Count number of paths with at-most k turns

4

Given a “m x n” matrix, count number of paths to reach bottom right from top left with maximum k turns allowed.

What is a turn? A movement is considered turn, if we were moving along row and now move along column. OR we were moving along column and now move along row.

There are two possible scenarios when a turn can occur
at point (i, j):

Turns Right: (i-1, j)  ->  (i, j)  ->  (i, j+1)
                      Down        Right

Turns Down:  (i, j-1)  ->  (i, j)  ->  (i+1, j)
                     Right        Down

Examples:

Input:  m = 3, n = 3, k = 2
Output: 4
See below diagram for four paths with 
maximum 2 turns.

Input:  m = 3, n = 3, k = 1
Output: 2 

pathswithkturns

Source: http://qa.geeksforgeeks.org/1993/no-of-paths-with-k-turns

We strongly recommend you to minimize your browser and try this yourself first.
This problem can be recursively computed using below recursive formula.

countPaths(i, j, k): Count of paths to reach (i,j) from (0, 0)
countPathsDir(i, j, k, 0): Count of paths if we reach (i, j) 
                           along row. 
countPathsDir(i, j, k, 1): Count of paths if we reach (i, j) 
                           along column. 
The fourth parameter in countPathsDir() indicates direction.

Value of countPaths() can be written as:
countPaths(i, j, k) = countPathsDir(i, j, k, 0) + 
                      countPathsDir(i, j, k, 1) 

And value of  countPathsDir() can be recursively defined as:

// Base cases

// If current direction is along row
If (d == 0) 
  // Count paths for two cases
  // 1) We reach here through previous row.
  // 2) We reach here through previous column, so number of 
  //    turns k reduce by 1.
  countPathsDir(i, j, k, d) = countPathsUtil(i, j-1, k, d) +
                              countPathsUtil(i-1, j, k-1, !d);

// If current direction is along column
Else 
  // Similar to above
  countPathsDir(i, j, k, d) =  countPathsUtil(i-1, j, k, d) +
                               countPathsUtil(i, j-1, k-1, !d);

We can solve this problem in Polynomial Time using Dynamic Programming. The idea is to use a 4 dimensional table dp[m][n][k][d] where m is number of rows, n is number of columns, k is number of allowed turns and d is direction.

Below is Dynamic Programming based C++ implementation.

// C++ program to count number of paths with maximum
// k turns allowed
#include<bits/stdc++.h>
using namespace std;
#define MAX 100

// table to store to store results of subproblems
int dp[MAX][MAX][MAX][2];

// Returns count of paths to reach (i, j) from (0, 0)
// using at-most k turns. d is current direction
// d = 0 indicates along row, d = 1 indicates along
// column.
int countPathsUtil(int i, int j, int k, int d)
{
    // If invalid row or column indexes
    if (i < 0 || j < 0)
        return 0;

    // If current cell is top left itself
    if (i == 0 && j == 0)
        return 1;

    // If 0 turns left
    if (k == 0)
    {
        // If direction is row, then we can reach here 
        // only if direction is row and row is 0.
        if (d == 0 && i == 0) return 1;

        // If direction is column, then we can reach here 
        // only if direction is column and column is 0.
        if (d == 1 && j == 0) return 1;

        return 0;
    }

    // If this subproblem is already evaluated
    if (dp[i][j][k][d] != -1)
        return dp[i][j][k][d];

    // If current direction is row, then count paths for two cases
    // 1) We reach here through previous row.
    // 2) We reach here through previous column, so number of 
    //    turns k reduce by 1.
    if (d == 0)
      return dp[i][j][k][d] = countPathsUtil(i, j-1, k, d) +
                              countPathsUtil(i-1, j, k-1, !d);

    // Similar to above if direction is column
    return dp[i][j][k][d] =  countPathsUtil(i-1, j, k, d) +
                             countPathsUtil(i, j-1, k-1, !d);
}

// This function mainly initializes 'dp' array as -1 and calls
// countPathsUtil()
int countPaths(int i, int j, int k)
{
    // If (0, 0) is target itself
    if (i == 0 && j == 0)
          return 1;

    // Initialize 'dp' array
    memset(dp, -1, sizeof dp);

    // Recur for two cases: moving along row and along column
    return countPathsUtil(i-1, j, k, 1) +  // Moving along row
           countPathsUtil(i, j-1, k, 0); // Moving along column
}

// Driver program
int main()
{
    int m = 3, n = 3, k = 2;
    cout << "Number of paths is "
         << countPaths(m-1, n-1, k) << endl;
    return 0;
}

Output:

Number of paths is 4

Time complexity of above solution is O(m*n*k)

Thanks to Gaurav Ahirwar for suggesting this solution.

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