Longest Substring of given characters by replacing at most K characters for Q queries

Given a string s of length N, and Q queries, each of type (K, C) where K is an integer C is a character, the task is to replace at most K characters of the string by C and have to print the maximum length of possible substring containing only character C. 

Examples : 

Input: s = "yamatonadeshiko", N = 15, Q = 10,  
queries[] = {{1, a}, {2, a}, {3, a}, {4, a}, {5, a}, {1, b}, {2, b}, {3, b}, {4, b}, {5, b}}
Output: 3, 4, 5, 7, 8, 1, 2, 3, 4, 5
Explanation: In the first query "ama" can be made "aaa" by 1 replacement 
In the second query "yama" can be made "aaaa" so answer is 4
In the third query "yamat" can be made "aaaaa" so answer is 5
In the 4th query "amatona" can be made "aaaaaaa" so answer is 7 
In the 5th query "yamatona" can be made "aaaaaaaa" so answer is 8
In the 6th query since there is no b any character say "y" can be made "b" 
In the 7th query  "ya" can be made "bb" 
In the 8th query "yam" can be made "bbb" 
In the 9th query "yama" can be made "bbbb"
In the 10th query "yamat" can be made "bbbbb" 

Input: s = "koyomi", N = 6, Q = 3, queries[] = {{1, o}, {4, o}, {4, m}}
Output: 3, 6, 5
Explanation : In the first query "oyo"  can be made "ooo" using 1 replacement.
In the second query  "koyomi"  can be made "ooooo".
In the third query "koyom" can be made "mmmmm" using 4 replacement "koyo".

Approach: The idea to efficiently solve this problem is based on the concept of dynamic programming:

If we can find out the longest substring of character 'ch' till i^th index by performing j replacements, we can easily find the longest substring of character ch till (i+1)^th index by performing j replacements. There are two cases: 

1. i^th character and ch are same and 
2. They are different.

If this is expressed mathematically as a function we can write it as f(ch, i, j). So the length in the two cases will be:

1. As they are same so the length will be one more than it was till i^th with j changes. So f(ch, i, j) = f(ch, i-1, j) + 1
2. As they are different, the length will be one more than it was till i^th index with j-1 changes. So f(ch, i, j) = f(ch, i-1, j-1) + 1

So for at most K changes by character C, it is the maximum of all the longest substring till any index for any number of changes in the range [1, K].

Follow the steps mentioned below to implement the idea: 

• Precompute the result using Dynamic programming and the relation as mentioned above.
• Create a 3-dimensional dp[][][] array to store the result of precomputation.
• Precompute the result for any number of changes for any character to answer the queries in linear time:
  • If the answer for ch with almost K changes is represented as g(K, ch) then it is the maximum of all the values of f(ch, i, K) where i varies from (0 to N-1).
  • Use another 2-dimensional array (say ans[][]) to store the result of g(K, ch).
• Traverse the queries and find the value from the ans[][] array.

Below is the implementation of the above approach:

// C++ code of above mentioned approach

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

int dp[1001][1001][26];
int ans[1001][26];

// Precomputation to get the values
// of the function f() and g()
void preprocess(string s)
{
    int n = s.length();
    memset(dp, 0, sizeof(dp));
    for (int i = 1; i <= n; i++) {

        // For the existing character in the string
        // the length will be 1
        dp[i][0][s[i - 1] - 'a'] = 1;
    }
    for (int i = 1; i <= n; i++) {
        for (int j = 0; j <= i; j++) {
            for (int k = 0; k < 26; k++) {

                // The ith character of the string
                // is equal to kth
                if (s[i - 1] == char(k + 'a'))
                    dp[i][j][k] = dp[i - 1][j][k] + 1;
                else if (j != 0)

                    // The ith character is not equal to
                    // kth, j != 0 as for j = 0
                    // we have done previously
                    dp[i][j][k] = dp[i - 1][j - 1][k] + 1;
            }
        }
    }

    // The ans array which stores length of
    // longest substring for any character
    memset(ans, 0, sizeof(ans));
    for (int i = 1; i <= n; i++) {
        for (int j = 0; j < 26; j++) {
            for (int k = 1; k <= n; k++)

                // Maximum among all indices
                ans[i][j] = max(ans[i][j], dp[k][i][j]);
        }
    }
}

// Function to find the longest substrings
vector<int> query(string s,
                  vector<pair<int, char> >& queries)
{
    preprocess(s);
    vector<int> res;
    for (auto u : queries) {
        res.push_back(ans[u.first][u.second - 'a']);
    }
    return res;
}

// Driver code
int main()
{
    string s = "yamatonadeshiko";
    vector<pair<int, char> > queries;
    queries.push_back({ 1, 'a' });
    queries.push_back({ 2, 'a' });
    queries.push_back({ 3, 'a' });
    queries.push_back({ 4, 'a' });
    queries.push_back({ 5, 'a' });
    queries.push_back({ 1, 'b' });
    queries.push_back({ 2, 'b' });
    queries.push_back({ 3, 'b' });
    queries.push_back({ 4, 'b' });
    queries.push_back({ 5, 'b' });

