Implementation of Dynamic Segment Trees with Poly Hash Tables

Introduction:

Dynamic data structures play a vital role in computer science by enabling efficient manipulation and querying of changing datasets. Dynamic Segment Trees and Poly Hash Tables offer a powerful combination for handling dynamic range queries on large datasets.

Dynamic Segment Trees:

Dynamic Segment Trees provide a flexible structure for performing range queries on dynamic datasets. Each Node in the tree represents a segment of the array, and by updating and querying these nodes, various range-based operations can be efficiently executed.

Dynamic Segment Trees are particularly useful when dealing with arrays that undergo frequent updates. While efficient for static arrays, traditional segment trees become impractical for dynamic datasets due to their static nature.

Basic Structure:

A Dynamic Segment Tree extends the idea of a binary search tree, where each node represents a range of elements. The root node covers the entire array, and each level of the tree subdivides the range further.

Dynamic Operations:

Insertions and deletions of elements require updating the corresponding nodes along the tree. When inserting a new element, the tree is adjusted to accommodate the new element's presence. Similarly, when deleting an element, the tree is modified to reflect its absence.

Lazy Propagation:

Lazy propagation is a technique used to optimize range update operations. Instead of immediately updating all affected nodes, the updates are postponed until the exact value of a node is required. This minimizes unnecessary updates and improves performance.

Memory Management:

Efficient memory management is crucial to prevent memory fragmentation and ensure optimal performance. Techniques like memory pooling or custom memory allocation strategies can be employed.

Complex Queries:

Dynamic Segment Trees can handle a variety of range queries, including sum, minimum, maximum, and other custom operations. These queries are efficiently performed by traversing the tree and combining the results from relevant nodes.

Poly Hash Tables:

Hash tables are fundamental data structures that allow efficient data retrieval based on keys. Poly Hash Tables enhance this concept by utilizing polynomial hash functions.

Hashing is a technique to convert data (keys) into a fixed-size value (hash) that can be used to index into an array or data structure for quick retrieval.

Polynomial Hash Functions:

Poly Hash Tables employ polynomial hash functions to distribute keys uniformly across the hash table. A polynomial is used to generate the hash value, reducing collisions and improving efficiency.

Collision Handling:

Collisions occur when two keys hash to the same value. Poly Hash Tables handle collisions using techniques like chaining or open addressing with probing.

Dynamic Resizing:

To maintain efficiency, Poly Hash Tables dynamically resize themselves as the number of stored elements grows. This involves creating a new, larger hash table and rehashing existing elements.

Hash Table Operations:

Standard hash table operations include insertion, deletion, and retrieval. Poly Hash Tables provide O(1) average-case time complexity for these operations when using a well-designed polynomial hash function.

Integration of Dynamic Segment Trees and Poly Hash Tables:

Combining Dynamic Segment Trees with Poly Hash Tables creates a powerful data structure that efficiently handles dynamic range queries.

Combining Principles:

The Dynamic Segment Tree can be mapped onto a Poly Hash Table, where each range corresponds to a key in the hash table. The values stored in the hash table represent the corresponding segment tree nodes.

Use Cases:

This integration is particularly useful for scenarios involving dynamic datasets with dynamic range queries. Use cases include real-time analytics, dynamic scheduling, and frequency tracking.

Advantages of Integration:

The integration of these structures allows for optimized range queries on dynamic datasets. It balances the strengths of both data structures to offer efficient updates and queries.

Implementation:

import java.util.*;

class DynamicSegmentTree {
    static class Node {
        long value;
        Node left, right;
        
        Node(long value) {
            this.value = value;
            this.left = null;
            this.right = null;
        }
    }
    
    static class PolyHashTable {
        static final int TABLE_SIZE = 1000000; // Adjust as needed
        Node[] table = new Node[TABLE_SIZE];
        
        void insert(long key, long value) {
            int hash = (int) (key % TABLE_SIZE);
            table[hash] = new Node(value);
        }
        
        Node find(long key) {
            int hash = (int) (key % TABLE_SIZE);
            return table[hash];
        }
    }
    
    static Node build(int left, int right) {
        if (left == right) {
            return new Node(0); // Initialize leaf node value
        }
        
        int mid = (left + right) / 2;
        Node leftChild = build(left, mid);
        Node rightChild = build(mid + 1, right);
        
        Node parent = new Node(leftChild.value + rightChild.value);
        parent.left = leftChild;
        parent.right = rightChild;
        
        return parent;
    }
    
    static void update(Node root, int left, int right, int index, long value) {
        if (left == right) {
            root.value = value;
            return;
        }
        
        int mid = (left + right) / 2;
        if (index <= mid) {
            if (root.left == null) root.left = new Node(0);
            update(root.left, left, mid, index, value);
        } else {
            if (root.right == null) root.right = new Node(0);
            update(root.right, mid + 1, right, index, value);
        }
        
        root.value = (root.left != null ? root.left.value : 0) + (root.right != null ? root.right.value : 0);
    }
    
    static long query(Node root, int left, int right, int queryLeft, int queryRight) {
        if (root == null || queryLeft > right || queryRight < left) {
            return 0;
        }
        
        if (queryLeft <= left && right <= queryRight) {
            return root.value;
        }
        
        int mid = (left + right) / 2;
        long leftSum = query(root.left, left, mid, queryLeft, queryRight);
        long rightSum = query(root.right, mid + 1, right, queryLeft, queryRight);
        
        return leftSum + rightSum;
    }
    
    public static void main(String[] args) {
        int n = 10; // Size of array
        Node root = build(1, n); // Build the Dynamic Segment Tree
        
        PolyHashTable hashTable = new PolyHashTable();
        
        // Example update operations
        update(root, 1, n, 5, 10); // Update element at index 5 to 10
        update(root, 1, n, 7, 5); // Update element at index 7 to 5
        
        // Example range query
        long sum = query(root, 1, n, 3, 8); // Query sum of elements from index 3 to 8
        
        // Insert into Poly Hash Table
        hashTable.insert(5, root.value); // Store the root value corresponding to index 5
        
        // Retrieve from Poly Hash Table
        Node retrievedNode = hashTable.find(5); // Retrieve root value using index 5
        
        System.out.println("Sum of elements from index 3 to 8: " + sum);
        System.out.println("Retrieved Node value: " + (retrievedNode != null ? retrievedNode.value : "Not found"));
    }
}

Output:

Implementation of Dynamic Segment Trees with Poly Hash Tables

Explanation:

Define a node class depicting the structure of our dynamic segment tree

Which consists of value, left and right pointers

Define a PolyHashTable class with a constant table size for the hashtable size and an array of Node objects to store the values based on their hash.

insert Method: Inserts a key-value pair into the hash table using polynomial hashing. It calculates the hash based on the key, creating a new Node with the provided value.

find Method: Retrieves a value from the hash table based on the key. It calculates the hash and returns the corresponding Node.

build Method: Recursively builds the Dynamic Segment Tree. It takes the range [left, right] and constructs the tree by dividing the range into smaller segments.

update Method: Updates a value in the Dynamic Segment Tree. It recursively traverses the tree and updates the relevant nodes based on the index and value provided.

query Method: Performs a range query in the Dynamic Segment Tree. It recursively traverses the tree, returning the sum of values within the specified query range.