Thecover tree is a type ofdata structure incomputer science that is specifically designed to facilitate the speed-up of anearest neighbor search. It is a refinement of the Navigating Net data structure, and related to a variety of other data structures developed for indexing intrinsically low-dimensional data.^[1]

The tree can be thought of as a hierarchy of levels with the top level containing the rootpoint and the bottom level containing every point in the metric space. Each levelC is associated with an integer valuei that decrements by one as the tree is descended. Each levelC in the cover tree has three important properties:

• Nesting:${\displaystyle C_i\subseteq C_{i-1}}$
• Covering: For every point${\displaystyle p\in C_{i-1}}$, there exists a point${\displaystyle q\in C_i}$ such that the distance from${\displaystyle p}$ to${\displaystyle q}$ is less than or equal to${\displaystyle 2^i}$ and exactly one such${\displaystyle q}$ is a parent of${\displaystyle p}$.
• Separation: For all points${\displaystyle p,q\in C_i}$, the distance from${\displaystyle p}$ to${\displaystyle q}$ is greater than${\displaystyle 2^i}$.

Like othermetric trees the cover tree allows for nearest neighbor searches in${\displaystyle O(\eta \ast \log n)}$ where${\displaystyle \eta }$ is a constant associated with the dimensionality of the dataset and n is the cardinality. To compare, a basic linear search requires${\displaystyle O(n)}$, which is a much worse dependence on${\displaystyle n}$. However, in high-dimensionalmetric spaces the${\displaystyle \eta }$ constant is non-trivial, which means it cannot be ignored in complexity analysis. Unlike other metric trees, the cover tree has a theoretical bound on its constant that is based on the dataset'sexpansion constant or doubling constant (in the case of approximate NN retrieval). The bound on search time is${\displaystyle O(c^{12}\log n)}$ where${\displaystyle c}$ is the expansion constant of the dataset.

Although cover trees provide faster searches than the naive approach, this advantage must be weighed with the additional cost of maintaining the data structure. In a naive approach adding a new point to the dataset is trivial because order does not need to be preserved, but in a cover tree it can take${\displaystyle O(c^6\log n)}$ time. However, this is an upper-bound, and some techniques have been implemented that seem to improve the performance in practice.^[2]

The cover tree uses implicit representation to keep track of repeated points. Thus, it only requires O(n) space.

• Nearest neighbor search
• kd-tree
• M-tree

Notes

Bibliography

• Alina Beygelzimer, Sham Kakade, and John Langford. Cover Trees for Nearest Neighbor. In Proc. International Conference on Machine Learning (ICML), 2006.
• JL's Cover Tree page. John Langford's page links to papers and code.
• A C++ Cover Tree implementation on GitHub.
• A cover tree implementation in Java.

• Trees (data structures)

• Articles with short description
• Short description matches Wikidata