Given a matrix of N rows and M columns. From m[i][j], we can move to m[i+1][j], if m[i+1][j] > m[i][j], or can move to m[i][j+1] if m[i][j+1] > m[i][j]. The task is print longest path length if we start from (0, 0).
Examples:
Input : N = 4, M = 4 m[][] = { { 1, 2, 3, 4 }, { 2, 2, 3, 4 }, { 3, 2, 3, 4 }, { 4, 5, 6, 7 } }; Output : 7 Longest path is 1 2 3 4 5 6 7. Input : N = 2, M =2 m[][] = { { 1, 2 }, { 3, 4 } }; Output :3 Longest path is either 1 2 4 or 1 3 4.
The idea is to use dynamic programming. Maintain the 2D matrix, dp[][], where dp[i][j] store the value of the length of the longest increasing sequence for submatrix starting from the ith row and jth column.
Let the longest increasing sub sequence values for m[i+1][j] and m[i][j+1] be known already as v1 and v2 respectively. Then the value for m[i][j] will be max(v1, v2) + 1.
We can start from m[n-1][m-1] as the base case with the length of longest increasing subsequence be 1, moving upwards and leftwards updating the value of cells. Then the LIP value for cell m[0][0] will be the answer.
Below is the implementation of this approach:
// CPP program to find longest increasing // path in a matrix. #include <bits/stdc++.h> #define MAX 10 using namespace std;
// Return the length of LIP in 2D matrix int LIP( int dp[][MAX], int mat[][MAX], int n, int m, int x, int y)
{ // If value not calculated yet.
if (dp[x][y] < 0) {
int result = 0;
// If reach bottom right cell, return 1.
if (x == n - 1 && y == m - 1)
return dp[x][y] = 1;
// If reach the corner of the matrix.
if (x == n - 1 || y == m - 1)
result = 1;
// If value greater than below cell.
if (x != n-1) // x reaches last row
if (mat[x][y] < mat[x + 1][y])
result = 1 + LIP(dp, mat, n, m, x + 1, y);
// If value greater than left cell.
if (y != m-1) // y reaches last column
if (mat[x][y] < mat[x][y + 1])
result = max(result, 1 + LIP(dp, mat, n, m, x, y + 1));
dp[x][y] = result;
}
return dp[x][y];
} // Wrapper function int wrapper( int mat[][MAX], int n, int m)
{ int dp[MAX][MAX];
memset (dp, -1, sizeof dp);
return LIP(dp, mat, n, m, 0, 0);
} // Driven Program int main()
{ int mat[][MAX] = {
{ 1, 2, 3, 4 },
{ 2, 2, 3, 4 },
{ 3, 2, 3, 4 },
{ 4, 5, 6, 7 },
};
int n = 4, m = 4;
cout << wrapper(mat, n, m) << endl;
return 0;
} |
// Java program to find longest increasing // path in a matrix. import java.util.*;
class GFG {
// Return the length of LIP in 2D matrix
static int LIP( int dp[][], int mat[][], int n,
int m, int x, int y)
{
// If value not calculated yet.
if (dp[x][y] < 0 ) {
int result = 0 ;
// If reach bottom right cell, return 1.
if (x == n - 1 && y == m - 1 )
return dp[x][y] = 1 ;
// If reach the corner of the matrix.
if (x == n - 1 || y == m - 1 )
result = 1 ;
// If value greater than below cell.
if (x + 1 < n && mat[x][y] < mat[x + 1 ][y])
result = 1 + LIP(dp, mat, n, m, x + 1 , y);
// If value greater than left cell.
if (y + 1 < m && mat[x][y] < mat[x][y + 1 ])
result = Math.max(result, 1 + LIP(dp, mat, n, m, x, y + 1 ));
dp[x][y] = result;
}
return dp[x][y];
}
// Wrapper function
static int wrapper( int mat[][], int n, int m)
{
int dp[][] = new int [ 10 ][ 10 ];
for ( int i = 0 ; i < 10 ; i++)
Arrays.fill(dp[i], - 1 );
return LIP(dp, mat, n, m, 0 , 0 );
}
/* Driver program to test above function */
public static void main(String[] args)
{
int mat[][] = {
{ 1 , 2 , 3 , 4 },
{ 2 , 2 , 3 , 4 },
{ 3 , 2 , 3 , 4 },
{ 4 , 5 , 6 , 7 },
};
int n = 4 , m = 4 ;
System.out.println(wrapper(mat, n, m));
