# Count Distinct Subsequences

Given a string, find the count of distinct subsequences of it.

Examples:

Input: str = “gfg”
Output: 7
Explanation: The seven distinct subsequences are “”, “g”, “f”, “gf”, “fg”, “gg” and “gfg”

Input: str = “ggg”
Output: 4
Explanation: The four distinct subsequences are “”, “g”, “gg” and “ggg”

The problem of counting distinct subsequences is easy if all characters of input string are distinct. The count is equal to nC0 + nC1 + nC2 + … nCn = 2n.
How to count distinct subsequences when there can be repetition in input string?
A Simple Solution to count distinct subsequences in a string with duplicates is to generate all subsequences. For every subsequence, store it in a hash table if it doesn’t exist already. The time complexity of this solution is exponential and it requires exponential extra space.

Method 1(Naive Approach): Using a set (without Dynamic Programming)

Approach: Generate all the possible subsequences of a given string. The subsequences of a string can be generated in the following manner:

1. Include a particular element(say ith) in the output array and recursively call the function for the rest of the input string. This results in the subsequences of a string having ith character.
2. Exclude a particular element(say ith) and recursively call the function for the rest of the input string. This contains all the subsequences which don’t have the ith character.
Once we have generated a subsequence, in the base case of the function we insert that generated subsequence in an unordered set. An unordered Set is a Data structure, that stores distinct elements in an unordered manner. This way we insert all the generated subsequences in the set and print the size of the set as our answer because at last, the set will contain only distinct subsequences.

Implementation:

## C++

 `// C++ program to print distinct``// subsequences of a given string``#include ``using` `namespace` `std;` `// Create an empty set to store the subsequences``unordered_set sn;` `// Function for generating the subsequences``void` `subsequences(``char` `s[], ``char` `op[], ``int` `i, ``int` `j)``{` `    ``// Base Case``    ``if` `(s[i] == ``'\0'``) {``        ``op[j] = ``'\0'``;` `        ``// Insert each generated``        ``// subsequence into the set``        ``sn.insert(op);``        ``return``;``    ``}` `    ``// Recursive Case``    ``else` `{``        ``// When a particular character is taken``        ``op[j] = s[i];``        ``subsequences(s, op, i + 1, j + 1);` `        ``// When a particular character isn't taken``        ``subsequences(s, op, i + 1, j);``        ``return``;``    ``}``}` `// Driver Code``int` `main()``{``    ``char` `str[] = ``"ggg"``;``    ``int` `m = ``sizeof``(str) / ``sizeof``(``char``);``    ``int` `n = ``pow``(2, m) + 1;` `    ``// Output array for storing``    ``// the generating subsequences``    ``// in each call``    ``char` `op[m+1]; ``//extra one for having \0 at the end` `    ``// Function Call``    ``subsequences(str, op, 0, 0);` `    ``// Output will be the number``    ``// of elements in the set``    ``cout << sn.size();``    ``sn.clear();``    ``return` `0;` `    ``// This code is contributed by Kishan Mishra``}`

