Prerequisites: Segment Tree
Given a number N which represents the size of the array initialized to 0 and Q queries to process where there are two types of queries:
- 1 P V: Put the value V at position P.
- 2 L R: Output the sum of values from L to R.
The task is to answer these queries.
Constraints:
- 1 ? N ? 1018
- Q ? 105
- 1 ? L ? R? N
Note: Queries are online. Therefore:
- L = (previousAnswer + L) % N + 1
- R = (previousAnswer + R) % N + 1
Examples:
Input: N = 5, Q = 5, arr[][] = {{1, 2, 3}, {1, 1, 4}, {1, 3, 5}, {1, 4, 7}, {2, 3, 4}}
Output: 12
Explanation:
There are five queries. Since N = 5, therefore, initially, the array is {0, 0, 0, 0, 0}
For query 1: 1 2 3 array = {0, 3, 0, 0, 0}
For query 2: 1 1 4 array = {4, 3, 0, 0, 0}
For query 3: 1 3 5 array = {4, 3, 5, 0, 0}
For query 4: 1 4 7 array = {4, 3, 5, 7, 0}
For query 5: 2 3 4 Sum from [3, 4] = 7 + 5 = 12.Input: N = 3, Q = 2, arr[][] = {{1, 1, 1}, {1, 2, 2}, {1, 3, 3}}
Output: 0
Approach: Here, since the updates are high, Kadane’s algorithm doesn’t work quite well. Moreover, since it is given that the queries are online, a simple segment tree would not be able to solve this problem because the constraints for the number of elements is very high. Therefore, a new type of data structure, a dynamic segment tree is used in this problem.
Dynamic Segment Tree: Dynamic segment tree is not a new data structure. It is very similar to the segment tree. The following are the properties of the dynamic segment tree:
- Instead of using an array to represent the intervals, a node is created whenever a new interval is to be updated.
- The following is the structure of the node of the dynamic segment tree:
C++
// Every node contains the value and // the left subtree and right subtree struct Node { long long value; struct Node *L, *R; }; struct Node* getnode() { struct Node* temp = new struct Node; temp->value = 0; temp->L = NULL; temp->R = NULL; return temp; } |
Java
// Every node contains the value and // the left subtree and right subtree static class Node { int value; Node L, R; } static Node getnode() { Node temp = new Node(); temp.value = 0; temp.L = null; temp.R = null; return temp; }// This code is contributed by Aman Kumar. |
Python3
# Define a class to represent a binary tree nodeclass Node: def __init__(self): # Each node contains a value and left and # right subtree references self.value = 0 self.L = None self.R = None# Define a function to create a new nodedef getnode(): # Allocate memory for the new node temp = Node() # Set the initial values for the node's value # and left and right subtree references temp.value = 0 temp.L = None temp.R = None # Return the newly created node return temp# this code is contributed by prajwal kandekar |
C#
// C# program for the above approachusing System;public class Node{ public int value; public Node L, R;}public class Program{ public static Node getnode() { Node temp = new Node(); temp.value = 0; temp.L = null; temp.R = null; return temp; }}// This code is contributed by rishab |
Javascript
// Javascript code for above approachclass Node { constructor() { // Each node contains a value and left and // right subtree references this.value = 0; this.L = null; this.R = null; }}// Define a function to create a new nodefunction getnode() { // Allocate memory for the new node let temp = new Node(); // Set the initial values for the node's value // and left and right subtree references temp.value = 0; temp.L = null; temp.R = null; // Return the newly created node return temp;}// This code is contributed by sdeadityasharma |
- Clearly, the above structure is the same as a Binary Search Tree. In every node, we are storing the node’s value and two pointers pointing to the left and right subtree.
- The Interval of the root is from [1, N], the interval of the left subtree will be [1, N/2] and the interval for the right subtree will be [N/2 + 1, N].
- Similarly, for every node, we can calculate the interval it is representing. Let’s say the interval of the current node is [L, R]. Then, the Interval of its left and right subtree are [L, (L + R)/2] and [(L + R)/2+1, R] respectively.
