Lexicographic ranking of binary strings

In this article, we will explore an interesting problem involving binary strings and lexicon ordering. Our task is to find the lexicographic ranking of a given binary string. We'll demonstrate our solution using C, a popular programming language known for its efficiency and flexibility.

Understanding dictionary order

Dictionary order (also called alphabetical order or dictionary order) is the arrangement of words in alphabetical order according to their constituent letters.

Problem Statement

Given a binary string, we need to determine its lexicographic ranking among all permutations. The lexicographic rank of a string is its position in the set of all permutations when they are listed lexicographically.

Solution method

Our approach includes the following key steps −

• Initialize count − Initialize a counter to store the number of '1's in the binary string.

• Ranking calculation − Iteratively traverses the binary string from left to right. If the current character is '1', calculate its rank using a combined formula and decrement the counter for each subsequent '1'.

• Return Result− The result will be the lexicographic order of the binary string.

C implementation

Example

The following C code outlines our solution −

#include <bits/stdc++.h>
using namespace std;

// Function to calculate factorial
int fact(int n) {
   int res = 1;
   for (int i = 1; i <= n; i++)
      res *= i;
   return res;
}

// Function to find lexicographic rank of a binary string
int lexRank(string str) {
   int rank = 1;
   int onesCount = count(str.begin(), str.end(), '1');
   
   for (char c : str) {
      if (c == '1') {
         onesCount--;
         rank += fact(onesCount);
      }
   }
   
   return rank;
}

int main() {
   string str = "110";
   
   int result = lexRank(str);
   cout << "Lexicographic rank of the binary string: " << result << endl;
   
   return 0;
}

Output

Lexicographic rank of the binary string: 3

The Chinese translation of

Explanation

is:

Explanation

Consider binary string −

str = "110"

The permutations and combinations of this binary string are: "011", "101", "110". In lexicographic order, these permutations are: "011", "101", "110".

The binary string "110" has a rank of 3, which is the output of our program.

in conclusion

The problem of finding the lexicographic ranking of a binary string is a very interesting problem, which is based on our understanding of binary strings, permutations and lexicographic order. This solution implemented in C shows how we can use basic programming constructs to solve this problem efficiently.

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