Abstract: We give an efficient local search
algorithm that computes a good vertex coloring of a graph $G$. In
order to better illustrate this local search method, we view local
moves as selfish moves in a suitably defined game. In particular,
given a graph $G=(V,E)$ of $n$ vertices and $m$ edges, we define
the \emph{graph coloring game} $\Gamma(G)$ as a strategic game
where the set of players is the set of vertices and the players
share the same action set, which is a set of $n$ colors. The
payoff that a vertex $v$ receives, given the actions chosen by all
vertices, equals the total number of vertices that have chosen the
same color as $v$, unless a neighbor of $v$ has also chosen the
same color, in which case the payoff of $v$ is 0. We show:
\begin{itemize}
\item The game $\Gamma(G)$ has always pure Nash equilibria. Each
pure equilibrium is a proper coloring of $G$. Furthermore, there
exists a pure equilibrium that corresponds to an optimum coloring.
\item We give a polynomial time algorithm $\mathcal{A}$ which
computes a pure Nash equilibrium of $\Gamma(G)$. \item The total
number, $k$, of colors used in \emph{any} pure Nash equilibrium
(and thus achieved by $\mathcal{A}$) is $k\leq\min\{\Delta_2+1,
\frac{n+\omega}{2}, \frac{1+\sqrt{1+8m}}{2}, n-\alpha+1\}$, where
$\omega, \alpha$ are the cliquenumber and the independence number
of $G$ and $\Delta_2$ is the maximum degree that a vertex can have
subject to the condition that it is adjacent to at least one
vertex of equal or greater degree. ($\Delta_2$ is no more than the
maximum degree $\Delta$ of $G$.) \item Thus, in fact, we propose
here a \emph{new}, \emph{efficient} coloring method that achieves
a number of colors \emph{satisfying (together) the known general
upper bounds on the chromatic number $\chi$}. Our method is also
an alternative general way of \emph{proving},
\emph{constructively}, all these bounds. \item Finally, we show
how to strengthen our method (staying in polynomial time) so that
it avoids ``bad'' pure Nash equilibria (i.e. those admitting a
number of colors $k$ far away from $\chi$). In particular, we show
that our enhanced method colors \emph{optimally} dense random
$q$-partite graphs (of fixed $q$) with high probability.
\end{itemize}
Abstract: Consider k particles, 1 red and k1 white, chasing each other on the nodes of a graph G. If the red one catches one of the white, it ldquoinfectsrdquo it with its color. The newly red particles are now available to infect more white ones. When is it the case that all white will become red? It turns out that this simple question is an instance of information propagation between random walks and has important applications to mobile computing where a set of mobile hosts acts as an intermediary for the spread of information.
In this paper we model this problem by k concurrent random walks, one corresponding to the red particle and k1 to the white ones. The infection time Tk of infecting all the white particles with red color is then a random variable that depends on k, the initial position of the particles, the number of nodes and edges of the graph, as well as on the structure of the graph.
We easily get that an upper bound on the expected value of Tk is the worst case (over all initial positions) expected meeting time m* of two random walks multiplied by THgr (log k). We demonstrate that this is, indeed, a tight bound; i.e. there is a graph G (a special case of the ldquolollipoprdquo graph), a range of values kclique or has nice expansion properties, we prove much smaller bounds for Tk. We have evaluated and validated all our results by large scale experiments which we also present and discuss here. In particular, the experiments demonstrate that our analytical results for these expander graphs are tight.
Abstract: An intersection graph of n vertices assumes that each vertex is equipped with a subset of a global label set. Two vertices share an edge
when their label sets intersect. Random Intersection Graphs (RIGs) (as defined in [18, 31]) consider label sets formed by the following experiment:
each vertex, independently and uniformly, examines all the labels (m in total) one by one. Each examination is independent and the vertex
succeeds to put the label in her set with probability p. Such graphs nicely capture interactions in networks due to sharing of resources among nodes.
We study here the problem of efficiently coloring (and of finding upper bounds to the chromatic number) of RIGs. We concentrate in a range
of parameters not examined in the literature, namely: (a) m = n{\'a} for less than 1 (in this range, RIGs differ substantially from the Erdos- Renyi random graphs) and (b) the selection probability p is quite high
(e.g. at least ln2 n m in our algorithm) and disallows direct greedy colouring methods.
We manage to get the following results:
For the case mp ln n, for any constant < 1 − , we prove that np colours are enough to colour most of the vertices of the graph with high probability (whp). This means that even for quite dense
graphs, using the same number of colours as those needed to properly colour the clique induced by any label suffices to colour almost all of the vertices of the graph. Note also that this range of values of m, p
is quite wider than the one studied in [4].