## Java

 `// java program to print distinct``// subsequences of a given string``import` `java.io.*;``import` `java.lang.Math;``import` `java.util.*;` `class` `GFG {``    ``// Function for generating the subsequences``    ``public` `static` `void` `subsequences(Set sn,``                                    ``char``[] s, ``char``[] op,``                                    ``int` `i, ``int` `j, ``int` `n)``    ``{``        ``// Base Case``        ``if` `(i == n) {``            ``op[j] = ``'\0'``;` `            ``// Insert each generated``            ``// subsequence into the set``            ``sn.add(String.valueOf(op));``            ``return``;``        ``}` `        ``// Recursive Case``        ``else` `{``            ``// When a particular character is taken``            ``op[j] = s[i];``            ``subsequences(sn, s, op, i + ``1``, j + ``1``, n);` `            ``// When a particular character isn't taken``            ``subsequences(sn, s, op, i + ``1``, j, n);``            ``return``;``        ``}``    ``}` `    ``public` `static` `void` `main(String[] args)``    ``{``        ``char``[] str = { ``'g'``, ``'g'``, ``'g'` `};``        ``int` `m = str.length;``        ``int` `n = (``int``)Math.pow(``2``, m) + ``1``;` `        ``// Create an empty set to store the subsequences``        ``Set sn = ``new` `HashSet();` `        ``// Output array for storing``        ``// the generating subsequences``        ``// in each call``        ``char``[] op = ``new` `char``[m + ``1``]; ``// extra one for having``                                     ``// \0 at the end` `        ``// Function Call``        ``subsequences(sn, str, op, ``0``, ``0``, m);` `        ``// Output will be the number``        ``// of elements in the set``        ``System.out.println(sn.size());``        ``sn.clear();` `        ``// This code is contributed by Aditya Sharma``    ``}``}`

## Python3

 `# Python3 program to print``# distinct subsequences of``# a given string``import` `math` `# Create an empty set``# to store the subsequences``sn ``=` `[]``global` `m``m ``=` `0` `# Function for generating``# the subsequences`  `def` `subsequences(s, op, i, j):` `    ``# Base Case``    ``if``(i ``=``=` `m):``        ``op[j] ``=` `None``        ``temp ``=` `"".join([i ``for` `i ``in` `op ``if` `i])` `        ``# Insert each generated``        ``# subsequence into the set``        ``sn.append(temp)``        ``return` `    ``# Recursive Case``    ``else``:` `        ``# When a particular``        ``# character is taken``        ``op[j] ``=` `s[i]` `        ``subsequences(s, op,``                     ``i ``+` `1``, j ``+` `1``)` `        ``# When a particular``        ``# character isn't taken``        ``subsequences(s, op,``                     ``i ``+` `1``, j)``        ``return`  `# Driver Code``str` `=` `"ggg"``m ``=` `len``(``str``)``n ``=` `int``(math.``pow``(``2``, m) ``+` `1``)` `# Output array for storing``# the generating subsequences``# in each call``op ``=` `[``None` `for` `i ``in` `range``(n)]` `# Function Call``subsequences(``str``, op, ``0``, ``0``)` `# Output will be the number``# of elements in the set``print``(``len``(``set``(sn)))` `# This code is contributed by avanitrachhadiya2155`

## C#

 `// C# program to print distinct``// subsequences of a given string``using` `System;``using` `System.Collections.Generic;` `class` `GFG ``{` `  ``// Function for generating the subsequences``  ``public` `static` `void` `subsequences(HashSet<``string``> sn,``                                  ``char``[] s, ``char``[] op,``                                  ``int` `i, ``int` `j, ``int` `n)``  ``{``    ``// Base Case``    ``if` `(i == n) {``      ``op[j] = ``'\0'``;` `      ``// Insert each generated``      ``// subsequence into the set``      ``sn.Add(``string``.Join(``""``, op));``      ``return``;``    ``}` `    ``// Recursive Case``    ``else` `{``      ``// When a particular character is taken``      ``op[j] = s[i];``      ``subsequences(sn, s, op, i + 1, j + 1, n);` `      ``// When a particular character isn't taken``      ``subsequences(sn, s, op, i + 1, j, n);``      ``return``;``    ``}``  ``}` `  ``public` `static` `void` `Main(``string``[] args)``  ``{``    ``char``[] str = { ``'g'``, ``'g'``, ``'g'` `};``    ``int` `m = str.Length;``    ``int` `n = (``int``)Math.Pow(2, m) + 1;` `    ``// Create an empty set to store the subsequences``    ``HashSet<``string``> sn = ``new` `HashSet<``string``>();` `    ``// Output array for storing``    ``// the generating subsequences``    ``// in each call``    ``char``[] op = ``new` `char``[m + 1]; ``// extra one for having``    ``// \0 at the end` `    ``// Function Call``    ``subsequences(sn, str, op, 0, 0, m);` `    ``// Output will be the number``    ``// of elements in the set``    ``Console.WriteLine(sn.Count);` `  ``}``}` `// This code is contributed by Abhijeet Kumar(abhijeet19403)`

## Javascript

 ``

Output
```4

```

Time Complexity: O(2^n)
Auxiliary Space: O(2^n)
where n is the length of the string.

