Paths with maximum number of ‘a’ from (1, 1) to (X, Y) vertically or horizontally

Given a N X N matrix consisting of characters. Also given are Q queries, where each query contains a co-ordinate (X, Y). For every query, find all paths from (1, 1) to (X, Y) by either moving vertically or horizontally and take the path which has the maximum number of in it. The task is to print the number of non-‘a’-characters along that path. 

Examples: 

Input: mat[][] = {{'a', 'b', 'a'}, 
                             {'a', 'c', 'd'}, 
                             {'b', 'a', 'b'}} 
            Queries: 
            X = 1, Y = 3
            X = 3, Y = 3
Output: 
1st query: 1
2nd query: 2

Query-1: There is only one path from (1, 1) to (1, 3)
i.e., "aba" and the number of characters 
which are not 'a' is 1. 
Query-2: The path which has the maximum number of 'a'
in it is "aabab", hence non 'a' 
characters are 2. 

The problem is a variant of Min-Cost path Dp problem. We need to pre-compute the DP[][] array and then the answer will be DP[X][Y], hence every query can be answered in O(1). If the index position (1, 1) character is not ‘a’, then increase the dp[1][1] value by 1. Then simply iterate for rows and columns, and do a min-cost DP considering ‘a’ as 1 and non-‘a’ character as 0. Since the DP[][] array stores the min-cost path from (1, 1) to any index (i, j), hence the query can be answered in O(1). 

Below is the implementation of the above approach:  

C++

// C++ program to find paths with maximum number
// of 'a' from (1, 1) to (X, Y) vertically
// or horizontally

#include <bits/stdc++.h>
using namespace std;

const int n = 3;
int dp[n][n];

// Function to answer queries
void answerQueries(pair<int, int> queries[], int q)
{
    // Iterate till query
    for (int i = 0; i < q; i++) {

        // Decrease to get 0-based indexing
        int x = queries[i].first;
        x--;
        int y = queries[i].second;
        y--;

        // Print answer
        cout << dp[x][y] << endl;
    }
}

// Function that pre-computes the dp array
void pre_compute(char a[][n])
{
    // Check for the first character
    if (a[0][0] == 'a')
        dp[0][0] = 0;
    else
        dp[0][0] = 1;

    // Iterate in row and columns
    for (int row = 0; row < n; row++) {
        for (int col = 0; col < n; col++) {
            // If not first row or not first column
            if (row != 0 || col != 0)
                dp[row][col] = INT_MAX;

            // Not first row
            if (row != 0) {
                dp[row][col] = min(dp[row][col],
                                   dp[row - 1][col]);
            }

            // Not first column
            if (col != 0) {
                dp[row][col] = min(dp[row][col],
                                   dp[row][col - 1]);
            }

            // If it is not 'a' then increase by 1
            if (a[row][col] != 'a' && (row != 0 || col != 0))
                dp[row][col] += 1;
        }
    }
}

// Driver code
int main()
{
    // character N X N array
    char a[][3] = { { 'a', 'b', 'a' },
                    { 'a', 'c', 'd' },
                    { 'b', 'a', 'b' } };

    // queries
    pair<int, int> queries[] = { { 1, 3 }, { 3, 3 } };

    // number of queries
    int q = 2;

    // function call to pre-compute
    pre_compute(a);

    // function call to answer every query
    answerQueries(queries, q);
}

Java

// Java program to find paths with maximum number
// of 'a' from (1, 1) to (X, Y) vertically
// or horizontally
class GFG
{
static class pair
{
    int first, second;
    public pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}

static int n = 3;
static int [][]dp = new int[n][n];

// Function to answer queries
static void answerQueries(pair queries[], int q)
{
    // Iterate till query
    for (int i = 0; i < q; i++)
    {

        // Decrease to get 0-based indexing
        int x = queries[i].first;
        x--;
        int y = queries[i].second;
        y--;

        // Print answer
        System.out.println(dp[x][y]);
    }
}

// Function that pre-computes the dp array
static void pre_compute(char a[][])
{
    // Check for the first character
    if (a[0][0] == 'a')
        dp[0][0] = 0;
    else
        dp[0][0] = 1;

    // Iterate in row and columns
    for (int row = 0; row < n; row++)
    {
        for (int col = 0; col < n; col++)
        {
            // If not first row or not first column
            if (row != 0 || col != 0)
                dp[row][col] = Integer.MAX_VALUE;

            // Not first row
            if (row != 0)
            {
                dp[row][col] = Math.min(dp[row][col],
                                        dp[row - 1][col]);
            }

            // Not first column
            if (col != 0)
            {
                dp[row][col] = Math.min(dp[row][col],
                                        dp[row][col - 1]);
            }

            // If it is not 'a' then increase by 1
            if (a[row][col] != 'a' && (row != 0 || col != 0))
                dp[row][col] += 1;
        }
    }
}

// Driver code
public static void main(String[] args)
{
    // character N X N array
    char a[][] = {{ 'a', 'b', 'a' },
                  { 'a', 'c', 'd' },
                  { 'b', 'a', 'b' }};

    // queries
    pair queries[] = { new pair( 1, 3 ),
                       new pair(3, 3 ) };

    // number of queries
    int q = 2;

    // function call to pre-compute
    pre_compute(a);

    // function call to answer every query
    answerQueries(queries, q);
}
}

// This code is contributed by 29AjayKumar

Python3

# Python3 program to find paths with maximum number
# of 'a' from (1, 1) to (X, Y) vertically
# or horizontally
n = 3
dp = [[0 for i in range(n)]
         for j in range(n)]

