Approximation of polygonal curves with minimum number of line segments

1992 ◽  
pp. 378-387 ◽  
Author(s):  
W. S. Chan ◽  
F. Chin
Keyword(s):  
Line Segments ◽  
Minimum Number ◽  
Polygonal Curves

scholarly journals APPROXIMATION OF POLYGONAL CURVES WITH MINIMUM NUMBER OF LINE SEGMENTS OR MINIMUM ERROR

1996 ◽  
Vol 06 (01) ◽  
pp. 59-77 ◽  
Author(s):  
W.S. CHAN ◽  
F. CHIN
Keyword(s):  
Time Complexity ◽  
Convex Polygon ◽  
Line Segments ◽  
Minimum Error ◽  
Curve Approximation ◽  
Polygonal Curve ◽  
Minimum Number ◽  
Polygonal Curves ◽  
Approximation Problems

We improve the time complexities for solving the polygonal curve approximation problems formulated by Imai and Iri. The time complexity for approximating any polygonal curve of n vertices with minimum number of line segments can be improved from O(n2 log n) to O(n2). The time complexity for approximating any polygonal curve with minimum error can also be improved from O(n2 log 2n) to O(n2 log n). We further show that if the curve to be approximated forms part of a convex polygon, the two problems can be solved in O(n) and O(n2) time respectively for both open and closed polygonal curves.

scholarly journals Minimum Cell Connection in Line Segment Arrangements

2017 ◽  
Vol 27 (03) ◽  
pp. 159-176
Author(s):  
Helmut Alt ◽  
Sergio Cabello ◽  
Panos Giannopoulos ◽  
Christian Knauer
Keyword(s):  
Linear Time ◽  
Optimal Solution ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Constant Number ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Segments ◽  
Connection Problems

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

OPTIMAL BINARY SPACE PARTITIONS FOR SEGMENTS IN THE PLANE

2012 ◽  
Vol 22 (03) ◽  
pp. 187-205 ◽  
Author(s):  
MARK DE BERG ◽  
AMIRALI KHOSRAVI
Keyword(s):  
Recursive Partitioning ◽  
Np Hard ◽  
Line Segments ◽  
Minimum Number ◽  
Disjoint Line

An optimal BSP for a set S of disjoint line segments in the plane is a BSP for S that produces the minimum number of cuts. We study optimal BSPs for three classes of BSPs, which differ in the splitting lines that can be used when partitioning a set of fragments in the recursive partitioning process: free BSPs can use any splitting line, restricted BSPs can only use splitting lines through pairs of fragment endpoints, and auto-partitions can only use splitting lines containing a fragment. We obtain the following two results: • It is NP-hard to decide whether a given set of segments admits an auto-partition that does not make any cuts. • An optimal restricted BSP makes at most 2 times as many cuts as an optimal free BSP for the same set of segments.

APPROXIMATING POLYGONS AND SUBDIVISIONS WITH MINIMUM-LINK PATHS

1993 ◽  
Vol 03 (04) ◽  
pp. 383-415 ◽  
Author(s):  
LEONIDAS J. GUIBAS ◽  
JOHN E. HERSHBERGER ◽  
JOSEPH S.B. MITCHELL ◽  
JACK SCOTT SNOEYINK
Keyword(s):  
Greedy Algorithms ◽  
Np Hard ◽  
Basic Approach ◽  
Line Segments ◽  
Minimum Number ◽  
Polygonal Chains ◽  
The Given ◽  
Plane Polygon

We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with. no self-intersections are NP-hard.

