Levelwise Alternating GCD and LCM of nodes in Segment Tree
A Levelwise GCD/LCM alternating segment tree is a segment tree, such that at every level the operations GCD and LCM alternate. In other words at Level 1 the left and right sub-trees combine together by the GCD operation i.e Parent node = Left Child GCD Right Child and on Level 2 the left
and right sub-trees combine together by the LCM operation i.e Parent node = Left Child LCM Right Child
Such a type of Segment tree has the following type of structure:
The operations (GCD) and (LCM) indicate which operation was carried out to merge the child nodes
Given N leaf nodes, the task is to build such a segment tree and print the root node.
Examples:
Input : arr[] = { 5, 4, 8, 10, 6 } Output : Value at Root Node = 2 Explanation : The image given above shows the segment tree corresponding to the given set leaf nodes.
Prerequisites: Segment Trees
In this Segment Tree, we carry two operations:- GCD and LCM.
Now, along with the information which is passed recursively for the sub-trees, information regarding the operation to be carried out at that level is also passed since these operations alternate levelwise. It is important to note that a parent node when calls its left and right children the same operation information is passed to both the children as they are on the same level.
Let’s represent the two operations i.e GCD and LCM by 0 and 1 respectively. Then, if at Level i GCD operation is performed then at Level (i + 1) LCM operation will be performed. Thus if Level i has 0 as operation then level (i + 1) will have 1 as operation.
Operation at Level (i + 1) = ! (Operation at Level i) where, Operation at Level i ? {0, 1}
Careful analysis of the image suggests that if the height of the tree is even then the root node is a result of LCM operation of its left and right children else a result of GCD operation.
Implementation:
C++
#include <bits/stdc++.h> using namespace std; // Recursive function to return gcd of a and b int gcd( int a, int b) { // Everything divides 0 if (a == 0 || b == 0) return 0; // base case if (a == b) return a; // a is greater if (a > b) return gcd(a-b, b); return gcd(a, b-a); } // A utility function to get the middle index from // corner indexes. int getMid( int s, int e) { return s + (e - s) / 2; } void STconstructUtill( int arr[], int ss, int se, int * st, int si, int op) { // 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 ; } // If there are more than one elements, then recur // for left and right subtrees and store the sum of // values in this node int mid = getMid(ss, se); // Build the left and the right subtrees by using // the fact that operation at level (i + 1) = ! // (operation at level i) STconstructUtill(arr, ss, mid, st, si * 2 + 1, !op); STconstructUtill(arr, mid + 1, se, st, si * 2 + 2, !op); // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1) { // GCD operation st[si] = __gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls STconstructUtil() to fill the allocated memory */ int * STconstruct( int arr[], int n) { // Allocate memory for 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]; // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node int opAtRoot = (x % 2 == 0 ? 0 : 1); // Fill the allocated memory st STconstructUtill(arr, 0, n - 1, st, 0, opAtRoot); // Return the constructed segment tree return st; } int main() { int arr[] = { 5, 4, 8, 10, 6 }; int n = sizeof (arr) / sizeof (arr[0]); // Build segment tree int * st = STconstruct(arr, n); // 0-based indexing in segment tree int rootIndex = 0; cout << "Value at Root Node = " << st[rootIndex]; return 0; } |
Java
import java.io.*; import java.util.*; class GFG { // Recursive function to return gcd of a and b static int gcd( int a, int b) { // Everything divides 0 if (a == 0 || b == 0 ) return 0 ; // base case if (a == b) return a; // a is greater if (a > b) return gcd(a - b, b); return gcd(a, b - a); } // A utility function to get the middle index from // corner indexes. static int getMid( int s, int e) { return s + (e - s) / 2 ; } static void STconstructUtill( int [] arr, int ss, int se, int [] st, int si, boolean op) { // 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 ; } // If there are more than one elements, then recur // for left and right subtrees and store the sum of // values in this node int mid = getMid(ss, se); // Build the left and the right subtrees by using // the fact that operation at level (i + 1) = ! // (operation at level i) STconstructUtill(arr, ss, mid, st, si * 2 + 1 , !op); STconstructUtill(arr, mid + 1 , se, st, si * 2 + 2 , !op); // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == true ) { // GCD operation st[si] = gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ]); } else { // LCM operation st[si] = (st[ 2 * si + 1 ] * st[ 2 * si + 2 ]) / (gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ])); } } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls STconstructUtil() to fill the allocated memory */ static int [] STconstruct( int arr[], int n) { // Allocate memory for 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 int [] st = new int [max_size]; boolean opAtRoot; // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node if (x % 2 == 0 ) { opAtRoot = false ; } else { opAtRoot = true ; } // Fill the allocated memory st STconstructUtill(arr, 0 , n - 1 , st, 0 , opAtRoot); // Return the constructed segment tree return st; } // Driver code public static void main (String[] args) { int [] arr = { 5 , 4 , 8 , 10 , 6 }; int n = arr.length; // Build segment tree int [] st=STconstruct(arr, n); // 0-based indexing in segment tree System.out.println( "Value at Root Node = " + st[ 0 ]); } } // This code is contributed by avanitrachhadiya2155 |
Python3
from math import ceil,floor,log # Recursive function to return gcd of a and b def gcd(a, b): # Everything divides 0 if (a = = 0 or b = = 0 ): return 0 # base case if (a = = b): return a # a is greater if (a > b): return gcd(a - b, b) return gcd(a, b - a) # A utility function to get the middle index from # corner indexes. def getMid(s, e): return s + (e - s) / / 2 def STconstructUtill(arr, ss, se, st, si, op): # 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 # If there are more than one elements, then recur # for left and right subtrees and store the sum of # values in this node mid = getMid(ss, se) # Build the left and the right subtrees by using # the fact that operation at level (i + 1) = ! # (operation at level i) STconstructUtill(arr, ss, mid, st, si * 2 + 1 , not op) STconstructUtill(arr, mid + 1 , se, st, si * 2 + 2 , not op) # merge the left and right subtrees by checking # the operation to be carried. If operation = 1, # then do GCD else LCM if (op = = 1 ): # GCD operation st[si] = gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ]) else : # LCM operation st[si] = (st[ 2 * si + 1 ] * st[ 2 * si + 2 ]) / / (gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ])) # # /* Function to construct segment tree from given array. # This function allocates memory for segment tree and # calls STconstructUtil() to fill the allocated memory */ def STconstruct(arr, n): # Allocate memory for segment tree # Height of segment tree x = ceil(log(n, 2 )) # maximum size of segment tree max_size = 2 * pow ( 2 , x) - 1 # allocate memory st = [ 0 ] * max_size # operation = 1(GCD) if Height of tree is # even else it is 0(LCM) for the root node if (x % 2 = = 0 ): opAtRoot = 0 else : opAtRoot = 1 # Fill the allocated memory st STconstructUtill(arr, 0 , n - 1 , st, 0 , opAtRoot) # Return the constructed segment tree return st # Driver code if __name__ = = '__main__' : arr = [ 5 , 4 , 8 , 10 , 6 ] n = len (arr) # Build segment tree st = STconstruct(arr, n) # 0-based indexing in segment tree rootIndex = 0 print ( "Value at Root Node = " ,st[rootIndex]) # This code is contributed by mohit kumar 29 |
C#
using System; class GFG { // Recursive function to return gcd of a and b static int gcd( int a, int b) { // Everything divides 0 if (a == 0 || b == 0) return 0; // base case if (a == b) return a; // a is greater if (a > b) return gcd(a - b, b); return gcd(a, b - a); } // A utility function to get the middle index from // corner indexes. static int getMid( int s, int e) { return s + (e - s) / 2; } static void STconstructUtill( int [] arr, int ss, int se, int [] st, int si, bool op) { // 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 ; } // If there are more than one elements, then recur // for left and right subtrees and store the sum of // values in this node int mid = getMid(ss, se); // Build the left and the right subtrees by using // the fact that operation at level (i + 1) = ! // (operation at level i) STconstructUtill(arr, ss, mid, st, si * 2 + 1, !op); STconstructUtill(arr, mid + 1, se, st, si * 2 + 2, !op); // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == true ) { // GCD operation st[si] = gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls STconstructUtil() to fill the allocated memory */ static int [] STconstruct( int [] arr, int n) { // Allocate memory for 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 int [] st = new int [max_size]; bool opAtRoot; // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node if (x % 2 == 0) { opAtRoot = false ; } else { opAtRoot = true ; } // Fill the allocated memory st STconstructUtill(arr, 0, n - 1, st, 0, opAtRoot); // Return the constructed segment tree return st; } // Driver code static public void Main () { int [] arr = { 5, 4, 8, 10, 6 }; int n = arr.Length; // Build segment tree int [] st=STconstruct(arr, n); // 0-based indexing in segment tree Console.WriteLine( "Value at Root Node = " + st[0]); } } // This code is contributed by rag2127 |
Javascript
<script> // Recursive function to return gcd of a and b function gcd(a , b) { // Everything divides 0 if (a == 0 || b == 0) return 0; // base case if (a == b) return a; // a is greater if (a > b) return gcd(a - b, b); return gcd(a, b - a); } // A utility function to get the middle index from // corner indexes. function getMid(s , e) { return s + parseInt((e - s) / 2); } function STconstructUtill(arr , ss , se, st , si, op) { // 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 ; } // If there are more than one elements, then recur // for left and right subtrees and store the sum of // values in this node var mid = getMid(ss, se); // Build the left and the right subtrees by using // the fact that operation at level (i + 1) = ! // (operation at level i) STconstructUtill(arr, ss, mid, st, si * 2 + 1, !op); STconstructUtill(arr, mid + 1, se, st, si * 2 + 2, !op); // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == true ) { // GCD operation st[si] = gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } /* * Function to construct segment tree from given array. This function allocates * memory for segment tree and calls STconstructUtil() to fill the allocated * memory */ function STconstruct(arr , n) { // Allocate memory for segment tree // Height of segment tree var x = parseInt( (Math.ceil((Math.log(n) / Math.log(2))))); // maximum size of segment tree var max_size = 2 * parseInt( Math.pow(2, x) - 1); // allocate memory var st = Array(max_size).fill(0); var opAtRoot; // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node if (x % 2 == 0) { opAtRoot = false ; } else { opAtRoot = true ; } // Fill the allocated memory st STconstructUtill(arr, 0, n - 1, st, 0, opAtRoot); // Return the constructed segment tree return st; } // Driver code var arr = [ 5, 4, 8, 10, 6 ]; var n = arr.length; // Build segment tree var st = STconstruct(arr, n); // 0-based indexing in segment tree document.write( "Value at Root Node = " + st[0]); // This code is contributed by umadevi9616 </script> |
Value at Root Node = 2
Time complexity for tree construction is O(n), as there are total 2*n-1 nodes and value at every node is calculated at once.
Now to perform point updates i.e. update the value with given index and value, can be done by traversing down the tree to the leaf node and performing the update.
While coming back to the root node we build the tree again similar to the build() function by passing the operation to be performed at every level and storing the result of applying that operation on values of its left and right children and storing the result into that node.
Consider the following Segment tree after performing the update,
arr[2] = 7
Now the updated segment tree looks like this:
Here nodes in black denote the fact that they are updated.
Implementation:
C++
#include <bits/stdc++.h> using namespace std; // Recursive function to return gcd of a and b int gcd( int a, int b) { // Everything divides 0 if (a == 0 || b == 0) return 0; // base case if (a == b) return a; // a is greater if (a > b) return gcd(a-b, b); return gcd(a, b-a); } // A utility function to get the middle index from // corner indexes. int getMid( int s, int e) { return s + (e - s) / 2; } void STconstructUtill( int arr[], int ss, int se, int * st, int si, int op) { // 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 ; } // If there are more than one elements, then recur // for left and right subtrees and store the sum of // values in this node int mid = getMid(ss, se); // Build the left and the right subtrees by using // the fact that operation at