Images of Linear Conditions on a Manhattan Plane

In this paper are considered planar point sets generated by linear conditions, which are realized in rectangular or Manhattan metric. Linear conditions are those expressed by the finite sum of the products of distances by numerical coefficients. Finite sets of points and lines are considered as figures defining linear conditions. It has been shown that linear conditions can be defined relative to other planar figures: lines, polygons, etc. The design solutions of the following general geometric problem are considered: for a finite set of figures (points, line segments, polygons...) specified on a plane with a rectangular metric, which are in a common position, it is necessary to construct sets that satisfy any linear condition. The problems in which the given sets are point and segment ones have been considered in detail, and linear conditions are represented as a sum or as relations of distances. It is proved that solution result can be isolated points, broken lines, and areas on the plane. Sets of broken lines satisfying the given conditions form families of isolines for the given condition. An algorithm for building isoline families is presented. The algorithm is based on the Hanan lattice construction and the isolines behavior in each node and each sub-region of the lattice. For isoline families defined by conditions for relation of distances, some of their properties allowing accelerate their construction process are proved. As an example for application of the described theory, the problem of plane partition into regions corresponding to a given set of points, lines and other figures is considered. The problem is generalized problem of Voronoi diagram construction, and considered in general formulation. It means the next: 1) the problem is considered in rectangular metric; 2) all given points may be integrated in various figures – separate points, line segments, triangles, quadrangles etc.; 3) the Voronoi diagram’s property of proximity is changed for property of proportionality. Have been represented examples for plane partition into regions, determined by two-point sets.

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ON THE STEINER RATIO IN $\mathcal{R}_{n}$

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001358 ◽

2011 ◽

Vol 03 (04) ◽

pp. 473-489

Author(s):

HAI DU ◽

WEILI WU ◽

ZAIXIN LU ◽

YINFENG XU

Keyword(s):

Euclidean Space ◽

Spanning Tree ◽

Minimum Spanning Tree ◽

Dimensional Euclidean Space ◽

Minimum Length ◽

Steiner Ratio ◽

Line Segments ◽

Finite Set ◽

Steiner Minimum Tree ◽

Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

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MINIMUM POLYGON TRANSVERSALS OF LINE SEGMENTS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195995000143 ◽

1995 ◽

Vol 05 (03) ◽

pp. 243-256 ◽

Cited By ~ 22

Author(s):

DAVID RAPPAPORT

Keyword(s):

Convex Hull ◽

General Problem ◽

Line Segment ◽

Simple Polygon ◽

Fixed Number ◽

Line Segments ◽

Geometric Objects ◽

Finite Set ◽

Minimum Perimeter ◽

Set Of Points

Let S be used to denote a finite set of planar geometric objects. Define a polygon transversal of S as a closed simple polygon that simultaneously intersects every object in S, and a minimum polygon transversal of S as a polygon transversal of S with minimum perimeter. If S is a set of points then the minimum polygon transversal of S is the convex hull of S. However, when the objects in S have some dimension then the minimum polygon transversal and the convex hull may no longer coincide. We consider the case where S is a set of line segments. If the line segments are constrained to lie in a fixed number of orientations we show that a minimum polygon transversal can be found in O(n log n) time. More explicitely, if m denotes the number of line segment orientations, then the complexity of the algorithm is given by O(3mn+log n). The general problem for line segments is not known to be polynomial nor is it known to be NP-hard.

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A Randomized Approach to Volume Constrained Polyhedronization Problem

Journal of Computing and Information Science in Engineering ◽

10.1115/1.4029559 ◽

2015 ◽

Vol 15 (1) ◽

Cited By ~ 1

Author(s):

Jiju Peethambaran ◽

Amal Dev Parakkat ◽

Ramanathan Muthuganapathy

Keyword(s):

Three Dimensional ◽

Greedy Heuristic ◽

Point Sets ◽

Maximal Volume ◽

Simple Polyhedron ◽

Practical Algorithms ◽

Finite Set ◽

Potential Applications ◽

Brute Force Method ◽

Set Of Points

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

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On Covering Points with Minimum Turns

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195915500016 ◽

2015 ◽

Vol 25 (01) ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Minghui Jiang

Keyword(s):

Np Hard ◽

Line Segments ◽

Polygonal Chain ◽

Axis Parallel ◽

A Chain ◽

Set Of Points ◽

The Given

We study the problem of finding a polygonal chain of line segments to cover a set of points in ℝd, d≥2, with the goal of minimizing the number of links or turns in the chain. A chain of line segments that covers all points in the given set is called a covering tour if the chain is closed, and is called a covering path if the chain is open. A covering tour or a covering path is rectilinear if all segments in the chain are axis-parallel. We prove that the two problems Minimum-Link Rectilinear Covering Tour and Minimum-Link Rectilinear Covering Path are both NP-hard in ℝ10.

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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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