192 research outputs found
Transversals to Line Segments in Three-Dimensional Space
We completely describe the structure of the connected components of transversals to a collection of n line segments in R3. We show that n \u3e 3 arbitrary line segments in R3 admit 0, 1, . . . , n or infinitely many line transversals. In the latter case, the transversals form up to n connected components
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
Orthogonal weighted linear L1 and L∞ approximation and applications
AbstractLet S={s1,s2,...,sn} be a set of sites in Ed, where every site si has a positive real weight ωi. This paper gives algorithms to find weighted orthogonal L∞ and L1 approximating hyperplanes for S. The algorithm for the weighted orthogonal L1 approximation is shown to require O(nd) worst-case time and O(n) space for d ≥ 2. The algorithm for the weighted orthogonal L∞ approximation is shown to require O(n log n) worst-case time and O(n) space for d = 2, and O(n⌊dl2 + 1⌋) worst-case time and O(n⌊(d+1)/2⌋) space for d > 2. In the latter case, the expected time complexity may be reduced to O(n⌊(d+1)/2⌋). The L∞ approximation algorithm can be modified to solve the problem of finding the width of a set of n points in Ed, and the problem of finding a stabbing hyperplane for a set of n hyperspheres in Ed with varying radii. The time and space complexities of the width and stabbing algorithms are seen to be the same as those of the L∞ approximation algorithm
Stabbing segments with rectilinear objects
Given a set S of n line segments in the plane, we say that a region R R2 is a
stabber for S if R contains exactly one endpoint of each segment of S. In this paper we provide
optimal or near-optimal algorithms for reporting all combinatorially di erent stabbers for
several shapes of stabbers. Speci cally, we consider the case in which the stabber can be
described as the intersection of axis-parallel halfplanes (thus the stabbers are halfplanes,
strips, quadrants, 3-sided rectangles, or rectangles). The running times are O(n) (for the
halfplane case), O(n log n) (for strips, quadrants, and 3-sided rectangles), and O(n2 log n)
(for rectangles).Junta de Andalucía PAI FQM-0164Ministerio de Economía y Competitividad MTM2014-60127-
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