Abstract: We present a parallel priority queue that supports the following operations in
constant time: parallel insertion of a sequence of elements ordered according
to key, parallel decrease key for a sequence of elements ordered according to
key, deletion of the minimum key element, and deletion of an arbitrary element.
Our data structure is the first to support multi-insertion and multi-decrease key in
constant time. The priority queue can be implemented on the EREW PRAM and
can perform any sequence of n operations in O(n) time and O(m log n) work, m
being the total number of keyes inserted and/or updated. A main application is a
parallel implementation of Dijkstra¢s algorithm for the single-source shortest path
problem, which runs in O(n) time and O(m log n) work on a CREW PRAM on
graphs with n vertices and m edges. This is a logarithmic factor improvement in
the running time compared with previous approaches.
Abstract: We present a simple parallel algorithm for the single-source shortest path
problem in planar digraphs with nonnegative real edge weights. The algorithm runs
on the EREW PRAM model of parallel computation in O((n2=+n1&=) log n)
time, performing O(n1+= log n) work for any 0<{\aa}<1/2. The strength of the
algorithm is its simplicity, making it easy to implement and presumable quite
efficient in practice. The algorithm improves upon the work of all previous
parallel algorithms. Our algorithm is based on a region decomposition of the
input graph and uses a well-known parallel implementation of Dijkstra's
algorithm. The logarithmic factor in both the work and the time can be
eliminated by plugging in a less practical, sequential planar shortest path
algorithm together with an improved parallel implementation of Dijkstra's
algorithm.
Abstract: The “small world” phenomenon, i.e., the fact that the
global social network is strongly connected in the sense
that every two persons are inter-related through a small
chain of friends, has attracted research attention and has
been strongly related to the results of the social
psychologist¢s Stanley Milgram experiments; properties
of social networks and relevant problems also emerge in
peer-to-peer systems and their study can shed light on
important modern network design properties.
In this paper, we have experimentally studied greedy
routing algorithms, i.e., algorithms that route information
using “long-range” connections that function as
shortcuts connecting “distant” network nodes. In
particular, we have implemented greedy routing
algorithms, and techniques from the recent literature in
networks of line and grid topology using parallelization
for increasing efficiency. To the best of our knowledge, no
similar attempt has been made so far
Abstract: We consider selfish routing over a network consisting of m parallellinks through which $n$ selfish users route their traffic trying tominimize their own expected latency. We study the class of mixedstrategies in which the expected latency through each link is at mosta constant multiple of the optimum maximum latency had globalregulation been available. For the case of uniform links it is knownthat all Nash equilibria belong to this class of strategies. We areinterested in bounding the coordination ratio (or price of anarchy) ofthese strategies defined as the worst-case ratio of the maximum (overall links) expected latency over the optimum maximum latency. The loadbalancing aspect of the problem immediately implies a lower boundO(ln m ln ln m) of the coordinationratio. We give a tight (up to a multiplicative constant) upper bound.To show the upper bound, we analyze a variant of the classical ballsand bins problem, in which balls with arbitrary weights are placedinto bins according to arbitrary probability distributions. At theheart of our approach is a new probabilistic tool that we call ballfusion; this tool is used to reduce the variant of the problem whereballs bear weights to the classical version (with no weights). Ballfusion applies to more general settings such as links with arbitrarycapacities and other latency functions.
Abstract: We examine the problem of assigning n independent jobs to m unrelated parallel machines, so that each job is processed without interruption on one of the machines, and at any time, every machine processes at most one job. We focus on the case where m is a fixed constant, and present a new rounding approach that yields approximation schemes for multi-objective minimum makespan scheduling with a fixed number of linear cost constraints. The same approach gives approximation schemes for covering problems like maximizing the minimum load on any machine, and for assigning specific or equal loads to the machines.
Abstract: We consider algorithmic questions concerning the existence,
tractability and quality of atomic congestion games, among users that
are considered to participate in (static) selfish coalitions. We carefully
define a coalitional congestion model among atomic players.
Our findings in this model are quite interesting, in the sense that we
demonstrate many similarities with the non–cooperative case. For example,
there exist potentials proving the existence of Pure Nash Equilibria
(PNE) in the (even unrelated) parallel links setting; the Finite Improvement
Property collapses as soon as we depart from linear delays, but
there is an exact potential (and thus PNE) for the case of linear delays,
in the network setting; the Price of Anarchy on identical parallel
links demonstrates a quite surprising threshold behavior: it persists on
being asymptotically equal to that in the case of the non–cooperative
KP–model, unless we enforce a sublogarithmic number of coalitions.