level (i + 1) = ! // (operation at level i) STconstructUtill(arr, ss, mid, st, si * 2 + 1, !op); STconstructUtill(arr, mid + 1, se, st, si * 2 + 2, !op); // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1) { // GCD operation st[si] = gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } void updateUtil( int * st, int ss, int se, int ind, int val, int si, int op) { // Base Case: If the input index lies outside // this segment if (ind < ss || ind > se) return ; // If the input index is in range of this node, // then update the value of the node and its // children // leaf node if (ss == se && ss == ind) { st[si] = val; return ; } int mid = getMid(ss, se); // Update the left and the right subtrees by // using the fact that operation at level // (i + 1) = ! (operation at level i) updateUtil(st, ss, mid, ind, val, 2 * si + 1, !op); updateUtil(st, mid + 1, se, ind, val, 2 * si + 2, !op); // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1) { // GCD operation st[si] = gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } void update( int arr[], int * st, int n, int ind, int val) { // Check for erroneous input index if (ind < 0 || ind > n - 1) { printf ( "Invalid Input" ); return ; } // Height of segment tree int x = ( int )( ceil (log2(n))); // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node int opAtRoot = (x % 2 == 0 ? 0 : 1); arr[ind] = val; // Update the values of nodes in segment tree updateUtil(st, 0, n - 1, ind, val, 0, opAtRoot); } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls STconstructUtil() to fill the allocated memory */ int * STconstruct( int arr[], int n) { // Allocate memory for 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]; // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node int opAtRoot = (x % 2 == 0 ? 0 : 1); // Fill the allocated memory st STconstructUtill(arr, 0, n - 1, st, 0, opAtRoot); // Return the constructed segment tree return st; } int main() { int arr[] = { 5, 4, 8, 10, 6 }; int n = sizeof (arr) / sizeof (arr[0]); // Build segment tree int * st = STconstruct(arr, n); // 0-based indexing in segment tree int rootIndex = 0; cout << "Old Value at Root Node = " << st[rootIndex] << endl; // perform update arr[2] = 7 update(arr, st, n, 2, 7); cout << "New Value at Root Node = " << st[rootIndex] << endl; return 0; } |
Java
import java.util.*; public class GFG { // Recursive function to return gcd of a and b static int gcd( int a, int b) { // Everything divides 0 if (a == 0 || b == 0 ) return 0 ; // base case if (a == b) return a; // a is greater if (a > b) return gcd(a - b, b); return gcd(a, b - a); } // A utility function to get the middle index from // corner indexes. static int getMid( int s, int e) { return s + (e - s) / 2 ; } static void STconstructUtill( int [] arr, int ss, int se, int [] st, int si, int op) { // 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 ; } // If there are more than one elements, then recur // for left and right subtrees and store the sum of // values in this node int mid = getMid(ss, se); // Build the left and the right subtrees by using // the fact that operation at level (i + 1) = ! // (operation at level i) if (op != 0 ) { STconstructUtill(arr, ss, mid, st, si * 2 + 1 , 0 ); } else { STconstructUtill(arr, ss, mid, st, si * 2 + 1 , 1 ); } if (op != 0 ) { STconstructUtill(arr, mid + 1 , se, st, si * 2 + 2 , 0 ); } else { STconstructUtill(arr, mid + 1 , se, st, si * 2 + 2 , 1 ); } // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1 ) { // GCD operation st[si] = gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ]); } else { // LCM operation st[si] = (st[ 2 * si + 1 ] * st[ 2 * si + 2 ]) / (gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ])); } } static void updateUtil( int [] st, int ss, int se, int ind, int val, int si, int op) { // Base Case: If the input index lies outside // this segment if (ind < ss || ind > se) return ; // If the input index is in range of this node, // then update the value of the node and its // children // leaf node if (ss == se && ss == ind) { st[si] = val; return ; } int mid = getMid(ss, se); // Update the left and the right subtrees by // using the fact that operation at level // (i + 1) = ! (operation at level i) if (op != 0 ) { updateUtil(st, ss, mid, ind, val, 2 * si + 1 , 0 ); } else { updateUtil(st, ss, mid, ind, val, 2 * si + 1 , 1 ); } if (op != 0 ) { updateUtil(st, mid + 1 , se, ind, val, 2 * si + 2 , 0 ); } else { updateUtil(st, mid + 1 , se, ind, val, 2 * si + 2 , 1 ); } // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1 ) { // GCD operation st[si] = gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ]); } else { // LCM operation st[si] = (st[ 2 * si + 1 ] * st[ 2 * si + 2 ]) / (gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ])); } } static void update( int [] arr, int [] st, int n, int ind, int val) { // Check for erroneous input index if (ind < 0 || ind > n - 1 ) { System.out.print( "Invalid Input" ); return ; } // Height of segment tree int x = ( int )(Math.ceil(Math.log(n) / Math.log( 2 ))); // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node int opAtRoot = (x % 2 == 0 ? 0 : 1 ); arr[ind] = val; // Update the values of nodes in segment tree updateUtil(st, 0 , n - 1 , ind, val, 0 , opAtRoot); } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls STconstructUtil() to fill the allocated memory */ static int [] STconstruct( int [] arr, int n) { // Allocate memory for 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 int [] st = new int [max_size]; // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node int opAtRoot = (x % 2 == 0 ? 0 : 1 ); // Fill the allocated memory st STconstructUtill(arr, 0 , n - 1 , st, 0 , opAtRoot); // Return the constructed segment tree return st; } // Driver code public static void main(String[] args) { int [] arr = { 5 , 4 , 8 , 10 , 6 }; int n = arr.length; // Build segment tree int [] st = STconstruct(arr, n); // 0-based indexing in segment tree int rootIndex = 0 ; System.out.println( "Old Value at Root Node = " + st[rootIndex]); // perform update arr[2] = 7 update(arr, st, n, 2 , 7 ); System.out.println( "New Value at Root Node = " + st[rootIndex]); } } // This code is contributed by divyeshrabadiya07. |
Python3
import math # Recursive function to return gcd of a and b def gcd(a , b): # Everything divides 0 if (a = = 0 or b = = 0 ): return 0 # base case if (a = = b): return a # a is greater if (a > b): return gcd(a - b, b) return gcd(a, b - a) # A utility function to get the middle index from # corner indexes. def getMid(s , e): return s + int ((e - s) / 2 ) def STconstructUtill(arr , ss , se, st , si , op): # 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 # If there are more than one elements, then recur # for left and right subtrees and store the sum of # values in this node mid = getMid(ss, se) # Build the left and the right subtrees by using # the fact that operation at level (i + 1) = ! # (operation at level i) if (op ! = 0 ): STconstructUtill(arr, ss, mid, st, si * 2 + 1 , 0 ) else : STconstructUtill(arr, ss, mid, st, si * 2 + 1 , 1 ) if (op ! = 0 ): STconstructUtill(arr, mid + 1 , se, st, si * 2 + 2 , 0 ) else : STconstructUtill(arr, mid + 1 , se, st, si * 2 + 2 , 1 ) # merge the left and right subtrees by checking # the operation to be carried. If operation = 1, # then do GCD else LCM if (op = = 1 ): # GCD operation st[si] = gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ]) else : # LCM operation st[si] = (st[ 2 * si + 1 ] * st[ 2 * si + 2 ]) / (gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ])) def updateUtil(st , ss , se , ind , val , si , op): # Base Case: If the input index lies outside # this segment if (ind < ss or ind > se): return # If the input index is in range of this node, # then update the value of the node and its # children # leaf node if (ss = = se and ss = = ind): st[si] = val return mid = getMid(ss, se) # Update the left and the right subtrees by # using the fact that operation at level # (i + 1) = ! (operation at level i) if (op ! = 0 ): updateUtil(st, ss, mid, ind, val, 2 * si + 1 , 0 ) else : updateUtil(st, ss, mid, ind, val, 2 * si + 1 , 1 ) if (op ! = 0 ): updateUtil(st, mid + 1 , se, ind, val, 2 * si + 2 , 0 ) else : updateUtil(st, mid + 1 , se, ind, val, 2 * si + 2 , 1 ) # merge the left and right subtrees by checking # the operation to be carried. If operation = 1, # then do GCD else LCM if (op = = 1 ): # GCD operation st[si] = gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ]) else : # LCM operation st[si] = (st[ 2 * si + 1 ] * st[ 2 * si + 2 ]) / (gcd(st[ 2 * si + 1 ], st[ 2 * si + 2 ])) def update(arr, st , n , ind , val): # Check for erroneous input index if (ind < 0 and ind > n - 1 ): print ( "Invalid Input" ) return # Height of segment tree x = int ((math.ceil(math.log(n) / math.log( 2 )))) # operation = 1(GCD) if Height of tree is # even else