- Since we are creating a new node only when required, the build() function from the segment tree is completely removed.
Before getting into the algorithm for the operations, let’s define the terms used in this article:
- Node’s interval: It is the interval the node is representing.
- Required interval: Interval for which the sum is to calculate.
- Required index: Index at which Update is required.
The following is the algorithm used for the operations on the tree with the above-mentioned properties:
- Point Update: The following algorithm is used for the point update:
- Start with the root node.
- If the interval at the node doesn’t overlap with the required index then return.
- If the node is a NULL entry then create a new node with the appropriate intervals and descend into that node by going back to step 2 for every new child created.
- If both, intervals and the index at which the value is to be stored are equal, then store the value into at that node.
- If the interval at the node overlaps partially with the required index then descend into its children and continue the execution from step 2.
- Finding the sum for every query: The following algorithm is used to find the sum for every query:
- Start with the root node.
- If the node is a NULL or the interval at that node doesn’t overlap with the required interval, then return 0.
- If the interval at the node completely overlaps with the required interval then return the value stored at the node.
- If the interval at the node overlaps partially with the required interval then descend into its children and continue the execution from step 2 for both of its children.
Example: Lets visualize the update and sum with an example. Let N = 10 and the operations needed to perform on the tree are as follows:
- Insert 10 at position 1.
- Find the sum of value of indices from 2 to 8.
- Insert 3 at position 5.
- Find the sum of value of indices from 3 to 6.
- Initially, for the value N = 10, the tree is empty. Therefore:
- Insert 10 at position 1. In order to do this, create a new node until we get the required interval. Therefore:
- Find the sum of value of indices from 2 to 8. In order to do this, the sum from [1, 8] is found and the value [1, 2] is subtracted from it. Since the node [1, 8] is not yet created, the value of [1, 8] is the value of the root [1, 10]. Therefore:
- Insert 3 at position 5. In order to do this, create a new node until we get the required interval. Therefore:
Below is the implementation of the above approach:
C++
// C++ program for the implementation// of the Dynamic segment tree and// perform the range updates on the// given queries#include <bits/stdc++.h>using namespace std;typedef long long ll;// Structure of the nodestruct Node { ll value; struct Node *L, *R;};// Structure to get the newly formed// nodestruct Node* getnode(){ struct Node* temp = new struct Node; temp->value = 0; temp->L = NULL; temp->R = NULL; return temp;}// Creating the Root nodestruct Node* root;// Function to perform the point update// on the dynamic segment treevoid UpdateHelper(struct Node* curr, ll index, ll L, ll R, ll val){ // If the index is not overlapping // with the index if (L > index || R < index) return; // If the index is completely overlapping // with the index if (L == R && L == index) { // Update the value of the node // to the given value curr->value = val; return; } // Computing the middle index if none // of the above base cases are satisfied ll mid = L - (L - R) / 2; ll sum1 = 0, sum2 = 0; // If the index is in the left subtree if (index <= mid) { // Create a new node if the left // subtree is null if (curr->L == NULL) curr->L = getnode(); // Recursively call the function // for the left subtree UpdateHelper(curr->L, index, L, mid, val); } // If the index is in the right subtree else { // Create a new node if the right // subtree is null if (curr->R == NULL) curr->R = getnode(); // Recursively call the function // for the right subtree UpdateHelper(curr->R, index, mid + 1, R, val); } // Storing the sum of the left subtree if (curr->L) sum1 = curr->L->value; // Storing the sum of the right subtree if (curr->R) sum2 = curr->R->value; // Storing the sum of the children into // the node's value curr->value = sum1 + sum2; return;}// Function to find the sum of the// values given by the rangell queryHelper(struct Node* curr, ll a, ll b, ll L, ll R){ // Return 0 if the root is null if (curr == NULL) return 