3,175 research outputs found
On the Maximum Crossing Number
Research about crossings is typically about minimization. In this paper, we
consider \emph{maximizing} the number of crossings over all possible ways to
draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009]
conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a
drawing with vertices in convex position, that maximizes the number of edge
crossings. We disprove this conjecture by constructing a planar graph on twelve
vertices that allows a non-convex drawing with more crossings than any convex
one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the
maximum number of crossings of a geometric graph and that the weighted
geometric case is NP-hard to approximate. We strengthen these results by
showing hardness of approximation even for the unweighted geometric case and
prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure
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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Directed Placement for mVLSI Devices
Continuous-flow microfluidic devices based on integrated channel networks are becoming increasingly prevalent in research in the biological sciences. At present, these devices are physically laid out by hand by domain experts who understand both the underlying technology and the biological functions that will execute on fabricated devices. The lack of a design science that is specific to microfluidic technology creates a substantial barrier to entry. To address this concern, this article introduces Directed Placement, a physical design algorithm that leverages the natural "directedness" in most modern microfluidic designs: fluid enters at designated inputs, flows through a linear or tree-based network of channels and fluidic components, and exits the device at dedicated outputs. Directed placement creates physical layouts that share many principle similarities to those created by domain experts. Directed placement allows components to be placed closer to their neighbors compared to existing layout algorithms based on planar graph embedding or simulated annealing, leading to an average reduction in laid-out fluid channel length of 91% while improving area utilization by 8% on average. Directed placement is compatible with both passive and active microfluidic devices and is compatible with a variety of mainstream manufacturing technologies
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