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. We study the aggregate group nearest neighbor searching for the MAX operator in the plane. For a set @math of @math points and a query set @math of @math points, the query asks for a point of @math whose maximum distance to the points in @math is minimized. We present data structures for answering such queries for both @math and @math distance measures. Previously, only heuristic and approximation algorithms were given for both versions. For the @math version, we build a data structure of O(n) size in @math time, such that each query can be answered in @math time. For the @math version, we build a data structure in @math time and @math space, such that each query can be answered in @math time, and alternatively, we build a data structure in @math time and space for any @math , such that each query can be answered in @math time. Further, we extend our result for the @math version to the top- @math queries where each query asks for the @math points of @math whose maximum distances to @math are the smallest for any @math with @math : We build a data structure of O(n) size in @math time, such that each top- @math query can be answered in @math time. Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure.