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

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

iin an array is said to beperfectif all the elements at indices smaller thaniare smaller than the element at indexi.

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**Examples:**

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

Input:N = 12, K = 4Output: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 <iostream>` `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

`<script>` `// javascript program for the above approach` `// Function to prvar the lexicographically` `// smallest permutation with K perfect indices` `function` `findPerfectIndex(N , K)` `{` ` ` ` ` `// Iterator to traverse the array` ` ` `var` `i = 0;` ` ` `// Traverse first K array indices` ` ` `for` `(; i < K; i++)` ` ` `{` ` ` `document.write((N - K + 1) + i+ ` `" "` `);` ` ` `}` ` ` `// Traverse remaining indices` ` ` `for` `(; i < N; i++)` ` ` `{` ` ` `document.write(i - K + 1+ ` `" "` `);` ` ` `}` `}` `// Driver Code` `var` `N = 10, K = 3;` `findPerfectIndex(N, K);` `// This code is contributed by 29AjayKumar` `</script>` |

**Output:**

8 9 10 1 2 3 4 5 6 7

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