Pattern Matching for Sets of Segments

Efrat, Alon; Indyk, Piotr; Venkatasubramanian, Suresh

doi:10.1007/s00453-004-1089-y

Cited by 5 publications

(6 citation statements)

References 17 publications

(24 reference statements)

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“…Moreover, we show that, by using pattern-matching techniques (notably, algorithms for the less-than-matching problem [AF95]), we can perform the preprocessing in less than O(M n) time per function g i . We mention that less-than matching has been earlier used for a geometric problem in [EIV01].…”

Section: Introductionmentioning

confidence: 99%

Efficient algorithms for substring near neighbor problem

Andoni¹,

Indyk²

2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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In this paper we consider the problem of finding the approximate nearest neighbor when the data set points are the substrings of a given text T . Specifically, for a string T of length n, we present a data structure which does the following: given a pattern P , if there is a substring of T within the distance R from P , it reports a (possibly different) substring of T within distance cR from P . The length of the pattern P , denoted by m, is not known in advance. For the case where the distances are measured using the Hamming distance, we present a data structure which usesÕ(n 1+1/c ) space 1 and withÕ n 1/c + mnquery time. This essentially matches the earlier bounds of [Ind98], which assumed that the pattern length m is fixed in advance. In addition, our data structure can be constructed in timeÕ n 1+1/c + n 1+o(1) M 1/3 , where M is an upper bound for m. This essentially matches the preprocessing bound of [Ind98] as long as the termÕ n 1+1/c dominates the running time, which is the case when, e.g., c < 3.We also extend our results to the case where the distances are measured according to the l1 distance. The query time and the space bound are essentially the same, while the preprocessing time becomesÕ n 1+1/c + n 1+o(1) M 2/3 .

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Section: Introductionmentioning

confidence: 99%

Efficient algorithms for substring near neighbor problem

Andoni¹,

Indyk²

2006

Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm - SODA '06

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“…Thus, this algorithm is also an arrangement-based approach. Similarly, the first algorithms [7,30] for the continuous Fréchet distance under translation as well as the weak continuous Fréchet distance under translation relied on arrangement constructions.…”

Section: Further Related Workmentioning

confidence: 99%

“…

Discrete Fréchet Distance Under Translation 25:3 any translation of σ , i.e., we minimize over all possible translations of σ . Clearly, this yields a translation-invariant distance measure, and thus enables the above application.The continuous Fréchet distance under translation was independently introduced by Efrat et al [30] and Alt et al [7], who designed algorithms in the plane with running timeÕ(n 10 ) and O(n 8 ), respectively. 2 Both groups of researchers also presented approximation algorithms, e.g., a (1 + ε)-approximation running in time O(n 2 /ε 2 ) in the plane [7].

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Discrete Fréchet Distance under Translation

Bringmann

Künnemann

Nusser

2021

ACM Trans. Algorithms

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The discrete Fréchet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fréchet distance under translation, which enables detection of similar movement patterns in different spatial domains. For polygonal curves of length n in the plane, the fastest known algorithm runs in time Õ( n 5 ) [12]. This is achieved by constructing an arrangement of disks of size Õ( n 4 ), and then traversing its faces while updating reachability in a directed grid graph of size N := Õ( n 5 ), which can be done in time Õ(√ N ) per update [27]. The contribution of this article is two-fold. First, although it is an open problem to solve dynamic reachability in directed grid graphs faster than Õ(√ N ), we improve this part of the algorithm: We observe that an offline variant of dynamic s - t -reachability in directed grid graphs suffices, and we solve this variant in amortized time Õ( N 1/3 ) per update, resulting in an improved running time of Õ( N 4.66 ) for the discrete Fréchet distance under translation. Second, we provide evidence that constructing the arrangement of size Õ( N 4 ) is necessary in the worst case by proving a conditional lower bound of n 4 - o(1) on the running time for the discrete Fréchet distance under translation, assuming the Strong Exponential Time Hypothesis.

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“…Subsequent to the first presentation of this work, other problems in geometric matching have been solved via a reduction to combinatorial pattern matching [17], [32].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial and Experimental Methods for Approximate Point Pattern Matching

et al. 2003

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Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modeling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high-degree polynomials, and require robust calculations of intersection points of high-degree surfaces.We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate.We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average-case analysis and a detailed empirical study of our methods. Introduction.Given point sets P and Q, the problem of pattern matching is to determine whether P is similar to some portion of Q. Similarity is typically defined in terms of a distance measure, and P and Q may undergo transformations like rotations, translations, and scaling before the distance between them is computed. This general formulation captures a wide variety of problems, arising in image registration [33], model-based object recognition [4], [21], computer vision [35], molecular modeling [27], and many other areas.In the domain of computational geometry, these problems were mostly studied in an exact formulation: point coordinates are given with infinite precision, and two points match if their coordinates are identical. The disadvantage with this formulation is twofold: Firstly, this is an inaccurate representation of reality, where the input points are typically noisy, and any match requires some notion of a tolerance parameter. Secondly, the algorithms designed to compute similarity were complex, with running times in the order of large polynomials, and required sophisticated algebraic manipulation of high-degree manifolds in a robust fashion.This provides the motivation for approximate pattern matching. In an approximate setting, one might replace points by balls of radius ε (where ε is a tolerance parameter)

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Pattern Matching for Sets of Segments

Cited by 5 publications

References 17 publications

Efficient algorithms for substring near neighbor problem

Efficient algorithms for substring near neighbor problem

Discrete Fréchet Distance under Translation

Combinatorial and Experimental Methods for Approximate Point Pattern Matching

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