}
} // This code is contributed by Arnav Kr. Mandal. |
# Python3 program to find longest # increasing path in a matrix. MAX = 20
# Return the length of # LIP in 2D matrix def LIP(dp, mat, n, m, x, y):
# If value not calculated yet.
if (dp[x][y] < 0 ):
result = 0
# // If reach bottom right cell, return 1
if (x = = n - 1 and y = = m - 1 ):
dp[x][y] = 1
return dp[x][y]
# If reach the corner
# of the matrix.
if (x = = n - 1 or y = = m - 1 ):
result = 1
# If value greater than below cell.
if (x + 1 < n and mat[x][y] < mat[x + 1 ][y]):
result = 1 + LIP(dp, mat, n,
m, x + 1 , y)
# If value greater than left cell.
if (y + 1 < m and mat[x][y] < mat[x][y + 1 ]):
result = max (result, 1 + LIP(dp, mat, n,
m, x, y + 1 ))
dp[x][y] = result
return dp[x][y]
# Wrapper function def wrapper(mat, n, m):
dp = [[ - 1 for i in range ( MAX )]
for i in range ( MAX )]
return LIP(dp, mat, n, m, 0 , 0 )
# Driver Code mat = [[ 1 , 2 , 3 , 4 ],
[ 2 , 2 , 3 , 4 ],
[ 3 , 2 , 3 , 4 ],
[ 4 , 5 , 6 , 7 ]]
n = 4
m = 4
print (wrapper(mat, n, m))
# This code is contributed # by Sahil Shelangia |
// C# program to find longest increasing // path in a matrix. using System;
public class GFG {
// Return the length of LIP in 2D matrix
static int LIP( int [, ] dp, int [, ] mat, int n,
int m, int x, int y)
{
// If value not calculated yet.
if (dp[x, y] < 0) {
int result = 0;
// If reach bottom right cell, return 1.
if (x == n - 1 && y == m - 1)
return dp[x, y] = 1;
// If reach the corner of the matrix.
if (x == n - 1 || y == m - 1)
result = 1;
// If value greater than below cell.
if (x + 1 < n && mat[x, y] < mat[x + 1, y])
result = 1 + LIP(dp, mat, n, m, x + 1, y);
// If value greater than left cell.
if (y + 1 < m && mat[x, y] < mat[x, y + 1])
result = Math.Max(result, 1 + LIP(dp, mat, n, m, x, y + 1));
dp[x, y] = result;
}
return dp[x, y];
}
// Wrapper function
static int wrapper( int [, ] mat, int n, int m)
{
int [, ] dp = new int [10, 10];
for ( int i = 0; i < 10; i++) {
for ( int j = 0; j < 10; j++) {
dp[i, j] = -1;
}
}
return LIP(dp, mat, n, m, 0, 0);
}
/* Driver code */
public static void Main()
{
int [, ] mat = {
{ 1, 2, 3, 4 },
{ 2, 2, 3, 4 },
{ 3, 2, 3, 4 },
{ 4, 5, 6, 7 },
};
int n = 4, m = 4;
Console.WriteLine(wrapper(mat, n, m));
}
} /* This code contributed by PrinciRaj1992 */ |
<script> // Javascript program to find longest increasing path in a matrix.
// Return the length of LIP in 2D matrix
function LIP(dp, mat, n, m, x, y)
{
// If value not calculated yet.
if (dp[x][y] < 0) {
let result = 0;
// If reach bottom right cell, return 1.
if (x == n - 1 && y == m - 1)
return dp[x][y] = 1;
// If reach the corner of the matrix.
if (x == n - 1 || y == m - 1)
result = 1;
// If value greater than below cell.
if (x + 1 < n && mat[x][y] < mat[x + 1][y])
result = 1 + LIP(dp, mat, n, m, x + 1, y);
// If value greater than left cell.
if (y + 1 < m && mat[x][y] < mat[x][y + 1])
result = Math.max(result, 1 + LIP(dp, mat, n, m, x, y + 1));
dp[x][y] = result;
}
return dp[x][y];
}
// Wrapper function
function wrapper(mat, n, m)
{
let dp = new Array(10);
for (let i = 0; i < 10; i++)
{
dp[i] = new Array(10);
for (let j = 0; j < 10; j++)
{
dp[i][j] = -1;
}
}
return LIP(dp, mat, n, m, 0, 0);
}
let mat = [
[ 1, 2, 3, 4 ],
[ 2, 2, 3, 4 ],
[ 3, 2, 3, 4 ],
[ 4, 5, 6, 7 ],
];
let n = 4, m = 4;
document.write(wrapper(mat, n, m));
</script> |
7
Time Complexity: O(N*M).
Space Complexity: O(N*M)