Subsequence queries after removing substrings

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Given two strings A and B, the problem is to find if string B will be a subsequence of string A if we remove the substring [A[i]..A[j]] from string A. Assume that there are Q queries giving the indices i and j and each query is independent of the other.

Examples:

Input : A = abcabcxy, B = acy
        Q = 2
        i = 2, j = 5
        i = 3, j = 6
Output :
Yes
No
Explanation :
In the first query we remove A[2]..A[5], 
getting acxy and acy is its subsequence.

In the second query we remove A[3]..A[6], 
getting abxy but acy is not its subsequence.

A brute force approach is, for each query remove the required substring from A and check if B is a subsequence of A, but is inefficient because we have to modify the string A for each query and also check if string B is its subsequence.

A more efficient approach is to do preprocessing on the strings as we have to encounter multiple queries. We can store the number of characters of string B that matches till each index of string A in both the forward and backward directions, in two separate arrays. Finally we can say the answer is Yes, if the following equation holds, otherwise No:
forward[i-1] + backward[j+1] >= length(B).
This works because we are removing A[i]..A[j] from A and want to know the sum of number of characters of B that match in A
from A[1]..A[i-1] and A[j+1]..A[len], which is a subsequence if this sum is atleast the length of string B.

Following is the implementation of the above approach:

C++

// CPP program for answering queries to check
// whether a string subsequence or not after
// removing a substring.
#include <bits/stdc++.h>
using namespace std;
 
// arrays to store results of preprocessing
int *fwd, *bwd;
 
// function to preprocess the strings
void preProcess(string a, string b)
{
    int n = a.size();
 
    // Allocate memory for fwd and bwd, and
    // initialize it as 0.
    fwd = new int[n]();
    bwd = new int[n](); 
 
    int j = 0;
 
    // store subsequence count in forward direction
    for (int i = 1; i <= a.size(); i++) {
        if (j < b.size() && a[i - 1] == b[j])
            j++;
 
        // store number of matches till now
        fwd[i] = j;
    }
 
    j = 0;
 
    // store subsequence count in backward direction
    for (int i = a.size(); i >= 1; i--) {
        if (j < b.size() &&
            a[i - 1] == b[b.size() - j - 1])
            j++;
 
        // store number of matches till now
        bwd[i] = j;
    }
}
 
// function that gives the output
void query(string a, string b, int x, int y)
{
    // length of remaining string A is less 
    // than B's length
    if ((x - 1 + a.size() - y) < b.size()) {
        cout << "No\n";
        return;
    }
 
    if (fwd[x - 1] + bwd[y + 1] >= b.size())
        cout << "Yes\n";
    else
        cout << "No\n";
}
 
// driver function
int main()
{
    string a = "abcabcxy", b = "acy";
    preProcess(a, b);
 
    // two queries
    int x = 2, y = 5;
    query(a, b, x, y);
 
    x = 3, y = 6;
    query(a, b, x, y);
 
    return 0;
}

Java

// Java program for answering 
// queries to check whether 
// a String subsequence or 
// not after removing a substring. 
 
class GFG 
{
    // arrays to store results 
    // of preprocessing 
 
    static int[] fwd = new int[100];
    static int[] bwd = new int[100];
 
    // function to preprocess 
    // the strings 
    static void preProcess(String a,
                            String b) 
    {
        int n = a.length();
 
        // initialize it as 0. 
        int j = 0;
 
        // store subsequence count 
        // in forward direction 
        for (int i = 1;
                i <= a.length(); i++) 
        {
            if (j < b.length() &&
                a.charAt(i - 1) == b.charAt(j))
            {
                j++;
            }
 
            // store number of 
            // matches till now 
            fwd[i] = j;
        }
 
        j = 0;
 
        // store subsequence count 
        // in backward direction 
        for (int i = a.length(); i >= 1; i--) 
        {
            if (j < b.length() && a.charAt(i - 1) ==
                    b.charAt(b.length() - j - 1)) 
            {
                j++;
            }
 
            // store number of 
            // matches till now 
            bwd[i] = j;
        }
    }
 
    // function that gives 
    // the output 
    static void query(String a, String b,
                            int x, int y) 
    {
        // length of remaining 
        // String A is less 
        // than B's length 
        if ((x - 1 + a.length() - y) < b.length()) 
        {
            System.out.print("No\n");
            return;
        }
 
        if (fwd[x - 1] + bwd[y + 1] >= b.length()) 
        {
            System.out.print("Yes\n");
        } 
        else
        {
            System.out.print("No\n");
        }
    }
 
    // Driver Code 
    public static void main(String[] args) 
    {
 
        String a = "abcabcxy", b = "acy";
        preProcess(a, b);
 