# Function to answer queries
def answerQueries(queries, q):

    # Iterate till query
    for i in range(q):

        # Decrease to get 0-based indexing
        x = queries[i][0]
        x -= 1
        y = queries[i][1]
        y -= 1

        # Print answer
        print(dp[x][y])

# Function that pre-computes the dp array
def pre_compute(a):

    # Check for the first character
    if a[0][0] == 'a':
        dp[0][0] = 0
    else:
        dp[0][0] = 1

    # Iterate in row and columns
    for row in range(n):
        for col in range(n):

            # If not first row or not first column
            if (row != 0 or col != 0):
                dp[row][col] = 9999

            # Not first row
            if (row != 0):
                dp[row][col] = min(dp[row][col],
                                   dp[row - 1][col])

            # Not first column
            if (col != 0):
                dp[row][col] = min(dp[row][col],
                                   dp[row][col - 1])

            # If it is not 'a' then increase by 1
            if (a[row][col] != 'a' and (row != 0 or
                                        col != 0)):
                dp[row][col] += 1

# Driver code
if __name__ == '__main__':

    # Character N X N array
    a = [ ('a', 'b', 'a'),
          ('a', 'c', 'd'),
          ('b', 'a', 'b') ]

    # Queries
    queries = [ (1, 3), (3, 3) ]

    # Number of queries
    q = 2

    # Function call to pre-compute
    pre_compute(a)

    # function call to answer every query
    answerQueries(queries, q)

# This code is contributed by kirtishsurangalikar

C#

// C# program to find paths with maximum number
// of 'a' from (1, 1) to (X, Y) vertically
// or horizontally
using System;

class GFG
{
class pair
{
    public int first, second;
    public pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}

static int n = 3;
static int [,]dp = new int[n, n];

// Function to answer queries
static void answerQueries(pair []queries, int q)
{
    // Iterate till query
    for (int i = 0; i < q; i++)
    {

        // Decrease to get 0-based indexing
        int x = queries[i].first;
        x--;
        int y = queries[i].second;
        y--;

        // Print answer
        Console.WriteLine(dp[x, y]);
    }
}

// Function that pre-computes the dp array
static void pre_compute(char [,]a)
{
    // Check for the first character
    if (a[0, 0] == 'a')
        dp[0, 0] = 0;
    else
        dp[0, 0] = 1;

    // Iterate in row and columns
    for (int row = 0; row < n; row++)
    {
        for (int col = 0; col < n; col++)
        {
            // If not first row or not first column
            if (row != 0 || col != 0)
                dp[row, col] = int.MaxValue;

            // Not first row
            if (row != 0)
            {
                dp[row, col] = Math.Min(dp[row, col],
                                        dp[row - 1, col]);
            }

            // Not first column
            if (col != 0)
            {
                dp[row, col] = Math.Min(dp[row, col],
                                        dp[row, col - 1]);
            }

            // If it is not 'a' then increase by 1
            if (a[row, col] != 'a' &&
               (row != 0 || col != 0))
                dp[row, col] += 1;
        }
    }
}

// Driver code
public static void Main(String[] args)
{
    // character N X N array
    char [,]a = {{ 'a', 'b', 'a' },
                 { 'a', 'c', 'd' },
                 { 'b', 'a', 'b' }};

    // queries
    pair []queries = { new pair(1, 3),
                       new pair(3, 3) };

    // number of queries
    int q = 2;

    // function call to pre-compute
    pre_compute(a);

    // function call to answer every query
    answerQueries(queries, q);
}
}

// This code is contributed by PrinciRaj1992

Javascript

<script>
// Javascript program to find paths with maximum number
// of 'a' from (1, 1) to (X, Y) vertically
// or horizontally

class pair
{
    constructor(first,second)
    {
        this.first = first;
        this.second = second;
    }
}

let n = 3;

let dp=new Array(n);
for(let i=0;i<n;i++)
{
    dp[i]=new Array(n);
    for(let j=0;j<n;j++)
    {
        dp[i][j]=0;
    }
}

// Function to answer queries
function answerQueries(queries,q)
{
    // Iterate till query
    for (let i = 0; i < q; i++)
    {

        // Decrease to get 0-based indexing
        let x = queries[i].first;
        x--;
        let y = queries[i].second;
        y--;

        // Print answer
        document.write(dp[x][y]+"<br>");
    }
}

// Function that pre-computes the dp array
function pre_compute(a)
{
    // Check for the first character
    if (a[0][0] == 'a')
        dp[0][0] = 0;
    else
        dp[0][0] = 1;

    // Iterate in row and columns
    for (let row = 0; row < n; row++)
    {
        for (let col = 0; col < n; col++)
        {
            // If not first row or not first column
            if (row != 0 || col != 0)
                dp[row][col] = Number.MAX_VALUE;

            // Not first row
            if (row != 0)
            {
                dp[row][col] = Math.min(dp[row][col],
                                        dp[row - 1][col]);
            }

            // Not first column
            if (col != 0)
            {
                dp[row][col] = Math.min(dp[row][col],
                                        dp[row][col - 1]);
            }

            // If it is not 'a' then increase by 1
            if (a[row][col] != 'a' && (row != 0 || col != 0))
                dp[row][col] += 1;
        }
    }
}

// Driver code
let a=[[ 'a', 'b', 'a' ],
                  [ 'a', 'c', 'd' ],
                  [ 'b', 'a', 'b' ]];
// queries
let queries = [ new pair( 1, 3 ),
new pair(3, 3 ) ];

// number of queries
let q = 2;

// function call to pre-compute
pre_compute(a);

// function call to answer every query
answerQueries(queries, q);

// This code is contributed by rag2127
</script>

Output

1
2

Time Complexity: O(N²) for pre-computation and O(1) for each query. 
Auxiliary Space: O(N²)