We also show crucial differences, mainly concerning the hardness of algorithmic
problems that are solved efficiently in the non–cooperative case.
Although we demonstrate convergence to robust PNE, we also prove the
hardness of computing them. On the other hand, we can easily construct
a generalized fully mixed Nash Equilibrium. Finally, we propose a new
improvement policy that converges to PNE that are robust against (even
dynamically forming) coalitions of small size, in pseudo–polynomial time.
Keywords. Game Theory, Atomic Congestion Games, Coalitions, Convergence
to Equilibria, Price of Anarchy.
Abstract: We consider algorithmic questions concerning the existence, tractability and quality of Nash equi-
libria, in atomic congestion games among users participating in sel sh coalitions.
We introduce a coalitional congestion model among atomic players and demonstrate many in-
teresting similarities with the non-cooperative case. For example, there exists a potential function
proving the existence of Pure Nash Equilibria (PNE) in the unrelated parallel links setting; in
the network setting, the Finite Improvement Property collapses as soon as we depart from linear
delays, but there is an exact potential (and thus PNE) for linear delays; the Price of Anarchy on
identical parallel links demonstrates a quite surprising threshold behavior: it persists on being
asymptotically equal to that in the case of the non-cooperative KP-model, unless the number of
coalitions is sublogarithmic.
We also show crucial di erences, mainly concerning the hardness of algorithmic problems that
are solved eciently in the non{cooperative case. Although we demonstrate convergence to robust
PNE, we also prove the hardness of computing them. On the other hand, we propose a generalized
fully mixed Nash Equilibrium, that can be eciently constructed in most cases. Finally, we
propose a natural improvement policy and prove its convergence in pseudo{polynomial time to
PNE which are robust against (even dynamically forming) coalitions of small size.
Abstract: We consider algorithmic questions concerning the existence,
tractability and quality of atomic congestion games, among users that
are considered to participate in (static) selfish coalitions. We carefully
define a coalitional congestion model among atomic players.
Our findings in this model are quite interesting, in the sense that we
demonstrate many similarities with the non–cooperative case. For example,
there exist potentials proving the existence of Pure Nash Equilibria
(PNE) in the (even unrelated) parallel links setting; the Finite Improvement
Property collapses as soon as we depart from linear delays, but
there is an exact potential (and thus PNE) for the case of linear delays,
in the network setting; the Price of Anarchy on identical parallel
links demonstrates a quite surprising threshold behavior: it persists on
being asymptotically equal to that in the case of the non–cooperative
KP–model, unless we enforce a sublogarithmic number of coalitions.
We also show crucial differences, mainly concerning the hardness of algorithmic
problems that are solved efficiently in the non–cooperative case.
Although we demonstrate convergence to robust PNE, we also prove the
hardness of computing them. On the other hand, we can easily construct
a generalized fully mixed Nash Equilibrium. Finally, we propose a new
improvement policy that converges to PNE that are robust against (even
dynamically forming) coalitions of small size, in pseudo–polynomial time.
Keywords. Game Theory, Atomic Congestion Games, Coalitions, Convergence
to Equilibria, Price of Anarchy.
Abstract: We investigate the existence of optimal tolls for atomic symmetric
network congestion games with unsplittable traffic and arbitrary non-negative and
non-decreasing latency functions.We focus on pure Nash equilibria and a natural
toll mechanism, which we call cost-balancing tolls. A set of cost-balancing tolls
turns every path with positive traffic on its edges into a minimum cost path. Hence
any given configuration is induced as a pure Nash equilibrium of the modified
game with the corresponding cost-balancing tolls. We show how to compute in
linear time a set of cost-balancing tolls for the optimal solution such that the total
amount of tolls paid by any player in any pure Nash equilibrium of the modified
game does not exceed the latency on the maximum latency path in the optimal
solution. Our main result is that for congestion games on series-parallel networks
with increasing latencies, the optimal solution is induced as the unique pure Nash
equilibrium of the game with the corresponding cost-balancing tolls. To the best
of our knowledge, only linear congestion games on parallel links were known to
admit optimal tolls prior to this work. To demonstrate the difficulty of computing
a better set of optimal tolls, we show that even for 2-player linear congestion
games on series-parallel networks, it is NP-hard to decide whether the optimal
solution is the unique pure Nash equilibrium or there is another equilibrium of
total cost at least 6/5 times the optimal cost.
Abstract: When one engineers distributed algorithms, some special characteristics
arise that are different from conventional (sequential or parallel)
computing paradigms. These characteristics include: the need for either a
scalable real network environment or a platform supporting a simulated
distributed environment; the need to incorporate asynchrony, where arbitrarya
synchrony is hard, if not impossible, to implement; and the generation
of “difficult” input instances which is a particular challenge. In this
work, we identifys ome of the methodological issues required to address
the above characteristics in distributed algorithm engineering and illustrate
certain approaches to tackle them via case studies. Our discussion
begins byad dressing the need of a simulation environment and how asynchronyis
incorporated when experimenting with distributed algorithms.
We then proceed bys uggesting two methods for generating difficult input
instances for distributed experiments, namelya game-theoretic one and another
based on simulations of adversarial arguments or lower bound proofs.
We give examples of the experimental analysis of a pursuit-evasion protocol
and of a shared memorypro blem in order to demonstrate these ideas.
We then address a particularlyi nteresting case of conducting experiments
with algorithms for mobile computing and tackle the important issue of
motion of processes in this context. We discuss the two-tier principle as
well as a concurrent random walks approach on an explicit representation
of motions in ad-hoc mobile networks, which allow at least for averagecase
analysis and measurements and may give worst-case inputs in some
cases. Finally, we discuss a useful interplay between theory and practice
that arise in modeling attack propagation in networks.
Abstract: We exploit the game-theoretic ideas presented in [12] to
study the vertex coloring problem in a distributed setting. The vertices
of the graph are seen as players in a suitably defined strategic game,
where each player has to choose some color, and the payoff of a vertex is
the total number of players that have chosen the same color as its own.
We extend here the results of [12] by showing that, if any subset of nonneighboring
vertices perform a selfish step (i.e., change their colors in order
to increase their payoffs) in parallel, then a (Nash equilibrium) proper
coloring, using a number of colors within several known upper bounds
on the chromatic number, can still be reached in polynomial time. We
also present an implementation of the distributed algorithm in wireless
networks of tiny devices and evaluate the performance in simulated and
experimental environments. The performance analysis indicates that it
is the first practically implementable distributed algorithm.
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: The problem of finding an implicit representation for a graph such that vertex adjacency can be tested quickly is fundamental to all graph algorithms. In particular, it is possible to represent sparse graphs on n vertices using O(n) space such that vertex adjacency is tested in O(l) time. We show here how to construct such a representation efficiently by providing simple and optimal algorithms, both in a sequential and a parallel setting. Our sequential algorithm runs in O(n) time. The parallel algorithm runs in O(log n) time using O(n/log n) CRCW PRAM processors, or inO(log n log* n) time using O(n/log n log* n) EREW PRAM processors. Previous results for this problem are based on matroid partitioning and thus have a high complexity.
Abstract: We consider the problem of computing minimum congestion,
fault-tolerant, redundant assignments of messages to faulty parallel de-
livery channels. In particular, we are given a set M of faulty channels,
each having an integer capacity ci and failing independently with proba-
bility fi. We are also given a set of messages to be delivered over M, and
a fault-tolerance constraint (1), and we seek a redundant assignment
that minimizes congestion Cong( ), i.e. the maximum channel load,
subject to the constraint that, with probability no less than (1 ), all
the messages have a copy on at least one active channel. We present a
4-approximation algorithm for identical capacity channels and arbitrary
message sizes, and a 2l ln(jMj=)
ln(1=fmax)m-approximation algorithm for related
capacity channels and unit size messages.
Both algorithms are based on computing a collection of disjoint chan-
nel subsets such that, with probability no less than (1 ), at least one
channel is active in each subset. The objective is to maximize the sum of
the minimum subset capacities. Since the exact version of this problem
is NP-complete, we present a 2-approximation algorithm for identical
capacities, and a (8 + o(1))-approximation algorithm for arbitrary ca-
pacities.
Abstract: Urban ecosystems are becoming one of the most potentially attractive scenarios for innovating new services and technologies. In parallel, city managers, urban utilities and other stakeholders are fostering the intensive use of advanced technologies aiming at improving present city performance and its sustainability. The deployment of such technology entails the generation of massive amounts of information which in many cases might become useful for other services and applications. Hence, aiming at taking advantage of such massive amounts of information and deployed technology as well as breaking the potential digital barrier that can be raised, some easy-to-use tools have to be made available to the urban stakeholders. These tools integrated in a platform, operated directly or indirectly by the city, provides a singular opportunity for exploiting the concept of connected city whilst fostering innovation in all city dimensions and making the co-creation concept a reality and eventually impacting on government policies.
