Given an array a[0 . . . n-1]. We should be able to efficiently find the GCD from index qs (query start) to qe (query end) where 0 <= qs <= qe <= n-1.
Example :
Input : a[] = {2, 3, 60, 90, 50}; Index Ranges : {1, 3}, {2, 4}, {0, 2} Output: GCDs of given ranges are 3, 10, 1
Method 1 (Simple)
A simple solution is to run a loop from qs to qe and find GCD in given range. This solution takes O(n) time in worst case.
Method 2 (2D Array)
Another solution is to create a 2D array where an entry [i, j] stores the GCD in range arr[i..j]. GCD of a given range can now be calculated in O(1) time, but preprocessing takes O(n^2) time. Also, this approach needs O(n^2) extra space which may become huge for large input arrays.
Method 3 (Segment Tree)
Prerequisites : ,
Segment tree can be used to do preprocessing and query in moderate time. With segment tree, preprocessing time is O(n) and time to for GCD query is O(Logn). The extra space required is O(n) to store the segment tree.
Representation of Segment trees
Array representation of tree is used to represent Segment Trees i.e., for each node at index i,
Construction of Segment Tree from given array
Query for GCD of given range
/ qs --> query start index, qe --> query end index int GCD(node, qs, qe) { if range of node is within qs and qe return value in node else if range of node is completely outside qs and qe return INFINITE else return GCD( GCD(node's left child, qs, qe), GCD(node's right child, qs, qe) ) } Below is Implementation of this method.
C++
// 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; /* 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; } // Driver program to test above functions int main() { int a[] = {2, 3, 6, 9, 5}; int n = sizeof(a)/sizeof(a[0]); // Build segment tree from given array constructSegmentTree(a, n); // Starting index of range. These indexes are 0 based. int l = 1; // Last index of range.These indexes are 0 based. int r = 3; cout << "GCD of the given range is:"; cout << findRangeGcd(l, r, a, n) << "\n"; return 0; } |
chevron_right
filter_none
Java
// Java Program to find GCD of a number in a given Range // using segment Trees import java.io.*; public class Main { private static int[] st; // Array to store segment tree /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls constructSTUtil() to fill the allocated memory */ public static int[] constructSegmentTree(int[] arr) { int height = (int)Math.ceil(Math.log(arr.length)/Math.log(2)); int size = 2*(int)Math.pow(2, height)-1; st = new int[size]; constructST(arr, 0, arr.length-1, 0); return st; } // A recursive function that constructs Segment // Tree for array[ss..se]. si is index of current // node in segment tree st public static 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 find gcd of 2 numbers. private static int gcd(int a, int b) { if (a < b) { // If b greater than a swap a and b int temp = b; b = a; a = temp; } if (b==0) return a; return gcd(b,a%b); } //Finding The gcd of given Range public static int findRangeGcd(int ss, int se, int[] arr) { int n = arr.length; if (ss<0 || se > n-1 || ss>se) throw new IllegalArgumentException("Invalid arguments"); return findGcd(0, n-1, ss, se, 0); } /* 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[si] qs & qe --> Starting and ending indexes of query range */ public static 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)); } // Driver Code public static void main(String[] args)throws IOException { int[] a = {2, 3, 6, 9, 5}; constructSegmentTree(a); int l = 1; // Starting index of range. int r = 3; //Last index of range. System.out.print("GCD of the given range is: "); System.out.print(findRangeGcd(l, r, a)); } } |
chevron_right
filter_none
C#
// C# Program to find GCD of a number in a given Range // using segment Trees using System; class GFG { private static int[] st; // Array to store segment tree /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls constructSTUtil() to fill the allocated memory */ public static int[] constructSegmentTree(int[] arr) { int height = (int)Math.Ceiling(Math.Log(arr.Length)/Math.Log(2)); int size = 2*(int)Math.Pow(2, height) - 1; st = new int[size]; constructST(arr, 0, arr.Length - 1, 0); return st; } // A recursive function that constructs Segment // Tree for array[ss..se]. si is index of current // node in segment tree st public static 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 find gcd of 2 numbers. private static int gcd(int a, int b) { if (a < b) { // If b greater than a swap a and b int temp = b; b = a; a = temp; } if (b == 0) return a; return gcd(b,a % b); } // Finding The gcd of given Range public static int findRangeGcd(int ss, int se, int[] arr) { int n = arr.Length; if (ss < 0 || se > n-1 || ss > se) { Console.WriteLine("Invalid arguments"); return int.MinValue; } return findGcd(0, n - 1, ss, se, 0); } /* 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[si] qs & qe --> Starting and ending indexes of query range */ public static 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)); } // Driver Code public static void Main(String[] args) { int[] a = {2, 3, 6, 9, 5}; constructSegmentTree(a); int l = 1; // Starting index of range. int r = 3; //Last index of range. Console.Write("GCD of the given range is: "); Console.Write(findRangeGcd(l, r, a)); } } // This code has been contributed by 29AjayKumar |
chevron_right
filter_none
Output:
GCD of the given range is: 3
Time Complexity: Time Complexity for tree construction is O(n * log(min(a, b))), where n is the number of modes and a and b are nodes whose GCD is calculated during merge operation. There are total 2n-1 nodes, and value of every node is calculated only once in tree construction. Time complexity to query is O(Log n * Log n).
This article is contributed by Nikhil Tekwani. , .
Recommended Posts:
Improved By :
Related Article
{module [317]}
destination source:https://www.geeksforgeeks.org/gcds-of-a-given-index-ranges-in-an-array/