� We propose and analyze an algorithm CliqueColour for finding a proper colouring of a random instance of Gn,m,p, for any mp >=ln2 n. The algorithm uses information of the label sets assigned to the
vertices of Gn,m,p and runs in O (n2mp2/ln n) time, which is polynomial in n and m. We also show by a reduction to the uniform random
intersection graphs model that the number of colours required by the algorithm are of the correct order of magnitude with the actual
chromatic number of Gn,m,p.
⋆ This work was partially supported by the ICT Programme of the European Union under contract number ICT-2008-215270 (FRONTS). Also supported by Research Training Group GK-693 of the Paderborn Institute for Scientific Computation
(PaSCo).
� We finally compare the problem of finding a proper colouring for Gn,m,p to that of colouring hypergraphs so that no edge is monochromatic.We show how one can find in polynomial time a k-colouring of the vertices of Gn,m,p, for any integer k, such that no clique induced by only one label in Gn,m,p is monochromatic. Our techniques are novel and try to exploit as much as possible the hidden structure of random intersection graphs in this interesting range.
Abstract: Andrews et al. [Automatic method for hiding latency in high bandwidth networks, in: Proceedings of the ACM Symposium on Theory of Computing, 1996, pp. 257–265; Improved methods for hiding latency in high bandwidth networks, in: Proceedings of the Eighth Annual ACM Symposium on Parallel Algorithms and Architectures, 1996, pp. 52–61] introduced a number of techniques for automatically hiding latency when performing simulations of networks with unit delay links on networks with arbitrary unequal delay links. In their work, they assume that processors of the host network are identical in computational power to those of the guest network being simulated. They further assume that the links of the host are able to pipeline messages, i.e., they are able to deliver P packets in time O(P+d) where d is the delay on the link.
In this paper we examine the effect of eliminating one or both of these assumptions. In particular, we provide an efficient simulation of a linear array of homogeneous processors connected by unit-delay links on a linear array of heterogeneous processors connected by links with arbitrary delay. We show that the slowdown achieved by our simulation is optimal. We then consider the case of simulating cliques by cliques; i.e., a clique of heterogeneous processors with arbitrary delay links is used to simulate a clique of homogeneous processors with unit delay links. We reduce the slowdown from the obvious bound of the maximum delay link to the average of the link delays. In the case of the linear array we consider both links with and without pipelining. For the clique simulation the links are not assumed to support pipelining.
The main motivation of our results (as was the case with Andrews et al.) is to mitigate the degradation of performance when executing parallel programs designed for different architectures on a network of workstations (NOW). In such a setting it is unlikely that the links provided by the NOW will support pipelining and it is quite probable the processors will be heterogeneous. Combining our result on clique simulation with well-known techniques for simulating shared memory PRAMs on distributed memory machines provides an effective automatic compilation of a PRAM algorithm on a NOW.
Abstract: We consider the problem of searching for a piece of
information in a fully interconnected computer network
(also called a complete network or clique) by exploiting
advice about its location from the network nodes. Each
node contains a database that ?knows? what kind of
documents or information are stored in other nodes
(e.g., a node could be a Web server that answers queries
about documents stored on the Web). The databases in
each node, when queried, provide a pointer that leads to
the node that contains the information. However, this
information is up-to-date (or correct) with some
bounded probability. While, in principle, one may always
locate the information by simply visiting the network
nodes in some prescribed ordering, this requires a time
complexity in the order of the number of nodes of the
network. In this paper, we provide algorithms for locating
an information node in the complete communication
network, which take advantage of advice given from
network nodes. The nodes may either give correct advice,
by pointing directly to the information node, or give
wrong advice, by pointing elsewhere. On the lowerbounds?
side, we show that no fixed-memory (i.e., with
memory independent of the network size) deterministic
algorithm may locate the information node in a constant
(independent of the network size) expected number of
steps. Moreover, if p (1/n) is the probability that a
node of an n-node clique gives correct advice, we show
that no algorithm may locate the information node in an
expected number of steps less than 1/p o(1). To study
how the expected number of steps is affected by the
amount of memory allowed to the algorithms, we give a
memoryless randomized algorithm with expected number
of steps 4/p o(1/p) o(1) and a 1-bit randomized
algorithm requiring on the average at most 2/p o(1)
steps. In addition, in the memoryless case, we also
prove a 4/p lower bound for the expected number of
steps in the case where the nodes giving faulty advice
may decide on the content of this advice in any possible
way and not merely at random (adversarial fault model).
Finally, for the case where faulty nodes behave randomly,
we give an optimal, unlimited memory deterministic
algorithm with expected number of steps bounded
from above by 1/p o(1/p) 1.
Abstract: We consider the problem of searching for a piece of information in a fully interconnected computer network or clique by exploiting
advice about its location from the network nodes Each node contains a
database that knows what kind of documents or information are stored
in other nodes e.g. a node could be a Web server that answers queries
about documents stored on the Web. The databases in each node when
queried provide a pointer that leads to the node that contains the information. However this information is up to date or correct with some
bounded probability. While in principle one may always locate the information by simply visiting the network nodes in some prescribed ordering
this requires a time complexity in the order of the number of nodes of the
network. In this paper we provide algorithms for locating an information node in the complete communication network that take advantage
of advice given from network nodes The nodes may either give correct
advice by pointing directly to the information node or give wrong advice
by pointing elsewhere We show that on the averageif the probability that a node provides correct advice is asymptotically larger than
where is the number of the computer nodes then the average time complexity for locating the information node is asymptotically or depending on the available memory.The probability may in general be a function of the number of network nodes . On the lower bounds
side we prove that noxed memory deterministic algorithm can locate
the information node in nite expected number of steps. We also prove
a lower bound of
for the expected number of steps of any algorithm
that locates the information node in the complete network.
Abstract: In this work we consider temporal graphs, i.e. graphs, each edge of which isassigned a set of discrete time-labels drawn from a set of integers. The labelsof an edge indicate the discrete moments in time at which the edge isavailable. We also consider temporal paths in a temporal graph, i.e. pathswhose edges are assigned a strictly increasing sequence of labels. Furthermore,we assume the uniform case (UNI-CASE), in which every edge of a graph isassigned exactly one time label from a set of integers and the time labelsassigned to the edges of the graph are chosen randomly and independently, withthe selection following the uniform distribution. We call uniform randomtemporal graphs the graphs that satisfy the UNI-CASE. We begin by deriving theexpected number of temporal paths of a given length in the uniform randomtemporal clique. We define the term temporal distance of two vertices, which isthe arrival time, i.e. the time-label of the last edge, of the temporal paththat connects those vertices, which has the smallest arrival time amongst alltemporal paths that connect those vertices. We then propose and study twostatistical properties of temporal graphs. One is the maximum expected temporaldistance which is, as the term indicates, the maximum of all expected temporaldistances in the graph. The other one is the temporal diameter which, looselyspeaking, is the expectation of the maximum temporal distance in the graph. Wederive the maximum expected temporal distance of a uniform random temporal stargraph as well as an upper bound on both the maximum expected temporal distanceand the temporal diameter of the normalized version of the uniform randomtemporal clique, in which the largest time-label available equals the number ofvertices. Finally, we provide an algorithm that solves an optimization problemon a specific type of temporal (multi)graphs of two vertices.
Abstract: Random scaled sector graphs were introduced as a generalization of random geometric graphs to model networks of sensors using optical communication. In the random scaled sector graph model vertices are placed uniformly at random into the [0, 1]2 unit square. Each vertex i is assigned uniformly at random sector Si, of central angle {\'a}i, in a circle of radius ri (with vertex i as the origin). An arc is present from vertex i to any vertex j, if j falls in Si. In this work, we study the value of the chromatic number {\^O}(Gn), directed cliquenumber {\`u}(Gn), and undirected cliquenumber {\`u}2 (Gn) for random scaled sector graphs with n vertices, where each vertex spans a sector of {\'a} degrees with radius rn = \~{a}ln n/n. We prove that for values {\'a} < {\^I}, as n w.h.p., {\^O}(Gn) and {\`u}2 (Gn) are {\`E}(ln n/ln ln n), while {\`u}(Gn) is O(1), showing a clear difference with the random geometric graph model. For {\'a} > {\^I} w.h.p., {\^O}(Gn) and {\`u}2 (Gn) are {\`E} (ln n), being the same for random scaled sector and random geometric graphs, while {\`u}(Gn) is {\`E}(ln n/ln ln n).
Abstract: Consider k particles, 1 red and k-1 white, chasing each other on the nodes of a graph G. If the red one catches one of the white, it infects it with its color. The newly red particles are now available to infect more white ones. When is it the case that all white will become red? It turns out that this simple question is an instance of information propagation between random walks and has important applications to mobile computing where a set of mobile hosts acts as an intermediary for the spread of information.
In this paper we model this problem by k concurrent random walks, one corresponding to the red particle and k-1 to the white ones. The infection time Tk of infecting all the white particles with red color is then a random variable that depends on k, the initial position of the particles, the number of nodes and edges of the graph, as well as on the structure of the graph.
In this work we develop a set of probabilistic tools that we use to obtain upper bounds on the (worst case w.r.t. initial positions of particles) expected value of Tk for general graphs and important special cases. We easily get that an upper bound on the expected value of Tk is the worst case (over all initial positions) expected meeting time m* of two random walks multiplied by . We demonstrate that this is, indeed, a tight bound; i.e. there is a graph G (a special case of the lollipop graph), a range of values kclique or has nice expansion properties, we prove much smaller bounds for Tk. We have evaluated and validated all our results by large scale experiments which we also present and discuss here. In particular, the experiments demonstrate that our analytical results for these expander graphs are tight.
