Minimum number of prefix reversals to sort permutation of first N numbers
Last Updated :
11 Jul, 2025
Given N numbers that have a permutation of first N numbers. In a single operation, any prefix can be reversed. The task is to find the minimum number of such operations such that the numbers in the array are in increasing order.
Examples:
Input : a[] = {3, 1, 2}
Output : 2
Step1: Reverse the complete array a, a[] = {2, 1, 3}
Step2: Reverse the prefix(0-1) in s, a[] = {1, 2, 3}
Input : a[] = {1, 2, 4, 3}
Output : 3
Step1: Reverse the complete array a, a[] = {3, 4, 2, 1}
Step2: Reverse the prefix(0-1) in s, a[] = {4, 3, 2, 1}
Step3: Reverse the complete array a, a[] = {1, 2, 3, 4}
The approach to solve this problem is to use BFS.
- Encode the given numbers in a string. Sort the array and encode it into a string destination.
- Then do a BFS from the initial permutation. Each time, check all permutations induced by reversing a prefix of current permutation.
- If it is not visited, put it into the queue with the count of reversals done.
- When the permutation of the encoded string is same as the destination string, return the numbers of reversals required to reach here.
- That is, all permutations of strings are done and the minimal of those is returned as the answer.
Below is the implementation of the above approach:
C++
// C++ program to find
// minimum number of prefix reversals
// to sort permutation of first N numbers
#include <bits/stdc++.h>
using namespace std;
// Function to return the minimum prefix reversals
int minimumPrefixReversals(int a[], int n)
{
string start = "";
string destination = "", t, r;
for (int i = 0; i < n; i++) {
// converts the number to a character
// and add to string
start += to_string(a[i]);
}
sort(a, a + n);
for (int i = 0; i < n; i++) {
destination += to_string(a[i]);
}
// Queue to store the pairs
// of string and number of reversals
queue<pair<string, int> > qu;
pair<string, int> p;
// Initially push the original string
qu.push(make_pair(start, 0));
// if original string is the destination string
if (start == destination) {
return 0;
}
// iterate till queue is empty
while (!qu.empty()) {
// pair at the top
p = qu.front();
// string
t = p.first;
// pop the top-most element
qu.pop();
// perform prefix reversals for all index and push
// in the queue and check for the minimal
for (int j = 2; j <= n; j++) {
r = t;
// reverse the string till prefix j
reverse(r.begin(), r.begin() + j);
// if after reversing the string from first i index
// it is the destination
if (r == destination) {
return p.second + 1;
}
// push the number of reversals for string r
qu.push(make_pair(r, p.second + 1));
}
}
}
// Driver Code
int main()
{
int a[] = { 1, 2, 4, 3 };
int n = sizeof(a) / sizeof(a[0]);
// Calling function
cout << minimumPrefixReversals(a, n);
return 0;
}
// Java program to find minimum
// number of prefix reversals to
// sort permutation of first N numbers
import java.util.*;
public class Main
{
// function to find minimum prefix reversal through BFS
public static int minimumPrefixReversals(int[] a)
{
// size of array
int n = a.length;
// string for initial and goal nodes
String start = "", destination = "";
// string for manipulation in while loop
String original = "",modified = "";
// node to store temporary values from front of queue
Node temp = null;
// create the starting string
for (int i = 0; i < n; i++)
start += a[i];
// sort the array and prepare final destination string
Arrays.sort(a);
for (int i = 0; i < n; i++)
destination += a[i];
// this queue will store all the BFS siblings
Queue<Node> q = new LinkedList<>();
// place the starting node in queue
q.add(new Node(start, 0));
//base case:- if array is already sorted
if (start == destination)
return 0;
// loop until the size of queue is empty
while (q.size() != 0)
{
// put front node of queue in temporary variable
temp = q.poll();
// store the original string at this step
original = temp.string;
for (int j = 2; j <= n; j++)
{
// modified will be used to generate all
// manipulation of original string
// like if original = 1342
// modified = 3142 , 4312 , 2431
modified = original;
// generate the permutation by reversing
modified = reverse (modified , j);
if (modified.equals(destination))
{
// if string match then return
// the height of the current node
return temp.steps + 1;
}
// else put this node into queue
q.add(new Node(modified,temp.steps + 1));
}
}
// if no case match then default value
return Integer.MIN_VALUE;
}
// function to reverse the string upto an index
public static String reverse (String s , int index)
{
char temp []= s.toCharArray();
int i = 0;
while (i < index)
{
char c = temp[i];
temp[i] = temp[index-1];
temp[index-1] = c;
i++;index--;
}
return String.valueOf(temp);
}
// Driver code
public static void main(String []args)
{
int a[] = new int []{1,2,4,3};
System.out.println(minimumPrefixReversals(a));
}
// Node class to store a combined set of values
static class Node
{
public String string ;
public int steps;
public Node(String string,int steps)
{
this.string = string;
this.steps= steps;
}
}
}
// This code is contributed by Sparsh Singhal
// C# program to find minimum
// number of prefix reversals to
// sort permutation of first N numbers
using System;
