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.

CONVEX GRID DRAWINGS OF FOUR-CONNECTED PLANE GRAPHS

2006 ◽  
Vol 17 (05) ◽  
pp. 1031-1060 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI
Keyword(s):  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Outer Face ◽  
Plane Graphs ◽  
Line Segments ◽  
Grid Drawing ◽  
Straight Line ◽  
Grid Points

A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.

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.

ON INTERSECTING A SET OF ISOTHETIC LINE SEGMENTS WITH A CONVEX POLYGON OF MINIMUM AREA

2009 ◽  
Vol 19 (06) ◽  
pp. 557-577 ◽  
Author(s):  
ASISH MUKHOPADHYAY ◽  
EUGENE GREENE ◽  
S. V. RAO
Keyword(s):  
Convex Polygon ◽  
Time Algorithm ◽  
Minimum Area ◽  
Line Segments

We describe an O(n2)-time algorithm for computing a minimum-area convex polygon that intersects a set of n isothetic line segments.

CONVEX DRAWINGS OF PLANE GRAPHS OF MINIMUM OUTER APICES

2006 ◽  
Vol 17 (05) ◽  
pp. 1115-1127 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
MACHIKO AZUMA ◽  
TAKAO NISHIZEKI
Keyword(s):  
Convex Polygon ◽  
Linear Time ◽  
Plane Graph ◽  
Decomposition Tree ◽  
Necessary And Sufficient Condition ◽  
Component Decomposition ◽  
Minimum Number ◽  
Convex Drawing ◽  
Number Of Leaves ◽  
Necessary And Sufficient

In a convex drawing of a plane graph G, every facial cycle of G is drawn as a convex polygon. A polygon for the outer facial cycle is called an outer convex polygon. A necessary and sufficient condition for a plane graph G to have a convex drawing is known. However, it has not been known how many apices of an outer convex polygon are necessary for G to have a convex drawing. In this paper, we show that the minimum number of apices of an outer convex polygon necessary for G to have a convex drawing is, in effect, equal to the number of leaves in a triconnected component decomposition tree of a new graph constructed from G, and that a convex drawing of G having the minimum number of apices can be found in linear time.

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