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Queries to find the length of the longest prefix from given array having all elements divisible by K

  • Difficulty Level : Easy
  • Last Updated : 27 May, 2021

Given an array arr[] consisting of N elements and an arrayQ[], the task for each query is to find the length of the longest prefix such that all the elements of this prefix are divisible by K.

Examples:

Input: arr[] = {12, 6, 15, 3, 10}, Q[] = {4, 3, 2}
Output: 1 4 2
Explanation:

  • Q[0] = 4: arr[0] (= 12) is divisible by 4. arr[1] is not divisible by 4. Therefore, the length of the longest prefix divisible by 4 is 1.
  • Q[1] = 3: The longest prefix which is divisible by 3 is {12, 6, 15, 3}. Therefore, print 4 as the required output.
  • Q[2] = 2: The longest prefix which is divisible by 2 is {12, 6}.

Input: arr[] = {4, 3, 2, 1}, Q[] = {1, 2, 3}
Output: 4 1 0

Approach: The idea is to observe that if the first i elements of the array are divisible by the given integer K, then their GCD will also be divisible by K. Follow the steps below to solve the problem:



  1. Before finding the answer to each query, pre-compute the gcd of the prefixes of the array in another array g[] such that g[0] = arr[0] and for the rest of the elements g[i] = GCD(arr[i], g[i – 1]).
  2. Then, for each element in the array Q[], perform Binary Search on the array g[] and find the last index of g[] which is divisible by Q[i].
  3. It has to be noted that all the elements in this index will be divisible by K, since the gcd of all the elements till this index is divisible by K.
  4. Print the length of the longest prefix.

Below is the implementation of the above approach.

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find index of the last
// element which is divisible the given element
int binSearch(int* g, int st, int en, int val)
{
    // Initially assume the
    // index to be -1
    int ind = -1;
    while (st <= en) {
 
        // Find the mid element
        // of the subarray
        int mid = st + (en - st) / 2;
 
        // If the mid element is divisible by
        // given element, then move to right
        // of mid and update ind to mid
        if (g[mid] % val == 0) {
            ind = mid;
            st = mid + 1;
        }
 
        // Otherwise, move to left of mid
        else {
            en = mid - 1;
        }
    }
 
    // Return the length of prefix
    return ind + 1;
}
 
// Function to compute and print for each query
void solveQueries(int* arr, int* queries,
                  int n, int q)
{
    int g[n];
 
    // Pre compute the gcd of the prefixes
    // of the input array in the array g[]
    for (int i = 0; i < n; i++) {
        if (i == 0) {
            g[i] = arr[i];
        }
        else {
            g[i] = __gcd(g[i - 1], arr[i]);
        }
    }
 
    // Perform binary search
    // for each query
    for (int i = 0; i < q; i++) {
        cout << binSearch(g, 0, n - 1,
                          queries[i])
             << " ";
    }
}
 
// Driver Code
int main()
{
    // Given array
    int arr[] = { 12, 6, 15, 3, 10 };
 
    // Size of array
    int n = sizeof(arr) / sizeof(arr[0]);
 
    // Given Queries
    int queries[] = { 4, 3, 2 };
 
    // Size of queries
    int q = sizeof(queries) / sizeof(queries[0]);
 
    solveQueries(arr, queries, n, q);
}

Java




// Java program for the above approach
import java.util.*;
class GFG
{
 
// Function to find index of the last
// element which is divisible the given element
static int binSearch(int[] g, int st, int en, int val)
{
   
    // Initially assume the
    // index to be -1
    int ind = -1;
    while (st <= en)
    {
 
        // Find the mid element
        // of the subarray
        int mid = st + (en - st) / 2;
 
        // If the mid element is divisible by
        // given element, then move to right
        // of mid and update ind to mid
        if (g[mid] % val == 0)
        {
            ind = mid;
            st = mid + 1;
        }
 
        // Otherwise, move to left of mid
        else
        {
            en = mid - 1;
        }
    }
 
    // Return the length of prefix
    return ind + 1;
}
 
// Function to compute and print for each query
static void solveQueries(int []arr, int []queries,
                  int n, int q)
{
    int []g = new int[n];
 
    // Pre compute the gcd of the prefixes
    // of the input array in the array g[]
    for (int i = 0; i < n; i++)
    {
        if (i == 0) {
            g[i] = arr[i];
        }
        else {
            g[i] = __gcd(g[i - 1], arr[i]);
        }
    }
 
    // Perform binary search
    // for each query
    for (int i = 0; i < q; i++)
    {
        System.out.print(binSearch(g, 0, n - 1,
                          queries[i])
            + " ");
    }
}
static int __gcd(int a, int b) 
    return b == 0? a:__gcd(b, a % b);    
}
   
// Driver Code
public static void main(String[] args)
{
   
    // Given array
    int arr[] = { 12, 6, 15, 3, 10 };
 
    // Size of array
    int n = arr.length;
 
    // Given Queries
    int queries[] = { 4, 3, 2 };
 
    // Size of queries
    int q = queries.length;
    solveQueries(arr, queries, n, q);
}
}
 
// This code is contributed by 29AjayKumar

Python3




# Python3 program for the above approach
from math import gcd as __gcd
 
# Function to find index of the last
# element which is divisible the
# given element
def binSearch(g, st, en, val):
     
    # Initially assume the
    # index to be -1
    ind = -1
     
    while (st <= en):
 
