COMPUTING THE DIAMETER OF A POINT SET

Given a finite set of points [Formula: see text] in ℝd, the diameter of [Formula: see text] is defined as the maximum distance between two points of [Formula: see text]. We propose a very simple algorithm to compute the diameter of a finite set of points. Although the algorithm is not worst-case optimal, an extensive experimental study has shown that it is extremely fast for a large variety of point distributions. In addition, we propose a comparison with the recent approach of Har-Peled5 and derive hybrid algorithms to combine advantages of both approaches.

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APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

International Journal of Computational Geometry & Applications ◽

10.1142/s021819591350009x ◽

2013 ◽

Vol 23 (06) ◽

pp. 461-477 ◽

Cited By ~ 7

Author(s):

MINATI DE ◽

GAUTAM K. DAS ◽

PAZ CARMI ◽

SUBHAS C. NANDY

Keyword(s):

Approximation Algorithms ◽

Simple Algorithm ◽

Constant Factor ◽

Performance Ratio ◽

Approximation Result ◽

Worst Case ◽

Approximation Factor ◽

Minimum Number ◽

Unit Disks ◽

Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

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Robust Universal Inference

Entropy ◽

10.3390/e23060773 ◽

2021 ◽

Vol 23 (6) ◽

pp. 773

Author(s):

Amichai Painsky ◽

Meir Feder

Keyword(s):

Robust Estimation ◽

Minimax Estimator ◽

Worst Case ◽

Capacity Theory ◽

Alternative Approach ◽

True Model ◽

Finite Set ◽

Universal Prediction ◽

Estimation Scheme

Learning and making inference from a finite set of samples are among the fundamental problems in science. In most popular applications, the paradigmatic approach is to seek a model that best explains the data. This approach has many desirable properties when the number of samples is large. However, in many practical setups, data acquisition is costly and only a limited number of samples is available. In this work, we study an alternative approach for this challenging setup. Our framework suggests that the role of the train-set is not to provide a single estimated model, which may be inaccurate due to the limited number of samples. Instead, we define a class of “reasonable” models. Then, the worst-case performance in the class is controlled by a minimax estimator with respect to it. Further, we introduce a robust estimation scheme that provides minimax guarantees, also for the case where the true model is not a member of the model class. Our results draw important connections to universal prediction, the redundancy-capacity theorem, and channel capacity theory. We demonstrate our suggested scheme in different setups, showing a significant improvement in worst-case performance over currently known alternatives.

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On extremums of sums of powered distances to a finite set of points

Geometriae Dedicata ◽

10.1007/s10711-012-9804-3 ◽

2012 ◽

Vol 167 (1) ◽

pp. 69-89 ◽

Cited By ~ 3

Author(s):

Nikolai Nikolov ◽

Rafael Rafailov

Keyword(s):

Finite Set ◽

Set Of Points

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MAINTAINING VISIBILITY INFORMATION OF PLANAR POINT SETS WITH A MOVING VIEWPOINT

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195907002343 ◽

2007 ◽

Vol 17 (04) ◽

pp. 297-304 ◽

Cited By ~ 2

Author(s):

OLIVIER DEVILLERS ◽

VIDA DUJMOVIĆ ◽

HAZEL EVERETT ◽

SAMUEL HORNUS ◽

SUE WHITESIDES ◽

...

Keyword(s):

Linear Space ◽

Point Sets ◽

Worst Case ◽

Planar Point ◽

Set Of Points

Given a set of n points in the plane, we consider the problem of computing the circular ordering of the points about a viewpoint q and efficiently maintaining this ordering information as q moves. In linear space, and after O(n log n) preprocessing time, our solution maintains the view at a cost of O( log n) amortized time (resp.O( log 2 n) worst case time) for each change. Our algorithm can also be used to maintain the set of points sorted according to their distance to q .

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Angle Bisector Algorithm and Modified Dynamic Programming Algorithm for Dubins Traveling Salesman Problem

10.36227/techrxiv.14767593.v1 ◽

2021 ◽

Author(s):

Eswara Venkata Kumar Dhulipala

Keyword(s):

Euclidean Distance ◽

Travelling Salesman Problem ◽

Dynamic Programming Algorithm ◽

Constant Factor ◽

Programming Algorithm ◽

Worst Case ◽

Angle Bisector ◽

Set Of Points ◽

The Given ◽

Tour Length

A Dubin's Travelling Salesman Problem (DTSP) of finding a minimum length tour through a given set of points is considered. DTSP has a Dubins vehicle, which is capable of moving only forward with constant speed. In this paper, first, a worst case upper bound is obtained on DTSP tour length by assuming DTSP tour sequence same as Euclidean Travelling Salesman Problem (ETSP) tour sequence. It is noted that, in the worst case, \emph{any algorithm that uses of ETSP tour sequence} is a constant factor approximation algorithm for DTSP. Next, two new algorithms are introduced, viz., Angle Bisector Algorithm (ABA) and Modified Dynamic Programming Algorithm (MDPA). In ABA, ETSP tour sequence is used as DTSP tour sequence and orientation angle at each point $i_k$ are calculated by using angle bisector of the relative angle formed between the rays $i_{k}i_{k-1}$ and $i_ki_{k+1}$. In MDPA, tour sequence and orientation angles are computed in an integrated manner. It is shown that the ABA and MDPA are constant factor approximation algorithms and ABA provides an improved upper bound as compared to Alternating Algorithm (AA) \cite{savla2008traveling}. Through numerical simulations, we show that ABA provides an improved tour length compared to AA, Single Vehicle Algorithm (SVA) \cite{rathinam2007resource} and Optimized Heading Algorithm (OHA) \cite{babel2020new,manyam2018tightly} when the Euclidean distance between any two points in the given set of points is at least $4\rho$ where $\rho$ is the minimum turning radius. The time complexity of ABA is comparable with AA and SVA and is better than OHA. Also we show that MDPA provides an improved tour length compared to AA and SVA and is comparable with OHA when there is no constraint on Euclidean distance between the points. In particular, ABA gives a tour length which is at most $4\%$ more than the ETSP tour length when the Euclidean distance between any two points in the given set of points is at least $4\rho$.

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BASELINE BOUNDED HALF-PLANE VORONOI DIAGRAM

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830913500213 ◽

2013 ◽

Vol 05 (03) ◽

pp. 1350021 ◽

Cited By ~ 1

Author(s):

BING SU ◽

YINFENG XU ◽

BINHAI ZHU

Keyword(s):

Voronoi Diagram ◽

Half Plane ◽

Line Segment ◽

Euclidean Plane ◽

Voronoi Region ◽

Point Set ◽

Set Of Points

Given a set of points P = {p1, p2, …, pn} in the Euclidean plane, with each point piassociated with a given direction vi∈ V. P(pi, vi) defines a half-plane and L(pi, vi) denotes the baseline that is perpendicular to viand passing through pi. Define a region dominated by piand vias a Baseline Bounded Half-Plane Voronoi Region, denoted as V or(pi, vi), if a point x ∈ V or(pi, vi), then (1) x ∈ P(pi, vi); (2) the line segment l(x, pi) does not cross any baseline; (3) if there is a point pj, such that x ∈ P(pj, vj), and the line segment l(x, pj) does not cross any baseline then d(x, pi) ≤ d(x, pj), j ≠ i. The Baseline Bounded Half-Plane Voronoi Diagram, denoted as V or(P, V), is the union of all V or(pi, vi). We show that V or(pi, vi) and V or(P, V) can be computed in O(n log n) and O(n2log n) time, respectively. For the heterogeneous point set, the same problem is also considered.

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