从行和列的排序矩阵中按排序顺序打印所有元素
原文： https://www.geeksforgeeks.org/print-elements-sorted-order-row-column-wise-sorted-matrix/
给定一个n x n矩阵，其中每一行和每一列均以非降序排列。 按排序顺序打印矩阵的所有元素。
示例：
Input: mat[][] = { {10, 20, 30, 40},
{15, 25, 35, 45},
{27, 29, 37, 48},
{32, 33, 39, 50},
};
Output:
Elements of matrix in sorted order
10 15 20 25 27 29 30 32 33 35 37 39 40 45 48 50
我们可以使用 Young Tableau 解决上述问题。 想法是将给定的 2D 数组视为 Young Tableau，并调用提取最小值O(n)。
C++
// A C++ program to Print all elements in sorted order from row and
// column wise sorted matrix
#include<iostream>
#include<climits>
using namespace std;
#define INF INT_MAX
#define N 4
// A utility function to youngify a Young Tableau. This is different
// from standard youngify. It assumes that the value at mat[0][0] is
// infinite.
void youngify(int mat[][N], int i, int j)
{
// Find the values at down and right sides of mat[i][j]
int downVal = (i+1 < N)? mat[i+1][j]: INF;
int rightVal = (j+1 < N)? mat[i][j+1]: INF;
// If mat[i][j] is the down right corner element, return
if (downVal==INF && rightVal==INF)
return;
// Move the smaller of two values (downVal and rightVal) to
// mat[i][j] and recur for smaller value
if (downVal < rightVal)
{
mat[i][j] = downVal;
mat[i+1][j] = INF;
youngify(mat, i+1, j);
}
else
{
mat[i][j] = rightVal;
mat[i][j+1] = INF;
youngify(mat, i, j+1);
}
}
// A utility function to extract minimum element from Young tableau
int extractMin(int mat[][N])
{
int ret = mat[0][0];
mat[0][0] = INF;
youngify(mat, 0, 0);
return ret;
}
// This function uses extractMin() to print elements in sorted order
void printSorted(int mat[][N])
{
cout << "Elements of matrix in sorted order n";
for (int i=0; i<N*N; i++)
cout << extractMin(mat) << " ";
}
// driver program to test above function
int main()
{
int mat[N][N] = { {10, 20, 30, 40},
{15, 25, 35, 45},
{27, 29, 37, 48},
{32, 33, 39, 50},
};
printSorted(mat);
return 0;
}
Java
// A Java program to Print all elements
// in sorted order from row and
// column wise sorted matrix
class GFG
{
static final int INF = Integer.MAX_VALUE;
static final int N = 4;
// A utility function to youngify a Young Tableau.
// This is different from standard youngify.
// It assumes that the value at mat[0][0] is infinite.
static void youngify(int mat[][], int i, int j)
{
// Find the values at down and right sides of mat[i][j]
int downVal = (i + 1 < N) ?
mat[i + 1][j] : INF;
int rightVal = (j + 1 < N) ?
mat[i][j + 1] : INF;
// If mat[i][j] is the down right corner element,
// return
if (downVal == INF && rightVal == INF)
{
return;
}
// Move the smaller of two values
// (downVal and rightVal) to mat[i][j]
// and recur for smaller value
if (downVal < rightVal)
{
mat[i][j] = downVal;
mat[i + 1][j] = INF;
youngify(mat, i + 1, j);
}
else
{
mat[i][j] = rightVal;
mat[i][j + 1] = INF;
youngify(mat, i, j + 1);
}
}
// A utility function to extract
// minimum element from Young tableau
static int extractMin(int mat[][])
{
int ret = mat[0][0];
mat[0][0] = INF;
youngify(mat, 0, 0);
return ret;
}
// This function uses extractMin()
// to print elements in sorted order
static void printSorted(int mat[][])
{
System.out.println("Elements of matrix in sorted order n");
for (int i = 0; i < N * N; i++)
{
System.out.print(extractMin(mat) + " ");
}
}
// Driver Code
public static void main(String args[])
{
int mat[][] = {{10, 20, 30, 40},
{15, 25, 35, 45},
{27, 29, 37, 48},
{32, 33, 39, 50}};
printSorted(mat);
