Queries to update a given index and find gcd in range

Given an array arr[] of N integers and queries Q. Queries are of two types:

  1. Update a given index ind by X.
  2. Find the gcd of the elements in the index range [L, R].

Examples:

Input: arr[] = {1, 3, 6, 9, 9, 11}
Type 2 query: L = 1, R = 3
Type 1 query: ind = 1, X = 10
Type 2 query: L = 1, R = 3
Output:
3
1

Input: arr[] = {1, 2, 4, 9, 3}
Type 2 query: L = 1, R = 2
Type 1 query: ind = 2, X = 7
Type 2 query: L = 1, R = 2
Type 2 query: L = 3, R = 4
Output:
2
1
3



Approach: The following problem can be solved using Segment Tree. 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

  • Leaf Nodes are the elements of the input array.
  • Each internal node represents GCD of all leaves under it.

Array representation of tree is used to represent Segment Trees i.e., for each node at index i

  • Left child is at index 2*i+1
  • Right child at 2*i+2 and the parent is at floor((i-1)/2).

Construction of Segment Tree from given array

  • Begin with a segment arr[0 . . . n-1] and keep dividing into two halves. Every time we divide the current segment into two halves (if it has not yet become a segment of length 1), then call the same procedure on both halves, and for each such segment, we store the GCD value in a segment tree node.
  • All levels of the constructed segment tree will be completely filled except the last level. Also, the tree will be a Full Binary Tree (every node has 0 or two children) because we always divide segments in two halves at every level.
  • Since the constructed tree is always full binary tree with n leaves, there will be n-1 internal nodes. So total number of nodes will be 2*n – 1.

Like tree construction and query operations, the update can also be done recursively. We are given an index which needs to be updated. Let diff be the value to be added. We start from root of the segment tree and add diff to all nodes which have given index in their range. If a node doesn’t have given index in its range, we don’t make any changes to that node.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// A utility function to get the
// middle index from corner indexes
int getMid(int s, int e)
{
    return (s + (e - s) / 2);
}
  
// A recursive function to get the gcd of values in given range
// of the array. 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
int getGcdUtil(int* st, int ss, int se, int qs, int qe, int si)
{
    // If segment of this node is a part of given range
    // then return the gcd of the segment
    if (qs <= ss && qe >= se)
        return st[si];
  
    // If segment of this node is outside the given range
    if (se < qs || ss > qe)
        return 0;
  
    // If a part of this segment overlaps with the given range
    int mid = getMid(ss, se);
    return __gcd(getGcdUtil(st, ss, mid, qs, qe, 2 * si + 1),
                 getGcdUtil(st, mid + 1, se, qs, qe, 2 * si + 2));
}
  
// A recursive function to update the nodes which have the given
// index in their range. The following are parameters
// st, si, ss and se are same as getSumUtil()
// i --> index of the element to be updated. This index is
// in the input array.
// diff --> Value to be added to all nodes which have i in range
void updateValueUtil(int* st, int ss, int se, int i, int diff, int si)
{
    // Base Case: If the input index lies outside the range of
    // this segment
    if (i < ss || i > se)
        return;
  
    // If the input index is in range of this node, then update
    // the value of the node and its children
    st[si] = st[si] + diff;
    if (se != ss) {
        int mid = getMid(ss, se);
        updateValueUtil(st, ss, mid, i, diff, 2 * si + 1);
        updateValueUtil(st, mid + 1, se, i, diff, 2 * si + 2);
    }
}
  
// The function to update a value in input array and segment tree.
// It uses updateValueUtil() to update the value in segment tree
void updateValue(int arr[], int* st, int n, int i, int new_val)
{
    // Check for erroneous input index
    if (i < 0 || i > n - 1) {
        cout << "Invalid Input";
        return;
    }
  
    // Get the difference between new value and old value
    int diff = new_val - arr[i];
  
    // Update the value in array
    arr[i] = new_val;
  
    // Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0);
}
  
// Function to return the sum of elements in range
// from index qs (query start) to qe (query end)
// It mainly uses getSumUtil()
int getGcd(int* st, int n, int qs, int qe)
{
  
    // Check for erroneous input values
    if (qs < 0 || qe > n - 1 || qs > qe) {
        cout << "Invalid Input";
        return -1;
    }
  
    return getGcdUtil(st, 0, n - 1, qs, qe, 0);
}
  
// A recursive function that constructs Segment Tree for array[ss..se].
// si is index of current node in segment tree st
int constructGcdUtil(int arr[], int ss, int se, int* st, int si)
{
    // If there is one element in array, store it in current node of
    // segment tree and return
    if (ss == se) {
        st[si] = arr[ss];
        return arr[ss];
    }
  
    // If there are more than one element then recur for left and
    // right subtrees and store the sum of values in this node
    int mid = getMid(ss, se);
    st[si] = __gcd(constructGcdUtil(arr, ss, mid, st, si * 2 + 1),
                   constructGcdUtil(arr, mid + 1, se, st, 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* constructGcd(int arr[], int n)
{
    // Allocate memory for the segment tree
  
    // Height of segment tree
    int x = (int)(ceil(log2(n)));
  
    // Maximum size of segment tree
    int max_size = 2 * (int)pow(2, x) - 1;
  
    // Allocate memory
    int* st = new int[max_size];
  
    // Fill the allocated memory st
    constructGcdUtil(arr, 0, n - 1, st, 0);
  
    // Return the constructed segment tree
    return st;
}
  
// Driver code
int main()
{
    int arr[] = { 1, 3, 6, 9, 9, 11 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    // Build segment tree from given array
    int* st = constructGcd(arr, n);
  
    // Print GCD of values in array from index 1 to 3
    cout << getGcd(st, n, 1, 3) << endl;
  
    // Update: set arr[1] = 10 and update corresponding
    // segment tree nodes
    updateValue(arr, st, n, 1, 10);
  
    // Find GCD after the value is updated
    cout << getGcd(st, n, 1, 3) << endl;
  
    return 0;
}

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Java

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// Java implementation of the approach 
class GFG
{
      
// segment tree
static int st[];
  
// Recursive function to return gcd of a and b 
static int __gcd(int a, int b) 
    if (b == 0
        return a; 
    return __gcd(b, a % b); 
      
  
// A utility function to get the 
// middle index from corner indexes 
static int getMid(int s, int e) 
    return (s + (e - s) / 2); 
  
// A recursive function to get the gcd of values in given range 
// of the array. 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 
static int getGcdUtil( int ss, int se, int qs, int qe, int si) 
    // If segment of this node is a part of given range 
    // then return the gcd of the segment 
    if (qs <= ss && qe >= se) 
        return st[si]; 
  
    // If segment of this node is outside the given range 
    if (se < qs || ss > qe) 
        return 0
  
    // If a part of this segment overlaps with the given range 
    int mid = getMid(ss, se); 
    return __gcd(getGcdUtil( ss, mid, qs, qe, 2 * si + 1), 
                getGcdUtil( mid + 1, se, qs, qe, 2 * si + 2)); 
  
// A recursive function to update the nodes which have the given 
// index in their range. The following are parameters 
// si, ss and se are same as getSumUtil() 
// i --> index of the element to be updated. This index is 
// in the input array. 
// diff --> Value to be added to all nodes which have i in range 
static void updateValueUtil( int ss, int se, int i, int diff, int si) 
    // Base Case: If the input index lies outside the range of 
    // this segment 
    if (i < ss || i > se) 
        return
  
    // If the input index is in range of this node, then update 
    // the value of the node and its children 
    st[si] = st[si] + diff; 
    if (se != ss) { 
        int mid = getMid(ss, se); 
        updateValueUtil( ss, mid, i, diff, 2 * si + 1); 
        updateValueUtil( mid + 1, se, i, diff, 2 * si + 2); 
    