using System.Collections.Generic;
class GFG
{
// Node class to store a combined set of values
public class Node
{
public String str;
public int steps;
public Node(String str,int steps)
{
this.str = str;
this.steps= steps;
}
}
// function to find minimum prefix reversal through BFS
public static int minimumPrefixReversals(int[] a)
{
// size of array
int n = a.Length;
// string for initial and goal nodes
String start = "", destination = "";
// string for manipulation in while loop
String original = "", modified = "";
// node to store temporary values
// from front of queue
Node temp = null;
// create the starting string
for (int i = 0; i < n; i++)
start += a[i];
// sort the array and prepare
// final destination string
Array.Sort(a);
for (int i = 0; i < n; i++)
destination += a[i];
// this queue will store all the BFS siblings
Queue<Node> q = new Queue<Node>();
// place the starting node in queue
q.Enqueue(new Node(start, 0));
//base case:- if array is already sorted
if (start == destination)
return 0;
// loop until the size of queue is empty
while (q.Count != 0)
{
// put front node of queue in temporary variable
temp = q.Dequeue();
// store the original string at this step
original = temp.str;
for (int j = 2; j <= n; j++)
{
// modified will be used to generate all
// manipulation of original string
// like if original = 1342
// modified = 3142 , 4312 , 2431
modified = original;
// generate the permutation by reversing
modified = reverse (modified , j);
if (modified.Equals(destination))
{
// if string match then return
// the height of the current node
return temp.steps + 1;
}
// else put this node into queue
q.Enqueue(new Node(modified, temp.steps + 1));
}
}
// if no case match then default value
return int.MinValue;
}
// function to reverse the string upto an index
public static String reverse (String s, int index)
{
char []temp = s.ToCharArray();
int i = 0;
while (i < index)
{
char c = temp[i];
temp[i] = temp[index - 1];
temp[index - 1] = c;
i++;index--;
}
return String.Join("", temp);
}
// Driver code
public static void Main(String []args)
{
int []a = new int []{1, 2, 4, 3};
Console.WriteLine(minimumPrefixReversals(a));
}
}
// This code is contributed by 29AjayKumar
<script>
class Node
{
constructor(string,steps)
{
this.string = string;
this.steps= steps;
}
}
function minimumPrefixReversals(a)
{
// size of array
let n = a.length;
// string for initial and goal nodes
let start = "", destination = "";
// string for manipulation in while loop
let original = "",modified = "";
// node to store temporary values from front of queue
let temp = null;
// create the starting string
for (let i = 0; i < n; i++)
start += a[i];
// sort the array and prepare final destination string
a.sort(function(a,b){return a-b;});
for (let i = 0; i < n; i++)
destination += a[i];
// this queue will store all the BFS siblings
let q = [];
// place the starting node in queue
q.push(new Node(start, 0));
//base case:- if array is already sorted
if (start == destination)
return 0;
// loop until the size of queue is empty
while (q.length != 0)
{
// put front node of queue in temporary variable
temp = q.shift();
// store the original string at this step
original = temp.string;
for (let j = 2; j <= n; j++)
{
// modified will be used to generate all
// manipulation of original string
// like if original = 1342
// modified = 3142 , 4312 , 2431
modified = original;
// generate the permutation by reversing
modified = reverse (modified , j);
if (modified == (destination))
{
// if string match then return
// the height of the current node
return temp.steps + 1;
}
// else put this node into queue
q.push(new Node(modified,temp.steps + 1));
}
}
// if no case match then default value
return Number.MIN_VALUE;
}
function reverse (s,index)
{
let temp = s.split("");
let i = 0;
while (i < index)
{
let c = temp[i];
temp[i] = temp[index-1];
temp[index-1] = c;
i++;index--;
}
return (temp).join("");
}
let a = [1, 2, 4, 3];
document.write(minimumPrefixReversals(a));
// This code is contributed by rag2127
</script>
# Python3 program to find
# minimum number of prefix reversals
# to sort permutation of [0] N numbers
from queue import Queue
# Function to return the minimum prefix reversals
def minimumPrefixReversals( a, n):
start = ""
destination = ""
for i in range(n):
# converts the number to a character
# and add to
start += str(a[i])
a.sort()
for i in range(n):
destination += str(a[i])
# Queue to store the pairs
# of and number of reversals
qu=Queue()
# Initially push the original
qu.put((start, 0))
# if original is the destination
if (start == destination) :
return 0
# iterate till queue is empty
while qu.not_empty :
# pair at the top
p = qu.get()
#
t = p[0]
# perform prefix reversals for all index and push
# in the queue and check for the minimal
for j in range(2,n+1) :
r = t
# reverse the till prefix j
tmpR=list(r)
tmpR[:j]=tmpR[j-1::-1]
r=''.join(tmpR)
# if after reversing the from [0] i index
# it is the destination
if (r == destination) :
return p[1] + 1
# push the number of reversals for r
qu.put((r, p[1] + 1))
# Driver Code
if __name__ == '__main__':
a = [ 1, 2, 4, 3]
n = len(a)
# Calling function
print(minimumPrefixReversals(a, n))
Time Complexity: O(N! * N2)
Auxiliary Space: O(N!)
