927 research outputs found
Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings
We provide linear-time algorithms for geometric graphs with sublinearly many
crossings. That is, we provide algorithms running in O(n) time on connected
geometric graphs having n vertices and k crossings, where k is smaller than n
by an iterated logarithmic factor. Specific problems we study include Voronoi
diagrams and single-source shortest paths. Our algorithms all run in linear
time in the standard comparison-based computational model; hence, we make no
assumptions about the distribution or bit complexities of edge weights, nor do
we utilize unusual bit-level operations on memory words. Instead, our
algorithms are based on a planarization method that "zeroes in" on edge
crossings, together with methods for extending planar separator decompositions
to geometric graphs with sublinearly many crossings. Incidentally, our
planarization algorithm also solves an open computational geometry problem of
Chazelle for triangulating a self-intersecting polygonal chain having n
segments and k crossings in linear time, for the case when k is sublinear in n
by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium
on Discrete Algorithms (SODA09
Escaping the Curse of Spatial Partitioning: Matchings with Low Crossing Numbers and Their Applications
Given a set system (X, S), constructing a matching of X with low crossing number is a key tool in combinatorics and algorithms. In this paper we present a new sampling-based algorithm which is applicable to finite set systems. Let n = |X|, m = | S| and assume that X has a perfect matching M such that any set in ? crosses at most ? = ?(n^?) edges of M. In the case ? = 1- 1/d, our algorithm computes a perfect matching of X with expected crossing number at most 10 ?, in expected time O? (n^{2+(2/d)} + mn^(2/d)).
As an immediate consequence, we get improved bounds for constructing low-crossing matchings for a slew of both abstract and geometric problems, including many basic geometric set systems (e.g., balls in ?^d). This further implies improved algorithms for many well-studied problems such as construction of ?-approximations. Our work is related to two earlier themes: the work of Varadarajan (STOC \u2710) / Chan et al. (SODA \u2712) that avoids spatial partitionings for constructing ?-nets, and of Chan (DCG \u2712) that gives an optimal algorithm for matchings with respect to hyperplanes in ?^d.
Another major advantage of our method is its simplicity. An implementation of a variant of our algorithm in C++ is available on Github; it is approximately 200 lines of basic code without any non-trivial data-structure. Since the start of the study of matchings with low-crossing numbers with respect to half-spaces in the 1980s, this is the first implementation made possible for dimensions larger than 2
Practical Low-Dimensional Halfspace Range Space Sampling
We develop, analyze, implement, and compare new algorithms for creating epsilon-samples of range spaces defined by halfspaces which have size sub-quadratic in 1/epsilon, and have runtime linear in the input size and near-quadratic in 1/epsilon. The key to our solution is an efficient construction of partition trees. Despite not requiring any techniques developed after the early 1990s, apparently such a result was never explicitly described. We demonstrate that our implementations, including new implementations of several variants of partition trees, do indeed run in time linear in the input, appear to run linear in output size, and observe smaller error for the same size sample compared to the ubiquitous random sample (which requires size quadratic in 1/epsilon). This result has direct applications in speeding up discrepancy evaluation, approximate range counting, and spatial anomaly detection
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
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