Indexing Point Sets for Approximate Bottleneck Distance Queries. (arXiv:1810.09482v1 [cs.CG])

The {\em 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 $m$ planar point sets (of varying sizes) to create a database $D$ that supports nearest bottleneck distance queries: given a query point set $Q$ of size $n$, the point sets $P \in D$ 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 $[0,1]^2$ in the plane. The main contribution of this work is a {\em trie}-based data structure that is space efficient and supports $8$-approximate nearest bottleneck queries in $O(-\lg(d_B(D,Q)) \min(m,{10}^n) n \lg^3 n)$ time, where $d_B(D,Q)$ is the minimum bottleneck distance from $Q$ to any point set in $D$. A direct consequence, of independent interest, is a simple $O(-\lg(d_B(P,Q)) n \lg^3 n)$ time algorithm to $2 \sqrt{2}$-approximate $d_B(P,Q)$, for any two point sets $P$ and $Q$. Finally, the querying algorithm proposed is easily adapted to support nearest subset and superset queries.

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