0; // If the index is not overlapping // with the index, then the node // is not created. So sum is 0 if (L > b || R < a) return 0; // If the index is completely overlapping // with the index, return the node's value if (L >= a && R <= b) return curr->value; ll mid = L - (L - R) / 2; // Return the sum of values stored // at the node's children return queryHelper(curr->L, a, b, L, mid) + queryHelper(curr->R, a, b, mid + 1, R);}// Function to call the queryHelper// function to find the sum for// the queryll query(int L, int R){ return queryHelper(root, L, R, 1, 10);}// Function to call the UpdateHelper// function for the point updatevoid update(int index, int value){ UpdateHelper(root, index, 1, 10, value);}// Function to perform the operations// on the treevoid operations(){ // Creating an empty tree root = getnode(); // Update the value at position 1 to 10 update(1, 10); // Update the value at position 3 to 5 update(3, 5); // Finding sum for the range [2, 8] cout << query(2, 8) << endl; // Finding sum for the range [1, 10] cout << query(1, 10) << endl;}// Driver codeint main(){ operations(); return 0;} |
Java
// Java program for the implementation// of the Dynamic segment tree and// perform the range updates on the// given queriesclass GFG { // Structure of the node static class Node { int value; Node L, R; } // Structure to get the newly formed // node static Node getnode() { Node temp = new Node(); temp.value = 0; temp.L = null; temp.R = null; return temp; } // Creating the Root node static Node root = new Node(); // Function to perform the point update // on the dynamic segment tree static void UpdateHelper(Node curr, int index, int L, int R, int val) { // If the index is not overlapping // with the index if (L > index || R < index) return; // If the index is completely overlapping // with the index if (L == R && L == index) { // Update the value of the node // to the given value curr.value = val; return; } // Computing the middle index if none // of the above base cases are satisfied int mid = L - (L - R) / 2; int sum1 = 0, sum2 = 0; // If the index is in the left subtree if (index <= mid) { // Create a new node if the left // subtree is null if (curr.L == null) curr.L = getnode(); // Recursively call the function // for the left subtree UpdateHelper(curr.L, index, L, mid, val); } // If the index is in the right subtree else { // Create a new node if the right // subtree is null if (curr.R == null) curr.R = getnode(); // Recursively call the function // for the right subtree UpdateHelper(curr.R, index, mid + 1, R, val); } // Storing the sum of the left subtree if (curr.L != null) sum1 = curr.L.value; // Storing the sum of the right subtree if (curr.R != null) sum2 = curr.R.value; // Storing the sum of the children into // the node's value curr.value = sum1 + sum2; return; } // Function to find the sum of the // values given by the range static int queryHelper(Node curr, int a, int b, int L, int R) { // Return 0 if the root is null if (curr == null) return 0; // If the index is not overlapping // with the index, then the node // is not created. So sum is 0 if (L > b || R < a) return 0; // If the index is completely overlapping // with the index, return the node's value if (L >= a && R <= b) return curr.value; int mid = L - (L - R) / 2; // Return the sum of values stored // at the node's children return queryHelper(curr.L, a, b, L, mid) + queryHelper(curr.R, a, b, mid + 1, R); } // Function to call the queryHelper // function to find the sum for // the query static int query(int L, int R) { return queryHelper(root, L, R, 1, 10); } // Function to call the UpdateHelper // function for the point update static void update(int index, int value) { UpdateHelper(root, index, 1, 10, value); } // Function to perform the operations // on the tree static void operations() { // Creating an empty tree root = getnode(); // Update the value at position 1 to 10 update(1, 10); // Update the value at position 3 to 5 update(3, 5); // Finding sum for the range [2, 8] System.out.println(query(2, 8)); // Finding sum for the range [1, 10] System.out.println(query(1, 10)); } // Driver code public static void main(String[] args) { operations(); }}// This code is contributed by sanjeev2552 |
Python3
