Damn Cool Algorithms, Part 1: BK-Trees - Nick's Blog
BK-Tree | Introduction & Implementation - GeeksforGeeks

A BK-Tree (Burkhard-Keller Tree) is a tree-based data structure that is specifically designed for efficiently performing approximate string matching, which is the process of finding strings that are close to a given query string. It was first introduced by Burkhard and Keller in 1973.

The BK-Tree works by taking advantage of the concept of metric space. A metric space is a set with a distance function defined between its elements, which should satisfy certain properties like non-negativity, symmetry, and the triangle inequality. In the context of string matching, a metric space can be represented by a set of strings with an associated distance function like Levenshtein distance (edit distance), which measures how many edits (insertions, deletions or substitutions) are required to transform one string into another.

The main idea behind BK-Trees is to organize similar strings close to each other in the tree structure. To build a BK-Tree:

  1. Select an arbitrary element from the input set as the root node.
  2. For each remaining element in the input set:
    • Calculate the distance between this element and the root node.
    • If there's no child node at that distance from the root node, add this element as a child node at that distance.
    • If there's already a child node at that distance, treat this child node as the new root and repeat step 2.

To search for approximate matches within a BK-Tree:

  1. Calculate the distance between the query string and the root node.
  2. Traverse all nodes within a certain range from this calculated distance.
  3. Repeat steps 1-2 recursively for each visited subtree until all nodes within range have been visited.

The advantage of using BK-Trees over other methods like brute force searching or trie-based structures is its efficiency in terms of both time complexity and memory usage when dealing with large datasets containing many similar strings. However, it's worth noting that this efficiency relies heavily on the metric space properties of the underlying distance function being used. If the distance function doesn't satisfy these properties, the performance of BK-Trees may be significantly reduced.