Method 2(Efficient Approach): Using Dynamic Programming

An Efficient Solution doesn’t require the generation of subsequences.

`Let countSub(n) be count of subsequences of first n characters in input string. We canrecursively write it as below. countSub(n) = 2*Count(n-1) - RepetitionIf current character, i.e., str[n-1] of str hasnot appeared before, then    Repetition = 0Else:   Repetition  =  Count(m)   Here m is index of previous occurrence of   current character. We basically remove all   counts ending with previous occurrence of   current character.`

How does this work?
If there are no repetitions, then count becomes double of count for n-1 because we get count(n-1) more subsequences by adding current character at the end of all subsequences possible with n-1 length.
If there are repetitions, then we find a count of all distinct subsequences ending with the previous occurrence. This count can be obtained by recursively calling for an index of the previous occurrence.
Since the above recurrence has overlapping subproblems, we can solve it using Dynamic Programming.

Below is the implementation of the above idea.

## C++

 `// C++ program to count number of distinct``// subsequences of a given string.``#include ``using` `namespace` `std;``const` `int` `MAX_CHAR = 256;` `// Returns count of distinct subsequences of str.``int` `countSub(string str)``{``    ``// Create an array to store index``    ``// of last``    ``vector<``int``> last(MAX_CHAR, -1);` `    ``// Length of input string``    ``int` `n = str.length();` `    ``// dp[i] is going to store count of distinct``    ``// subsequences of length i.``    ``int` `dp[n + 1];` `    ``// Empty substring has only one subsequence``    ``dp[0] = 1;` `    ``// Traverse through all lengths from 1 to n.``    ``for` `(``int` `i = 1; i <= n; i++) {``        ``// Number of subsequences with substring``        ``// str[0..i-1]``        ``dp[i] = 2 * dp[i - 1];` `        ``// If current character has appeared``        ``// before, then remove all subsequences``        ``// ending with previous occurrence.``        ``if` `(last[str[i - 1]] != -1)``            ``dp[i] = dp[i] - dp[last[str[i - 1]]];` `        ``// Mark occurrence of current character``        ``last[str[i - 1]] = (i - 1);``    ``}` `    ``return` `dp[n];``}` `// Driver code``int` `main()``{``    ``cout << countSub(``"gfg"``);``    ``return` `0;``}`

## Java

 `// Java program to count number of distinct``// subsequences of a given string.``import` `java.util.ArrayList;``import` `java.util.Arrays;``public` `class` `Count_Subsequences {` `    ``static` `final` `int` `MAX_CHAR = ``256``;` `    ``// Returns count of distinct subsequences of str.``    ``static` `int` `countSub(String str)``    ``{``        ``// Create an array to store index``        ``// of last``        ``int``[] last = ``new` `int``[MAX_CHAR];``        ``Arrays.fill(last, -``1``);` `        ``// Length of input string``        ``int` `n = str.length();` `        ``// dp[i] is going to store count of distinct``        ``// subsequences of length i.``        ``int``[] dp = ``new` `int``[n + ``1``];` `        ``// Empty substring has only one subsequence``        ``dp[``0``] = ``1``;` `        ``// Traverse through all lengths from 1 to n.``        ``for` `(``int` `i = ``1``; i <= n; i++) {``            ``// Number of subsequences with substring``            ``// str[0..i-1]``            ``dp[i] = ``2` `* dp[i - ``1``];` `            ``// If current character has appeared``            ``// before, then remove all subsequences``            ``// ending with previous occurrence.``            ``if` `(last[(``int``)str.charAt(i - ``1``)] != -``1``)``                ``dp[i] = dp[i] - dp[last[(``int``)str.charAt(i - ``1``)]];` `            ``// Mark occurrence of current character``            ``last[(``int``)str.charAt(i - ``1``)] = (i - ``1``);``        ``}` `        ``return` `dp[n];``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String args[])``    ``{``        ``System.out.println(countSub(``"gfg"``));``    ``}``}``// This code is contributed by Sumit Ghosh`

