10,601 research outputs found
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
Graph Theory with Applications to Statistical Mechanics
This work will have two parts. The first will be related to various types of graph connectivity, and will consist of some exposition on the work of Andreas Holtkamp on local variants of vertex connectivity and edge connectivity in graphs. The second part will consist of an introduction to the field of physics known as percolation theory, which has to do with infinite connected components in certain types of graphs, which has numerous physical applications, especially in the field of statistical mechanics
A Universal Point Set for 2-Outerplanar Graphs
A point set is universal for a class if
every graph of has a planar straight-line embedding on . It is
well-known that the integer grid is a quadratic-size universal point set for
planar graphs, while the existence of a sub-quadratic universal point set for
them is one of the most fascinating open problems in Graph Drawing. Motivated
by the fact that outerplanarity is a key property for the existence of small
universal point sets, we study 2-outerplanar graphs and provide for them a
universal point set of size .Comment: 23 pages, 11 figures, conference version at GD 201
On the Distributed Complexity of Large-Scale Graph Computations
Motivated by the increasing need to understand the distributed algorithmic
foundations of large-scale graph computations, we study some fundamental graph
problems in a message-passing model for distributed computing where
machines jointly perform computations on graphs with nodes (typically, ). The input graph is assumed to be initially randomly partitioned among
the machines, a common implementation in many real-world systems.
Communication is point-to-point, and the goal is to minimize the number of
communication {\em rounds} of the computation.
Our main contribution is the {\em General Lower Bound Theorem}, a theorem
that can be used to show non-trivial lower bounds on the round complexity of
distributed large-scale data computations. The General Lower Bound Theorem is
established via an information-theoretic approach that relates the round
complexity to the minimal amount of information required by machines to solve
the problem. Our approach is generic and this theorem can be used in a
"cookbook" fashion to show distributed lower bounds in the context of several
problems, including non-graph problems. We present two applications by showing
(almost) tight lower bounds for the round complexity of two fundamental graph
problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our
approach, as demonstrated in the case of PageRank, can yield tight lower bounds
for problems (including, and especially, under a stochastic partition of the
input) where communication complexity techniques are not obvious.
Our approach, as demonstrated in the case of triangle enumeration, can yield
stronger round lower bounds as well as message-round tradeoffs compared to
approaches that use communication complexity techniques
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