3,283 research outputs found
Evasion Paths in Mobile Sensor Networks
Suppose that ball-shaped sensors wander in a bounded domain. A sensor doesn't
know its location but does know when it overlaps a nearby sensor. We say that
an evasion path exists in this sensor network if a moving intruder can avoid
detection. In "Coordinate-free coverage in sensor networks with controlled
boundaries via homology", Vin deSilva and Robert Ghrist give a necessary
condition, depending only on the time-varying connectivity data of the sensors,
for an evasion path to exist. Using zigzag persistent homology, we provide an
equivalent condition that moreover can be computed in a streaming fashion.
However, no method with time-varying connectivity data as input can give
necessary and sufficient conditions for the existence of an evasion path.
Indeed, we show that the existence of an evasion path depends not only on the
fibrewise homotopy type of the region covered by sensors but also on its
embedding in spacetime. For planar sensors that also measure weak rotation and
distance information, we provide necessary and sufficient conditions for the
existence of an evasion path
Recognizing Planar Laman Graphs
Laman graphs are the minimally rigid graphs in the plane. We present two algorithms for recognizing planar Laman graphs. A simple algorithm with running time O(n^(3/2)) and a more complicated algorithm with running time O(n log^3 n) based on involved planar network flow algorithms. Both improve upon the previously fastest algorithm for general graphs by Gabow and Westermann [Algorithmica, 7(5-6):465 - 497, 1992] with running time O(n sqrt{n log n}).
To solve this problem we introduce two algorithms (with the running times stated above) that check whether for a directed planar graph G, disjoint sets S, T subseteq V(G), and a fixed k the following connectivity condition holds: for each vertex s in S there are k directed paths from s to T pairwise having only vertex s in common. This variant of connectivity seems interesting on its own
Rectangular Layouts and Contact Graphs
Contact graphs of isothetic rectangles unify many concepts from applications
including VLSI and architectural design, computational geometry, and GIS.
Minimizing the area of their corresponding {\em rectangular layouts} is a key
problem. We study the area-optimization problem and show that it is NP-hard to
find a minimum-area rectangular layout of a given contact graph. We present
O(n)-time algorithms that construct -area rectangular layouts for
general contact graphs and -area rectangular layouts for trees.
(For trees, this is an -approximation algorithm.) We also present an
infinite family of graphs (rsp., trees) that require (rsp.,
) area.
We derive these results by presenting a new characterization of graphs that
admit rectangular layouts using the related concept of {\em rectangular duals}.
A corollary to our results relates the class of graphs that admit rectangular
layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi
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