    // Function call
    vector<int> sol = query(s, queries);
    for (int x : sol)
        cout << x << " ";
    return 0;
}

// Java code of above mentioned approach
import java.io.*;
import java.util.*;

class Pair {
    int first;
    char second;
    Pair(int fi, char sec)
    {
        first = fi;
        second = sec;
    }
}
class GFG {
    static int dp[][][] = new int[1001][1001][26];
    static int ans[][] = new int[1001][26];

    // Precomputation to get the values
    // of the function f() and g()
    public static void preprocess(String s)
    {
        int n = s.length();
        for (int i = 1; i <= n; i++) {

            // For the existing character in the string
            // the length will be 1
            dp[i][0][s.charAt(i - 1) - 'a'] = 1;
        }
        for (int i = 1; i <= n; i++) {
            for (int j = 0; j <= i; j++) {
                for (int k = 0; k < 26; k++) {

                    // The ith character of the string
                    // is equal to kth
                    if (s.charAt(i - 1) == (char)(k + 'a'))
                        dp[i][j][k] = dp[i - 1][j][k] + 1;
                    else if (j != 0)

                        // The ith character is not equal to
                        // kth, j != 0 as for j = 0
                        // we have done previously
                        dp[i][j][k]
                            = dp[i - 1][j - 1][k] + 1;
                }
            }
        }

        // The ans array which stores length of
        // longest substring for any character
        for (int i = 1; i <= n; i++) {
            for (int j = 0; j < 26; j++) {
                for (int k = 1; k <= n; k++)

                    // Maximum among all indices
                    ans[i][j]
                        = Math.max(ans[i][j], dp[k][i][j]);
            }
        }
    }

    // Function to find the longest substrings
    public static ArrayList<Integer> query(String s,
                                           Pair queries[])
    {
        preprocess(s);
        ArrayList<Integer> res = new ArrayList<Integer>();
        for (int i = 0; i < queries.length; i++) {
            res.add(ans[queries[i].first]
                       [queries[i].second - 'a']);
        }
        return res;
    }

    // Driver Code
    public static void main(String[] args)
    {
        String s = "yamatonadeshiko";
        Pair queries[]
            = { new Pair(1, 'a'), new Pair(2, 'a'),
                new Pair(3, 'a'), new Pair(4, 'a'),
                new Pair(5, 'a'), new Pair(1, 'b'),
                new Pair(2, 'b'), new Pair(3, 'b'),
                new Pair(4, 'b'), new Pair(5, 'b') };

        // Function call
        ArrayList<Integer> sol = query(s, queries);
        for (Integer x : sol)
            System.out.print(x + " ");
    }
}

// This code is contributed by Rohit Pradhan

# Python code of the above Metioned approach
dp = [[[0 for i in range(26)] for j in range(101)] for k in range(101)]
ans = [[0 for i in range(26)] for j in range(101)]

# Precomputation to get the values
# of the function f() and g()
def preprocess(s):
    n = len(s)
    for i in range(n+1):
        for j in range(n+1):
            for k in range(26):
                dp[i][j][k] = 0
    for i in range(1, n+1):
      
        # For the existing character in the string
        # the length will be 1
        dp[i][0][ord(s[i-1])-ord('a')] = 1
    for i in range(1, n+1):
        for j in range(i+1):
            for k in range(26):
              
                # The ith character of the string
                # is equal to kth
                if s[i-1] == chr(k+ord('a')):
                    dp[i][j][k] = dp[i-1][j][k]+1
                elif j != 0:
                  
                    # The ith character is not equal to
                    # kth, j != 0 as for j = 0
                    # we have done previously
                    dp[i][j][k] = dp[i-1][j-1][k]+1
                    
    # The ans array which stores length of
    # longest substring for any character
    for i in range(n+1):
        for j in range(26):
            ans[i][j] = 0

    for i in range(1, n+1):
        for j in range(26):
            for k in range(1, n+1):
                ans[i][j] = max(ans[i][j], dp[k][i][j])

# Function to find the longest substrings
def query(s, queries):
    preprocess(s)
    res = []
    for u in queries:
        res.append(ans[u[0]][ord(u[1])-ord('a')])
    return res

# Driver Code
if __name__ == "__main__":
    s = "yamatonadeshiko"
    queries = []
    queries.append([1, 'a'])
    queries.append([2, 'a'])
    queries.append([3, 'a'])
    queries.append([4, 'a'])
    queries.append([5, 'a'])
    queries.append([1, 'b'])
    queries.append([2, 'b'])
    queries.append([3, 'b'])
    queries.append([4, 'b'])
    queries.append([5, 'b'])
    