# C++ program for the implementation# of the Dynamic segment tree and# perform the range updates on the# given queries# Structure of the nodeclass Node: def __init__(self): self.value=-1 self.L, self.R=None,None# Structure to get the newly formed# nodedef getnode(): temp = Node() temp.value = 0 temp.L = None temp.R = None return temp# Creating the Root noderoot=None# Function to perform the point update# on the dynamic segment treedef UpdateHelper(curr, index, L, R, val): # If the index is not overlapping # with the index if (L > index or R < index): return # If the index is completely overlapping # with the index if (L == R and L == index) : # Update the value of the node # to the given value curr.value = val return # Computing the middle index if none # of the above base cases are satisfied mid = int(L - (L - R) / 2) sum1 = 0; sum2 = 0 # If the index is in the left subtree if (index <= mid) : # Create a new node if the left # subtree is None if (curr.L == None): curr.L = getnode() # Recursively call the function # for the left subtree UpdateHelper(curr.L, index, L, mid, val) # If the index is in the right subtree else : # Create a new node if the right # subtree is None if (curr.R == None): curr.R = getnode() # Recursively call the function # for the right subtree UpdateHelper(curr.R, index, mid + 1, R, val) # Storing the sum of the left subtree if (curr.L): sum1 = curr.L.value # Storing the sum of the right subtree if (curr.R): sum2 = curr.R.value # Storing the sum of the children into # the node's value curr.value = sum1 + sum2 return# Function to find the sum of the# values given by the rangedef queryHelper(curr, a, b, L, R): # Return 0 if the root is None if (curr == None): return 0 # If the index is not overlapping # with the index, then the node # is not created. So sum is 0 if (L > b or R < a): return 0 # If the index is completely overlapping # with the index, return the node's value if (L >= a and R <= b): return curr.value mid = int(L - (L - R) / 2) # Return the sum of values stored # at the node's children return queryHelper(curr.L, a, b, L, mid) + queryHelper(curr.R, a, b, mid + 1, R)# Function to call the queryHelper# function to find the sum for# the querydef query(L, R): return queryHelper(root, L, R, 1, 10)# Function to call the UpdateHelper# function for the point updatedef update(index, value): UpdateHelper(root, index, 1, 10, value)# Function to perform the operations# on the treedef operations(): global root # Creating an empty tree root = getnode() # Update the value at position 1 to 10 update(1, 10) # Update the value at position 3 to 5 update(3, 5) # Finding sum for the range [2, 8] print(query(2, 8)) # Finding sum for the range [1, 10] print(query(1, 10))# Driver codeif __name__ == '__main__': operations() |
C#
using System;// C# program for the implementation// of the Dynamic segment tree and// perform the range updates on the// given queriesclass GFG { // Structure of the node public class Node { public int value; public Node L, R; } // Structure to get the newly formed // node static Node getnode() { Node temp = new Node(); temp.value = 0; temp.L = null; temp.R = null; return temp; } // Creating the Root node static Node root = new Node(); // Function to perform the point update // on the dynamic segment tree static void UpdateHelper(Node curr, int index, int L, int R, int val) { // If the index is not overlapping // with the index if (L > index || R < index) return; // If the index is completely overlapping // with the index if (L == R && L == index) { // Update the value of the node // to the given value curr.value = val; return; } // Computing the middle index if none // of the above base cases are satisfied int mid = L - (L - R) / 2; int sum1 = 0, sum2 = 0; // If the index is in the left subtree if (index <= mid) { // Create a new node if the left // subtree is null if (curr.L == null) curr.L = getnode(); // Recursively call the function // for the left subtree UpdateHelper(curr.L, index, L, mid, val); } // If the index is in the right subtree else { // Create a new node if the right // subtree is null if (curr.R == null) curr.R = getnode(); // Recursively call the function // for the right subtree UpdateHelper(curr.R, index, mid + 1, R, val); } // Storing the sum of the left subtree if (curr.L != null) sum1 = curr.L.value; // Storing the sum of the right subtree if (curr.R != null) sum2 = curr.R.value; // Storing the sum of the children into // the node's value curr.value = sum1 + sum2; return; } // Function to find the sum of the // values given by the range static int queryHelper(Node curr, int a, int b, int L, int R) { // Return 0 if the root is null if (curr == null) return 0; // If the index is not overlapping // with the index, then the node // is not created. So sum is 0 if (L > b || R < a) return 0; // If the index is completely overlapping // with the index, return the node's value if (L >= a && R <= b) return curr.value; int mid = L - (L - R) / 2; // Return the sum of values stored // at the node's children return queryHelper(curr.L, a, b, L, mid) + queryHelper(curr.R, a, b, mid + 1, R); } // Function to call the queryHelper // function to find the sum for // the query static int query(int L, int R) { return queryHelper(root, L, R, 1, 10); } // Function to call the UpdateHelper // function for the point update static void update(int index, int value) { UpdateHelper(root, index, 1, 10, value); } // Function to perform the operations // on the tree static void operations() { // Creating an empty tree root = getnode(); // Update the value at position 1 to 10 update(1, 10); // Update the value at position 3 to 5 update(3, 5); // Finding sum for the range [2, 8] Console.WriteLine(query(2, 8)); // Finding sum for the range [1, 10] Console.WriteLine(query(1, 10)); } // Driver code public static void Main(String[] args) { operations(); }}// This code is contributed by jana_sayantan. |
Javascript
<script>// Javascript program for the implementation// of the Dynamic segment tree and perform // the range updates on the given queries// Structure of the nodeclass Node{ constructor() { this.L = null; this.R = null; this.value = 0; }}// Structure to get the newly formed// nodefunction getnode() { let temp = new Node(); return temp;}// Creating the Root nodelet root = new Node();// Function to perform the point update// on the dynamic segment treefunction UpdateHelper(curr, index, L, R, val){ // If the index is not overlapping // with the index if (L > index || R < index) return; // If the index is completely overlapping // with the index if (L == R && L == index) { // Update the value of the node // to the given value curr.value = val; return; } // Computing the middle index if none // of the above base cases are satisfied let mid = L - parseInt((L - R) / 2, 10); let sum1 = 0, sum2 = 0; // If the index is in the left subtree if (index <= mid) { // Create a new node if the left // subtree is null if (curr.L == null) curr.L = getnode(); // Recursively call the function // for the left subtree UpdateHelper(curr.L, index, L, mid, val); } // If the index is in the right subtree else { // Create a new node if the right // subtree is null if (curr.R == null) curr.R = getnode(); // Recursively call the function // for the right subtree UpdateHelper(curr.R, index, mid + 1, R, val); } // Storing the sum of the left subtree if (curr.L != null) sum1 = curr.L.value; // Storing the sum of the right subtree if (curr.R != null) sum2 = curr.R.value; // Storing the sum of the children into // the node's value curr.value = sum1 + sum2; return;}// Function to find the sum of the// values given by the rangefunction queryHelper(curr, a, b, L, R){ // Return 0 if the root is null if (curr == null) return 0; // If the index is not overlapping // with the index, then the node // is not created. So sum is 0 if (L > b || R < a) return 0; // If the index is completely overlapping // with the index, return the node's value if (L >= a && R <= b) return curr.value; let mid = L - parseInt((L - R) / 2, 10); // Return the sum of values stored // at the node's children return queryHelper(curr.L, a, b, L, mid) + queryHelper(curr.R, a, b, mid + 1, R);}// Function to call the queryHelper// function to find the sum for// the queryfunction query(L, R){ return queryHelper(root, L, R, 1, 10);}// Function to call the UpdateHelper// function for the point updatefunction update(index, value) { UpdateHelper(root, index, 1, 10, value);}// Function to perform the operations// on the treefunction operations() { // Creating an empty tree root = getnode(); // Update the value at position 1 to 10 update(1, 10); // Update the value at position 3 to 5 update(3, 5); // Finding sum for the range [2, 8] document.write(query(2, 8) + "</br>"); // Finding sum for the range [1, 10] document.write(query(1, 10) + "</br>");}// Driver codeoperations();// This code is contributed by mukesh07</script> |
Output:
5 15
Time Complexity: O(Q * logN)
Auxiliary Space: O(N)
Related Topic: Segment Tree
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