Let us consider the following problem to understand Segment Trees.
We have an array arr[0 . . . n-1]. We should be able to
1 Find the product of elements from index l to r where 0 <= l <= r <= n-1 take its modulus by an integer m.
2 Change value of a specified element of the array to a new value x. We need to do arr[i] = x where 0 <= i <= n-1.
A simple solution is to run a loop from l to r and calculate product of elements in a given range and modulo it by m. To update a value, simply do arr[i] = x. The first operation takes O(n) time and second operation takes O(1) time.
Another solution is to create two arrays and store the product modulo m from start to l-1 in first array and the product from r+1 to end of the array modulo m in another array. Product of a given range can now be calculated in O(1) time, but update operation takes O(n) time now.
Lets say the product of all the elements be P, then product P from a given range l to r can be calculated as :
P: Product of all the elements of the array modulo m.
A: Product of all the elements till l-1 modulo m.
B: Product of all the elements till r+1 modulo m.
PDT = P*(modInverse(A))*(modInverse(B))
This works well if the number of query operations are large and very few updates.
Segment Tree Solution :
If the number of query and updates are equal, we can perform both the operations in O(log n) time. We can use a Segment Tree to do both operations in O(Logn) time.
Representation of Segment trees
1. Leaf Nodes are the elements of the input array.
2. Each internal node represents some merging of the leaf nodes. The merging may be different for different problems. For this problem, merging is product of leaves under a node.
An array representation of tree is used to represent Segment Trees. 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 (i-1)/2.
Query for Product of given range
Once the tree is constructed, how to get the product using the constructed segment tree. Following is algorithm to get the product of elements.
int getPdt(node, l, r)
{
if range of node is within l and r
return value in node
else if range of node is completely outside l and r
return 1
else
return (getPdt(node's left child, l, r)%mod *
getPdt(node's right child, l, r)%mod)%mod
}
Update a value
Like tree construction and query operations, update can also be done recursively. We are given an index which needs to update. We start from root of the segment tree, and multiply the range product with new value and divide the range product with previous value. If a node doesn’t have given index in its range, we don’t make any changes to that node.
Implementation:
Following is the implementation of segment tree. The program implements construction of segment tree for any given array. It also implements query and updates operations.
C++
// C++ program to show segment tree operations like // construction, query and update#include <bits/stdc++.h>#include <math.h>using namespace std;int mod = 1000000000; // 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 Pdt 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 getPdtUtil(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 Pdt 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 1; // If a part of this segment overlaps with the // given range int mid = getMid(ss, se); return (getPdtUtil(st, ss, mid, qs, qe, 2*si+1)%mod * getPdtUtil(st, mid+1, se, qs, qe, 2*si+2)%mod)%mod;} /* 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 getPdtUtil() i --> index of the element to be updated. This index is in input array.*/ void updateValueUtil(int *st, int ss, int se, int i, int prev_val, 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 the input index is in range of this node, then // update the value of the node and its children st[si] = (st[si]*new_val)/prev_val; if (se != ss) { int mid = getMid(ss, se); updateValueUtil(st, ss, mid, i, prev_val, new_val, 2*si + 1); updateValueUtil(st, mid+1, se, i, prev_val, new_val, 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; } int temp = 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, temp, new_val, 0);} // Return Pdt of elements in range from index qs// (query start)to qe (query end). It mainly // uses getPdtUtil()int getPdt(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 getPdtUtil(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 constructSTUtil(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 elements, then // recur for left and right subtrees and store // the Pdt of values in