## Python3

 `# Python3 program to count number of ``# distinct subsequences of a given string` `MAX_CHAR ``=` `256` `def` `countSub(ss):` `    ``# create an array to store index of last``    ``last ``=` `[``-``1` `for` `i ``in` `range``(MAX_CHAR ``+` `1``)]``    ` `    ``# length of input string``    ``n ``=` `len``(ss)``    ` `    ``# dp[i] is going to store count of ``    ``# discount subsequence of length of i``    ``dp ``=` `[``-``2` `for` `i ``in` `range``(n ``+` `1``)]``     ` `    ``# empty substring has only ``    ``# one subsequence``    ``dp[``0``] ``=` `1``    ` `    ``# Traverse through all lengths``    ``# from 1 to n ``    ``for` `i ``in` `range``(``1``, n ``+` `1``):``        ` `        ``# number of subsequence with ``        ``# substring str[0...i-1]``        ``dp[i] ``=` `2` `*` `dp[i ``-` `1``]` `        ``# if current character has appeared``        ``# before, then remove all subsequences``        ``# ending with previous occurrence.``        ``if` `last[``ord``(ss[i ``-` `1``])] !``=` `-``1``:``            ``dp[i] ``=` `dp[i] ``-` `dp[last[``ord``(ss[i ``-` `1``])]]``        ``last[``ord``(ss[i ``-` `1``])] ``=` `i ``-` `1``    ` `    ``return` `dp[n]``    ` `# Driver code``print``(countSub(``"gfg"``))` `# This code is contributed ``# by mohit kumar 29`

## C#

 `// C# program to count number of distinct``// subsequences of a given string.``using` `System;` `public` `class` `Count_Subsequences {` `    ``static` `readonly` `int` `MAX_CHAR = 256;` `    ``// Returns count of distinct subsequences of str.``    ``static` `int` `countSub(String str)``    ``{``        ``// Create an array to store index``        ``// of last``        ``int``[] last = ``new` `int``[MAX_CHAR];` `        ``for` `(``int` `i = 0; i < MAX_CHAR; i++)``            ``last[i] = -1;` `        ``// Length of input string``        ``int` `n = str.Length;` `        ``// dp[i] is going to store count of``        ``// distinct subsequences of length i.``        ``int``[] dp = ``new` `int``[n + 1];` `        ``// Empty substring has only one subsequence``        ``dp[0] = 1;` `        ``// Traverse through all lengths from 1 to n.``        ``for` `(``int` `i = 1; i <= n; i++) {``            ``// Number of subsequences with substring``            ``// str[0..i-1]``            ``dp[i] = 2 * dp[i - 1];` `            ``// If current character has appeared``            ``// before, then remove all subsequences``            ``// ending with previous occurrence.``            ``if` `(last[(``int``)str[i - 1]] != -1)``                ``dp[i] = dp[i] - dp[last[(``int``)str[i - 1]]];` `            ``// Mark occurrence of current character``            ``last[(``int``)str[i - 1]] = (i - 1);``        ``}``        ``return` `dp[n];``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``Console.WriteLine(countSub(``"gfg"``));``    ``}``}` `// This code is contributed 29AjayKumar`

## Javascript

 ``

Output
```7

```

Time Complexity: O(n)
Auxiliary Space: O(n)

Method 3: Using Map

Idea:

Let’s say we have 2 variables : `allCount` which adds up total distinct subsequence count and `levelCount` which stores the count of subsequences ending at index i. To find repetitions we will store the most recent levelCount for each character. Finally we will see how we can determine `allCount` using the `levelCount` variable.