it is 0(LCM) for the root node if x % 2 = = 0 : opAtRoot = 0 else : opAtRoot = 1 arr[ind] = val # Update the values of nodes in segment tree updateUtil(st, 0 , n - 1 , ind, val, 0 , opAtRoot) """ Function to construct segment tree from given array. This function allocates memory for segment tree and calls STconstructUtil() to fill the allocated memory """ def STconstruct(arr , n): # Allocate memory for segment tree # Height of segment tree x = int ((math.ceil(math.log(n) / math.log( 2 )))) # maximum size of segment tree max_size = 2 * int (math. pow ( 2 , x) - 1 ) # allocate memory st = [ 0 ] * (max_size) # operation = 1(GCD) if Height of tree is # even else it is 0(LCM) for the root node if x % 2 = = 0 : opAtRoot = 0 else : opAtRoot = 1 # Fill the allocated memory st STconstructUtill(arr, 0 , n - 1 , st, 0 , opAtRoot) # Return the constructed segment tree return st arr = [ 5 , 4 , 8 , 10 , 6 ] n = len (arr) # Build segment tree st = STconstruct(arr, n) # 0-based indexing in segment tree rootIndex = 0 print ( "Old Value at Root Node =" , int (st[rootIndex])) # perform update arr[2] = 7 update(arr, st, n, 2 , 7 ) print ( "New Value at Root Node =" , int (st[rootIndex])) # This code is contributed by decode2207. |
C#
using System; class GFG { // Recursive function to return gcd of a and b static int gcd( int a, int b) { // Everything divides 0 if (a == 0 || b == 0) return 0; // base case if (a == b) return a; // a is greater if (a > b) return gcd(a - b, b); return gcd(a, b - a); } // A utility function to get the middle index from // corner indexes. static int getMid( int s, int e) { return s + (e - s) / 2; } static void STconstructUtill( int [] arr, int ss, int se, int [] st, int si, int op) { // 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 ; } // If there are more than one elements, then recur // for left and right subtrees and store the sum of // values in this node int mid = getMid(ss, se); // Build the left and the right subtrees by using // the fact that operation at level (i + 1) = ! // (operation at level i) if (op != 0) { STconstructUtill(arr, ss, mid, st, si * 2 + 1, 0); } else { STconstructUtill(arr, ss, mid, st, si * 2 + 1, 1); } if (op != 0) { STconstructUtill(arr, mid + 1, se, st, si * 2 + 2, 0); } else { STconstructUtill(arr, mid + 1, se, st, si * 2 + 2, 1); } // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1) { // GCD operation st[si] = gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } static void updateUtil( int [] st, int ss, int se, int ind, int val, int si, int op) { // Base Case: If the input index lies outside // this segment if (ind < ss || ind > se) return ; // If the input index is in range of this node, // then update the value of the node and its // children // leaf node if (ss == se && ss == ind) { st[si] = val; return ; } int mid = getMid(ss, se); // Update the left and the right subtrees by // using the fact that operation at level // (i + 1) = ! (operation at level i) if (op != 0) { updateUtil(st, ss, mid, ind, val, 2 * si + 1, 0); } else { updateUtil(st, ss, mid, ind, val, 2 * si + 1, 1); } if (op != 0) { updateUtil(st, mid + 1, se, ind, val, 2 * si + 2, 0); } else { updateUtil(st, mid + 1, se, ind, val, 2 * si + 2, 1); } // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1) { // GCD operation st[si] = gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } static void update( int [] arr, int [] st, int n, int ind, int val) { // Check for erroneous input index if (ind < 0 || ind > n - 1) { Console.Write( "Invalid Input" ); return ; } // Height of segment tree int x = ( int )(Math.Ceiling(Math.Log(n, 2))); // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node int opAtRoot = (x % 2 == 0 ? 0 : 1); arr[ind] = val; // Update the values of nodes in segment tree updateUtil(st, 0, n - 1, ind, val, 0, opAtRoot); } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls STconstructUtil() to fill the allocated memory */ static int [] STconstruct( int [] arr, int n) { // Allocate memory for segment tree // Height of segment tree int x = ( int )(Math.Ceiling(Math.Log(n, 2))); // maximum size of segment tree int max_size = 2 * ( int )Math.Pow(2, x) - 1; // allocate memory int [] st = new int [max_size]; // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node int opAtRoot = (x % 2 == 0 ? 0 : 1); // Fill the allocated memory st STconstructUtill(arr, 0, n - 1, st, 0, opAtRoot); // Return the constructed segment tree return st; } // Driver code static void Main() { int [] arr = { 5, 4, 8, 10, 6 }; int n = arr.Length; // Build segment tree int [] st = STconstruct(arr, n); // 0-based indexing in segment tree int rootIndex = 0; Console.WriteLine( "Old Value at Root Node = " + st[rootIndex]); // perform update arr[2] = 7 update(arr, st, n, 2, 7); Console.WriteLine( "New Value at Root Node = " + st[rootIndex]); } } // This code is contributed by divyesh072019. |
