# Minimum number of given operations required to convert a permutation into an identity permutation

• Difficulty Level : Hard
• Last Updated : 26 May, 2021

Given a permutation P (P1, P2, P3, … Pn) of first n natural numbers. Find the minimum number of operations to convert it into an identity permutation i.e. 1, 2, 3, …, n where each operation is defined as:
P[i] = P[P[P[i]]] i from 1 to n (1 based indexing). If there is no way to convert then print -1.
Examples:

Input: arr[] = {2, 3, 1}
Output:
After 1 operation:
P = P[P[P]] = P[P] = P = 1
P = P[P[P]] = P[P] = P = 2
P = P[P[P]] = P[P] = P = 3
Thus after 1 operation we obtain an identity permutation.
Input: arr[] = {2, 1, 3}
Output: -1
There is no way to obtain identity permutation
no matter how many operations we apply.

Approach: First, find all the cycles in the given permutation. Here, a cycle is a directed graph in which there is an edge from an element e to the element on position e.
For example, Here’s the graph for permutation {4, 6, 5, 3, 2, 1, 8, 7} Now in one operation, each cycle of length 3k breaks into 3 cycles each of length k while cycles of length 3k+1 or 3k+2 do not break. Since in the end, we need all cycles of length 1, therefore, all cycles must be a power of 3 otherwise answer doesn’t exist. The answer would then be the maximum of log(base 3) of all cycle lengths.
Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;` `int` `calculateCycleOperations(``int` `len)``{``    ``int` `cycle_operations = 0;``    ``while` `(len) {``        ``len /= 3;``        ``++cycle_operations;``    ``}``    ``return` `--cycle_operations;``}` `// Function to return the minimum operations required``int` `minimumOperations(``int` `p[], ``int` `n)``{` `    ``// Array to keep track of visited elements``    ``bool` `visited[n + 1] = { 0 };` `    ``// To store the final answer``    ``int` `ans = 0;` `    ``// Looping through all the elements``    ``for` `(``int` `i = 1; i <= n; i++) {` `        ``// Current element``        ``int` `ele = p[i];` `        ``// If current element is not present in the``        ``// previous cycles then only consider this``        ``if` `(!visited[ele]) {` `            ``// Mark current element visited so that it``            ``// will not be considered in other cycles``            ``visited[ele] = 1;` `            ``// To store the length of each cycle``            ``int` `len = 1;``            ``ele = p[ele];` `            ``// Calculating cycle length``            ``while` `(!visited[ele]) {``                ``visited[ele] = 1;``                ``++len;``                ``ele = p[ele];``            ``}` `            ``// Operations needed for this cycle to reduce``            ``// to length 1 (if possible)``            ``int` `operations = calculateCycleOperations(len);` `            ``// Checking cycle length to be power of 3``            ``// if not, then return -1``            ``int` `num = ``pow``(3, operations);``            ``if` `(num != len) {``                ``return` `-1;``            ``}` `            ``// Taking maximum of the operations``            ``ans = max(ans, operations);``        ``}``    ``}``    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``// 1-based indexing``    ``int` `P[] = { -1, 4, 6, 5, 3, 2, 7, 8, 9, 1 };``    ``int` `n = (``sizeof``(P) / ``sizeof``(P)) - 1;` `    ``// Calling function``    ``cout << minimumOperations(P, n);` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;` `class` `GFG``{` `static` `int` `calculateCycleOperations(``int` `len)``{``    ``int` `cycle_operations = ``0``;``    ``while` `(len > ``0``)``    ``{``        ``len /= ``3``;``        ``++cycle_operations;``    ``}``    ``return` `--cycle_operations;``}` `// Function to return the minimum operations required``static` `int` `minimumOperations(``int` `p[], ``int` `n)``{` `    ``// Array to keep track of visited elements``    ``int` `[]visited = ``new` `int``[n+``1``];``    ``Arrays.fill(visited,``0``);` `    ``// To store the final answer``    ``int` `ans = ``0``;` `    ``// Looping through all the elements``    ``for` `(``int` `i = ``1``; i <= n; i++)``    ``{` `        ``// Current element``        ``int` `ele = p[i];` `        ``// If current element is not present in the``        ``// previous cycles then only consider this``        ``if` `(visited[ele] == ``0``)``        ``{` `            ``// Mark current element visited so that it``            ``// will not be considered in other cycles``            ``visited[ele] = ``1``;` `            ``// To store the length of each cycle``            ``int` `len = ``1``;``            ``ele = p[ele];` `            ``// Calculating cycle length``            ``while` `(visited[ele] == ``0``)``            ``{``                ``visited[ele] = ``1``;``                ``++len;``                ``ele = p[ele];``            ``}` `            ``// Operations needed for this cycle to reduce``            ``// to length 1 (if possible)``            ``int` `operations = calculateCycleOperations(len);` `            ``// Checking cycle length to be power of 3``            ``// if not, then return -1``            ``int` `num = (``int``)Math.pow(``3``, operations);``            ``if` `(num != len) {``                ``return` `-``1``;``            ``}` `            ``// Taking maximum of the operations``            ``ans = Math.max(ans, operations);``        ``}``    ``}``    ``return` `ans;``}` `// Driver code``public` `static` `void` `main(String args[])``{``    ``// 1-based indexing``    ``int` `P[] = { -``1``, ``4``, ``6``, ``5``, ``3``, ``2``, ``7``, ``8``, ``9``, ``1` `};``    ``int` `n = P.length-``1``;` `    ``// Calling function``    ``System.out.println(minimumOperations(P, n));``}``}` `// This code is contributed by``// Surendra_Gangwar`

