Weight-Balanced Tree

A weight-balanced tree is a search tree that rebalances based on subtree weights rather than height. Weights represent node values or importances. This provides more flexibility than height balancing.

Some examples:

Weight-balanced BST
Weight-balanced B-tree
Weight-balanced quadtree

Java - Weight-balanced BST:

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class Node {
  int weight;
  int value;
  Node left, right;
}

void insert(Node root, Node node) {
  if (node.value < root.value) {
    root.left = insert(root.left, node);
  } else {
    root.right = insert(root.right, node);
  }
  
  fixImbalance(root); // Rebalance
}

void fixImbalance(Node root) {
  if (root.left.weight > T*root.right.weight) {
    // Rotate right  
  } else if (root.right.weight > T*root.left.weight) {
    // Rotate left
  }
}

C++ - Weight-balanced B-tree:

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template<typename T>
class WBTree {
public:

  void insert(const T& value) {
    // Insert logic
    
    rebalance(); // Rebalance on weight  
  }

private:

  void rebalance() {
    // Split or merge pages
    // Based on weight thresholds
  }

};

Python - Weight-balanced quadtree:

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class QuadNode:

  def __init__(self):
    self.weight = 0
    # Children nodes  

  def insert(self, point):
    # Insert logic

    self.rebalance() # Check weight balance

  def rebalance(self):
    if self.weight > THRESHOLD:
      # Split node
    elif self.weight < MERGE_THRESHOLD:
      # Merge node

Weight balancing generalizes balanced trees using non-height metrics like node values or weights.

A weight-balanced tree is a binary tree where for every node, the sizes of the left and right subtrees differ by at most a constant factor. The primary goal of such balancing is to ensure that the tree remains approximately balanced, thereby maintaining efficient performance for operations like add, delete, and find. It’s a variant of balanced trees like AVL trees and Red-Black trees.

A weight-balanced tree is a type of binary search tree in which for each node, the sizes of the left and right subtrees are balanced by a certain weight factor. This ensures more balanced tree operations.

Suppose we have a simple binary tree:

In a weight-balanced tree, every node would hold additional information about the size of its left and right subtree. For example, node ( A ) might hold information that its left subtree has 2 nodes and its right subtree has 3 nodes.

Textually, the tree might look like:

         A(2,3)
        /     \
     B(1,0)  C(1,1)
    /        /     \
  D(0,0)  E(0,0)  F(0,0)

The numbers in the parentheses next to each node represent the sizes of the left and right subtrees. In a weight-balanced tree, these sizes should not differ by more than a specific weight factor ( \alpha ).

For instance, if ( \alpha = 2 ), then for each node, the size of its left subtree should be at most twice the size of its right subtree and vice versa.

In our example, ( A ) holds ( B ) and ( C ) with sizes 2 and 3 respectively, meeting the ( \alpha = 2 ) criteria (since ( 3 <= 2 \times 2 ) and ( 2 <= 2 \times 3 )).

This visual understanding helps in grasping the concept of how weight-balanced trees maintain their balance and how they are different from regular binary search trees. This balance is critical for achieving good performance in operations like insertions, deletions, and lookups.

Solution

Java

In Java, you can define a weight-balanced tree by keeping track of the sizes of the left and right subtrees for each node.

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class Node {
    int value, size;
    Node left, right;
    public Node(int value) {
        this.value = value;
        this.size = 1;
    }
}

class WeightBalancedTree {
    Node root;

    // Check if tree is balanced at node n
    boolean isBalanced(Node n) {
        int leftSize = (n.left != null) ? n.left.size : 0;
        int rightSize = (n.right != null) ? n.right.size : 0;
        return Math.abs(leftSize - rightSize) <= 1;
    }

    // Placeholder for add, delete and find methods
}

C++

In C++, similar to Java, you can keep track of the subtree sizes to check for balance.

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struct Node {
    int value, size;
    Node *left, *right;
    Node(int value) : value(value), size(1), left(nullptr), right(nullptr) {}
};

class WeightBalancedTree {
public:
    Node* root;

    // Check if tree is balanced at node n
    bool isBalanced(Node* n) {
        int leftSize = (n->left != nullptr) ? n->left->size : 0;
        int rightSize = (n->right != nullptr) ? n->right->size : 0;
        return abs(leftSize - rightSize) <= 1;
    }

    // Placeholder for add, delete and find methods
};

Python

Python’s dynamic nature makes it easier to implement a weight-balanced tree.

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class Node:
    def __init__(self, value):
        self.value = value
        self.size = 1
        self.left = None
        self.right = None

class WeightBalancedTree:
    def __init__(self):
        self.root = None

    def is_balanced(self, node):
        left_size = node.left.size if node.left else 0
        right_size = node.right.size if node.right else 0
        return abs(left_size - right_size) <= 1

    # Placeholder for add, delete and find methods

Key Takeaways

A weight-balanced tree keeps the sizes of left and right subtrees at each node approximately balanced.
You can implement such a tree in Java, C++, and Python by adding a size field to each node and checking for balance as you modify the tree.
Useful operations like add, delete, and find can be implemented efficiently with this kind of tree.