Problems and Results on Geometric Patterns

Braß, Peter; Pach, János

doi:10.1007/0-387-25592-3_2

Cited by 22 publications

(22 citation statements)

References 42 publications

(47 reference statements)

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“…In fact, such bounds directly yield a lower bound on the computational problem of listing all occurrences of the pattern. A prototypal example is Erdős' unit distance problem; see Braß and Pach [12] for more examples. It is known, in particular, that there can be Θ(n 2 ) similar copies of a pattern in an n-point set [18,3,4].…”

Section: Problem 2 (Affine Matching)mentioning

confidence: 99%

Geometric Pattern Matching Reduces to k -SUM

Aronov

Cardinal

2021

Discrete Comput Geom

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We prove that some exact geometric pattern matching problems reduce in linear time to k-SUM when the pattern has a fixed size k. This holds in the real RAM model for searching for a similar copy of a set of k ≥ 3 points within a set of n points in the plane, and for searching for an affine image of a set of k ≥ d + 2 points within a set of n points in d-space.As corollaries, we obtain improved real RAM algorithms and decision trees for the two problems. In particular, they can be solved by algebraic decision trees of near-linear height. ACM Subject ClassificationTheory of computation → Design and analysis of algorithms; Theory of computation → Computational geometry Keywords and phrases Geometric pattern matching, k-SUM problem, Linear decision trees

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Section: Problem 2 (Affine Matching)mentioning

confidence: 99%

Geometric Pattern Matching Reduces to k -SUM

Aronov

Cardinal

2021

Discrete Comput Geom

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“…This implies that S △ (4) ≥ 4. However, S △ (4) is also bounded above by 4 3 = 4. Hence, S △ (4) = 4.…”

Section: Small Casesmentioning

confidence: 99%

“…Apart from being a natural question in Discrete Geometry, this problem also arose in connection to optimization of algorithms designed to look for patterns among data obtained from scanners, digital cameras, telescopes, etc. (See [2,3,4] for further references.) Our paper considers the case when Q is the set of vertices of an isosceles right triangle.…”

Section: Introductionmentioning

confidence: 99%

On the maximum number of isosceles right triangles in a finite point set

Ábrego

Fernández

Roberts

2011

Involve

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Let Q be a finite set of points in the plane. For any set P of points in the plane, S Q (P ) denotes the number of similar copies of Q contained in P . For a fixed n, Erdős and Purdy asked to determine the maximum possible value of S Q (P ), denoted by S Q (n), over all sets P of n points in the plane. We consider this problem when Q = △ is the set of vertices of an isosceles right triangle. We give exact solutions when n ≤ 9, and provide new upper and lower bounds for S △ (n).

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“…These problems trace their inspiration to Erdős' question [17] about the maximum number of pairs of points at unit distance that can be determined by an n-subset of the plane, and other related questions, which have led to a rich literature (see [9,24] for an overview). Apart from being a well-known and important question in discrete geometry, the problem of determining the number of instances of a pattern in an n-subset of the plane is relevant to the problem of pattern recognition in data from scanners, cameras, and telescopes [8,9,10].…”

Section: Introductionmentioning

confidence: 99%

On The Number of Similar Instances of a Pattern in a Finite Set

Ábrego¹,

Fernández-Merchant²,

Katz³

et al. 2015

Preprint

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New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an n-point subset of the plane is shown to be no more than ⌊(4n − 1)(n − 1)/18⌋. The number of k-term arithmetic progressions that lie within an n-point subset of the line is shown to be at most (n − r)(n + r − k + 1)/(2k − 2), where r is the remainder when n is divided by k − 1. This upper bound is achieved when the n points themselves form an arithmetic progression, but for some values of k and n, it can also be achieved for other configurations of the n points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.

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Problems and Results on Geometric Patterns

Abstract: Many interesting problems in combinatorial and computational geometry can be reformulated as questions about occurrences of certain patterns in finite point sets. We illustrate this framework by a few typical results and list a number of unsolved problems.

Cited by 22 publications

References 42 publications

Geometric Pattern Matching Reduces to k -SUM

Geometric Pattern Matching Reduces to k -SUM

On the maximum number of isosceles right triangles in a finite point set

On The Number of Similar Instances of a Pattern in a Finite Set

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