Below is the steps to solve the problem:

• Declare a map .
• Start a loop to iterate through the characters of the input string s.
• Inside the loop, when i (the current index) is 0, this is the first character in the string.
• Set allCount to 1 since the first character is always unique.
• Update the map mp with the index 1 for the first character c.
• For characters at positions other than 0:
• Calculate the current levelCount as allCount + 1, representing the number of unique substrings at the current level.
• If char c is not present in map, it means the character is new (has not been seen before in this substring). In this case:
• Increment allCount by levelCount to account for the new character.
• If char c is present in map, it means the character has been seen before in this substring. In this case:
• Adjust allCount by adding levelCount – mp to account for the fact that some substrings may have been counted already.
• Update the map mp with the current levelCount for the character c since this is the latest level of uniqueness.

## C++

 `#include ``#include ``using` `namespace` `std;` `int` `countSub(string s) {``    ``map<``char``, ``int``> mp;` `    ``int` `n = s.size();``    ``int` `allCount = 0, levelCount = 0;` `    ``for` `(``int` `i = 0; i < n; i++) {``        ``char` `c = s[i];` `        ``if` `(i == 0) {``            ``allCount = levelCount = 1;``            ``mp = 1; ``// Initialize the map with the first character``            ``continue``;``        ``}` `        ``levelCount = allCount + 1;` `        ``if` `(mp.find(c) == mp.end()) {``            ``allCount += levelCount;``        ``} ``else` `{``            ``allCount += levelCount - mp;``        ``}``        ``mp = levelCount; ``// Update the count for the current character``    ``}` `    ``return` `allCount;``}` `int` `main() {``    ``string list[] = {``"abab"``, ``"gfg"``};` `    ``for` `(string s : list) {``        ``int` `cnt = countSub(s);``        ``int` `withEmptyString = cnt + 1;` `        ``cout << ``"With empty string count for "` `<< s << ``" is "` `<< withEmptyString << endl;``        ``cout << ``"Without empty string count for "` `<< s << ``" is "` `<< cnt << endl;``    ``}``    ``return` `0;``}`

## Java

 `// Java Program for above approach``import` `java.io.*;``import` `java.util.*;``class` `SubsequenceCount ``{` `  ``// Returns count of distinct ``  ``// subsequences of str. ``  ``public` `static` `int` `countSub(String s) ``  ``{``    ``HashMap map = ``new` `HashMap();` `    ``// Iterate from 0 to s.length()``    ``for``(``int` `i = ``0``; i < s.length(); i++) ``    ``{``      ``map.put(s.charAt(i), -``1``);``    ``}``    ` `    ``int` `allCount = ``0``;``    ``int` `levelCount = ``0``;``    ` `    ``// Iterate from 0 to s.length()``    ``for``(``int` `i=``0``;i list = Arrays.asList(``"abab"``,``"gfg"``);``    ` `    ``for``(String s : list)``    ``{``      ``int` `cnt = countSub(s);``      ``int` `withEmptyString = cnt+``1``;``      ``System.out.println(``"With empty string count for "` `+``                         ``s +``" is "` `+ withEmptyString);``      ``System.out.println(``"Without empty string count for "` `+``                         ``s + ``" is "` `+ cnt);``    ``}``  ``}``}``//Code is contributed by abhisht7`

