Smallest subarray which upon repetition gives the original array

Given an array arr[] of N integers, the task is to find the smallest subarray brr[] of size at least 2 such that by performing repeating operation on the array brr[] gives the original array arr[]. Print "-1" if it is not possible to find such a subarray.

A repeating operation on an array is to append all the current element of the array to the same array again. 
For Example, if an array arr[] = {1, 2} then on repeating operation array becomes {1, 2, 1, 2}.

Examples:

Input: arr[] = {1, 2, 3, 3, 1, 2, 3, 3}
Output: {1, 2, 3, 3}
Explanation:
{1, 2, 3, 3} is the smallest subarray which when repeated 2 times gives the original array {1, 2, 3, 3, 1, 2, 3, 3}

Input: arr[] = {1, 1, 6, 1, 1, 7}
Output: -1
Explanation:
There doesn't exist any subarray.

Naive Approach: The idea is to generate all possible subarrays of length at least 2 and check whether repeating those subarrays gives the original array or not.

Time Complexity: O(N³)
Auxiliary Space: O(N)

Efficient Approach: The above approach can be optimized by observing the fact that the resultant subarray brr[] must start from the 1st index of the original array to generate arr[] on repeat. Therefore, generate only those subarrays which start from the 1st index and have a length of at least 2 and check whether repeating those subarrays gives the original array or not. Below are the steps:

1. Create an auxiliary array brr[] and insert the first two elements of the original array into it as the resulting array must be of at least two in size.
2. Traverse over the possible length of subarray [2, N/2 + 1] and check if the array brr[] of length i on repeating gives the original array arr[] or not.
3. If yes then print this subarray and break the loop.
4. Otherwise, insert the current element into the subarray and check again.
5. Repeat the above steps until all the subarrays are checked.
6. Print "-1" if the array brr[] is not found.

Below is the implementation of the above approach:

// C++ program for the above approach
#include <iostream>
#include <vector>
using namespace std;

// Function to print the array
void printArray(vector<int>& brr)
{
    for (auto& it : brr) {
        cout << it << ' ';
    }
}

// Function to find the smallest subarray
void RepeatingSubarray(int arr[], int N)
{
    // Corner Case
    if (N < 2) {
        cout << "-1";
    }

    // Initialize the auxiliary subarray
    vector<int> brr;

    // Push the first 2 elements into
    // the subarray brr[]
    brr.push_back(arr[0]);
    brr.push_back(arr[1]);

    // Iterate over the length of
    // subarray
    for (int i = 2; i < N / 2 + 1; i++) {

        // If array can be divided into
        // subarray of i equal length
        if (N % i == 0) {

            bool a = false;
            int n = brr.size();
            int j = i;

            // Check if on repeating the
            // current subarray gives the
            // original array or not
            while (j < N) {
                int K = j % i;
                if (arr[j] == brr[K]) {
                    j++;
                }
                else {
                    a = true;
                    break;
                }
            }

            // Subarray found
            if (!a && j == N) {
                printArray(brr);
                return;
            }
        }

        // Add current element into
        // subarray
        brr.push_back(arr[i]);
    }

    // No subarray found
    cout << "-1";
    return;
}

// Driver Code
int main()
{
    int arr[] = { 1, 2, 2, 1, 2,
                  2, 1, 2, 2 };

    int N = sizeof(arr) / sizeof(arr[0]);

    // Function call
    RepeatingSubarray(arr, N);
    return 0;
}

// Java program for the above approach
import java.util.*;

class GFG{

// Function to print the array
static void printArray(Vector<Integer> brr)
{
    for(int it : brr)
    {
        System.out.print(it + " ");
    }
}

// Function to find the smallest subarray
static void RepeatingSubarray(int arr[], int N)
{
    
    // Corner Case
    if (N < 2)
    {
        System.out.print("-1");
    }

    // Initialize the auxiliary subarray
    Vector<Integer> brr = new Vector<Integer>();

    // Push the first 2 elements into
    // the subarray brr[]
    brr.add(arr[0]);
    brr.add(arr[1]);

    // Iterate over the length of
    // subarray
    for(int i = 2; i < N / 2 + 1; i++) 
    {
        
        // If array can be divided into
        // subarray of i equal length
        if (N % i == 0)
        {
            boolean a = false;
            int n = brr.size();
            int j = i;

            // Check if on repeating the
            // current subarray gives the
            // original array or not
            while (j < N) 
            {
                int K = j % i;
                if (arr[j] == brr.get(K))
                {
                    j++;
                }
                else 
                {
                    a = true;
                    break;
                }
            }

            // Subarray found
            if (!a && j == N)
            {
                printArray(brr);
                return;
            }
        }

