Lexicographic rank of a Binary String

Given a binary string S of length N, the task is to find the lexicographic rank of the given string.

Examples:

Input: S = “001”
Output: 8
Explanation:
Strings in order of their increasing rank:
“0” = 1, 
“1” = 2, 
“00” = 3, 
“01” = 4, 
“10” = 5, 
“11” = 6, 
“000” = 7, 
“001” = 8.

Input: S = “01”
Output: 4

Naive Approach: The simplest approach is to generate all possible binary strings (consisting of 0s and 1s) up to length N and store them in an array. Once done, sort the array of strings in lexicographic order and print the index of the given string in the array. 
Time Complexity: O(2^N)
Auxiliary Space: O(1)

Efficient Approach: The idea is to generate a binary string consisting of 0s and 1s by replacing every occurrence of ‘a’ with 0 and ‘b’ with 1. Therefore, the strings in the list become:

“a” = 0, 
“b” = 1, 
“aa” = 00, 
“ab” = 01, 
“ba” = 10, 
“bb” = 11, 
“aaa” = 000, 
“aab” = 001 and so on.

It can be observed that for a string of size N, there exist (2^N – 2) strings of length less than N before that given string. Let it be equal to X. Now, its lexicographic position among strings of length N can be calculated by converting the string into its decimal equivalent number and adding 1 to it. Let it be equal to Y.

Rank of a string = X + Y + 1
                          = (2^N – 2) + Y + 1
                          = 2^N + Y – 1

Below is the implementation of the above approach:

C++

//C++ program for the above approach
#include <iostream>
using namespace std;

// Function to find the rank of a string
void findRank(string s)
{

  // Store the length of the string
  int N = s.size();

  // Stores its equivalent decimal value
  string bin;

  // Traverse the string
  for (int i = 0; i < N; i++)
  {
    if (s[i] == '0')
      bin += "0";
    else
      bin += "1";
  }

  // Store the number of strings of
  // length less than N occurring
  // before the given string
  long long X = 1ll<<N;

  // Store the decimal equivalent
  // number of string bin
  long long res = 0, val = 1;
  for(int i = N - 1; i >= 0; i--)
  {
    if (bin[i] == '1')
      res += (val);

    val *= 2ll;
  }

  long long Y = res;

  // Store the rank in answer
  long ans = X + Y - 1;

  // Print the answer
  cout << ans;
}

// Driver program
int main()
{
  string S = "001";
  findRank(S);
  return 0;
}

// This code is contributed by jyoti369.

Java

// Java program for the above approach
import java.io.*;
import java.lang.*;
import java.util.*;
public class Main {

    // Function to find the rank of a string
    static void findRank(String s)
    {
        // Store the length of the string
        int N = s.length();

        // Stores its equivalent decimal value
        StringBuilder sb = new StringBuilder("");

        // Traverse the string
        for (int i = 0; i < N; i++) {

            if (s.charAt(i) == '0')
                sb.append('0');
            else
                sb.append('1');
        }

        String bin = sb.toString();

        // Store the number of strings of
        // length less than N occurring
        // before the given string
        long X = (long)Math.pow(2, N);

        // Store the decimal equivalent
        // number of string bin
        long Y = Long.parseLong(bin, 2);

        // Store the rank in answer
        long ans = X + Y - 1;

        // Print the answer
        System.out.println(ans);
    }

    // Driver Code
    public static void main(String[] args)
    {
        String S = "001";
        findRank(S);
    }
}

Python3

# Python program for the above approach


# Function to find the rank of a string
def findRank(s):

    # Store the length of the string
    N = len(s);

    # Stores its equivalent decimal value
    sb = "";

    # Traverse the string
    for i in range(0,N):

        if (s[i] == '0'):
            sb += str('0');
        else:
            sb += str('1');

    bin = str(sb);

    # Store the number of strings of
    # length less than N occurring
    # before the given string
    X = pow(2, N);

    # Store the decimal equivalent
    # number of string bin
    Y = int(bin);

    # Store the rank in answer
    ans = X + Y - 1;

    # Print the answer
    print(ans);

# Driver Code
if __name__ == '__main__':
    S = "001";
    findRank(S);

# This code is contributed by shikhasingrajput

C#

// C# program to implement
// the above approach
using System;

class GFG
{

// Function to find the rank of a String
static void findRank(string s)
{

  // Store the length of the String
  int N = s.Length;

  // Stores its equivalent decimal value
  string bin = "";

  // Traverse the String
  for (int i = 0; i < N; i++)
  {
    if (s[i] == '0')
      bin += "0";
    else
      bin += "1";
  }

  // Store the number of string s of
  // length less than N occurring
  // before the given String
  int X = 1<<N;

  // Store the decimal equivalent
  // number of String bin
  int res = 0, val = 1;
  for(int i = N - 1; i >= 0; i--)
  {
    if (bin[i] == '1')
      res += (val);

    val *= 2;
  }
  int Y = res;

  // Store the rank in answer
  int ans = X + Y - 1;

  // Print the answer
  Console.Write(ans);
}

// Driver Code
public static void Main()
{
   string S = "001";
    findRank(S);
}
}

// This code is contributed by splevel62.

Javascript

<script>

// Javascript program for the above approach

// Function to find the rank of a string
function findRank( s)
{

// Store the length of the string
var N = s.length;

// Stores its equivalent decimal value
var bin = "";

// Traverse the string
for (var i = 0; i < N; i++)
{
    if (s[i] == '0')
        bin += "0";
    else
        bin += "1";
}

// Store the number of strings of
// length less than N occurring
// before the given string
var X = 1<<N;

// Store the decimal equivalent
// number of string bin
var res = 0, val = 1;
for(var i = N - 1; i >= 0; i--)
{
    if (bin[i] == '1')
        res += (val);

    val *= 2;
}

var Y = res;

// Store the rank in answer
var ans = X + Y - 1;

// Print the answer
document.write( ans);
}

// Driver program
var S = "001";
findRank(S);

</script>

Output:

8

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