}
}
// This code is contributed by Rajput-Ji
Python 3
# Python 3 program to Print all elements
# in sorted order from row and column
# wise sorted matrix
import sys
INF = sys.maxsize
N = 4
# A utility function to youngify a Young
# Tableau. This is different from standard
# youngify. It assumes that the value at
# mat[0][0] is infinite.
def youngify(mat, i, j):
# Find the values at down and
# right sides of mat[i][j]
downVal = mat[i + 1][j] if (i + 1 < N) else INF
rightVal = mat[i][j + 1] if (j + 1 < N) else INF
# If mat[i][j] is the down right
# corner element, return
if (downVal == INF and rightVal == INF):
return
# Move the smaller of two values
# (downVal and rightVal) to mat[i][j]
# and recur for smaller value
if (downVal < rightVal):
mat[i][j] = downVal
mat[i + 1][j] = INF
youngify(mat, i + 1, j)
else:
mat[i][j] = rightVal
mat[i][j + 1] = INF
youngify(mat, i, j + 1)
# A utility function to extract minimum
# element from Young tableau
def extractMin(mat):
ret = mat[0][0]
mat[0][0] = INF
youngify(mat, 0, 0)
return ret
# This function uses extractMin() to
# print elements in sorted order
def printSorted(mat):
print("Elements of matrix in sorted order n")
i = 0
while i < N * N:
print(extractMin(mat), end = " ")
i += 1
# Driver Code
if __name__ == "__main__":
mat = [[10, 20, 30, 40],
[15, 25, 35, 45],
[27, 29, 37, 48],
[32, 33, 39, 50]]
printSorted(mat)
# This code is contributed by ita_c
C#
// A C# program to Print all elements
// in sorted order from row and
// column wise sorted matrix
using System;
class GFG
{
static int INF = int.MaxValue;
static int N = 4;
// A utility function to youngify a Young Tableau.
// This is different from standard youngify.
// It assumes that the value at mat[0][0] is infinite.
static void youngify(int [,]mat, int i, int j)
{
// Find the values at down and right sides of mat[i][j]
int downVal = (i + 1 < N) ?
mat[i + 1,j] : INF;
int rightVal = (j + 1 < N) ?
mat[i,j + 1] : INF;
// If mat[i][j] is the down right corner element,
// return
if (downVal == INF && rightVal == INF)
{
return;
}
// Move the smaller of two values
// (downVal and rightVal) to mat[i][j]
// and recur for smaller value
if (downVal < rightVal)
{
mat[i,j] = downVal;
mat[i + 1,j] = INF;
youngify(mat, i + 1, j);
}
else
{
mat[i, j] = rightVal;
mat[i, j + 1] = INF;
youngify(mat, i, j + 1);
}
}
// A utility function to extract
// minimum element from Young tableau
static int extractMin(int [,]mat)
{
int ret = mat[0,0];
mat[0, 0] = INF;
youngify(mat, 0, 0);
return ret;
}
// This function uses extractMin()
// to print elements in sorted order
static void printSorted(int [,]mat)
{
Console.WriteLine("Elements of matrix in sorted order n");
for (int i = 0; i < N * N; i++)
{
Console.Write(extractMin(mat) + " ");
}
}
// Driver Code
static public void Main ()
{
int [,]mat = {{10, 20, 30, 40},
{15, 25, 35, 45},
{27, 29, 37, 48},
{32, 33, 39, 50}};
printSorted(mat);
}
}
// This code is contributed by ajit.