  
// The function to update a value in input array and segment tree. 
// It uses updateValueUtil() to update the value in segment tree 
static void updateValue(int arr[], int n, int i, int new_val) 
    // Check for erroneous input index 
    if (i < 0 || i > n - 1
    
        System.out.println("Invalid Input"); 
        return
    
  
    // Get the difference between new value and old value 
    int diff = new_val - arr[i]; 
  
    // Update the value in array 
    arr[i] = new_val; 
  
    // Update the values of nodes in segment tree 
    updateValueUtil( 0, n - 1, i, diff, 0); 
  
// Function to return the sum of elements in range 
// from index qs (query start) to qe (query end) 
// It mainly uses getSumUtil() 
static int getGcd( int n, int qs, int qe) 
  
    // Check for erroneous input values 
    if (qs < 0 || qe > n - 1 || qs > qe)
    
        System.out.println( "Invalid Input"); 
        return -1
    
  
    return getGcdUtil( 0, n - 1, qs, qe, 0); 
  
// A recursive function that constructs Segment Tree for array[ss..se]. 
// si is index of current node in segment tree st 
static int constructGcdUtil(int arr[], int ss, int se, int si) 
    // If there is one element in array, store it in current node of 
    // segment tree and return 
    if (ss == se) 
    
        st[si] = arr[ss]; 
        return arr[ss]; 
    
  
    // If there are more than one element then recur for left and 
    // right subtrees and store the sum of values in this node 
    int mid = getMid(ss, se); 
    st[si] = __gcd(constructGcdUtil(arr, ss, mid, si * 2 + 1), 
                constructGcdUtil(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 
static void constructGcd(int arr[], int n) 
    // Allocate memory for the segment tree 
  
    // Height of segment tree 
    int x = (int)(Math.ceil(Math.log(n)/Math.log(2))); 
  
    // Maximum size of segment tree 
    int max_size = 2 * (int)Math.pow(2, x) - 1
  
    // Allocate memory 
    st = new int[max_size];
  
    // Fill the allocated memory st 
    constructGcdUtil(arr, 0, n - 1, 0); 
  
  
// Driver code 
public static void main(String args[])
    int arr[] = { 1, 3, 6, 9, 9, 11 }; 
    int n = arr.length; 
  
    // Build segment tree from given array 
    constructGcd(arr, n); 
  
    // Print GCD of values in array from index 1 to 3 
    System.out.println( getGcd( n, 1, 3) ); 
  
    // Update: set arr[1] = 10 and update corresponding 
    // segment tree nodes 
    updateValue(arr, n, 1, 10); 
  
    // Find GCD after the value is updated 
    System.out.println( getGcd( n, 1, 3) ); 
}
  
// This code is constructed by Arnab Kundu

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Python3

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# Python 3 implementation of the approach
  
from math import gcd,ceil,log2,pow
  
# A utility function to get the
# middle index from corner indexes
def getMid(s, e):
    return (s + int((e - s) / 2))
  
# A recursive function to get the gcd of values in given range
# of the array. 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
def getGcdUtil(st,ss,se,qs,qe,si):
      
    # If segment of this node is a part of given range
    # then return the gcd of the segment
    if (qs <= ss and qe >= se):
        return st[si]
  
    # If segment of this node is outside the given range
    if (se < qs or ss > qe):
        return 0
  
    # If a part of this segment overlaps with the given range
    mid = getMid(ss, se)
    return gcd(getGcdUtil(st, ss, mid, qs, qe, 2 * si + 1),
            getGcdUtil(st, mid + 1, se, qs, qe, 2 * si + 2))
  
# A recursive function to update the nodes which have the given
# index in their range. The following are parameters
# st, si, ss and se are same as getSumUtil()
# i --> index of the element to be updated. This index is
# in the input array.
# diff --> Value to be added to all nodes which have i in range
def updateValueUtil(st,ss,se,i,diff,si):
      
