool same_val = (M[i][j]==M[i][j-1]);
int pop_height = (same_val) ? heights[i][j] : 0;
int pop_j = (same_val) ? j : j-1;
pop_sub_m (subs, i, pop_j, pop_height, max_m);
push_sub_m(subs, i, j, heights[i][j], max_m);
}
pop_sub_m(subs, i, width(M)-1, 0, max_m);
}
}
matrix_t M1{
{10, 9, 9, 9, 80},
{ 5, 9, 9, 9, 10},
{85, 86, 54, 45, 45},
{15, 21, 5, 1, 0},
{ 5, 6, 88, 11, 10},
};
matrix_t M2{
{10, 19, 9, 29, 80},
{ 5, 9, 9, 9, 10},
{ 9, 9, 54, 45, 45},
{ 9, 9, 5, 1, 0},
{ 5, 6, 88, 11, 10},
};
int main() {
sub_matrix_t answer;
std::cout << M1 << std::endl;
solve(M1, answer);
std::cout << '(' << (answer.w*answer.h)
<< ',' << (answer.j+1) << ',' << (answer.i+1) << ')'
<< std::endl;
answer = sub_matrix_t();
std::cout << M2 << std::endl;
solve(M2, answer);
std::cout << '(' << (answer.w*answer.h)
<< ',' << (answer.j+1) << ',' << (answer.i+1) << ')'
<< std::endl;
}
答案 1 :(得分：0)
这是订单行*列解决方案
它的工作原理是
这是一个有效的python实现。道歉，因为我不确定如何使语法突出显示正常工作
# this program finds the largest area in an array where all the elements have the same value
# It solves in O(rows * columns) time using O(rows*columns) space using dynamic programming
def max_area_subarray(array):
rows = len(array)
if (rows == 0):
return [[]]
columns = len(array[0])
# initialize a blank new array
# this will hold max elements of the same value in a column
new_array = []
for i in range(0,rows-1):
new_array.append([0] * columns)
# start with the bottom row, these all of 1 element of the same type
# below them, including themselves
new_array.append([1] * columns)
# go from the second to bottom row up, finding how many contiguous
# elements of the same type there are
for i in range(rows-2,-1,-1):
for j in range(columns-1,-1,-1):
if ( array[i][j] == array[i+1][j]):
new_array[i][j] = new_array[i+1][j]+1
else:
new_array[i][j] = 1
# go left to right and match up the max areas
max_area = 0
top = 0
bottom = 0
left = 0
right = 0
for i in range(0,rows):
running_height =[[0,0,0]]
for j in range(0,columns):
matched = False
if (j > 0): # if this isn't the leftmost column
if (array[i][j] == array[i][j-1]):
# this matches the array to the left
# keep track of if this is a longer column, shorter column, or same as
# the one on the left
matched = True
while( new_array[i][j] < running_height[-1][0]):
# this is less than the one on the left, pop that running
# height from the list, and add it's columns to the smaller
# running height below it
if (running_height[-1][1] > max_area):
max_area = running_height[-1][1]
top = i
right = j-1
bottom = i + running_height[-1][0]-1
left = j - running_height[-1][2]
previous_column = running_height.pop()
num_columns = previous_column[2]
if (len(running_height) > 0):
running_height[-1][1] += running_height[-1][0] * (num_columns)
running_height[-1][2] += num_columns
else:
# for instance, if we have heights 2,2,1
# this will trigger on the 1 after we pop the 2 out, and save the current
# height of 1, the running area of 3, and running columsn of 3
running_height.append([new_array[i][j],new_array[i][j]*(num_columns),num_columns])
if (new_array[i][j] > running_height[-1][0]):
# longer then the one on the left
# append this height and area
running_height.append([new_array[i][j],new_array[i][j],1])
elif (new_array[i][j] == running_height[-1][0]):
# same as the one on the left, add this area to the one on the left
running_height[-1][1] += new_array[i][j]
running_height[-1][2] += 1
if (matched == False or j == columns -1):
while(running_height):
# unwind the maximums & see if this is the new max area
if (running_height[-1][1] > max_area):
max_area = running_height[-1][1]
top = i
right = j
bottom = i + running_height[-1][0]-1
left = j - running_height[-1][2]+1
# this wasn't a match, so move everything one bay to the left
if (matched== False):
right = right-1
left = left-1
previous_column = running_height.pop()
num_columns = previous_column[2]
if (len(running_height) > 0):
running_height[-1][1] += running_height[-1][0] * num_columns
running_height[-1][2] += num_columns
if (matched == False):
# this is either the left column, or we don't match to the column to the left, so reset
running_height = [[new_array[i][j],new_array[i][j],1]]
if (running_height[-1][1] > max_area):
max_area = running_height[-1][1]
top = i
right = j
bottom = i + running_height[-1][0]-1
left = j - running_height[-1][2]+1
max_array = []
for i in range(top,bottom+1):
max_array.append(array[i][left:right+1])
return max_array
numbers = [[6,4,1,9],[5,2,2,7],[2,2,2,1],[2,3,1,5]]
for row in numbers:
print row
print
print
max_array = max_area_subarray(numbers)
max_area = len(max_array) * len(max_array[0])
print 'max area is ',max_area
print
for row in max_array:
print row