    # Function call
    sol = query(s, queries)
    print(*sol)
    
# This code is contributed by nebula_242

// Include namespace system
using System;
using System.Collections.Generic;

public class Pair
{
  public int first;
  public char second;
  public Pair(int fi, char sec)
  {
    this.first = fi;
    this.second = sec;
  }
}
public class GFG
{
  public static int[,,] dp = new int[1001,1001,26];
  public static int[,] ans = new int[1001,26];
  // Precomputation to get the values
  // of the function f() and g()
  public static void preprocess(String s)
  {
    var n = s.Length;
    for (int i = 1; i <= n; i++)
    {
      // For the existing character in the string
      // the length will be 1
      GFG.dp[i,0,(int)(s[i - 1]) - (int)('a')] = 1;
    }
    for (int i = 1; i <= n; i++)
    {
      for (int j = 0; j <= i; j++)
      {
        for (int k = 0; k < 26; k++)
        {
          // The ith character of the string
          // is equal to kth
          if (s[i - 1] == (char)(k + (int)('a')))
          {
            GFG.dp[i,j,k] = GFG.dp[i - 1,j,k] + 1;
          }
          else if (j != 0)
          {
            // The ith character is not equal to
            // kth, j != 0 as for j = 0
            // we have done previously
            GFG.dp[i,j,k] = GFG.dp[i - 1,j - 1,k] + 1;
          }
        }
      }
    }
    // The ans array which stores length of
    // longest substring for any character
    for (int i = 1; i <= n; i++)
    {
      for (int j = 0; j < 26; j++)
      {
        for (int k = 1; k <= n; k++)
        {
          // Maximum among all indices
          GFG.ans[i,j] = Math.Max(GFG.ans[i,j],GFG.dp[k,i,j]);
        }
      }
    }
  }
  // Function to find the longest substrings
  public static List<int> query(String s, Pair[] queries)
  {
    GFG.preprocess(s);
    var res = new List<int>();
    for (int i = 0; i < queries.Length; i++)
    {
      res.Add(GFG.ans[queries[i].first,(int)(queries[i].second) - (int)('a')]);
    }
    return res;
  }

  // Driver Code
  public static void Main(String[] args)
  {
    var s = "yamatonadeshiko";
    Pair[] queries = {new Pair(1, 'a'), new Pair(2, 'a'), new Pair(3, 'a'), new Pair(4, 'a'), new Pair(5, 'a'), new Pair(1, 'b'), new Pair(2, 'b'), new Pair(3, 'b'), new Pair(4, 'b'), new Pair(5, 'b')};

    // Function call
    var sol = GFG.query(s, queries);
    foreach (int x in sol)
    {            Console.Write(string.Join(", ",x) + " ");
    }
  }
}

// This code is contributed by aadityaburujwale.

    <script>
        // JavaScript code of above mentioned approach


        let dp = new Array(1001).fill(0).map(() => new Array(1001).fill(0).map(() => new Array(26).fill(0)));
        let ans = new Array(1001).fill(0).map(() => new Array(26).fill(0));

        // Precomputation to get the values
        // of the function f() and g()
        const preprocess = (s) => {
            let n = s.length;
            for (let i = 1; i <= n; i++) {

                // For the existing character in the string
                // the length will be 1
                dp[i][0][s.charCodeAt(i - 1) - 'a'.charCodeAt(0)] = 1;
            }
            for (let i = 1; i <= n; i++) {
                for (let j = 0; j <= i; j++) {
                    for (let k = 0; k < 26; k++) {

                        // The ith character of the string
                        // is equal to kth
                        if (s[i - 1] == String.fromCharCode(k + 'a'.charCodeAt(0)))
                            dp[i][j][k] = dp[i - 1][j][k] + 1;
                        else if (j != 0)

                            // The ith character is not equal to
                            // kth, j != 0 as for j = 0
                            // we have done previously
                            dp[i][j][k] = dp[i - 1][j - 1][k] + 1;
                    }
                }
            }

            // The ans array which stores length of
            // longest substring for any character

            for (let i = 1; i <= n; i++) {
                for (let j = 0; j < 26; j++) {
                    for (let k = 1; k <= n; k++)

                        // Maximum among all indices
                        ans[i][j] = Math.max(ans[i][j], dp[k][i][j]);
                }
            }
        }

        // Function to find the longest substrings
        const query = (s, queries) => {
            preprocess(s);
            let res = [];
            for (let u in queries) {
                res.push(ans[queries[u][0]][queries[u][1].charCodeAt(0) - 'a'.charCodeAt(0)]);
            }
            return res;
        }

        // Driver code

        let s = "yamatonadeshiko";
        let queries = [];
        queries.push([1, 'a']);
        queries.push([2, 'a']);
        queries.push([3, 'a']);
        queries.push([4, 'a']);
        queries.push([5, 'a']);
        queries.push([1, 'b']);
        queries.push([2, 'b']);
        queries.push([3, 'b']);
        queries.push([4, 'b']);
        queries.push([5, 'b']);

        // Function call
        let sol = query(s, queries);
        for (let x in sol)
            document.write(`${sol[x]} `);

        // This code is contributed by rakeshsahni

    </script>

3 4 5 7 8 1 2 3 4 5 

Time complexity: O(26*N² + Q)
Auxiliary Space: O(26*N²)