284 research outputs found
Decremental Single-Source Reachability in Planar Digraphs
In this paper we show a new algorithm for the decremental single-source
reachability problem in directed planar graphs. It processes any sequence of
edge deletions in total time and explicitly
maintains the set of vertices reachable from a fixed source vertex. Hence, if
all edges are eventually deleted, the amortized time of processing each edge
deletion is only , which improves upon a previously
known solution. We also show an algorithm for decremental
maintenance of strongly connected components in directed planar graphs with the
same total update time. These results constitute the first almost optimal (up
to polylogarithmic factors) algorithms for both problems.
To the best of our knowledge, these are the first dynamic algorithms with
polylogarithmic update times on general directed planar graphs for non-trivial
reachability-type problems, for which only polynomial bounds are known in
general graphs
Structural liveness of petri nets is ExpSpace-hard and decidable
Place/transition Petri nets are a standard model for a class of distributed systems whose reachability spaces might be infinite. One of well-studied topics is verification of safety and liveness properties in this model; despite an extensive research effort, some basic problems remain open, which is exemplified by the complexity status of the reachability problem that is still not fully clarified. The liveness problems are known to be closely related to the reachability problem, and various structural properties of nets that are related to liveness have been studied. Somewhat surprisingly, the decidability status of the problem of determining whether a net is structurally live, i.e. whether there is an initial marking for which it is live, remained open for some time; e.g. Best and Esparza (Inf Process Lett 116(6):423–427, 2016. https://doi.org/10.1016/j.ipl.2016.01.011) emphasize this open question. Here we show that the structural liveness problem for Petri nets is ExpSpace-hard and decidable. In particular, given a net N and a semilinear set S, it is decidable whether there is an initial marking of N for which the reachability set is included in S; this is based on results by Leroux (28th annual ACM/IEEE symposium on logic in computer science, LICS 2013, New Orleans, LA, USA, June 25–28, 2013, IEEE Computer Society, pp 23–32, 2013. https://doi.org/10.1109/LICS.2013.7)
On Petri Nets with Hierarchical Special Arcs
We investigate the decidability of termination, reachability, coverability and deadlock-freeness of Petri nets endowed with a hierarchy of places, and with inhibitor arcs, reset arcs and transfer arcs that respect this hierarchy. We also investigate what happens when we have a mix of these special arcs, some of which respect the hierarchy, while others do not. We settle the decidability status of the above four problems for all combinations of hierarchy, inhibitor, reset and transfer arcs, except the termination problem for two combinations. For both these combinations, we show that deciding termination is as hard as deciding the positivity problem on linear recurrence sequences -- a long-standing open problem
Engineering Art Galleries
The Art Gallery Problem is one of the most well-known problems in
Computational Geometry, with a rich history in the study of algorithms,
complexity, and variants. Recently there has been a surge in experimental work
on the problem. In this survey, we describe this work, show the chronology of
developments, and compare current algorithms, including two unpublished
versions, in an exhaustive experiment. Furthermore, we show what core
algorithmic ingredients have led to recent successes
Clemson Newsletter, 1969-1971
Information for the faculty and staff of Clemson Universityhttps://tigerprints.clemson.edu/clemson_newsletter/1004/thumbnail.jp
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