The bottleneck distance is a natural measure of the distance between two finite point sets of equal cardinality. In this work, we consider the problem of indexing a set of @math planar point sets (of varying sizes) to create a database @math that supports nearest bottleneck distance queries: given a query point set @math of size @math , the point sets @math that are closest in terms of bottleneck distance are returned. Without loss of generality, we assume that all point sets belong to the unit box @math in the plane. The main contribution of this work is a trie -based data structure that is space efficient and supports @math -approximate nearest bottleneck queries in @math time, where @math is the minimum bottleneck distance from @math to any point set in @math . A direct consequence, of independent interest, is a simple @math time algorithm to @math -approximate @math , for any two point sets @math and @math . Finally, the querying algorithm proposed is easily adapted to support nearest subset and superset queries.
Bottleneck distance arises naturally in the comparison of persistence diagrams in topological data analysis @cite_4 . @cite_0 consider the related problem of building a database of persistence diagrams that permits approximate querying in @math time ( @math is the number of point sets stored in @math ). Their approach is also based on representing point sets by snap-rounding each point to neighboring grid points at each level in a multilevel grid. All combinations of snap-roundings are considered and the resulting grid point distributions are stored in a database. Binary search is used on the hashed values to query the database and resulting matches are shown to provide a @math -approximation to the nearest point set in the database. They also observe that the approximation ratio can be improved to @math , if more snap-roundings are done ( @math per point). Rather than using a hashing scheme, it is possible to use a trie data structure (described in ) to achieve @math time queries.
@cite_0 @cite_4
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Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, current approaches for comparing a persistence diagram to a set of other persistence diagrams is linear in the number of diagrams or do not offer performance guarantees. In this paper, we apply concepts from locality-sensitive hashing to support approximate nearest neighbor search in the space of persistence diagrams. Given a set @math of @math @math -bounded persistence diagrams, each with at most @math points, we snap-round the points of each diagram to points on a cubical lattice and produce a key for each possible snap-rounding. Specifically, we fix a grid over each diagram at several resolutions and consider the snap-roundings of each diagram to the four nearest lattice points. Then, we propose a data structure with @math levels @math that stores all snap-roundings of each persistence diagram in @math at each resolution. This data structure has size @math to account for varying lattice resolutions as well as snap-roundings and the deletion of points with low persistence. To search for a persistence diagram, we compute a key for a query diagram by snapping each point to a lattice and deleting points of low persistence. Furthermore, as the lattice parameter decreases, searching our data structure yields a six-approximation of the nearest diagram in @math in @math time and a constant factor approximation of the @math th nearest diagram in @math time. We address the problem of replicating a Voronoi diagram @math of a planar point set @math by making proximity queries, which are of three possible (in decreasing order of information content): 1. the exact location of the nearest site(s) in @math ; 2. the distance to and label(s) of the nearest site(s) in @math ; 3. a unique label for every nearest site in @math . We provide algorithms showing how queries of Type 1 and Type 2 allow an exact cloning of @math with @math queries and @math processing time. We also prove that queries of Type 3 can never exactly clone @math , but we show that with @math queries we can construct an @math -approximate cloning of @math . In addition to showing the limits of nearest-neighbor database security, our methods also provide one of the first natural algorithmic applications of retroactive data structures.