7,154 research outputs found
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
Correlation function diagnostics for type-I fracton phases
Fracton phases are recent entrants to the roster of topological phases in
three dimensions. They are characterized by subextensively divergent
topological degeneracy and excitations that are constrained to move along lower
dimensional subspaces, including the eponymous fractons that are immobile in
isolation. We develop correlation function diagnostics to characterize Type I
fracton phases which build on their exhibiting {\it partial deconfinement}.
These are inspired by similar diagnostics from standard gauge theories and
utilize a generalized gauging procedure that links fracton phases to classical
Ising models with subsystem symmetries. En route, we explicitly construct the
spacetime partition function for the plaquette Ising model which, under such
gauging, maps into the X-cube fracton topological phase. We numerically verify
our results for this model via Monte Carlo calculations
How to Walk Your Dog in the Mountains with No Magic Leash
We describe a -approximation algorithm for computing the
homotopic \Frechet distance between two polygonal curves that lie on the
boundary of a triangulated topological disk. Prior to this work, algorithms
were known only for curves on the Euclidean plane with polygonal obstacles.
A key technical ingredient in our analysis is a -approximation
algorithm for computing the minimum height of a homotopy between two curves. No
algorithms were previously known for approximating this parameter.
Surprisingly, it is not even known if computing either the homotopic \Frechet
distance, or the minimum height of a homotopy, is in NP
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