ALTERNATING HAMILTON CYCLES WITH MINIMUM NUMBER OF CROSSINGS IN THE PLANE

2000 ◽  
Vol 10 (01) ◽  
pp. 73-78 ◽  
Author(s):  
ATSUSHI KANEKO ◽  
M. KANO ◽  
KIYOSHI YOSHIMOTO
Keyword(s):  
Upper Bound ◽  
Time Algorithm ◽  
Hamilton Cycle ◽  
Crossing Number ◽  
Hamilton Cycles ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Disjoint Sets

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n2 log n) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

A POLYNOMIAL-TIME ALGORITHM FOR COMPUTING THE RESILIENCE OF ARRANGEMENTS OF RAY SENSORS

2014 ◽  
Vol 24 (03) ◽  
pp. 225-236 ◽  
Author(s):  
DAVID KIRKPATRICK ◽  
BOTING YANG ◽  
SANDRA ZILLES
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Np Hard ◽  
Sensor Coverage ◽  
Entire Plane ◽  
Line Segments ◽  
Minimum Number

Given an arrangement A of n sensors and two points s and t in the plane, the barrier resilience of A with respect to s and t is the minimum number of sensors whose removal permits a path from s to t such that the path does not intersect the coverage region of any sensor in A. When the surveillance domain is the entire plane and sensor coverage regions are unit line segments, even with restricted orientations, the problem of determining the barrier resilience is known to be NP-hard. On the other hand, if sensor coverage regions are arbitrary lines, the problem has a trivial linear time solution. In this paper, we study the case where each sensor coverage region is an arbitrary ray, and give an O(n2m) time algorithm for computing the barrier resilience when there are m ⩾ 1 sensor intersections.

scholarly journals Drawing a Graph in a Hypercube

10.37236/1099 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
David R. Wood
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Line Segments ◽  
Sidon Sets ◽  
Minimum Number

A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections.

SIMPLE OPTIMAL ALGORITHMS FOR RECTILINEAR LINK PATH AND POLYGON SEPARATION PROBLEMS

1999 ◽  
Vol 09 (01) ◽  
pp. 31-42 ◽  
Author(s):  
ANIL MAHESHWARI ◽  
JÖRG-RÜDIGER SACK
Keyword(s):  
Parallel Algorithms ◽  
Random Access ◽  
Optimal Solution ◽  
Simple Polygon ◽  
Separation Problem ◽  
Line Segments ◽  
Point Pair ◽  
Erew Pram ◽  
Rectilinear Polygons ◽  
Minimum Number

The link metric, defined on a constrained region R of the plane, sets the distance between a pair of points in R to equal the minimum number of line segments or links needed to construct a path in R between the point pair. The minimum rectilinear link path problem considered here is to compute a rectilinear path consisting of the minimum number of links between two points in R, when R is inside an n-sided rectilinear simple polygon. In this paper we present optimal sequential and parallel algorithms to compute a minimum rectilinear link path in a trapezoided region R. Our parallel algorithm requires O( log n) time using a total of O(n) operations. The complexity of our algorithm matches that of the algorithm of McDonald and Peters [19]. By exploiting the dual structure of the trapezoidation of R, we obtain a conceptually simple and easy to implement algorithm. As applications of our techniques we provide an optimal solution to the minimum nested polygon problem and the minimum polygon separation problem. The minimum nested polygon problem asks for finding a rectilinear polygon, with minimum number of sides, that is nested between two given rectilinear polygons one of which is contained in the other. The minimum polygon separation problem asks for computing a minimum number of orthogonal lines and line segments that separate two given non-intersecting simple rectilinear polygons. All parallel algorithms are deterministic, designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM), and are optimal.

Is It FPT to Cover Points with Tours on Minimum Number of Bends (Errata)?

2015 ◽  
Vol 25 (01) ◽  
pp. 11-14
Author(s):  
Vladimir Estivill-Castro
Keyword(s):  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Formal Process ◽  
Line Segments ◽  
Fpt Algorithms ◽  
Peer Reviews ◽  
Minimum Number ◽  
Number Of Bends

Recent communication by Minghui Jiang has brought to my attention that I overlooked faults in the arguments built while collaborating closely with my PhD student Apichat Heednacram and his co-supervisor Francis Suraweera. These errors unfortunately also escaped the scrutiny of peer-reviews and the formal process of examination. Some results in Apichat's dissertation that were published in this journal (and other outlets) are actually incorrect. In particular, we had reported an FPT-algorithms for the k-Bends Traveling Salesman Problem in ℜ2 and some variants that result from adding constraints to the line-segments that constitute the tour. While the reduction rules to kernelize the problem produce reduced instances, a solution of the kernel instance does not lead directly to a solution of the original instance.

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