        # Find the mid element
        # of the subarray
        mid = st + (en - st) // 2
 
        # If the mid element is divisible by
        # given element, then move to right
        # of mid and update ind to mid
        if (g[mid] % val == 0):
            ind = mid
            st = mid + 1
 
        # Otherwise, move to left of mid
        else:
            en = mid - 1
 
    # Return the length of prefix
    return ind + 1
 
# Function to compute and prfor each query
def solveQueries(arr, queries, n, q):
     
    g = [0 for i in range(n)]
 
    # Pre compute the gcd of the prefixes
    # of the input array in the array g[]
    for i in range(n):
        if (i == 0):
            g[i] = arr[i]
        else:
            g[i] = __gcd(g[i - 1], arr[i])
 
    # Perform binary search
    # for each query
    for i in range(q):
        print(binSearch(g, 0, n - 1,
                        queries[i]), end = " ")
 
# Driver Code
if __name__ == '__main__':
     
    # Given array
    arr = [ 12, 6, 15, 3, 10 ]
 
    # Size of array
    n = len(arr)
 
    # Given Queries
    queries = [ 4, 3, 2 ]
 
    # Size of queries
    q = len(queries)
 
    solveQueries(arr, queries, n, q)
 
# This code is contributed by mohit kumar 29

C#




// C# program to implement
// the above approach 
using System;
class GFG
{
  
// Function to find index of the last
// element which is divisible the given element
static int binSearch(int[] g, int st, int en, int val)
{
   
  // Initially assume the
  // index to be -1
  int ind = -1;
  while (st <= en)
  {
 
    // Find the mid element
    // of the subarray
    int mid = st + (en - st) / 2;
 
    // If the mid element is divisible by
    // given element, then move to right
    // of mid and update ind to mid
    if (g[mid] % val == 0)
    {
      ind = mid;
      st = mid + 1;
    }
 
    // Otherwise, move to left of mid
    else
    {
      en = mid - 1;
    }
  }
 
  // Return the length of prefix
  return ind + 1;
}
 
  // Recursive function to return
  // gcd of a and b
  static int gcd(int a, int b)
  {
 
    // Everything divides 0
    if (a == 0)
      return b;
    if (b == 0)
      return a;
 
    // base case
    if (a == b)
      return a;
 
    // a is greater
    if (a > b)
      return gcd(a - b, b);
 
    return gcd(a, b - a);
  }
 
  // Function to compute and print for each query
  static void solveQueries(int[] arr, int[] queries,
                           int n, int q)
  {
    int[] g = new int[n];
 
    // Pre compute the gcd of the prefixes
    // of the input array in the array g[]
    for (int i = 0; i < n; i++)
    {
      if (i == 0)
      {
        g[i] = arr[i];
      }
      else
      {
        g[i] = gcd(g[i - 1], arr[i]);
      }
    }
 
    // Perform binary search
    // for each query
    for (int i = 0; i < q; i++)
    {
      Console.Write(binSearch(g, 0, n - 1,
                              queries[i]) + " ");
    }
  }
  
// Driver code
public static void Main()
{
   
    // Given array
    int[] arr = { 12, 6, 15, 3, 10 };
  
    // Size of array
    int n = arr.Length;
  
    // Given Queries
    int[] queries = { 4, 3, 2 };
  
    // Size of queries
    int q = queries.Length;
    solveQueries(arr, queries, n, q);
}
}
 
// This code is contributed by susmitakundugoaldanga

Javascript




<script>
//Javascript program for the above approach
 
//function for GCD
function __gcd( a, b) 
    return b == 0? a:__gcd(b, a % b);    
}
 
// Function to find index of the last
// element which is divisible the given element
function binSearch(g, st, en, val)
{
    // Initially assume the
    // index to be -1
    var ind = -1;
    while (st <= en) {
 
        // Find the mid element
        // of the subarray
        var mid = st + parseInt((en - st) / 2);
 
        // If the mid element is divisible by
        // given element, then move to right
        // of mid and update ind to mid
        if (g[mid] % val == 0) {
            ind = mid;
            st = mid + 1;
        }
 
        // Otherwise, move to left of mid
        else {
            en = mid - 1;
        }
    }
 
    // Return the length of prefix
    return ind + 1;
}
 
// Function to compute and print for each query
function solveQueries(arr, queries, n, q)
{
    var g = new Array(n);
 
    // Pre compute the gcd of the prefixes
    // of the input array in the array g[]
    for (var i = 0; i < n; i++) {
        if (i == 0) {
            g[i] = arr[i];
        }
        else {
            g[i] = __gcd(g[i - 1], arr[i]);
        }
    }
 
    // Perform binary search
    // for each query
    for (var i = 0; i < q; i++) {
        document.write( binSearch(g, 0, n - 1, queries[i]) + " ");
    }
}
 
var arr = [ 12, 6, 15, 3, 10 ];
 
// Size of array
var n = arr.length;
// Given Queries
var queries = [ 4, 3, 2 ];
// Size of queries
var q = queries.length;
solveQueries(arr, queries, n, q);
 
//This code is contributed by SoumikMondal
</script>
Output: 
1 4 2

 

Time Complexity: O(NlogN + QlogN)
Auxiliary Space: O(N)

 

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