Pattern Matching for sets of segments

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a \emph{coverage measure}, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over the general algorithm of Chew et al. that takes time $O(n^4 \log^2 n)$. In the second part of this paper we address the problem of matching polygonal chains. We study the well known \Frd, and present the first algorithm for computing the \Frd under general translations. Our methods also yield algorithms for computing a generalization of the \Fr distance, and we also present a simple approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200

Deterministic Sparse Pattern Matching via the Baur-Strassen Theorem

How fast can you test whether a constellation of stars appears in the night sky? This question can be modeled as the computational problem of testing whether a set of points $P$ can be moved into (or close to) another set $Q$ under some prescribed group of transformations. Consider, as a simple representative, the following problem: Given two sets of at most $n$ integers $P,Q\subseteq[N]$, determine whether there is some shift $s$ such that $P$ shifted by $s$ is a subset of $Q$, i.e., $P+s=\{p+s:p\in P\}\subseteq Q$. This problem, to which we refer as the Constellation problem, can be solved in near-linear time $O(n\log n)$ by a Monte Carlo randomized algorithm [Cardoze, Schulman; FOCS'98] and time $O(n\log^2 N)$ by a Las Vegas randomized algorithm [Cole, Hariharan; STOC'02]. Moreover, there is a deterministic algorithm running in time $n\cdot2^{O(\sqrt{\log n\log\log N})}$ [Chan, Lewenstein; STOC'15]. An interesting question left open by these previous works is whether Constellation is in deterministic near-linear time (i.e., with only polylogarithmic overhead). We answer this question positively by giving an $n\cdot(\log N)^{O(1)}$-time deterministic algorithm for the Constellation problem. Our algorithm extends to various more complex Point Pattern Matching problems in higher dimensions, under translations and rigid motions, and possibly with mismatches, and also to a near-linear-time derandomization of the Sparse Wildcard Matching problem on strings. We find it particularly interesting how we obtain our deterministic algorithm. All previous algorithms are based on the same baseline idea, using additive hashing and the Fast Fourier Transform. In contrast, our algorithms are based on new ideas, involving a surprising blend of combinatorial and algebraic techniques. At the heart lies an innovative application of the Baur-Strassen theorem from algebraic complexity theory.Comment: Abstract shortened to fit arxiv requirement

Lower bounds for the complexity of the Hausdorff distance

We describe new lower bounds for the complexity of the directed Hausdorff distance under translation and rigid motion. We exhibit lower bound constructions of \Omega\Gamma n 3 ) for point sets under translation, for the L 1 , L 2 and L1 norms, \Omega\Gamma n 4 ) for line segments under translation, for any L p norm, \Omega\Gamma n 5 ) for point sets under rigid motion and \Omega\Gamma n 6 ) for line segments under rigid motion, both for the L 2 norm. The results for point sets can also be extended to the undirected Hausdorff distance