89 research outputs found
Degree Four Plane Spanners: Simpler and Better
Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p, q is realized as the line segment [pq], and is assigned a weight equal to the Euclidean distance |pq|. In this paper, we show how to construct in O(nlg{n}) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance
On the Subexponential Time Complexity of CSP
A CSP with n variables ranging over a domain of d values can be solved by
brute-force in d^n steps (omitting a polynomial factor). With a more careful
approach, this trivial upper bound can be improved for certain natural
restrictions of the CSP. In this paper we establish theoretical limits to such
improvements, and draw a detailed landscape of the subexponential-time
complexity of CSP.
We first establish relations between the subexponential-time complexity of
CSP and that of other problems, including CNF-Sat. We exploit this connection
to provide tight characterizations of the subexponential-time complexity of CSP
under common assumptions in complexity theory. For several natural CSP
parameters, we obtain threshold functions that precisely dictate the
subexponential-time complexity of CSP with respect to the parameters under
consideration.
Our analysis provides fundamental results indicating whether and when one can
significantly improve on the brute-force search approach for solving CSP
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
How to Navigate Through Obstacles?
Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? This is a fundamental problem that has undergone a tremendous amount of work by researchers in various areas, including computational geometry, graph theory, wireless computing, and motion planning. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be formulated and generalized into the following graph problem: Given a planar graph G whose vertices are colored by color sets, two designated vertices s, t in V(G), and k in N, is there an s-t path in G that uses at most k colors? If each obstacle is connected, the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph.
We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, among which a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms, as without it, the problem is W[SAT]-hard parameterized by k. Previous results only implied that the problem is W[2]-hard. A corollary of this result is that, unless W[2] = FPT, the problem cannot be approximated in FPT time to within a factor that is a function of k. By describing a generic plane embedding of the graph instances, we show that our hardness results translate to the geometric instances of the problem.
We then focus on graphs satisfying the color-connectivity property. By exploiting the planarity of the graph and the connectivity of the colors, we develop topological results that allow us to prove that, for any vertex v, there exists a set of paths whose cardinality is upper bounded by a function of k, that "represents" the valid s-t paths containing subsets of colors from v. We employ these structural results to design an FPT algorithm for the problem parameterized by both k and the treewidth of the graph, and extend this result further to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result generalizes and explains previous FPT results for various obstacle shapes, such as unit disks and fat regions
There are Plane Spanners of Maximum Degree 4
Let E be the complete Euclidean graph on a set of points embedded in the
plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a
t-spanner, or simply a spanner, if for any pair of vertices u,v in E the
distance between u and v in G is at most t times their distance in E. A spanner
is plane if its edges do not cross.
This paper considers the question: "What is the smallest maximum degree that
can always be achieved for a plane spanner of E?" Without the planarity
constraint, it is known that the answer is 3 which is thus the best known lower
bound on the degree of any plane spanner. With the planarity requirement, the
best known upper bound on the maximum degree is 6, the last in a long sequence
of results improving the upper bound. In this paper we show that the complete
Euclidean graph always contains a plane spanner of maximum degree at most 4 and
make a big step toward closing the question. Our construction leads to an
efficient algorithm for obtaining the spanner from Chew's L1-Delaunay
triangulation
On Geometric Spanners of Euclidean and Unit Disk Graphs
We consider the problem of constructing bounded-degree planar geometric
spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay
subgraph is a planar geometric spanner with stretch factor C_{del\approx
2.42; however, its degree may not be bounded. Our first result is a very
simple linear time algorithm for constructing a subgraph of the Delaunay graph
with stretch factor \rho =1+2\pi(k\cos{\frac{\pi{k)^{-1 and degree bounded by
, for any integer parameter . This result immediately implies an
algorithm for constructing a planar geometric spanner of a Euclidean graph with
stretch factor \rho \cdot C_{del and degree bounded by , for any integer
parameter . Moreover, the resulting spanner contains a Euclidean
Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in
developing the structural results necessary to transfer our analysis and
algorithm from Euclidean graphs to unit disk graphs, the usual model for
wireless ad-hoc networks. We obtain a very simple distributed, {\em
strictly-localized algorithm that, given a unit disk graph embedded in the
plane, constructs a geometric spanner with the above stretch factor and degree
bound, and also containing an EMST as a subgraph. The obtained results
dramatically improve the previous results in all aspects, as shown in the
paper
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