124 research outputs found
On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs
We tackle the problem of the randomized generation of slowly synchronizing
deterministic automata (DFAs) by generating random primitive sets of matrices.
We show that when the randomized procedure is too simple the exponent of the
generated sets is O(n log n) with high probability, thus the procedure fails to
return DFAs with large reset threshold. We extend this result to random
nondeterministic automata (NDFAs) by showing, in particular, that a uniformly
sampled NDFA has both a 2-directing word and a 3-directing word of length O(n
log n) with high probability. We then present a more involved randomized
algorithm that manages to generate DFAs with large reset threshold and we
finally leverage this finding for exhibiting new families of DFAs with reset
threshold of order .Comment: 31 pages, 9 figures. arXiv admin note: text overlap with
arXiv:1805.0672
Karp's patching algorithm on random perturbations of dense digraphs
We consider the following question. We are given a dense digraph with
minimum in- and out-degree at least , where is a constant.
We then add random edges to to create a digraph . Here an edge
is placed independently into with probability where
is a small positive constant. The edges of are given edge
costs , where is an independent copy of the exponential
mean one random variable i.e. . Let
be the associated cost matrix where
if . We show that w.h.p. the patching
algorithm of Karp finds a tour for the asymmetric traveling salesperson problem
that is asymptotically equal to that of the associated assignment problem.
Karp's algorithm runs in polynomial time.Comment: Fixed the proof of a lemm
Tilings in randomly perturbed dense graphs
A perfect -tiling in a graph is a collection of vertex-disjoint copies
of a graph in that together cover all the vertices in . In this
paper we investigate perfect -tilings in a random graph model introduced by
Bohman, Frieze and Martin in which one starts with a dense graph and then adds
random edges to it. Specifically, for any fixed graph , we determine the
number of random edges required to add to an arbitrary graph of linear minimum
degree in order to ensure the resulting graph contains a perfect -tiling
with high probability. Our proof utilises Szemer\'edi's Regularity lemma as
well as a special case of a result of Koml\'os concerning almost perfect
-tilings in dense graphs.Comment: 19 pages, to appear in CP
Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology
Topological data analysis is an emerging area in exploratory data analysis
and data mining. Its main tool, persistent homology, has become a popular
technique to study the structure of complex, high-dimensional data. In this
paper, we propose a novel method using persistent homology to quantify
structural changes in time-varying graphs. Specifically, we transform each
instance of the time-varying graph into metric spaces, extract topological
features using persistent homology, and compare those features over time. We
provide a visualization that assists in time-varying graph exploration and
helps to identify patterns of behavior within the data. To validate our
approach, we conduct several case studies on real world data sets and show how
our method can find cyclic patterns, deviations from those patterns, and
one-time events in time-varying graphs. We also examine whether
persistence-based similarity measure as a graph metric satisfies a set of
well-established, desirable properties for graph metrics
On Randomized Generation of Slowly Synchronizing Automata
Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank n-1 . We present a constructive randomized procedure to generate synchronizing automata of that kind with (potentially) large alphabet size based on recent results on primitive sets of matrices. We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of Omega(n^2/4) . We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a 0 entry changed into a 1 is primitive and has exponent of O(n log n) with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to 1 with probability p and equal to 0 with probability 1-pwhen np-log n - > infty as n - > infty
Discretized Distributed Optimization over Dynamic Digraphs
We consider a discrete-time model of continuous-time distributed optimization
over dynamic directed-graphs (digraphs) with applications to distributed
learning. Our optimization algorithm works over general strongly connected
dynamic networks under switching topologies, e.g., in mobile multi-agent
systems and volatile networks due to link failures. Compared to many existing
lines of work, there is no need for bi-stochastic weight designs on the links.
The existing literature mostly needs the link weights to be stochastic using
specific weight-design algorithms needed both at the initialization and at all
times when the topology of the network changes. This paper eliminates the need
for such algorithms and paves the way for distributed optimization over
time-varying digraphs. We derive the bound on the gradient-tracking step-size
and discrete time-step for convergence and prove dynamic stability using
arguments from consensus algorithms, matrix perturbation theory, and Lyapunov
theory. This work, particularly, is an improvement over existing
stochastic-weight undirected networks in case of link removal or packet drops.
This is because the existing literature may need to rerun time-consuming and
computationally complex algorithms for stochastic design, while the proposed
strategy works as long as the underlying network is weight-symmetric and
balanced. The proposed optimization framework finds applications to distributed
classification and learning
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