5 research outputs found
Local Mixing Time: Distributed Computation and Applications
The mixing time of a graph is an important metric, which is not only useful
in analyzing connectivity and expansion properties of the network, but also
serves as a key parameter in designing efficient algorithms. We introduce a new
notion of mixing of a random walk on a (undirected) graph, called local mixing.
Informally, the local mixing with respect to a given node , is the mixing of
a random walk probability distribution restricted to a large enough subset of
nodes --- say, a subset of size at least for a given parameter
--- containing . The time to mix over such a subset by a random walk
starting from a source node is called the local mixing time with respect to
. The local mixing time captures the local connectivity and expansion
properties around a given source node and is a useful parameter that determines
the running time of algorithms for partial information spreading, gossip etc.
Our first contribution is formally defining the notion of local mixing time
in an undirected graph. We then present an efficient distributed algorithm
which computes a constant factor approximation to the local mixing time with
respect to a source node in rounds, where is
the local mixing time w.r.t in an -node regular graph. This bound holds
when is significantly smaller than the conductance of the local mixing
set (i.e., the set where the walk mixes locally); this is typically the
interesting case where the local mixing time is significantly smaller than the
mixing time (with respect to ). We also present a distributed algorithm that
computes the exact local mixing time in rounds,
where and is the diameter of the graph. We
further show that local mixing time tightly characterizes the complexity of
partial information spreading.Comment: 16 page
Property testing of graphs and the role of neighborhood distributions
Property testing considers decision problems in the regime of sublinear complexity. Most classical decision problems require at least linear time complexity in order to read the whole input. Hence, decision problems are relaxed by introducing a gap between “yes” and “no” instances: A property tester for a property Π (e. g., planarity) is a randomized algorithm with constant error probability that accepts objects that have Π (planar graphs) and that rejects objects that have linear edit distance to any object from Π (graphs with a linear number of crossing edges in every planar embedding). For property testers, locality is a natural and crucial concept because they cannot obtain a global view of their input. In this thesis, we investigate property testing in graphs and how testers leverage the information contained in the neighborhoods of randomly sampled vertices: We provide some structural insights regarding properties with constant testing complexity in graphs with bounded (maximum vertex) degree and a connection between testers with constant complexity for general graphs and testers with logarithmic space complexity for random-order streams. We also present testers for some minor-freeness properties and a tester for conductance in the distributed CONGEST model