Count of subarrays of size K having at least one pair with absolute difference divisible by K-1
Given an arr[] consisting of N elements, the task is to count all subarrays of size K having atleast one pair whose absolute difference is divisible by K – 1.
Examples:
Input: arr[] = {1, 5, 3, 2, 17, 18}, K = 4
Output: 3
Explanation:
The three subarrays of size 4 are:
{1, 5, 3, 2}: Pair {5, 2} have difference divisible by 3
{5, 3, 2, 17}: Pairs {5, 2}, {5, 17}, {2, 17} have difference divisible by 3
{3, 2, 17, 18}: Pairs {3, 18}, {2, 17} have difference divisible by 3Input: arr[] = {1, 2, 3, 4, 5}, K = 5
Output: 1
Explanation:
{1, 2, 3, 4, 5}: Pair {1, 5} is divisble by 4
Naive Approach:
The simplest approach to solve the problem is to iterate over all subarrays of size K and check if there exists any pair whose difference is divisible by K – 1.
Time Complexity: O(N * K * K)
Efficient Approach: The above approach can be optimized using Pigeonhole Principle. Follow the steps below to solve the problem:
- Consider K-1 boxes labeled 0, 1, 2, …, K-2 respectively. They represent the remainders when any number x from the array is divided by K-1, which means the boxes store the modulo K-1 of array elements.
- Now, in a subarray of size K, according to the Pigeonhole Principle, there must be atleast one pair of boxes with same remainders. It means that there is atleast one pair whose difference or even the summation will be divisible by K.
- From this theorem we can conclude that every subarray of size K, will always have atleast one pair whose difference is divisible by K-1.
- So, the answer will be equal to the number of subarrays of size K possible from the given array, which is equal to N – K + 1.
Below is the implementation of the above approach:
C++
// C++ implementation of the // above approach #include <bits/stdc++.h> using namespace std; // Function to return the required // number of subarrays int findSubarrays( int arr[], int N, int K) { // Return number of possible // subarrays of length K return N - K + 1; } // Driver Code int main() { int arr[] = { 1, 5, 3, 2, 17, 18 }; int K = 4; int N = sizeof (arr) / sizeof (arr[0]); cout << findSubarrays(arr, N, K); return 0; } |
Java
// Java implementation of the // above approach class GFG{ // Function to return the required // number of subarrays static int findSubarrays( int arr[], int N, int K) { // Return number of possible // subarrays of length K return N - K + 1 ; } // Driver Code public static void main(String[] args) { int arr[] = { 1 , 5 , 3 , 2 , 17 , 18 }; int K = 4 ; int N = arr.length; System.out.print(findSubarrays(arr, N, K)); } } // This code is contributed by shivanisinghss2110 |
3
Time complexity: O(1)
Auxiliary Space: O(1)
Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.
Recommended Posts:
- Maximum absolute difference between sum of subarrays of size K
- Count pairs in an array whose absolute difference is divisible by K
- Minimum absolute difference of XOR values of two subarrays
- Pair of prime numbers with a given sum and minimum absolute difference
- Longest subsequence such that absolute difference between every pair is atmost 1
- Count of longest possible subarrays with sum not divisible by K
- Count of subarrays of size K which is a permutation of numbers from 1 to K
- Count of subarrays of size K with elements having even frequencies
- Count non decreasing subarrays of size N from N Natural numbers
- Subsequence with maximum pairwise absolute difference and minimum size
- Minimum count of increment of K size subarrays required to form a given Array
- Count of sub-sets of size n with total element sum divisible by 3
- Absolute difference between set and unset bit count in N
- Count pairs in an array such that the absolute difference between them is ≥ K
- Count subarrays with sum as difference of squares of two numbers
- Count maximum elements of an array whose absolute difference does not exceed K
- Count all subarrays whose sum can be split as difference of squares of two Integers
- Count of subarrays whose product is equal to difference of two different numbers
- Count ways of choosing a pair with maximum difference
- Maximum subarray size, such that all subarrays of that size have sum less than k
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.
Improved By : shivanisinghss2110