Distinct Subsequences

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Average time to solve is 10m
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Problem statement

You have been given string 'S' of length 'N' that may contain duplicate alphabets. Your task is to return the count of distinct subsequences of it.

For example:

For the given string “deed” :
The possible subsequences are {“”}, {“d”}, {“e”}, {“de”}, {“e”}, {“de”}, {“ee”}, {“dee”}, {“d”}, {“dd”}, {“ed”}, {“ded”}, {“ed”}, {“ded”}, {“eed”} and {“deed”}.

As, {“d”}, {“e”}, {“de”}, {“ed”} and {“ded”} are repeated. 

The distinct subsequences are {“”}, {“d”}, {“e”}, {“de”}, {“ee”}, {“dee”}, {“dd”}, {“ed”}, {“ded”}, {“eed”} and {“deed”}

Thus, the output will be 11.

Note:

As the answer can be large, return your answer modulo 10^9  + 7.
Detailed explanation ( Input/output format, Notes, Images )
Input format: 
The first line contains an integer 'T' which denotes the number of test cases or queries to be run. Then, the T test cases follow.

The first and only line of each test case or query contains the string 'S'.
Output format: 
For each test case, print a single line containing the count of distinct subsequences modulo 10^9+7.

The output of each test case will be printed in a separate line.

Note:

You do not need to print anything, it has already been taken care of. Just implement the given function.
Constraints: 
1 <= T <= 10 
1 <= N <= 10 ^ 4 

where 'T' is the number of test cases and 'N' is the length of the given string 'S'.

Time limit: 1 sec.
Sample input 1:
2
abcd
dad
Sample output 1:
16
7
Explanation
For test case 1: 


Distinct subsequence are: {“”}, {a}, {b}, {ab}, {c}, {ac}, {bc}, {abc}, {d}, {ad}, {bd}, {abd}, {cd}, {acd}, {bcd} and {abcd}. Thus, the answer will be 16. 

For test case 2:

Distinct subsequences are: {“”}, {d}, {a}, {da}, {dd}, {ad} and {dad}. Thus, the answer is 7.    

For better understanding, let us take all the subsequences of string dad. They will be: 

{“ ”}, {d}, {a}, {da}, {d}, {dd}, {ad} and {dad}
Now, {d} occurs twice and thus we will count it only once.
Sample input 2:
2
xyzpqrs
abba
Sample output 2:
128
11
Hint

Try creating all the possible subsequences. Along with creating all the subsequences and maintain the count of all the distinct subsequences. 

Approaches (3)
Brute force
  • The IDea is to call a helper function so as to create all possible subsequences of the string which will contain:
    • String CUR = the possible subsequence of the string.
    • Int ID = the index of character which will either be included or excluded.
  • We maintain a set to count the distinct subsequences, each time a new subsequence is created, we add it into the set. Remember, the set does not contain duplicate elements.
  • We return the size of the set which is the count of distinct subsequences received from the helper function.

Algorithm:

void helper(STR, set, ID, CUR) :

  • Base Case: if ID >= STR.size()’, this means that CUR is a possible subsequence, thus, add CUR to set and return.
  • Call helper(STR, set, ID+1, CUR), this is where you exclude the current character.
  • Call helper(STR, set, ID+1, CUR + STR[ID]), this is where you include the current character.

 

int distinctSubsequences(S) :

  • Initialise an empty set.
  • Call helper function, helper(STR, set, 0, “”).
  • Return the size of the set.
Time Complexity

O(2 ^ N), where ‘N’ is the size of the given string. 

 

In the worst case, we are generating all the subsequences of a string. The possible number of subsequences can be 2 ^ N. Therefore, overall time complexity will be O(2 ^ N). 

Space Complexity

O(2 ^ N), where ‘N’ is the size of the given string.

 

In the worst case, extra space is required to store all the subsequences in a set and there can a maximum of 2 ^ N distinct subsequences when all characters of the input string are diffetrent and also an extra space is required used by a recursive stack which goes to a maximum depth of N.

Code Solution
(100% EXP penalty)
Distinct Subsequences
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