        // two queries 
        int x = 2, y = 5;
        query(a, b, x, y);
 
        x = 3;
        y = 6;
        query(a, b, x, y);
    }
} 
 
// This code is contributed by 29AjayKumar

Python3

# Python3 program for answering
# queries to check whether
# a String subsequence or
# not after removing a substring.
 
# arrays to store results
# of preprocessing
fwd = [0] * 100
bwd = [0] * 100
 
# function to preprocess
# the strings
def preProcess(a, b):
    n = len(a)
 
    # initialize it as 0.
    j = 0
 
    # store subsequence count
    # in forward direction
    for i in range(1, len(a) + 1):
        if j < len(b) and a[i - 1] == b[j]:
            j += 1
 
        # store number of
        # matches till now
        fwd[i] = j
 
    j = 0
 
    # store subsequence count
    # in backward direction
    for i in range(len(a), 0, -1):
        if (j < len(b) and
            a[i - 1] == b[len(b) - j - 1]):
            j += 1
 
        # store number of
        # matches till now
        bwd[i] = j
 
# function that gives
# the output
def query(a, b, x, y):
 
    # length of remaining
    # String A is less
    # than B's length
    if (x - 1 + len(a) - y) < len(b):
        print("No")
        return
 
    if (fwd[x - 1] + bwd[y + 1]) >= len(b):
        print("Yes")
    else:
        print("No")
 
# Driver Code
if __name__ == "__main__":
    a = "abcabcxy"
    b = "acy"
    preProcess(a, b)
 
    x = 2
    y = 5
 
    query(a, b, x, y)
 
    x = 3
    y = 6
    query(a, b, x, y)
 
# This code is contributed by
# sanjeev2552

C#

// C# program for answering 
// queries to check whether 
// a string subsequence or 
// not after removing a substring.
using System;
 
class GFG
{
    // arrays to store results
    // of preprocessing
    static int []fwd = new int[100];
    static int []bwd = new int[100];
     
    // function to preprocess
    // the strings
    static void preProcess(string a, 
                           string b)
    {
        int n = a.Length;
     
        // initialize it as 0.
        int j = 0;
     
        // store subsequence count
        // in forward direction
        for (int i = 1; 
                 i <= a.Length; i++) 
        {
            if (j < b.Length && 
                    a[i - 1] == b[j])
                j++;
     
            // store number of 
            // matches till now
            fwd[i] = j;
        }
     
        j = 0;
     
        // store subsequence count 
        // in backward direction
        for (int i = a.Length;
                 i >= 1; i--) 
        {
            if (j < b.Length &&
                a[i - 1] == b[b.Length - j - 1])
                j++;
     
            // store number of
            // matches till now
            bwd[i] = j;
        }
    }
     
    // function that gives
    // the output
    static void query(string a, string b, 
                      int x, int y)
    {
        // length of remaining 
        // string A is less 
        // than B's length
        if ((x - 1 + a.Length - y) < b.Length) 
        {
            Console.Write("No\n");
            return;
        }
     
        if (fwd[x - 1] + 
            bwd[y + 1] >= b.Length)
            Console.Write("Yes\n");
        else
            Console.Write("No\n");
    }
     
    // Driver Code
    static void Main()
    {
        string a = "abcabcxy", b = "acy";
        preProcess(a, b);
     
        // two queries
        int x = 2, y = 5;
        query(a, b, x, y);
     
        x = 3; y = 6;
        query(a, b, x, y);
    }
}
 
// This code is contributed by 
// Manish Shaw(manishshaw1)

Javascript

<script>
// Javascript program for answering 
// queries to check whether 
// a String subsequence or 
// not after removing a substring. 
 
// arrays to store results 
// of preprocessing 
let fwd = new Array(100);
let bwd = new Array(100);
 
// function to preprocess 
// the strings 
function preProcess(a,b)
{
    let n = a.length;
   
        // initialize it as 0. 
        let j = 0;
   
        // store subsequence count 
        // in forward direction 
        for (let i = 1;
                i <= a.length; i++) 
        {
            if (j < b.length &&
                a[i - 1] == b[j])
            {
                j++;
            }
   
            // store number of 
            // matches till now 
            fwd[i] = j;
        }
   
        j = 0;
   
        // store subsequence count 
        // in backward direction 
        for (let i = a.length; i >= 1; i--) 
        {
            if (j < b.length && a[i-1] ==
                    b[b.length - j - 1]) 
            {
                j++;
            }
   
            // store number of 
            // matches till now 
            bwd[i] = j;
        }
}
 
// function that gives 
// the output 
function query(a,b,x,y)
{
    // length of remaining 
        // String A is less 
        // than B's length 
        if ((x - 1 + a.length - y) < b.length) 
        {
            document.write("No<br>");
            return;
        }
   
        if (fwd[x - 1] + bwd[y + 1] >= b.length) 
        {
            document.write("Yes<br>");
        } 
        else
        {
            document.write("No<br>");
        }
}
 
// Driver Code 
let a = "abcabcxy", b = "acy";
preProcess(a, b);
 
// two queries 
let x = 2, y = 5;
query(a, b, x, y);
 
x = 3;
y = 6;
query(a, b, x, y);
 
// This code is contributed by rag2127
</script>

Output

Yes
No

The time complexity of the above approach is O(n + q), where q is the number of queries and n is the length of string A.

Auxiliary Space: O(n)

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