Abstract: In this work we study the problem of scheduling tasks with dependencies in multiprocessor architectures where processors have different speeds.
We present the preemptive algorithm "Save-Energy" that given a schedule of tasks it post processes it to improve the energy efficiency without any deterioration of the makespan. In terms of time efficiency, we show that preemptive scheduling in an asymmetric system can achieve the same or better optimal makespan than in a symmetric system. Motivited by real multiprocessor systems, we investigate architectures that exhibit limited asymmetry: there are two essentially different speeds. Interestingly, this special case has not been studied in the field of parallel computing and scheduling theory; only the general case was studied where processors have K essentially different speeds. We present the non-preemptive algorithm "Remnants'' that achieves almost optimal makespan. We provide a refined analysis of a recent scheduling method. Based on this analysis, we specialize the scheduling policy and provide an algorithm of (3 + o(1)) expected approximation factor. Note that this improves the previous best factor (6 for two speeds). We believe that our work will convince researchers to revisit this well studied scheduling problem for these simple, yet realistic, asymmetric multiprocessor architectures.
Abstract: We study the combinatorial structure and computational complexity of extreme Nash equilibria, ones that maximize or minimize a certain objective function, in the context of a selfish routing game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user routes its traffic on links that minimize its expected latency cost.
Our structural results provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria and that under a certain condition, the social cost of any Nash equilibrium is within a factor of 6 + epsi, of that of the fully mixed Nash equilibrium, assuming that link capacities are identical.
Our complexity results include hardness, approximability and inapproximability ones. Here we show, that for identical link capacities and under a certain condition, there is a randomized, polynomial-time algorithm to approximate the worst social cost within a factor arbitrarily close to 6 + epsi. Furthermore, we prove that for any arbitrary integer k > 0, it is -hard to decide whether or not any given allocation of users to links can be transformed into a pure Nash equilibrium using at most k selfish steps. Assuming identical link capacities, we give a polynomial-time approximation scheme (PTAS) to approximate the best social cost over all pure Nash equilibria. Finally we prove, that it is -hard to approximate the worst social cost within a multiplicative factor . The quantity is the tight upper bound on the ratio of the worst social cost and the optimal cost in the model of identical capacities.
Abstract: The simplex method has been successfully used in solving linear programming problems for many years. Parallel approaches have also extensively been studied due to the intensive computations required, especially for the solution of large linear problems (LPs). In this paper we present a highly scalable parallel implementation framework of the standard full tableau simplex method on a highly parallel (distributed memory) environment. Specifically, we have designed and implemented a suitable column distribution scheme as well as a row distribution scheme and we have entirely tested our implementations over a considerably powerful distributed platform (linux cluster with myrinet interface). We then compare our approaches (a) among each other for variable number of problem size (number of rows and columns) and (b) to other recent and valuable corresponding efforts in the literature. In most cases, the column distribution scheme performs quite/much better than the row distribution scheme. Moreover, both schemes (even the row distribution scheme over large-scale problems) lead to particularly high speedup and efficiency values, which are considerably better in all cases than the ones achieved in other similar research efforts and implementations. Moreover, we further evaluate our basic parallelization scheme over very large LPs in order to validate more reliably the high efficiency and scalability achieved.
Abstract: We consider the problem of computing minimum congestion, fault-tolerant, redundant assignments of messages to faulty, parallel delivery channels. In particular, we are given a set K of faulty channels, each having an integer capacity ci and failing independently with probability fi. We are also given a set M of messages to be delivered over K, and a fault-tolerance constraint (1 - {\aa}), and we seek a redundant assignment {\"o} that minimizes congestion Cong({\"o}), i.e. the maximum channel load, subject to the constraint that, with probability no less than (1 - e), all the messages have a copy on at least one active channel. We present a polynomial-time 4-approximation algorithm for identical capacity channels and arbitrary message sizes, and a 2[ln(|K|/{\aa})/ln(1/fmax)]-approximation algorithm for related capacity channels and unit size messages. Both algorithms are based on computing a collection (K1,., K{\'i}} of disjoint channel subsets such that, with probability no less than (1 - {\aa}), at least one channel is active in each subset. The objective is to maximize the sum of the minimum subset capacities. Since the exact version of this problem is NP-complete, we provide a 2-approximation algorithm for identical capacities, and a polynomial-time (8+o(1))-approximation algorithm for arbitrary capacities.