this node int mid = getMid(ss, se); st[si] = (constructSTUtil(arr, ss, mid, st, si*2+1)%mod * constructSTUtil(arr, mid+1, se, st, si*2+2)%mod)%mod; 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 *constructST(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]; // Fill the allocated memory st constructSTUtil(arr, 0, n-1, st, 0); // Return the constructed segment tree return st;} // Driver program to test above functionsint main(){ int arr[] = {1, 2, 3, 4, 5, 6}; int n = sizeof(arr)/sizeof(arr[0]); // Build segment tree from given array int *st = constructST(arr, n); // Print Product of values in array from index 1 to 3 cout << "Product of values in given range = " << getPdt(st, n, 1, 3) << endl; // Update: set arr[1] = 10 and update corresponding // segment tree nodes updateValue(arr, st, n, 1, 10); // Find Product after the value is updated cout << "Updated Product of values in given range = " << getPdt(st, n, 1, 3) << endl; return 0;} |
Java
// Java program to show segment tree operations// like construction, query and updateclass GFG{static final int mod = 1000000000;// 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 Pdt 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 getPdtUtil(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 Pdt 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 1; // If a part of this segment overlaps // with the given range int mid = getMid(ss, se); return (getPdtUtil(st, ss, mid, qs, qe, 2 * si + 1) % mod * getPdtUtil(st, mid + 1, se, qs, qe, 2 * si + 2) % mod) % mod;}/* * 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 getPdtUtil() * i --> index of the element to be updated. * This index is in input array. */static void updateValueUtil(int[] st, int ss, int se, int i, int prev_val, 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 the input index is in range of this node, then // update the value of the node and its children st[si] = (st[si] * new_val) / prev_val; if (se != ss) { int mid = getMid(ss, se); updateValueUtil(st, ss, mid, i, prev_val, new_val, 2 * si + 1); updateValueUtil(st, mid + 1, se, i, prev_val, new_val, 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[] st, int n, int i, int new_val){ // Check for erroneous input index if (i < 0 || i > n - 1) { System.out.println("Invalid Input"); return; } int temp = 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, temp, new_val, 0);}// Return Pdt of elements in range from index qs// (query start)to qe (query end). It mainly// uses getPdtUtil()static int getPdt(int[] st, 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 getPdtUtil(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 ststatic int constructSTUtil(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 elements, then // recur for left and right subtrees and store // the Pdt of values in this node int mid = getMid(ss, se); st[si] = (constructSTUtil(arr, ss, mid, st, si * 2 + 1) % mod * constructSTUtil(arr, mid + 1, se, st, si * 2 + 2) % mod) % mod; 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 int[] constructST(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]; // Fill the allocated memory st constructSTUtil(arr, 0, n - 1, st, 0); // Return the constructed segment tree return st;}// Driver codepublic static void main(String[] args) { int arr[] = { 1, 2, 3, 4, 5, 6 }; int n = arr.length; // Build segment tree from given array int[] st = constructST(arr, n); // Print Product of values in array from // index 1 to 3 System.out.printf("Product of values in " + "given range = %d\n", getPdt(st, n, 1, 3)); // Update: set arr[1] = 10 and update // corresponding segment tree nodes updateValue(arr, st, n, 1, 10); // Find Product after the value is updated System.out.printf("Updated Product of values " + "in given range = %d\n", getPdt(st, n, 1, 3));}}// This code is contributed by sanjeev2552 |
Python3
