90 research outputs found
Polylogarithmic Approximation for Generalized Minimum Manhattan Networks
Given a set of terminals, which are points in -dimensional Euclidean
space, the minimum Manhattan network problem (MMN) asks for a minimum-length
rectilinear network that connects each pair of terminals by a Manhattan path,
that is, a path consisting of axis-parallel segments whose total length equals
the pair's Manhattan distance. Even for , the problem is NP-hard, but
constant-factor approximations are known. For , the problem is
APX-hard; it is known to admit, for any \eps > 0, an
O(n^\eps)-approximation.
In the generalized minimum Manhattan network problem (GMMN), we are given a
set of terminal pairs, and the goal is to find a minimum-length
rectilinear network such that each pair in is connected by a Manhattan
path. GMMN is a generalization of both MMN and the well-known rectilinear
Steiner arborescence problem (RSA). So far, only special cases of GMMN have
been considered.
We present an -approximation algorithm for GMMN (and, hence,
MMN) in dimensions and an -approximation algorithm for 2D.
We show that an existing -approximation algorithm for RSA in 2D
generalizes easily to dimensions.Comment: 14 pages, 5 figures; added appendix and figure
Approximate range searching☆☆A preliminary version of this paper appeared in the Proc. of the 11th Annual ACM Symp. on Computational Geometry, 1995, pp. 172–181.
AbstractThe range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and ε>0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance εw of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in Rd can be preprocessed in O(n+logn) time and O(n) space, such that approximate queries can be answered in O(logn(1/ε)d) time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in constant time (depending on dimension). For convex ranges, we tighten this to O(logn+(1/ε)d−1) time. We also present a lower bound for approximate range searching based on partition trees of Ω(logn+(1/ε)d−1), which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times
On Covering Segments with Unit Intervals
We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem.
We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration.
We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise
Cutting Polygons into Small Pieces with Chords: Laser-Based Localization
Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem
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