Javascript
<script> // Recursive function to return gcd of a and b function gcd(a , b) { // Everything divides 0 if (a == 0 || b == 0) return 0; // base case if (a == b) return a; // a is greater if (a > b) return gcd(a - b, b); return gcd(a, b - a); } // A utility function to get the middle index from // corner indexes. function getMid(s , e) { return s + parseInt((e - s) / 2); } function STconstructUtill(arr , ss , se, st , si , op) { // 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 ; } // If there are more than one elements, then recur // for left and right subtrees and store the sum of // values in this node var mid = getMid(ss, se); // Build the left and the right subtrees by using // the fact that operation at level (i + 1) = ! // (operation at level i) if (op != 0) { STconstructUtill(arr, ss, mid, st, si * 2 + 1, 0); } else { STconstructUtill(arr, ss, mid, st, si * 2 + 1, 1); } if (op != 0) { STconstructUtill(arr, mid + 1, se, st, si * 2 + 2, 0); } else { STconstructUtill(arr, mid + 1, se, st, si * 2 + 2, 1); } // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1) { // GCD operation st[si] = gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } function updateUtil(st , ss , se , ind , val , si , op) { // Base Case: If the input index lies outside // this segment if (ind < ss || ind > se) return ; // If the input index is in range of this node, // then update the value of the node and its // children // leaf node if (ss == se && ss == ind) { st[si] = val; return ; } var mid = getMid(ss, se); // Update the left and the right subtrees by // using the fact that operation at level // (i + 1) = ! (operation at level i) if (op != 0) { updateUtil(st, ss, mid, ind, val, 2 * si + 1, 0); } else { updateUtil(st, ss, mid, ind, val, 2 * si + 1, 1); } if (op != 0) { updateUtil(st, mid + 1, se, ind, val, 2 * si + 2, 0); } else { updateUtil(st, mid + 1, se, ind, val, 2 * si + 2, 1); } // merge the left and right subtrees by checking // the operation to be carried. If operation = 1, // then do GCD else LCM if (op == 1) { // GCD operation st[si] = gcd(st[2 * si + 1], st[2 * si + 2]); } else { // LCM operation st[si] = (st[2 * si + 1] * st[2 * si + 2]) / (gcd(st[2 * si + 1], st[2 * si + 2])); } } function update(arr, st , n , ind , val) { // Check for erroneous input index if (ind < 0 || ind > n - 1) { document.write( "Invalid Input" ); return ; } // Height of segment tree var x = parseInt( (Math.ceil(Math.log(n) / Math.log(2)))); // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node var opAtRoot = (x % 2 == 0 ? 0 : 1); arr[ind] = val; // Update the values of nodes in segment tree updateUtil(st, 0, n - 1, ind, val, 0, opAtRoot); } /* Function to construct segment tree from given array. This function allocates memory for segment tree and calls STconstructUtil() to fill the allocated memory */ function STconstruct(arr , n) { // Allocate memory for segment tree // Height of segment tree var x = parseInt( (Math.ceil(Math.log(n) / Math.log(2)))); // maximum size of segment tree var max_size = 2 * parseInt( Math.pow(2, x) - 1); // allocate memory var st = Array(max_size).fill(0); // operation = 1(GCD) if Height of tree is // even else it is 0(LCM) for the root node var opAtRoot = (x % 2 == 0 ? 0 : 1); // Fill the allocated memory st STconstructUtill(arr, 0, n - 1, st, 0, opAtRoot); // Return the constructed segment tree return st; } // Driver code var arr = [ 5, 4, 8, 10, 6 ]; var n = arr.length; // Build segment tree var st = STconstruct(arr, n); // 0-based indexing in segment tree var rootIndex = 0; document.write( "Old Value at Root Node = " + st[rootIndex]); // perform update arr[2] = 7 update(arr, st, n, 2, 7); document.write( "<br/>New Value at Root Node = " + st[rootIndex]); // This code contributed by Rajput-Ji </script> |
Old Value at Root Node = 2 New Value at Root Node = 1
The time complexity of update is also O(Logn). To update a leaf value, one node is processed at every level and number of levels is O(Logn).