Combinatorial Framework for Similarity Search

Abstract

We present an overview of combinatorial framework for similarity search. An algorithm is combinatorial if only direct comparisons between two pairwise similarity values are allowed. Namely, the input dataset is represented by a comparison oracle that given any three points X,Y,Z answers whether Y or Z is closer to X. We assume that the similarity order of the dataset satisfies the four variations of the following disorder inequality: if X is the A'th most similar object to Y and Y is the B'th most similar object to Z, then X is among the D(A+B) most similar objects to Z, where D is a relatively small disorder constant. Combinatorial algorithms for nearest neighbor search have two important advantages: (1) they do not map similarity values to artificial distance values and do not use triangle inequality for the latter, and (2) they work for arbitrarily complicated data representations and similarity functions. Ranwalk, the first known combinatorial solution for nearest neighbors, is randomized, exact, zero-error algorithm with query time that is logarithmic in number of objects. But Ranwalk preprocessing time is quadratic. Later on, another solution, called combinatorial nets, was discovered. It is deterministic and exact algorithm with almost linear time and space complexity of preprocessing, and near-logarithmic time complexity of search. Combinatorial nets also have a number of side applications. For near-duplicate detection they lead to the first known deterministic algorithm that requires just near-linear time + time proportional to the size of output. For any dataset with small disorder combinatorial nets can be used to construct a visibility graph: the one in which greedy routing deterministically converges to the nearest neighbor of a target in logarithmic number of steps. The later result is the first known work-around for Navarro's impossibility of generalizing Delaunay graphs.