## Python3

 `# Python3 implementation of the approach``def` `calculateCycleOperations(length):` `    ``cycle_operations ``=` `0``    ``while` `length > ``0``:``        ``length ``/``/``=` `3``        ``cycle_operations ``+``=` `1``    ` `    ``return` `cycle_operations ``-` `1` `# Function to return the minimum``# operations required``def` `minimumOperations(p, n):` `    ``# Array to keep track of visited elements``    ``visited ``=` `[``0``] ``*` `(n ``+` `1``)` `    ``# To store the final answer``    ``ans ``=` `0` `    ``# Looping through all the elements``    ``for` `i ``in` `range``(``1``, n ``+` `1``):` `        ``# Current element``        ``ele ``=` `p[i]` `        ``# If current element is not present in the``        ``# previous cycles then only consider this``        ``if` `not` `visited[ele]:` `            ``# Mark current element visited so that it``            ``# will not be considered in other cycles``            ``visited[ele] ``=` `1` `            ``# To store the length of each cycle``            ``length ``=` `1``            ``ele ``=` `p[ele]` `            ``# Calculating cycle length``            ``while` `not` `visited[ele]:``                ``visited[ele] ``=` `1``                ``length ``+``=` `1``                ``ele ``=` `p[ele]``            ` `            ``# Operations needed for this cycle to``            ``# reduce to length 1 (if possible)``            ``operations ``=` `calculateCycleOperations(length)` `            ``# Checking cycle length to be power``            ``# of 3 if not, then return -1``            ``num ``=` `pow``(``3``, operations)``            ``if` `num !``=` `length:``                ``return` `-``1` `            ``# Taking maximum of the operations``            ``ans ``=` `max``(ans, operations)``        ` `    ``return` `ans` `# Driver code``if` `__name__ ``=``=` `"__main__"``:` `    ``# 1-based indexing``    ``P ``=` `[``-``1``, ``4``, ``6``, ``5``, ``3``, ``2``, ``7``, ``8``, ``9``, ``1``]``    ``n ``=` `len``(P) ``-` `1` `    ``# Calling function``    ``print``(minimumOperations(P, n))` `# This code is contributed by Rituraj Jain`

## C#

 `// C# implementation of the above approach``using` `System;` `class` `GFG``{` `    ``static` `int` `calculateCycleOperations(``int` `len)``    ``{``        ``int` `cycle_operations = 0;``        ``while` `(len > 0)``        ``{``            ``len /= 3;``            ``++cycle_operations;``        ``}``        ``return` `--cycle_operations;``    ``}``    ` `    ``// Function to return the minimum operations required``    ``static` `int` `minimumOperations(``int` `[]p, ``int` `n)``    ``{``    ` `        ``// Array to keep track of visited elements``        ``int` `[]visited = ``new` `int``[n+1];` `        ``// To store the final answer``        ``int` `ans = 0;``    ` `        ``// Looping through all the elements``        ``for` `(``int` `i = 1; i <= n; i++)``        ``{``    ` `            ``// Current element``            ``int` `ele = p[i];``    ` `            ``// If current element is not present in the``            ``// previous cycles then only consider this``            ``if` `(visited[ele] == 0)``            ``{``    ` `                ``// Mark current element visited so that it``                ``// will not be considered in other cycles``                ``visited[ele] = 1;``    ` `                ``// To store the length of each cycle``                ``int` `len = 1;``                ``ele = p[ele];``    ` `                ``// Calculating cycle length``                ``while` `(visited[ele] == 0)``                ``{``                    ``visited[ele] = 1;``                    ``++len;``                    ``ele = p[ele];``                ``}``    ` `                ``// Operations needed for this cycle to reduce``                ``// to length 1 (if possible)``                ``int` `operations = calculateCycleOperations(len);``    ` `                ``// Checking cycle length to be power of 3``                ``// if not, then return -1``                ``int` `num = (``int``)Math.Pow(3, operations);``                ``if` `(num != len)``                ``{``                    ``return` `-1;``                ``}``    ` `                ``// Taking maximum of the operations``                ``ans = Math.Max(ans, operations);``            ``}``        ``}``        ``return` `ans;``    ``}``    ` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ``// 1-based indexing``        ``int` `[]P = { -1, 4, 6, 5, 3, 2, 7, 8, 9, 1 };``        ``int` `n = P.Length-1;``    ` `        ``// Calling function``        ``Console.WriteLine(minimumOperations(P, n));``    ``}``}` `// This code is contributed by Ryuga`

## PHP

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## Javascript

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Output:

`2`

Time Complexity : O(N*LogN)

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