    # Base Case: If the input index lies outside the range of
    # this segment
    if (i < ss or i > se):
        return
  
    # If the input index is in range of this node, then update
    # the value of the node and its children
    st[si] = st[si] + diff
    if (se != ss):
        mid = getMid(ss, se)
        updateValueUtil(st, ss, mid, i, diff, 2 * si + 1)
        updateValueUtil(st, mid + 1, se, i, diff, 2 * si + 2)
  
# The function to update a value in input array and segment tree.
# It uses updateValueUtil() to update the value in segment tree
def updateValue(arr, st, n, i, new_val):
      
    # Check for erroneous input index
    if (i < 0 or i > n - 1):
        print("Invalid Input")
        return
  
    # Get the difference between new value and old value
    diff = new_val - arr[i]
  
    # Update the value in array
    arr[i] = new_val
  
    # Update the values of nodes in segment tree
    updateValueUtil(st, 0, n - 1, i, diff, 0)
  
# Function to return the sum of elements in range
# from index qs (query start) to qe (query end)
# It mainly uses getSumUtil()
def getGcd(st,n,qs,qe):
      
    # Check for erroneous input values
    if (qs < 0 or qe > n - 1 or qs > qe):
        cout << "Invalid Input"
        return -1
  
    return getGcdUtil(st, 0, n - 1, qs, qe, 0)
  
# A recursive function that constructs Segment Tree for array[ss..se].
# si is index of current node in segment tree st
def constructGcdUtil(arr, ss,se, st, si):
      
    # If there is one element in array, store it in current node of
    # segment tree and return
    if (ss == se):
        st[si] = arr[ss]
        return arr[ss]
  
    # If there are more than one element then recur for left and
    # right subtrees and store the sum of values in this node
    mid = getMid(ss, se)
    st[si] = gcd(constructGcdUtil(arr, ss, mid, st, si * 2 + 1),
                constructGcdUtil(arr, mid + 1, se, st, 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
def constructGcd(arr, n):
      
    # Allocate memory for the segment tree
  
    # Height of segment tree
    x = int(ceil(log2(n)))
  
    # Maximum size of segment tree
    max_size = 2 * int(pow(2, x) - 1)
  
    # Allocate memory
    st = [0 for i in range(max_size)]
  
    # Fill the allocated memory st
    constructGcdUtil(arr, 0, n - 1, st, 0)
  
    # Return the constructed segment tree
    return st
  
# Driver code
if __name__ == '__main__':
    arr = [1, 3, 6, 9, 9, 11]
    n = len(arr)
  
    # Build segment tree from given array
    st = constructGcd(arr, n)
  
    # Print GCD of values in array from index 1 to 3
    print(getGcd(st, n, 1, 3))
  
    # Update: set arr[1] = 10 and update corresponding
    # segment tree nodes
    updateValue(arr, st, n, 1, 10)
  
    # Find GCD after the value is updated
    print(getGcd(st, n, 1, 3))
  
# This code is contributed by
# SURENDRA_GANGWAR

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C#

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// C# implementation of the approach.
using System;
      
class GFG
{
      
// segment tree
static int []st;
  
// Recursive function to return gcd of a and b 
static int __gcd(int a, int b) 
    if (b == 0) 
        return a; 
    return __gcd(b, a % b); 
      
  
// A utility function to get the 
// middle index from corner indexes 
static int getMid(int s, int e) 
    return (s + (e - s) / 2); 
  
// A recursive function to get the gcd of values in given range 
// of the array. 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 
static int getGcdUtil( int ss, int se, int qs, int qe, int si) 
    // If segment of this node is a part of given range 
    // then return the gcd of the segment 
    if (qs <= ss && qe >= se) 
        return st[si]; 
  
    // If segment of this node is outside the given range 
    if (se < qs || ss > qe) 
        return 0; 
  
