# Lexicographically smallest permutation of first N natural numbers having K perfect indices

• Last Updated : 14 May, 2021

Given two positive integers N and K, the task is to find lexicographically the smallest permutation of first N natural numbers such that there are exactly K perfect indices.

An index i in an array is said to be perfect if all the elements at indices smaller than i are smaller than the element at index i.

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Examples:

Input: N = 10, K = 3
Output: 8 9 10 1 2 3 4 5 6 7
Explanation: There are exactly 3 perfect indices 0, 1 and 2.

Input: N = 12, K = 4
Output: 9 10 11 12 1 2 3 4 5 6 7 8

Naive Approach: The idea is to generate all the possible permutations of first N natural numbers and print that permutation which is lexicographically smallest and has K perfect indices
Time Complexity: O(N*N!)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is to observe that the smallest permutation will have the last K elements of the range [1, N] in the front in increasing order. The remaining elements can be appended in increasing order. Follow the steps below to solve the problem:

• Initialize an array A[] to store the lexicographically smallest permutation of first N natural numbers.
• Iterate over the range [0, K – 1] using a variable, say i, and update the array element A[i] to store (N – K + 1) + i.
• Iterate over the range [K, N – 1] using the variable i and update the array element A[i] to (i – K + 1).
• After completing the above steps, print the array A[] that contains lexicographically the smallest permutation with K perfect indices.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `// Function to print the lexicographically``// smallest permutation with K perfect indices``void` `findPerfectIndex(``int` `N, ``int` `K)``{``    ``// Iterator to traverse the array``    ``int` `i = 0;` `    ``// Traverse first K array indices``    ``for` `(; i < K; i++) {``        ``cout << (N - K + 1) + i << ``" "``;``    ``}` `    ``// Traverse remaining indices``    ``for` `(; i < N; i++) {``        ``cout << i - K + 1 << ``" "``;``    ``}``}` `// Driver Code``int` `main()``{``    ``int` `N = 10, K = 3;``    ``findPerfectIndex(N, K);` `    ``return` `0;``}`

## Java

 `// Java program for the above approach``import` `java.util.*;``class` `GFG``{` `// Function to print the lexicographically``// smallest permutation with K perfect indices``static` `void` `findPerfectIndex(``int` `N, ``int` `K)``{``  ` `    ``// Iterator to traverse the array``    ``int` `i = ``0``;` `    ``// Traverse first K array indices``    ``for` `(; i < K; i++)``    ``{``        ``System.out.print((N - K + ``1``) + i+ ``" "``);``    ``}` `    ``// Traverse remaining indices``    ``for` `(; i < N; i++)``    ``{``        ``System.out.print(i - K + ``1``+ ``" "``);``    ``}``}` `// Driver Code``public` `static` `void` `main(String[] args)``{``    ``int` `N = ``10``, K = ``3``;``    ``findPerfectIndex(N, K);``}``}` `// This code is contributed by shikhasingrajput`

## Python3

 `# Python program for the above approach` `# Function to print the lexicographically``# smallest permutation with K perfect indices``def` `findPerfectIndex(N, K) :``    ` `    ``# Iterator to traverse the array``    ``i ``=` `0` `    ``# Traverse first K array indices``    ``for` `i ``in` `range``(K):``        ``print``((N ``-` `K ``+` `1``) ``+` `i, end ``=` `" "``)``    `  `    ``# Traverse remaining indices``    ``for` `i ``in` `range``(``3``, N):``        ``print``( i ``-` `K ``+` `1``, end ``=` `" "``)``    ` `# Driver Code` `N ``=` `10``K ``=` `3``findPerfectIndex(N, K)` `# This code is contributed by code_hunt.`

## C#

 `// C# program for the above approach``using` `System;``class` `GFG``{` `// Function to print the lexicographically``// smallest permutation with K perfect indices``static` `void` `findPerfectIndex(``int` `N, ``int` `K)``{``  ` `    ``// Iterator to traverse the array``    ``int` `i = 0;` `    ``// Traverse first K array indices``    ``for` `(; i < K; i++)``    ``{``        ``Console.Write((N - K + 1) + i+ ``" "``);``    ``}` `    ``// Traverse remaining indices``    ``for` `(; i < N; i++)``    ``{``        ``Console.Write(i - K + 1+ ``" "``);``    ``}``}` `// Driver Code``public` `static` `void` `Main(String[] args)``{``    ``int` `N = 10, K = 3;``    ``findPerfectIndex(N, K);``}``}` `// This code is contributed by susmitakundugoaldanga.`

## Javascript

 ``
Output:
`8 9 10 1 2 3 4 5 6 7`

Time Complexity: O(N)
Auxiliary Space: O(1)

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