Vantage point tree

History

Peter Yianilos claimed that the vantage-point tree was discovered independently by him (Peter Yianilos) and by Jeffrey Uhlmann. Yet, Uhlmann published this method before Yianilos in 1991. Uhlmann called the data structure a metric tree, the name VP-tree was proposed by Yianilos. Vantage-point trees have been generalized to non-metric spaces using Bregman divergences by Nielsen et al.

This iterative partitioning process is similar to that of a k-d tree, but uses circular (or spherical, hyperspherical, etc.) rather than rectilinear partitions. In 2D Euclidean space, this can be visualized as a series of circles segregating the data.

The vantage-point tree is particularly useful in dividing data in a non-standard metric space into a metric tree.

Understanding a vantage-point tree

The way a vantage-point tree stores data can be represented by a circle. First, understand that each node of this tree contains an input point and a radius. All the left children of a given node are the points inside the circle and all the right children of a given node are outside of the circle. The tree itself does not need to know any other information about what is being stored. All it needs is the distance function that satisfies the properties of the metric space. Just imagine a circle with a radius. The left children are all located inside the circle and the right children are located outside the circle.

Searching through a vantage-point tree

A vantage-point tree can be used to find the nearest neighbor of a point x. The search algorithm is recursive. At any given step we are working with a node of the tree that has a vantage point v and a threshold distance t. The point of interest x will be some distance from the vantage point v. If that distance d is less than t then use the algorithm recursively to search the subtree of the node that contains the points closer to the vantage point than the threshold t; otherwise recurse to the subtree of the node that contains the points that are farther than the vantage point than the threshold t. If the recursive use of the algorithm finds a neighboring point n with distance to x that is less than | t − d | then it cannot help to search the other subtree of this node; the discovered node n is returned. Otherwise, the other subtree also needs to be searched recursively.

Advantages of a vantage-point tree

1. Instead of inferring multidimensional points for domain before the index being built, we build the index directly based on the distance. Doing this, avoids pre-processing steps.
2. Updating a vantage-point tree is relatively easy compared to the fast-map approach. For fast maps, after inserting or deleting data, there will come a time when fast-map will have to rescan itself. That takes up too much time and it is unclear to know when the rescanning will start.
3. Distance based methods are flexible. It is “able to index objects that are represented as feature vectors of a fixed number of dimensions."

Implementation examples

1. In Python
2. In C
3. In Java
4. In Java (alternative implementation)
5. In JavaScript
6. In Ruby