Abstract: We study network load games, a class of routing games in
networks which generalize sel{\^A}¯sh routing games on networks consisting
of parallel links. In these games, each user aims to route some tra{\^A}±c from
a source to a destination so that the maximum load she experiences in the
links of the network she occupies is minimum given the routing decisions
of other users. We present results related to the existence, complexity,
and price of anarchy of Pure Nash Equilibria for several network load
games. As corollaries, we present interesting new statements related to
the complexity of computing equilibria for sel{\^A}¯sh routing games in net-
works of restricted parallel links.
Abstract: A constraint network is arc consistent if any value of any of its variables is compatible with at
least one value of any other variable. The Arc Consistency Problem (ACP) consists in filtering out values of
the variables of a given network to obtain one that is arc consistent, without eliminating any solution. ACP is
known to be inherently sequential, or P-complete, so in this paper we examine some weaker versions of it and
their parallel complexity. We propose several natural approximation schemes for ACP and show that they are also
P-complete. In an attempt to overcome these negative results, we turn our attention to the problem of filtering
out values from the variables so that each value in the resulting network is compatible with at least one value of
not necessarily all, but a constant fraction of the other variables. We call such a network partially arc consistent.
We give a parallel algorithm that, for any constraint network, outputs a partially arc consistent subnetwork of it in
sublinear (O.pn log n/) parallel time using O.n2/ processors. This is the first (to our knowledge) sublinear-time
parallel algorithm with polynomially many processors that guarantees that in the resulting network every value is
compatible with at least one value in at least a constant fraction of the remaining variables. Finally, we generalize
the notion of partiality to the k-consistency problem.
Abstract: In routing games, the network performance at equilibrium can be significantly improved if we remove some edges from the network. This counterintuitive fact, widely known as Braess's paradox, gives rise to the (selfish) network design problem, where we seek to recognize routing games suffering from the paradox, and to improve the equilibrium performance by edge removal. In this work, we investigate the computational complexity and the approximability of the network design problem for non-atomic bottleneck routing games, where the individual cost of each player is the bottleneck cost of her path, and the social cost is the bottleneck cost of the network. We first show that bottleneck routing games do not suffer from Braess's paradox either if the network is series-parallel, or if we consider only subpath-optimal Nash flows. On the negative side, we prove that even for games with strictly increasing linear latencies, it is NP-hard not only to recognize instances suffering from the paradox, but also to distinguish between instances for which the Price of Anarchy (PoA) can decrease to 1 and instances for which the PoA is as large as \Omega(n^{0.121}) and cannot improve by edge removal. Thus, the network design problem for such games is NP-hard to approximate within a factor of O(n^{0.121-\eps}), for any constant \eps > 0. On the positive side, we show how to compute an almost optimal subnetwork w.r.t. the bottleneck cost of its worst Nash flow, when the worst Nash flow in the best subnetwork routes a non-negligible amount of flow on all used edges. The running time is determined by the total number of paths, and is quasipolynomial when the number of paths is quasipolynomial.
Abstract: We investigate the practical merits of a parallel priority queue
through its use in the development of a fast and work-efficient parallel
shortest path algorithm, originally designed for an EREW PRAM. Our
study reveals that an efficient implementation on a real supercomputer
requires considerable effort to reduce the communication performance
(which in theory is assumed to take constant time). It turns out that the
most crucial part of the implementation is the mapping of the logical
processors to the physical processing nodes of the supercomputer. We
achieve the requested efficient mapping through a new graph-theoretic
result of independent interest: computing a Hamiltonian cycle on a directed
hyper-torus. No such algorithm was known before for the case of
directed hypertori. Our Hamiltonian cycle algorithm allows us to considerably
improve the communication cost and thus the overall performance
of our implementation.
Abstract: Switching in core optical networks is currently being
performed using high-speed electronic or all-optical
circuit switches. Switching with high-speed electronics
requires optical-to-electronic (O/E) conversion of the
data stream, making the switch a potential bottleneck
of the network: any effort (including parallelization) for
electronics to approach the optical speeds seems to be
already reaching its practical limits. Furthermore, the
store-and-forward approach of packet-switching does
not seem suitable for all-optical implementation due to
the lack of practical optical random-access-memories
to buffer and resolve contentions. Circuit switching on
the other hand, involves a pre-transmission delay for
call setup and requires the aggregation of microlows
into circuits, sacriicing the granularity and the control
over individual lows, and is ineficient for bursty traf-
ic. Optical burst switching (OBS) has been proposed
by Qiao and Yoo (1999) to combine the advantages of
both packet and circuit switching and is considered a
promising technology for the next generation optical
internet.
