Smallest Subarray with given GCD

4.1

Given an array arr[] of n numbers and an integer k, find length of the minimum sub-array with gcd equals to k.

Example:

Input: arr[] = {6, 9, 7, 10, 12, 
                24, 36, 27}, 
           K = 3
Output: 2
Explanation: GCD of subarray {6,9} is 3.
GCD of subarray {24,36,27} is also 3,
but {6,9} is the smallest 

Note: Time complexity analysis of below approaches assume that numbers are fixed size and finding GCD of two elements take constant time.

Method 1

Find GCD of all subarrays and keep track of the minimum length subarray with gcd k. Time Complexity of this is O(n3), O(n2) for each subarray and O(n) for finding gcd of a subarray.

Method 2

Find GCD of all subarrays using segment tree based approach discussed here. Time complexity of this solution is O(n2logn), O(n2) for each subarray and O(logn) for finding GCD of subarray using segment tree.

Method 3

The idea is to use Segment Tree and Binary Search to achieve time complexity O(n (logn)2).

  1. If we have any number equal to ‘k’ in the array then the answer is 1 as GCD of k is k. Return 1.
  2. If there is no number which is divisible by k, then GCD doesn’t exist. Return -1.
  3. If none of the above cases is true, the length of minimum subarray is either greater than 1 or GCD doesn’t exist. In this case, we follow following steps.
    • Build segment tree so that we can quicky find GCD of any subarray using the approach discussed here
    • After building Segment Tree, we consider every index as starting point and do binary search for ending point such that the subarray between these two points has GCD k

Following is C++ implementation of above idea.

// C++ Program to find GCD of a number in a given Range
// using segment Trees
#include <bits/stdc++.h>
using namespace std;

// To store segment tree
int *st;

// Function to find gcd of 2 numbers.
int gcd(int a, int b)
{
    if (a < b)
        swap(a, b);
    if (b==0)
        return a;
    return gcd(b, a%b);
}

/*  A recursive function to get gcd of given
    range of array indexes. The following are parameters for
    this function.

    st    --> Pointer to segment tree
    si --> Index of current node in the segment tree. Initially
               0 is passed as root is always at index 0
    ss & se  --> Starting and ending indexes of the segment
                 represented by current node, i.e., st[index]
    qs & qe  --> Starting and ending indexes of query range */
int findGcd(int ss, int se, int qs, int qe, int si)
{
    if (ss>qe || se < qs)
        return 0;
    if (qs<=ss && qe>=se)
        return st[si];
    int mid = ss+(se-ss)/2;
    return gcd(findGcd(ss, mid, qs, qe, si*2+1),
               findGcd(mid+1, se, qs, qe, si*2+2));
}

//Finding The gcd of given Range
int findRangeGcd(int ss, int se, int arr[], int n)
{
    if (ss<0 || se > n-1 || ss>se)
    {
        cout << "Invalid Arguments" << "\n";
        return -1;
    }
    return findGcd(0, n-1, ss, se, 0);
}

// A recursive function that constructs Segment Tree for
// array[ss..se]. si is index of current node in segment
// tree st
int constructST(int arr[], int ss, int se, int si)
{
    if (ss==se)
    {
        st[si] = arr[ss];
        return st[si];
    }
    int mid = ss+(se-ss)/2;
    st[si] = gcd(constructST(arr, ss, mid, si*2+1),
                 constructST(arr, mid+1, se, si*2+2));
    return st[si];
}

/* Function to construct segment tree from given array.
   This function allocates memory for segment tree and
   calls constructSTUtil() to fill the allocated memory */
int *constructSegmentTree(int arr[], int n)
{
    int height = (int)(ceil(log2(n)));
    int size = 2*(int)pow(2, height)-1;
    st = new int[size];
    constructST(arr, 0, n-1, 0);
    return st;
}

// Returns size of smallest subarray of arr[0..n-1]
// with GCD equal to k.
int findSmallestSubarr(int arr[], int n, int k)
{
    // To check if a multiple of k exists.
    bool found = false;

    // Find if k or its multiple is present
    for (int i=0; i<n; i++)
    {
        // If k is present, then subarray size is 1.
        if (arr[i] == k)
            return 1;

        // Break the loop to indicate presence of a
        // multiple of k.
        if (arr[i] % k == 0)
            found = true;
    }

    // If there was no multiple of k in arr[], then
    // we can't get k as GCD.
    if (found == false)
        return -1;

    // If there is a multiple of k in arr[], build
    // segment tree from given array
    constructSegmentTree(arr, n);

    // Initialize result
    int res = n+1;

    // Now consider every element as starting point
    // and search for ending point using Binary Search
    for (int i=0; i<n-1; i++)
    {
        // a[i] cannot be a starting point, if it is
        // not a multiple of k.
        if (arr[i] % k != 0)
            continue;

        // Initialize indexes for binary search of closest
        // ending point to i with GCD of subarray as k.
        int low = i+1;
        int high = n-1;

        // Initialize closest ending point for i.
        int closest = 0;

        // Binary Search for closest ending point
        // with GCD equal to k.
        while (true)
        {
            // Find middle point and GCD of subarray
            // arr[i..mid]
            int mid = low + (high-low)/2;
            int gcd = findRangeGcd(i, mid, arr, n);

            // If GCD is more than k, look further
            if (gcd > k)
                low = mid;

            // If GCD is k, store this point and look for
            // a closer point
            else if (gcd == k)
            {
                high = mid;
                closest = mid;
                break;
            }

            // If GCD is less than, look closer
            else
                high = mid;

            // If termination condition reached, set
            // closest
            if (abs(high-low) <= 1)
            {
                if (findRangeGcd(i, low, arr, n) == k)
                    closest = low;
                else if (findRangeGcd(i, high, arr, n)==k)
                    closest = high;
                break;
            }
        }

        if (closest != 0)
            res = min(res, closest - i + 1);
    }

    // If res was not changed by loop, return -1,
    // else return its value.
    return (res == n+1) ? -1 : res;
}

// Driver program to test above functions
int main()
{
    int n = 8;
    int k = 3;
    int arr[] = {6, 9, 7, 10, 12, 24, 36, 27};
    cout << "Size of smallest sub-array with given"
         << " size is " << findSmallestSubarr(arr, n, k);
    return 0;
}

Output:

2

Example:

arr[] = {6, 9, 7, 10, 12, 24, 36, 27}, K = 3

// Initial value of minLen is equal to size 
// of array
minLen = 8 

No element is equal to k so result is either 
greater than 1 or doesn't exist. 

First index

  • GCD of subarray from 1 to 5 is 1.
  • GCD < k
  • GCD of subarray from 1 to 3 is 1.
  • GCD < k
  • GCD of subarray from 1 to 2 is 3
  • minLen = minimum(8, 2) = 2

Second Index

  • GCD of subarray from 2 to 5 is 1
  • GCD < k
  • GCD of subarray from 2 to 4 is 1
  • GCD < k
  • GCD of subarray from 6 to 8 is 3
  • minLen = minimum(2, 3) = 2.

.

.

.

Sixth Index

  • GCD of subarray from 6 to 7 is 12
  • GCD > k
  • GCD of subarray from 6 to 8 is 3
  • minLen = minimum(2, 3) = 2

Time Complexity: O(n (logn)2), O(n) for traversing to each index, O(logn) for each subarray and O(logn) for GCD of each subarray.

This article is contributed by Nikhil Tekwani. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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