Find number of times a string occurs as a subsequence in given string

Given two strings, find the number of times the second string occurs in the first string, whether continuous or discontinuous.

Examples: 

Input:  
string a = "neveropen"
string b = "Gks"

Output: 4

Explanation:  
The four strings are - (Check characters marked in bold)
GeeksforGeeks
GeeksforGeeks
GeeksforGeeks
GeeksforGeeks

If we carefully analyze the given problem, we can see that it can be easily divided into sub-problems. The idea is to process all characters of both strings one by one starting from either from left or right side. Let us traverse from right corner, there are two possibilities for every pair of character being traversed. 

m: Length of str1 (first string)
n: Length of str2 (second string)

If last characters of two strings are same, 
   1. We consider last characters and get count for remaining 
      strings. So we recur for lengths m-1 and n-1. 
   2. We can ignore last character of first string and 
      recurse for lengths m-1 and n.
else 
If last characters are not same, 
   We ignore last character of first string and 
   recurse for lengths m-1 and n.

Below is the implementation of above Naive recursive solution – 

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The time complexity of above solution is exponential. If we carefully analyze, we can see that many sub-problems are solved again and again. Since same sub-problems are called again, this problem has Overlapping sub-problems property. So the problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming problems, re-computations of same sub-problems can be avoided by constructing a temporary array that stores results of sub-problems.

Space Complexity: O(n)

We can solve it in 2 ways

1) Top-down DP

We can extend the recursion solution and stores the states of dp in 2d vector so that we don’t recompute the sub problems again.

4

Time complexity of above solutions is O(MN) excluding recursion stack space.

 Auxiliary space used by the program is O(MN).

2) Bottom-Up DP

Below is its implementation using Dynamic Programming – 

4

Time complexity of above solutions is O(MN). 
Auxiliary space used by the program is O(MN).

Another Method : 

This method was contributed by Kunal Hotwani

We can solve this problem by considering the fact that, in the final subsequences that will be chosen in the count, the character at b[i] will always follow the character at b[i – 1] (by the definition of a subsequence). 

Using this fact, we count the total number of occurrences of string b, using an array of size equal to the length of string b.

We can create an array of size equal to n (b.size()),  the j^th element in this array stores the number of subsequences in the string a which are equal to b[0]b[1]…b[j]. We will traverse the string a, each time updating all the values in this array. Let’s call this array count.

While traversing the string a, we will traverse the string b in reverse for each character of a, let’s say we are at the  i^th character of a (a[i]) and we start the reverse traversal of the b, if the j^th character of b (b[j]) is equal to a[i], a[i] can be appended to all the subsequences of a (traversed until now) which are equal to b[0]b[1]…b[j – 1] (number of these type of subsequences is stored in count[j -1]), making subsequences equal to b[0]b[1]…b[j] (number of these type of subsequences is stored in count[j]), thus count[j] can be increased by count[j – 1], when we encounter a character in string b which is equal to a[i].

NOTE : When j is equal to 0, the character a[i] cannot be appended to any subsequences, instead it can start a new subsequence equal to b[0], thus when a[i] is equal to b[0], we increase count[0] by 1 (As it stores the count all the subsequences of type b[0]).

Thus, in this method we are counting all the subsequences of a type b[0]b[1]…b[j] that can created using a particular character of a, a[i].

Finally, after traversing the complete string a, the final answer will be stored in count[n – 1], as we require the number of subsequences of type b[0]b[1]…b[n – 1]. 

The reason for traversal of b in reverse

Let us consider b = neveropen,  we will create the count array of size 5 and start the traversal of b string from front for each a[i].

When a[i] = ‘e’, we will first increase count[1](representing the first e in string b) by count[0], and then increase count[2](representing the second e in string b) by count[1], this not ideal because we always the need the value of count[j – 1] to represent the number of subsequences of type b[0]b[1]…b[j – 1] in the string a[0]a[1]…a[i – 1] as we cannot use the same ‘e’ twice in one subsequence (In this case count[1] will represent a[i] appended to all the subsequences of type b[0] and using this count[1] to update count[2] which will mean we use the same ‘e’ twice). 

Below is the code for this approach

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Time complexity of above solutions is O(MN). 
Auxiliary space used by the program is O(N).

This article is contributed by Aditya Goel. If you like neveropen and would like to contribute, you can also write an article using contribute.neveropen.org or mail your article to contribute@neveropen.org. See your article appearing on the neveropen main page and help other Geeks.
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