Abstract: The objective of this research is to propose two new optical procedures for packet routing and forwarding in the framework of transparent optical networks. The single-wavelength label-recognition and packet-forwarding unit, which represents the central physical constituent of the switching node, is fully described in both cases. The first architecture is a hybrid opto-electronic structure relying on an optical serial-to-parallel converter designed to slow down the label processing. The remaining switching operations are done electronically. The routing system remains transparent for the packet payloads. The second architecture is an all-optical architecture and is based on the implementation of all-optical decoding of the parallelized label. The packet-forwarding operations are done optically. The major subsystems required in both of the proposed architectures are described on the basis of nonlinear effects in semiconductor optical amplifiers. The experimental results are compatible with the integration of the whole architecture. Those subsystems are a 4-bit time-to-wavelength converter, a pulse extraction circuit, a an optical wavelength generator, a 3 x 8 all-optical decoder and a packet envelope detector.
Abstract: Two simple and work-efficient parallel algorithms for the minimum spanning tree problem are presented. Both algorithms perform O( m log n ) work. The first algorithm
runs in O( log^2 n ) time on an EREW PRAM while the second algorithm runs in O( log n ) time on a Common CRCW PRAM.
Abstract: We consider in this paper the problem of scheduling a set of independent
parallel tasks (jobs) with respect to two criteria, namely,
the makespan (time of the last finishing job) and the minsum (average
completion time). There exist several algorithms with a good
performance guaranty for one of these criteria. We are interested
here in studying the optimization of both criteria simultaneously.
The numerical values are given for the moldable task model, where
the execution time of a task depends on the number of processors
alloted to it. The main result of this paper is to derive explicitly
a family of algorithms guaranteed for both the minsum and the
makespan. The performance guaranty of these algorithms is better
than the best algorithms known so far. The Guaranty curve
of the family is the set of all points (x; y) such that there is an
algorithm with guarantees x on makespan and y on the minsum.
When the ratio on the minsum increases, the curve tends to the
best ratio known for the makespan for moldable tasks (3=2). One
extremal point of the curves is a (3;6)-approximation algorithm.
Finally a randomized version is given, which improves this results
to (3;4.08).
Abstract: We consider applications of probabilistic techniques in the
framework of algorithmic game theory. We focus on three distinct case
studies: (i) The exploitation of the probabilistic method to demonstrate
the existence of approximate Nash equilibria of logarithmic support sizes
in bimatrix games; (ii) the analysis of the statistical conflict that mixed
strategies cause in network congestion games; (iii) the effect of coalitions
in the quality of congestion games on parallel links.
Abstract: In this paper we present an efficient general simulation strategy for
computations designed for fully operational BSP machines of n ideal processors,
on n-processor dynamic-fault-prone BSP machines. The fault occurrences are failstop
and fully dynamic, i.e., they are allowed to happen on-line at any point of the
computation, subject to the constraint that the total number of faulty processors
may never exceed a known fraction. The computational paradigm can be exploited
for robust computations over virtual parallel settings with a volatile underlying
infrastructure, such as a NETWORK OF WORKSTATIONS (where workstations may be
taken out of the virtual parallel machine by their owner).
Our simulation strategy is Las Vegas (i.e., it may never fail, due to backtracking
operations to robustly stored instances of the computation, in case of locally
unrecoverable situations). It adopts an adaptive balancing scheme of the workload
among the currently live processors of the BSP machine.
Our strategy is efficient in the sense that, compared with an optimal off-line
adversarial computation under the same sequence of fault occurrences, it achieves an O
¡
.log n ¢ log log n/2¢
multiplicative factor times the optimal work (namely, this
measure is in the sense of the “competitive ratio” of on-line analysis). In addition,
our scheme is modular, integrated, and considers many implementation points.
We comment that, to our knowledge, no previous work on robust parallel computations
has considered fully dynamic faults in the BSP model, or in general distributed
memory systems. Furthermore, this is the first time an efficient Las Vegas
simulation in this area is achieved.
Abstract: We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give parallel algorithms for the EREW PRAM model of computation that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O({\'a}(n)) time using a single processor, after a preprocessing of O(log2n) time and O(n) work, where {\'a}(n) is the inverse of Ackermann's function. The class of constant treewidth graphs contains outerplanar graphs and series-parallel graphs, among others. To the best of our knowledge, these are the first parallel algorithms which achieve these bounds for any class of graphs except trees. We also give a dynamic algorithm which, after a change in an edge weight, updates our data structures in O(log n) time using O(n{\^a}) work, for any constant 0 < {\^a} < 1. Moreover, we give an algorithm of independent interest: computing a shortest path tree, or finding a negative cycle in O(log2n) time using O(n) work.