# Python3 program to show segment tree operations like# construction, query and updatefrom math import ceil,logmod = 1000000000# A utility function to get the middle index from# corner indexes.def getMid(s, e): return s + (e -s)//2"""A recursive function to get the Pdt 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 getPdtUtil(st, ss, se, qs, qe,si): # If segment of this node is a part of given # range, then return the Pdt 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 1 # If a part of this segment overlaps with the # given range mid = getMid(ss, se) return (getPdtUtil(st, ss, mid, qs, qe, 2*si+1)%mod* getPdtUtil(st, mid+1, se, qs, qe, 2*si+2)%mod)%mod"""A recursive function to update the nodes which havethe given index in their range. The following areparameters st, si, ss and se are same as getPdtUtil() i --> index of the element to be updated. This index is in input array."""def updateValueUtil(st, ss, se, i, prev_val, new_val, 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]*new_val)//prev_val if (se != ss): mid = getMid(ss, se) updateValueUtil(st, ss, mid, i, prev_val, new_val, 2*si + 1) updateValueUtil(st, mid+1, se, i, prev_val, new_val, 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): cout<<"Invalid Input" return temp = 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, temp, new_val, 0)# Return Pdt of elements in range from index qs# (query start)to qe (query end). It mainly# uses getPdtUtil()def getPdt(st, n, qs, qe): # Check for erroneous input values if (qs < 0 or qe > n-1 or qs > qe): print("Invalid Input") return -1 return getPdtUtil(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 constructSTUtil(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 elements, then # recur for left and right subtrees and store # the Pdt of values in this node mid = getMid(ss, se) st[si] = (constructSTUtil(arr, ss, mid, st, si*2+1)%mod* constructSTUtil(arr, mid+1, se, st, si*2+2)%mod)%mod return st[si]""" Function to construct segment tree from given array.This function allocates memory for segment tree andcalls constructSTUtil() to fill the allocated memory"""def constructST(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 # Fill the allocated memory st constructSTUtil(arr, 0, n-1, st, 0) # Return the constructed segment tree return st# Driver program to test above functionsif __name__ == '__main__': arr=[1, 2, 3, 4, 5, 6] n = len(arr) # Build segment tree from given array st = constructST(arr, n) # Print Product of values in array from index 1 to 3 print("Product of values in given range = ",getPdt(st, n, 1, 3)) # Update: set arr[1] = 10 and update corresponding # segment tree nodes updateValue(arr, st, n, 1, 10) # Find Product after the value is updated print("Updated Product of values in given range = ",getPdt(st, n, 1, 3))# This code is contributed by mohit kumar 29 |
C#
// C# program to show segment tree operations// like construction, query and updateusing System;class GFG{ static int mod = 1000000000; // A utility function to get the middle // index from corner indexes. public static int getMid(int s, int e) { return s + (e - s) / 2; } /* * A recursive function to get the Pdt 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 */ public static int getPdtUtil(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 Pdt 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 1; } // If a part of this segment overlaps // with the given range int mid=getMid(ss, se); return (getPdtUtil(st, ss, mid, qs,qe, 2 * si + 1) % mod * getPdtUtil(st, mid + 1, se, qs,qe, 2 * si + 2) % mod) % mod; } /* * 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 getPdtUtil() * i --> index of the element to be updated. * This index is in input array. */ public static void updateValueUtil(int[] st, int ss, int se, int i, int prev_val, 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 the input index is in range of this node, then // update the value of the node and its children st[si] = (st[si] * new_val) / prev_val; if (se != ss) { int mid = getMid(ss, se); updateValueUtil(st, ss, mid, i, prev_val,new_val, 2 * si + 1); updateValueUtil(st, mid + 1, se, i, prev_val,new_val, 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 public static void updateValue(int[] arr, int[] st, int n,int i, int new_val) { // Check for erroneous input index if(i < 0 || i > n - 1) { Console.WriteLine("Invalid Input"); return; } int temp = 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, temp, new_val, 0); } // Return Pdt of elements in range from index qs // (query start)to qe (query end). It mainly // uses getPdtUtil() public static int getPdt(int[] st, 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 getPdtUtil(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 public static int constructSTUtil(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 elements, then // recur for left and right subtrees and store // the Pdt of values in this node int mid = getMid(ss, se); st[si] = (constructSTUtil(arr, ss, mid, st, si * 2 + 1) % mod * constructSTUtil(arr, mid + 1, se, st,si * 2 + 2) % mod) % mod; 