        // Add current element into
        // subarray
        brr.add(arr[i]);
    }

    // No subarray found
    System.out.print("-1");
    return;
}

// Driver Code
public static void main(String[] args)
{
    int arr[] = { 1, 2, 2, 1, 2,
                  2, 1, 2, 2 };

    int N = arr.length;

    // Function call
    RepeatingSubarray(arr, N);
}
}

// This code is contributed by Amit Katiyar 

# Python3 program for the above approach

# Function to print the array
def printArray(brr):

    for it in brr:
        print(it, end = ' ')

# Function to find the smallest subarray
def RepeatingSubarray(arr, N):

    # Corner Case
    if (N < 2):
        print("-1")
        
    # Initialize the auxiliary subarray
    brr = []

    # Push the first 2 elements into
    # the subarray brr[]
    brr.append(arr[0])
    brr.append(arr[1])

    # Iterate over the length of
    # subarray
    for i in range(2, N // 2 + 1):

        # If array can be divided into
        # subarray of i equal length
        if (N % i == 0):
            a = False
            n = len(brr)
            j = i

            # Check if on repeating the
            # current subarray gives the
            # original array or not
            while (j < N):
                K = j % i
                
                if (arr[j] == brr[K]):
                    j += 1
                else:
                    a = True
                    break

            # Subarray found
            if (not a and j == N):
                printArray(brr)
                return
            
        # Add current element into
        # subarray
        brr.append(arr[i])
    
    # No subarray found
    print("-1")
    return

# Driver Code
if __name__ =="__main__":

    arr = [ 1, 2, 2, 1, 2,
            2, 1, 2, 2 ]

    N = len(arr)

    # Function call
    RepeatingSubarray(arr, N)

# This code is contributed by chitranayal

// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG{

// Function to print the array
static void printArray(List<int> brr)
{
    foreach(int it in brr)
    {
        Console.Write(it + " ");
    }
}

// Function to find the smallest subarray
static void RepeatingSubarray(int []arr, int N)
{    
    // Corner Case
    if (N < 2)
    {
        Console.Write("-1");
    }

    // Initialize the auxiliary subarray
    List<int> brr = new List<int>();

    // Push the first 2 elements into
    // the subarray brr[]
    brr.Add(arr[0]);
    brr.Add(arr[1]);

    // Iterate over the length of
    // subarray
    for(int i = 2; i < N / 2 + 1; i++) 
    {        
        // If array can be divided into
        // subarray of i equal length
        if (N % i == 0)
        {
            bool a = false;
            int n = brr.Count;
            int j = i;

            // Check if on repeating the
            // current subarray gives the
            // original array or not
            while (j < N) 
            {
                int K = j % i;
                if (arr[j] == brr[K])
                {
                    j++;
                }
                else 
                {
                    a = true;
                    break;
                }
            }

            // Subarray found
            if (!a && j == N)
            {
                printArray(brr);
                return;
            }
        }

        // Add current element into
        // subarray
        brr.Add(arr[i]);
    }

    // No subarray found
    Console.Write("-1");
    return;
}

// Driver Code
public static void Main(String[] args)
{
    int []arr = {1, 2, 2, 1, 
                 2, 2, 1, 2, 2};

    int N = arr.Length;

    // Function call
    RepeatingSubarray(arr, N);
}
}

// This code is contributed by 29AjayKumar 

<script>
// JavaScript program for the above approach

// Function to print the array
function printArray(brr)
{
    for (let it of brr) {
        document.write(it + ' ');
    }
}

// Function to find the smallest subarray
function RepeatingSubarray(arr, N)
{
    // Corner Case
    if (N < 2) {
        document.write("-1");
    }

    // Initialize the auxiliary subarray
    let brr = [];

    // Push the first 2 elements into
    // the subarray brr[]
    brr.push(arr[0]);
    brr.push(arr[1]);

    // Iterate over the length of
    // subarray
    for (let i = 2; i < Math.floor(N / 2) + 1; i++) {

        // If array can be divided into
        // subarray of i equal length
        if (N % i == 0) {

            let a = false;
            let n = brr.length;
            let j = i;

            // Check if on repeating the
            // current subarray gives the
            // original array or not
            while (j < N) {
                let K = j % i;
                if (arr[j] == brr[K]) {
                    j++;
                }
                else {
                    a = true;
                    break;
                }
            }

            // Subarray found
            if (!a && j == N) {
                printArray(brr);
                return;
            }
        }

        // Add current element into
        // subarray
        brr.push(arr[i]);
    }

    // No subarray found
    document.write("-1");
    return;
}

// Driver Code
    let arr = [ 1, 2, 2, 1, 2,
                2, 1, 2, 2 ];

    let N = arr.length;

    // Function call
    RepeatingSubarray(arr, N);



// This code is contributed by Surbhi Tyagi.
</script>

1 2 2

Time Complexity: O(N²)
Auxiliary Space: O(N)

Related Topic: Subarrays, Subsequences, and Subsets in Array