Abstract: We study extreme Nash equilibria in the context of a selfish routing game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel identical links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.
We provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria. Furthermore, we show, that under a certain condition, the social cost of any Nash equilibrium is within a factor of 2h(1+ɛ) of that of the fully mixed Nash equilibrium, where h is the factor by which the largest user traffic deviates from the average user traffic.
Considering pure Nash equilibria, we provide a PTAS to approximate the best social cost, we give an upper bound on the worst social cost and we show that it is View the MathML source-hard to approximate the worst social cost within a multiplicative factor better than 2-2/(m+1).
Abstract: We study extreme Nash equilibria in the context of a selfish routing game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel identical links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.We provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria. Furthermore, we show, that under a certain condition, the social cost of any Nash equilibrium is within a factor of 2h(1 + {\aa}) of that of the fully mixed Nash equilibrium, where h is the factor by which the largest user traffic deviates from the average user traffic.Considering pure Nash equilibria, we provide a PTAS to approximate the best social cost, we give an upper bound on the worst social cost and we show that it is N P-hard to approximate the worst social cost within a multiplicative factor better than 2 - 2/(m + 1).
Abstract: We study computational and coordination efficiency issues of
Nash equilibria in symmetric network congestion games. We first propose
a simple and natural greedy method that computes a pure Nash equilibrium
with respect to traffic congestion in a network. In this algorithm
each user plays only once and allocates her traffic to a path selected via
a shortest path computation. We then show that this algorithm works
for series-parallel networks when users are identical or when users are of
varying demands but have the same best response strategy for any initial
network traffic. We also give constructions where the algorithm fails if
either the above condition is violated (even for series-parallel networks)
or the network is not series-parallel (even for identical users). Thus, we
essentially indicate the limits of the applicability of this greedy approach.
We also study the price of anarchy for the objective of maximum
latency. We prove that for any network of m uniformly related links and
for identical users, the price of anarchy is {\`E}( logm
log logm).
Abstract: Let M be a single s-t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nash equilibrium with unbounded Coordination Ratio [E. Koutsoupias, C. Papadimitriou, Worst-case equilibria, in: 16th Annual Symposium on Theoretical Aspects of Computer Science, STACS, vol. 1563, 1999, pp. 404-413; T. Roughgarden, E. Tardos, How bad is selfish routing? in: 41st IEEE Annual Symposium of Foundations of Computer Science, FOCS, 2000, pp. 93-102]. A Leader can decrease the coordination ratio by assigning flow {\'a}r on M, and then all Followers assign selfishly the (1-{\'a})r remaining flow. This is a Stackelberg Scheduling Instance(M,r,{\'a}),0≤{\'a}≤1. It was shown [T. Roughgarden, Stackelberg scheduling strategies, in: 33rd Annual Symposium on Theory of Computing, STOC, 2001, pp. 104-113] that it is weakly NP-hard to compute the optimal Leader's strategy. For any such network M we efficiently compute the minimum portion @b"M of flow r>0 needed by a Leader to induce M's optimum cost, as well as her optimal strategy. This shows that the optimal Leader's strategy on instances (M,r,@a>=@b"M) is in P. Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden presented a modification of Braess's Paradox graph, such that no strategy controlling {\'a}r flow can induce ≤1/{\'a} times the optimum cost. However, we show that our main result also applies to any s-t net G. We take care of the Braess's graph explicitly, as a convincing example. Finally, we extend this result to k commodities. A conference version of this paper has appeared in [A. Kaporis, P. Spirakis, The price of optimum in stackelberg games on arbitrary single commodity networks and latency functions, in: 18th annual ACM symposium on Parallelism in Algorithms and Architectures, SPAA, 2006, pp. 19-28]. Some preliminary results have also appeared as technical report in [A.C. Kaporis, E. Politopoulou, P.G. Spirakis, The price of optimum in stackelberg games, in: Electronic Colloquium on Computational Complexity, ECCC, (056), 2005].
