7,961 research outputs found
Clearing Contamination in Large Networks
In this work, we study the problem of clearing contamination spreading
through a large network where we model the problem as a graph searching game.
The problem can be summarized as constructing a search strategy that will leave
the graph clear of any contamination at the end of the searching process in as
few steps as possible. We show that this problem is NP-hard even on directed
acyclic graphs and provide an efficient approximation algorithm. We
experimentally observe the performance of our approximation algorithm in
relation to the lower bound on several large online networks including
Slashdot, Epinions and Twitter. The experiments reveal that in most cases our
algorithm performs near optimally
Superpatterns and Universal Point Sets
An old open problem in graph drawing asks for the size of a universal point
set, a set of points that can be used as vertices for straight-line drawings of
all n-vertex planar graphs. We connect this problem to the theory of
permutation patterns, where another open problem concerns the size of
superpatterns, permutations that contain all patterns of a given size. We
generalize superpatterns to classes of permutations determined by forbidden
patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the
213-avoiding permutations, half the size of known superpatterns for
unconstrained permutations. We use our superpatterns to construct universal
point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16
factor. We prove that every proper subclass of the 213-avoiding permutations
has superpatterns of size O(n log^O(1) n), which we use to prove that the
planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA
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