Similar Reads
Basics & Prerequisites
Data Structures
Array Data StructureIn this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
3 min read
String in Data StructureA string is a sequence of characters. The following facts make string an interesting data structure.Small set of elements. Unlike normal array, strings typically have smaller set of items. For example, lowercase English alphabet has only 26 characters. ASCII has only 256 characters.Strings are immut
2 min read
Hashing in Data StructureHashing is a technique used in data structures that efficiently stores and retrieves data in a way that allows for quick access. Hashing involves mapping data to a specific index in a hash table (an array of items) using a hash function. It enables fast retrieval of information based on its key. The
2 min read
Linked List Data StructureA linked list is a fundamental data structure in computer science. It mainly allows efficient insertion and deletion operations compared to arrays. Like arrays, it is also used to implement other data structures like stack, queue and deque. Here’s the comparison of Linked List vs Arrays Linked List:
2 min read
Stack Data StructureA Stack is a linear data structure that follows a particular order in which the operations are performed. The order may be LIFO(Last In First Out) or FILO(First In Last Out). LIFO implies that the element that is inserted last, comes out first and FILO implies that the element that is inserted first
2 min read
Queue Data StructureA Queue Data Structure is a fundamental concept in computer science used for storing and managing data in a specific order. It follows the principle of "First in, First out" (FIFO), where the first element added to the queue is the first one to be removed. It is used as a buffer in computer systems
2 min read
Tree Data StructureTree Data Structure is a non-linear data structure in which a collection of elements known as nodes are connected to each other via edges such that there exists exactly one path between any two nodes. Types of TreeBinary Tree : Every node has at most two childrenTernary Tree : Every node has at most
4 min read
Graph Data StructureGraph Data Structure is a collection of nodes connected by edges. It's used to represent relationships between different entities. If you are looking for topic-wise list of problems on different topics like DFS, BFS, Topological Sort, Shortest Path, etc., please refer to Graph Algorithms. Basics of
3 min read
Trie Data StructureThe Trie data structure is a tree-like structure used for storing a dynamic set of strings. It allows for efficient retrieval and storage of keys, making it highly effective in handling large datasets. Trie supports operations such as insertion, search, deletion of keys, and prefix searches. In this
15+ min read
Algorithms
Searching AlgorithmsSearching algorithms are essential tools in computer science used to locate specific items within a collection of data. In this tutorial, we are mainly going to focus upon searching in an array. When we search an item in an array, there are two most common algorithms used based on the type of input
2 min read
Sorting AlgorithmsA Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read
Introduction to RecursionThe process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. A recursive algorithm takes one step toward solution and then recursively call itself to further move. The algorithm stops once we reach the solution
14 min read
Greedy AlgorithmsGreedy algorithms are a class of algorithms that make locally optimal choices at each step with the hope of finding a global optimum solution. At every step of the algorithm, we make a choice that looks the best at the moment. To make the choice, we sometimes sort the array so that we can always get
3 min read
Graph AlgorithmsGraph is a non-linear data structure like tree data structure. The limitation of tree is, it can only represent hierarchical data. For situations where nodes or vertices are randomly connected with each other other, we use Graph. Example situations where we use graph data structure are, a social net
3 min read
Dynamic Programming or DPDynamic Programming is an algorithmic technique with the following properties.It is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of
3 min read
Bitwise AlgorithmsBitwise algorithms in Data Structures and Algorithms (DSA) involve manipulating individual bits of binary representations of numbers to perform operations efficiently. These algorithms utilize bitwise operators like AND, OR, XOR, NOT, Left Shift, and Right Shift.BasicsIntroduction to Bitwise Algorit
4 min read
Advanced
Segment TreeSegment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array. It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array. The tree
3 min read
Pattern SearchingPattern searching algorithms are essential tools in computer science and data processing. These algorithms are designed to efficiently find a particular pattern within a larger set of data. Patten SearchingImportant Pattern Searching Algorithms:Naive String Matching : A Simple Algorithm that works i
2 min read
GeometryGeometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Interview Preparation
Practice Problem