Abstract: We study the problem of routing traffic through a congested network. We focus on the simplest case of a network consisting of m parallel links. We assume a collection of n network users; each user employs a mixed strategy, which is a probability distribution over links, to control the shipping of its own assigned traffic. Given a capacity for each link specifying the rate at which the link processes traffic, the objective is to route traffic so that the maximum (over all links) latency is minimized. We consider both uniform and arbitrary link capacities. How much decrease in global performace is necessary due to the absence of some central authority to regulate network traffic and implement an optimal assignment of traffic to links? We investigate this fundamental question in the context of Nash equilibria for such a system, where each network user selfishly routes its traffic only on those links available to it that minimize its expected latency cost, given the network congestion caused by the other users. We use the Coordination Ratio, originally defined by Koutsoupias and Papadimitriou, as a measure of the cost of lack of coordination among the users; roughly speaking, the Coordination Ratio is the ratio of the expectation of the maximum (over all links) latency in the worst possible Nash equilibrium, over the least possible maximum latency had global regulation been available. Our chief instrument is a set of combinatorial Minimum Expected Latency Cost Equations, one per user, that characterize the Nash equilibria of this system. These are linear equations in the minimum expected latency costs, involving the user traffics, the link capacities, and the routing pattern determined by the mixed strategies. In turn, we solve these equations in the case of fully mixed strategies, where each user assigns its traffic with a strictly positive probability to every link, to derive the first existence and uniqueness results for fully mixed Nash equilibria in this setting. Through a thorough analysis and characterization of fully mixed Nash equilibria, we obtain tight upper bounds of no worse than O(ln n/ln ln n) on the Coordination Ratio for (i) the case of uniform capacities and arbitrary traffics and (ii) the case of arbitrary capacities and identical traffics.
Abstract: In this work, we study the combinatorial structure and the
computational complexity of Nash equilibria for a certain game that
models selfish routing over a network consisting of m parallel links. We
assume a collection of n users, each employing a mixed strategy, which
is a probability distribution over links, to control the routing of its own
assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic
on those links that minimize its expected latency cost, given the network
congestion caused by the other users. The social cost of a Nash equilibrium
is the expectation, over all random choices of the users, of the
maximum, over all links, latency through a link.
We embark on a systematic study of several algorithmic problems related
to the computation of Nash equilibria for the selfish routing game we consider.
In a nutshell, these problems relate to deciding the existence of a
Nash equilibrium, constructing a Nash equilibrium with given support
characteristics, constructing the worst Nash equilibrium (the one with
maximum social cost), constructing the best Nash equilibrium (the one
with minimum social cost), or computing the social cost of a (given) Nash
equilibrium. Our work provides a comprehensive collection of efficient algorithms,
hardness results (both as NP-hardness and #P-completeness
results), and structural results for these algorithmic problems. Our results
span and contrast a wide range of assumptions on the syntax of the
Nash equilibria and on the parameters of the system.
Abstract: We study computationally hard combinatorial problems arising from the important engineering question of how to maximize the number of connections that can be simultaneously served in a WDM optical network. In such networks, WDM technology can satisfy a set of connections by computing a route and assigning a wavelength to each connection so that no two connections routed through the same fiber are assigned the same wavelength. Each fiber supports a limited number of w wavelengths and in order to fully exploit the parallelism provided by the technology, one should select a set connections of maximum cardinality which can be satisfied using the available wavelengths. This is known as the maximum routing and path coloring problem (maxRPC).
Our main contribution is a general analysis method for a class of iterative algorithms for a more general coloring problem. A lower bound on the benefit of such an algorithm in terms of the optimal benefit and the number of available wavelengths is given by a benefit-revealing linear program. We apply this method to maxRPC in both undirected and bidirected rings to obtain bounds on the approximability of several algorithms. Our results also apply to the problem maxPC where paths instead of connections are given as part of the input. We also study the profit version of maxPC in rings where each path has a profit and the objective is to satisfy a set of paths of maximum total profit.
Abstract: A Nash equilibrium of a routing network represents a stable state of the network where no user finds it beneficial to unilaterally deviate from its routing strategy. In this work, we investigate the structure of such equilibria within the context of a certain game that models selfish routing for a set of n users each shipping its traffic over a network consisting of m parallel links. In particular, we are interested in identifying the worst-case Nash equilibrium – the one that maximizes social cost. Worst-case Nash equilibria were first introduced and studied in the pioneering work of Koutsoupias and Papadimitriou [9].
More specifically, we continue the study of the Conjecture of the Fully Mixed Nash Equilibrium, henceforth abbreviated as FMNE Conjecture, which asserts that the fully mixed Nash equilibrium, when existing, is the worst-case Nash equilibrium. (In the fully mixed Nash equilibrium, the mixed strategy of each user assigns (strictly) positive probability to every link.) We report substantial progress towards identifying the validity, methodologies to establish, and limitations of, the FMNE Conjecture.
