Written on: October 30, 2023
Problem Description
This is a problem of finding the permutation of a given number through an input with issues and determining its order in the sequence. For example, if the given array is [1, 2, 3], the permutations of this array are as follows:
- [1, 2, 3]
- [1, 3, 2]
- [2, 1, 3]
- [2, 3, 1]
- [3, 1, 2]
- [3, 2, 1]
In this case, if the given order of the array is [2, 3, 1], it corresponds to the 4th position in total.
Input and Output Format
Input
The first line contains two integers N (1 ≤ N ≤ 9) and K (1 ≤ K ≤ N!). This represents the total number of integers in the array and the specific order of the permutation to find.
Output
Print the K-th permutation that is to be found.
Problem Solving
To solve this problem, the following steps should be followed:
1. Problem Analysis
Using the given N and K, we need to create an array consisting of N numbers and find the K-th permutation of that array. By utilizing the concept of permutations, we can create several combinations that reveal the order of all numbers. For example, when N is 3 and the array is [1, 2, 3], finding the permutations of this array is very intuitive.
2. Understanding Permutation Generation Algorithms
There are various approaches to generating permutations. One of them is a recursive method. However, since the problem is to find a specific K-th permutation, rather than using exhaustive search, we will use the properties of permutations to directly calculate the K-th permutation. To do this, we can implement a method that recursively generates possible number combinations to check for the K-th combination.
3. C# Implementation
Now, let’s implement the above approach in C#. Below is the code written in C#:
using System;
using System.Collections.Generic;
class Program
{
static void Main(string[] args)
{
// Read N and K
string[] input = Console.ReadLine().Split();
int N = int.Parse(input[0]);
int K = int.Parse(input[1]);
// Initialize the numbers list
List numbers = new List();
for (int i = 1; i <= N; i++)
{
numbers.Add(i);
}
// Result list to store the sequence
List result = new List();
bool[] used = new bool[N];
// Recursive call to find the K-th permutation
FindKthPermutation(numbers, used, result, K, 0);
}
static void FindKthPermutation(List numbers, bool[] used, List result, int K, int depth)
{
// Base condition: if depth is N
if (depth == numbers.Count)
{
// If we found the K-th permutation, print it
K--;
if (K == 0)
{
Console.WriteLine(string.Join(" ", result));
}
return;
}
for (int i = 0; i < numbers.Count; i++)
{
if (!used[i])
{
// Mark as used
result.Add(numbers[i]);
used[i] = true;
// Recursive call
FindKthPermutation(numbers, used, result, K, depth + 1);
// Backtrack
used[i] = false;
result.RemoveAt(result.Count - 1);
}
}
}
}
4. Code Explanation
The above C# code works on the following principles:
Input Handling
The code reads the values of N and K from the user. It then generates a list containing numbers from 1 to N.
Recursive Permutation Generation
The FindKthPermutation method generates permutations recursively. It continues until the current depth equals N, adding unused numbers to the list and making recursive calls with these values.
K-th Permutation Check
If K becomes 0, it means we have found the K-th permutation in the current list, so we print this list. Afterward, it backtracks to restore the unused numbers in the process.
5. Time Complexity
The base time complexity of this algorithm is O(N!). This is because it can consider all N! permutations through recursive calls. However, we can reduce the implementation time in many cases because we are directly finding the K-th permutation.
6. Conclusion
In this tutorial, we learned about the principles of generating permutations and how to find the K-th permutation among them. We verified the process of solving the problem by invoking functions recursively using C#. The method of finding a specific order in coding tests can be applied to various problems. It is essential to continuously practice to master these strategies.