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# 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:

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:

3

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

A segment tree can be used to do preprocessing and query in moderate time. With the segment tree, preprocessing time is O(n) and the time for the 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 the GCD of all leaves under it.

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

• The 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 the 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 into two halves at every level.
• Since the constructed tree is always a full binary tree with n leaves, there will be n-1 internal nodes. So the 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 that needs to be updated. Let diff be the value to be added. We start from the root of the segment tree and add diff to all nodes which have given index in their range. If a node doesn’t have a given index in its range, we don’t make any changes to that node.

Below is the implementation of the above approach:

## C++

 // C++ implementation of the approach#include using namespace std; // A utility function to get the// middle index from corner indexesint 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 rangeint 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 rangevoid updateValueUtil(int* st, int ss, int se, int i, int new_val, int si){    // Base Case: If the input index lies outside the range of    // this segment    if (i < ss || i > se)        return;     // If only single element is left in the range    if(ss == se)    {        st[si] = new_val;        return;    }         int mid = getMid(ss, se);    updateValueUtil(st, ss, mid, i, new_val, 2 * si + 1);    updateValueUtil(st, mid + 1, se, i, new_val, 2 * si + 2);         st[si] = __gcd(st[2*si + 1], st[2*si + 2]);} // The function to update a value in input array and segment tree.// It uses updateValueUtil() to update the value in segment treevoid 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;    }     // Update the values of nodes in segment tree    updateValueUtil(st, 0, n - 1, i, new_val, 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 stint 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 memoryint* 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 codeint 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;}

## Java

 // Java implementation of the approachclass GFG{     // segment treestatic int st[]; // Recursive function to return gcd of a and bstatic 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 indexesstatic 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 rangestatic 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 rangestatic void updateValueUtil( int ss, int se, int i, int new_val, int si){    // Base Case: If the input index lies outside the range of    // this segment    if (i < ss || i > se)        return;     // If only single element is left in the range    if(ss == se)    {        st[si] = new_val;        return;    }         int mid = getMid(ss, se);    updateValueUtil(ss, mid, i, new_val, 2 * si + 1);    updateValueUtil(mid + 1, se, i, new_val, 2 * si + 2);         st[si] = __gcd(st[2*si + 1], st[2*si + 2]);} // The function to update a value in input array and segment tree.// It uses updateValueUtil() to update the value in segment treestatic 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;    }       // Update the values of nodes in segment tree    updateValueUtil( 0, n - 1, i, new_val, 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 ststatic 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 memorystatic 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 codepublic 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

## Python3

 # Python 3 implementation of the approach from math import gcd,ceil,log2,pow # A utility function to get the# middle index from corner indexesdef 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 rangedef 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 rangedef updateValueUtil(st,ss,se,i,new_val,si):         # Base Case: If the input index lies outside the range of    # this segment    if (i < ss or i > se):        return         if(ss == se):        st[si] = new_val        return     # If the input index is in range of this node, then update    # the value of the node and its children         mid = getMid(ss, se)    updateValueUtil(st, ss, mid, i, new_val, 2 * si + 1)    updateValueUtil(st, mid + 1, se, i, new_val, 2 * si + 2)         st[si] = gcd(st[2*si + 1], st[2*si + 2]) # The function to update a value in input array and segment tree.# It uses updateValueUtil() to update the value in segment treedef updateValue(arr, st, n, i, new_val):         # Check for erroneous input index    if (i < 0 or i > n - 1):        print("Invalid Input")        return     # Update the values of nodes in segment tree    updateValueUtil(st, 0, n - 1, i, new_val, 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 stdef 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 memorydef 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 codeif __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

## C#

 // C# implementation of the approach.using System;     class GFG{     // segment treestatic int []st; // Recursive function to return gcd of a and bstatic 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 indexesstatic 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 rangestatic 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 rangestatic void updateValueUtil( int ss, int se, int i, int new_val, int si){    // Base Case: If the input index lies outside the range of    // this segment    if (i < ss || i > se)        return;     // If only single element is left in the range    if(ss == se)    {        st[si] = new_val;        return;    }         int mid = getMid(ss, se);    updateValueUtil(ss, mid, i, new_val, 2 * si + 1);    updateValueUtil(mid + 1, se, i, new_val, 2 * si + 2);         st[si] = __gcd(st[2*si + 1], st[2*si + 2]);} // The function to update a value in input array and segment tree.// It uses updateValueUtil() to update the value in segment treestatic 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;    }     // Update the values of nodes in segment tree    updateValueUtil( 0, n - 1, i, new_val, 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 ststatic 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 memorystatic 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 codepublic 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

## Javascript



Output

3
1

Time Complexity: O(n log n), as segment tree construction will take O(n log n) time. Where n is the number of elements in the array.
Auxiliary Space: O(n log n), as we are using extra space for the segment tree. Where n is the number of elements in the array.

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