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 */ public static int[] constructST(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]; // Fill the allocated memory st constructSTUtil(arr, 0, n - 1, st, 0); // Return the constructed segment tree return st; } // Driver code static public void Main () { int[] arr = { 1, 2, 3, 4, 5, 6 }; int n = arr.Length; // Build segment tree from given array int[] st = constructST(arr, n); // Print Product of values in array from // index 1 to 3 Console.WriteLine("Product of values in " + "given range = " + getPdt(st, n, 1, 3)); // Update: set arr[1] = 10 and update // corresponding segment tree nodes updateValue(arr, st, n, 1, 10); // Find Product after the value is updated Console.WriteLine("Updated Product of values " + "in given range = " + getPdt(st, n, 1, 3)); }}// This code is contributed by avanitrachhadiya2155 |
Javascript
<script>// JavaScript program to // show segment tree operations// like construction, query and update let mod = 1000000000;// A utility function to get the middle// index from corner indexes.function getMid(s,e){ return s + Math.floor((e - s) / 2);}/* * A recursive function to get the Pdt 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 */function getPdtUtil(st,ss,se,qs,qe,si){ // If segment of this node is a part of given // range, then return the Pdt 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 1; // If a part of this segment overlaps // with the given range let mid = getMid(ss, se); return (getPdtUtil(st, ss, mid, qs, qe, 2 * si + 1) % mod * getPdtUtil(st, mid + 1, se, qs, qe, 2 * si + 2) % mod) % mod;}/* * 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 getPdtUtil() * i --> index of the element to be updated. * This index is in input array. */function updateValueUtil(st,ss,se,i,prev_val,new_val,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] = Math.floor((st[si] * new_val) / prev_val); if (se != ss) { let mid = getMid(ss, se); updateValueUtil(st, ss, mid, i, prev_val, new_val, 2 * si + 1); updateValueUtil(st, mid + 1, se, i, prev_val, new_val, 2 * si + 2); }}// The function to update a value in input array// and segment tree. It uses updateValueUtil() to// update the value in segment treefunction updateValue(arr,st,n,i,new_val){ // Check for erroneous input index if (i < 0 || i > n - 1) { document.write("Invalid Input<br>"); return; } let temp = 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, temp, new_val, 0);}// Return Pdt of elements in range from index qs// (query start)to qe (query end). It mainly// uses getPdtUtil()function getPdt(st,n,qs,qe){ // Check for erroneous input values if (qs < 0 || qe > n - 1 || qs > qe) { document.write("Invalid Input<br>"); return -1; } return getPdtUtil(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 stfunction constructSTUtil(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 elements, then // recur for left and right subtrees and store // the Pdt of values in this node let mid = getMid(ss, se); st[si] = (constructSTUtil(arr, ss, mid, st, si * 2 + 1) % mod * constructSTUtil(arr, mid + 1, se, st, si * 2 + 2) % mod) % mod; 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 */function constructST(arr,n){ // Allocate memory for segment tree // Height of segment tree let x = (Math.ceil(Math.log(n) / Math.log(2))); // Maximum size of segment tree let max_size = 2 * Math.pow(2, x) - 1; // Allocate memory let st = new Array(max_size); // Fill the allocated memory st constructSTUtil(arr, 0, n - 1, st, 0); // Return the constructed segment tree return st;}// Driver codelet arr=[1, 2, 3, 4, 5, 6 ];let n = arr.length;// Build segment tree from given arraylet st = constructST(arr, n);// Print Product of values in array from// index 1 to 3document.write("Product of values in " + "given range = ", getPdt(st, n, 1, 3)+"<br>");// Update: set arr[1] = 10 and update// corresponding segment tree nodesupdateValue(arr, st, n, 1, 10);// Find Product after the value is updateddocument.write("Updated Product of values " + "in given range = ", getPdt(st, n, 1, 3)+"<br>"); // This code is contributed by patel2127</script> |
Output:
Product of values in given range = 24 Updated Product of values in given range = 120
Time Complexity: O(n*log(n). The time complexity of constructing the segment tree is O(n*logn). For the query operation, the time complexity is O(logn), and for the update operation, the time complexity is also O(logn).
Auxiliary Space: O(n)
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