## Python3

 `def` `count_sub(s):``    ``mp ``=` `{}` `    ``n ``=` `len``(s)``    ``all_count ``=` `level_count ``=` `0` `    ``for` `i ``in` `range``(n):``        ``c ``=` `s[i]` `        ``if` `i ``=``=` `0``:``            ``all_count ``=` `mp ``=` `level_count ``=` `1``            ``continue` `        ``level_count ``=` `all_count ``+` `1` `        ``if` `c ``not` `in` `mp:``            ``all_count ``+``=` `level_count``        ``else``:``            ``all_count ``+``=` `level_count ``-` `mp``        ``mp ``=` `level_count` `    ``return` `all_count` `if` `__name__ ``=``=` `"__main__"``:``    ``strings ``=` `[``"abab"``, ``"gfg"``]` `    ``for` `s ``in` `strings:``        ``cnt ``=` `count_sub(s)``        ``with_empty_string ``=` `cnt ``+` `1` `        ``print``(f``"With empty string count for {s} is {with_empty_string}"``)``        ``print``(f``"Without empty string count for {s} is {cnt}"``)`

## C#

 `using` `System;``using` `System.Collections.Generic;` `class` `GFG``{``    ``// Returns count of distinct subsequences of str.``    ``public` `static` `int` `countSub(``string` `s)``    ``{``        ``Dictionary<``char``, ``int``> map = ``new` `Dictionary<``char``, ``int``>();` `        ``// Iterate from 0 to s.length()``        ``for` `(``int` `i = 0; i < s.Length; i++)``        ``{``            ``if` `(!map.ContainsKey(s[i]))``            ``{``                ``map.Add(s[i], -1);``            ``}``        ``}` `        ``int` `allCount = 0;``        ``int` `levelCount = 0;` `        ``// Iterate from 0 to s.length()``        ``for` `(``int` `i = 0; i < s.Length; i++)``        ``{``            ``char` `c = s[i];` `            ``// Check if i equal to 0``            ``if` `(i == 0)``            ``{``                ``allCount = 1;``                ``if` `(!map.ContainsKey(c))``                ``{``                    ``map.Add(c, 1);``                ``}``                ``else``                ``{``                    ``map = 1;``                ``}``                ``levelCount = 1;``                ``continue``;``            ``}` `            ``// Replace levelCount with allCount + 1``            ``levelCount = allCount + 1;` `            ``// If map is less than 0``            ``if` `(map < 0)``            ``{``                ``allCount = (allCount + levelCount);``            ``}``            ``else``            ``{``                ``allCount = (allCount + levelCount - map);``            ``}` `            ``if` `(!map.ContainsKey(c))``            ``{``                ``map.Add(c, levelCount);``            ``}``            ``else``            ``{``                ``map = levelCount;``            ``}``        ``}` `        ``// Return answer``        ``return` `allCount;``    ``}` `    ``// Driver Code``    ``static` `void` `Main()``    ``{``        ``List<``string``> list = ``new` `List<``string``>();``        ``list.Add(``"abab"``);``        ``list.Add(``"gfg"``);` `        ``foreach` `(``string` `s ``in` `list)``        ``{``            ``int` `cnt = countSub(s);``            ``int` `withEmptyString = cnt + 1;` `            ``Console.WriteLine(``"With empty string count for "` `+``                                ``s + ``" is "` `+ withEmptyString);``            ``Console.WriteLine(``"Without empty string count for "` `+``                                ``s + ``" is "` `+ cnt);``        ``}``    ``}``}`

## Javascript

 `// Javascript Program for above approach``    ` `    ``// Returns count of distinct``  ``// subsequences of str.``    ``function` `countSub(s)``    ``{``        ``let map = ``new` `Map();``        ``// Iterate from 0 to s.length()``    ``for``(let i = 0; i < s.length; i++)``    ``{``      ``map.set(s[i], -1);``    ``}``     ` `    ``let allCount = 0;``    ``let levelCount = 0;``     ` `    ``// Iterate from 0 to s.length()``    ``for``(let i=0;i

Output
```With empty string count for abab is 12
Without empty string count for abab is 11
With empty string count for gfg is 7
Without empty string count for gfg is 6

```

Time Complexity: O(n)
Space Complexity: O(1)

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