输出：
Elements of matrix in sorted order
10 15 20 25 27 29 30 32 33 35 37 39 40 45 48 50
提取最小值的时间复杂度为O(n)，称为 O(n^2)倍。 因此，总时间复杂度为O(N ^ 3)。
更好的解决方案是使用方法来合并k个排序的数组。 想法是使用大小为N的最小堆来存储第一列的元素。 做最小提取。 在最小提取中，将最小元素替换为要从中提取元素的行的下一个元素。 该解决方案的时间复杂度为O(N ^ 2 LogN)。
// C++ program to merge k sorted arrays of size n each.
#include<iostream>
#include<climits>
using namespace std;
#define N 4
// A min heap node
struct MinHeapNode
{
int element; // The element to be stored
int i; // index of the row from which the element is taken
int j; // index of the next element to be picked from row
};
// Prototype of a utility function to swap two min heap nodes
void swap(MinHeapNode *x, MinHeapNode *y);
// A class for Min Heap
class MinHeap
{
MinHeapNode *harr; // pointer to array of elements in heap
int heap_size; // size of min heap
public:
// Constructor: creates a min heap of given size
MinHeap(MinHeapNode a[], int size);
// to heapify a subtree with root at given index
void MinHeapify(int );
// to get index of left child of node at index i
int left(int i) { return (2*i + 1); }
// to get index of right child of node at index i
int right(int i) { return (2*i + 2); }
// to get the root
MinHeapNode getMin() { return harr[0]; }
// to replace root with new node x and heapify() new root
void replaceMin(MinHeapNode x) { harr[0] = x; MinHeapify(0); }
};
// This function prints elements of a given matrix in non-decreasing
// order. It assumes that ma[][] is sorted row wise sorted.
void printSorted(int mat[][N])
{
// Create a min heap with k heap nodes. Every heap node
// has first element of an array
MinHeapNode *harr = new MinHeapNode[N];
for (int i = 0; i < N; i++)
{
harr[i].element = mat[i][0]; // Store the first element
harr[i].i = i; // index of row
harr[i].j = 1; // Index of next element to be stored from row
}
MinHeap hp(harr, N); // Create the min heap
// Now one by one get the minimum element from min
// heap and replace it with next element of its array
for (int count = 0; count < N*N; count++)
{
// Get the minimum element and store it in output
MinHeapNode root = hp.getMin();
cout << root.element << " ";
// Find the next elelement that will replace current
// root of heap. The next element belongs to same
// array as the current root.
if (root.j < N)
{
root.element = mat[root.i][root.j];
root.j += 1;
}
// If root was the last element of its array
else root.element = INT_MAX; //INT_MAX is for infinite
// Replace root with next element of array
hp.replaceMin(root);
}
}
// FOLLOWING ARE IMPLEMENTATIONS OF STANDARD MIN HEAP METHODS
// FROM CORMEN BOOK
// Constructor: Builds a heap from a given array a[] of given size
MinHeap::MinHeap(MinHeapNode a[], int size)
{
heap_size = size;
harr = a; // store address of array
int i = (heap_size - 1)/2;
while (i >= 0)
{
MinHeapify(i);
i--;
}
}
// A recursive method to heapify a subtree with root at given index
// This method assumes that the subtrees are already heapified
void MinHeap::MinHeapify(int i)
{
int l = left(i);
int r = right(i);
int smallest = i;
if (l < heap_size && harr[l].element < harr[i].element)
smallest = l;
if (r < heap_size && harr[r].element < harr[smallest].element)
smallest = r;
if (smallest != i)
{
swap(&harr[i], &harr[smallest]);
MinHeapify(smallest);
}
}
// A utility function to swap two elements
void swap(MinHeapNode *x, MinHeapNode *y)
{
MinHeapNode temp = *x; *x = *y; *y = temp;
}
// driver program to test above function
int main()
{
int mat[N][N] = { {10, 20, 30, 40},
{15, 25, 35, 45},
{27, 29, 37, 48},
{32, 33, 39, 50},
};
printSorted(mat);
return 0;
}
输出：
10 15 20 25 27 29 30 32 33 35 37 39 40 45 48 50
练习：
上述解决方案适用于方阵。 扩展上述解决方案以用于M * N矩形矩阵。
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