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Lexicographically smallest permutation of [1, N] based on given Binary string

  • Last Updated : 13 Sep, 2021
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Given a binary string S of size (N – 1), the task is to find the lexicographically smallest permutation P of the first N natural numbers such that for every index i, if S[i] equals ‘0‘ then P[i + 1] must be greater than P[i] and if S[i] equals ‘1‘ then P[i + 1] must be less than P[i].

Examples:

Input: N = 7, S = 100101
Output: 2 1 3 5 4 7 6
Explanation:
Consider the permutation as {2, 1, 3, 5, 4, 7, 6} that satisfy the given criteria as:
For index 0, S[0] = 1, P[1] < P[0], i.e. 1 < 2
For index 1, S[1] = 0, P[2] < P[1], i.e. 3 > 1
For index 2, S[2] = 0, P[3] < P[2], i.e. 5 > 3
For index 3, S[3] = 1, P[4] < P[3], i.e. 4 < 5
For index 4, S[4] = 0, P[5] < P[4], i.e. 7 > 4
For index 5, S[5] = 1, P[6] < P[5], i.e. 6 < 7

Input: N = 4, S = 000
Output: 1 2 3 4

Approach: The given problem can be solved by using the Greedy Approach by using smaller numbers at lower indices as much as possible will create the lexicographically smallest permutations. Follow the steps below to solve the problem:



  • Initialize an array, say ans[] of size N that stores the resultant permutation.
  • Traverse the given string S and perform the following steps:
    • If the value of S[i] equals ‘0‘ then assign the number greater than the last assigned number.
    • Otherwise, assign the smallest number which is larger than all currently used numbers.
  • After completing the above steps, print the resultant permutation formed in the array ans[].

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to generate the lexicographically
// smallest permutation according to the
// given criteria
void constructPermutation(string S, int N)
{
    // Stores the resultant permutation
    int ans[N];
 
    // Initialize the first elements to 1
    ans[0] = 1;
 
    // Traverse the given string S
    for (int i = 1; i < N; ++i) {
        if (S[i - 1] == '0') {
 
            // Number greater than last
            // number
            ans[i] = i + 1;
        }
        else {
            // Number equal to the last
            // number
            ans[i] = ans[i - 1];
        }
 
        // Correct all numbers to the left
        // of the current index
        for (int j = 0; j < i; ++j) {
            if (ans[j] >= ans[i]) {
                ans[j]++;
            }
        }
    }
 
    // Printing the permutation
    for (int i = 0; i < N; i++) {
        cout << ans[i];
        if (i != N - 1) {
            cout << " ";
        }
    }
}
 
// Driver Code
int main()
{
    string S = "100101";
    constructPermutation(S, S.length() + 1);
 
    return 0;
}

Java




// Java program for the above approach
import java.util.*;
 
class GFG {
 
    // Function to generate the lexicographically
    // smallest permutation according to the
    // given criteria
    static void constructPermutation(String S, int N)
    {
       
        // Stores the resultant permutation
        int[] ans = new int[N];
 
        // Initialize the first elements to 1
        ans[0] = 1;
 
        // Traverse the given string S
        for (int i = 1; i < N; ++i) {
            if (S.charAt(i - 1) == '0') {
 
                // Number greater than last
                // number
                ans[i] = i + 1;
            }
            else {
                // Number equal to the last
                // number
                ans[i] = ans[i - 1];
            }
 
            // Correct all numbers to the left
            // of the current index
            for (int j = 0; j < i; ++j) {
                if (ans[j] >= ans[i]) {
                    ans[j]++;
                }
            }
        }
 
        // Printing the permutation
        for (int i = 0; i < N; i++) {
            System.out.print(ans[i]);
            if (i != N - 1) {
                System.out.print(" ");
            }
        }
    }
 
    // Driver Code
    public static void main(String[] args) {
         
        String S = "100101";
        constructPermutation(S, S.length() + 1);
    }
}
 
// This code is contributed by code_hunt.

Python3




# Python Program to implement
# the above approach
 
# Function to generate the lexicographically
# smallest permutation according to the
# given criteria
def constructPermutation(S, N):
   
    # Stores the resultant permutation
    ans = [0] * N
 
    # Initialize the first elements to 1
    ans[0] = 1
 
    # Traverse the given string S
    for i in range(1, N):
        if (S[i - 1] == '0'):
 
            # Number greater than last
            # number
            ans[i] = i + 1
        else :
            # Number equal to the last
            # number
            ans[i] = ans[i - 1]
         
 
        # Correct all numbers to the left
        # of the current index
        for j in range(i):
            if (ans[j] >= ans[i]):
                ans[j] += 1
            
    # Printing the permutation
    for i in range(N):
        print(ans[i], end="")
        if (i != N - 1):
            print(" ", end="")
         
# Driver Code
S = "100101"
constructPermutation(S, len(S) + 1)
 
# This code is contributed by Saurabh Jaiswal

C#




// C# program for the above approach
using System;
 
class GFG {
 
    // Function to generate the lexicographically
    // smallest permutation according to the
    // given criteria
    static void constructPermutation(string S, int N)
    {
       
        // Stores the resultant permutation
        int[] ans = new int[N];
 
        // Initialize the first elements to 1
        ans[0] = 1;
 
        // Traverse the given string S
        for (int i = 1; i < N; ++i) {
            if (S[i - 1] == '0') {
 
                // Number greater than last
                // number
                ans[i] = i + 1;
            }
            else {
                // Number equal to the last
                // number
                ans[i] = ans[i - 1];
            }
 
            // Correct all numbers to the left
            // of the current index
            for (int j = 0; j < i; ++j) {
                if (ans[j] >= ans[i]) {
                    ans[j]++;
                }
            }
        }
 
        // Printing the permutation
        for (int i = 0; i < N; i++) {
            Console.Write(ans[i]);
            if (i != N - 1) {
                Console.Write(" ");
            }
        }
    }
 
    // Driver Code
    public static void Main()
    {
        string S = "100101";
        constructPermutation(S, S.Length + 1);
    }
}
 
// This code is contributed by ukasp.

Javascript




<script>
        // JavaScript Program to implement
        // the above approach
 
 
        // Function to generate the lexicographically
        // smallest permutation according to the
        // given criteria
        function constructPermutation(S, N) {
            // Stores the resultant permutation
            let ans = new Array(N);
 
            // Initialize the first elements to 1
            ans[0] = 1;
 
            // Traverse the given string S
            for (let i = 1; i < N; ++i) {
                if (S[i - 1] == '0') {
 
                    // Number greater than last
                    // number
                    ans[i] = i + 1;
                }
                else {
                    // Number equal to the last
                    // number
                    ans[i] = ans[i - 1];
                }
 
                // Correct all numbers to the left
                // of the current index
                for (let j = 0; j < i; ++j) {
                    if (ans[j] >= ans[i]) {
                        ans[j]++;
                    }
                }
            }
 
            // Printing the permutation
            for (let i = 0; i < N; i++) {
                document.write(ans[i]);
                if (i != N - 1) {
                    document.write(" ");
                }
            }
        }
 
        // Driver Code
 
        let S = "100101";
        constructPermutation(S, S.length + 1);
 
 
 
// This code is contributed by Potta Lokesh
    </script>
Output: 
2 1 3 5 4 7 6

 

Time Complexity: O(N2)
Auxiliary Space: O(N)

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