    // If a part of this segment overlaps with the given range 
    int mid = getMid(ss, se); 
    return __gcd(getGcdUtil( ss, mid, qs, qe, 2 * si + 1), 
                getGcdUtil( mid + 1, se, qs, qe, 2 * si + 2)); 
  
// A recursive function to update the nodes which have the given 
// index in their range. The following are parameters 
// si, ss and se are same as getSumUtil() 
// i --> index of the element to be updated. This index is 
// in the input array. 
// diff --> Value to be added to all nodes which have i in range 
static void updateValueUtil( int ss, int se, int i, int diff, int si) 
    // Base Case: If the input index lies outside the range of 
    // this segment 
    if (i < ss || i > se) 
        return
  
    // If the input index is in range of this node, then update 
    // the value of the node and its children 
    st[si] = st[si] + diff; 
    if (se != ss) { 
        int mid = getMid(ss, se); 
        updateValueUtil( ss, mid, i, diff, 2 * si + 1); 
        updateValueUtil( mid + 1, se, i, diff, 2 * si + 2); 
    
  
// The function to update a value in input array and segment tree. 
// It uses updateValueUtil() to update the value in segment tree 
static void updateValue(int []arr, int n, int i, int new_val) 
    // Check for erroneous input index 
    if (i < 0 || i > n - 1) 
    
        Console.WriteLine("Invalid Input"); 
        return
    
  
    // Get the difference between new value and old value 
    int diff = new_val - arr[i]; 
  
    // Update the value in array 
    arr[i] = new_val; 
  
    // Update the values of nodes in segment tree 
    updateValueUtil( 0, n - 1, i, diff, 0); 
  
// Function to return the sum of elements in range 
// from index qs (query start) to qe (query end) 
// It mainly uses getSumUtil() 
static int getGcd( int n, int qs, int qe) 
  
    // Check for erroneous input values 
    if (qs < 0 || qe > n - 1 || qs > qe)
    
        Console.WriteLine( "Invalid Input"); 
        return -1; 
    
  
    return getGcdUtil( 0, n - 1, qs, qe, 0); 
  
// A recursive function that constructs Segment Tree for array[ss..se]. 
// si is index of current node in segment tree st 
static int constructGcdUtil(int []arr, int ss, int se, int si) 
    // If there is one element in array, store it in current node of 
    // segment tree and return 
    if (ss == se) 
    
        st[si] = arr[ss]; 
        return arr[ss]; 
    
  
    // If there are more than one element then recur for left and 
    // right subtrees and store the sum of values in this node 
    int mid = getMid(ss, se); 
    st[si] = __gcd(constructGcdUtil(arr, ss, mid, si * 2 + 1), 
                constructGcdUtil(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 
static void constructGcd(int []arr, int n) 
    // Allocate memory for the segment tree 
  
    // Height of segment tree 
    int x = (int)(Math.Ceiling(Math.Log(n)/Math.Log(2))); 
  
    // Maximum size of segment tree 
    int max_size = 2 * (int)Math.Pow(2, x) - 1; 
  
    // Allocate memory 
    st = new int[max_size];
  
    // Fill the allocated memory st 
    constructGcdUtil(arr, 0, n - 1, 0); 
  
  
// Driver code 
public static void Main(String []args)
    int []arr = { 1, 3, 6, 9, 9, 11 }; 
    int n = arr.Length; 
  
    // Build segment tree from given array 
    constructGcd(arr, n); 
  
    // Print GCD of values in array from index 1 to 3 
    Console.WriteLine( getGcd( n, 1, 3) ); 
  
    // Update: set arr[1] = 10 and update corresponding 
    // segment tree nodes 
    updateValue(arr, n, 1, 10); 
  
    // Find GCD after the value is updated 
    Console.WriteLine( getGcd( n, 1, 3) ); 
}
}
  
// This code contributed by Rajput-Ji

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Output:

3
1


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