Abstract: In this work we study the important problem of colouring squares of planar graphs (SQPG). We design and implement two new algorithms that colour in a different way SQPG. We call these algorithms MDsatur and RC. We have also implemented and experimentally evaluated the performance of most of the known approximation colouring algorithms for SQPG [14, 6, 4, 10]. We compare the quality of the colourings achieved by these algorithms, with the colourings obtained by our algorithms and with the results obtained from two well-known greedy colouring heuristics. The heuristics are mainly used for comparison reasons and unexpectedly give very good results. Our algorithm MDsatur outperforms the known algorithms as shown by the extensive experiments we have carried out.
The planar graph instances whose squares are used in our experiments are “non-extremal” graphs obtained by LEDA and hard colourable graph instances that we construct.
The most interesting conclusions of our experimental study are:
1) all colouring algorithms considered here have almost optimal performance on the squares of “non-extremal” planar graphs. 2) all known colouring algorithms especially designed for colouring SQPG, give significantly better results, even on hard to colour graphs, when the vertices of the input graph are randomly named. On the other hand, the performance of our algorithm, MDsatur, becomes worse in this case, however it still has the best performance compared to the others. MDsatur colours the tested graphs with 1.1 OPT colours in most of the cases, even on hard instances, where OPT denotes the number of colours in an optimal colouring. 3) we construct worst case instances for the algorithm of Fotakis el al. [6], which show that its theoretical analysis is tight.

Abstract: An ad hoc mobile network is a collection of mobile hosts, with wireless communication capabilities, forming a temporary network without the aid of any established fixed infrastructure. In such networks, topological connectivity is subject to frequent, unpredictable change. Our work focuses on networks with high rate of such changes to connectivity. For such dynamically changing networks we propose protocols which exploit the co-ordinated (by the protocol) motion of a small part of the network. We show that such protocols can be designed to work correctly and efficiently even in the case of arbitrary (but not malicious) movements of the hosts not affected by the protocol. We also propose a methodology for the analysis of the expected behavior of protocols for such networks, based on the assumption that mobile hosts (those whose motion is not guided by the protocol) conduct concurrent random walks in their motion space. In particular, our work examines the fundamental problem of communication and proposes distributed algorithms for it. We provide rigorous proofs of their correctness, and also give performance analyses by combinatorial tools. Finally, we have evaluated these protocols by experimental means.

Abstract: An ad hoc mobile network is a collection of mobile hosts, with wireless communication capabilities, forming a temporary network without the aid of any established fixed infrastructure. In such networks, topological connectivity is subject to frequent, unpredictable change. Our work focuses on networks with high rate of such changes to connectivity. For such dynamically changing networks we propose protocols which exploit the co-ordinated (by the protocol) motion of a small part of the network. We show that such protocols can be designed to work correctly and efficiently even in the case of arbitrary (but not malicious) movements of the hosts not affected by the protocol. We also propose a methodology for the analysis of the expected behavior of protocols for such networks, based on the assumption that mobile hosts (those whose motion is not guided by the protocol) conduct concurrent random walks in their motion space. In particular, our work examines the fundamental problem of communication and proposes distributed algorithms for it. We provide rigorous proofs of their correctness, and also give performance analyses by combinatorial tools. Finally, we have evaluated these protocols by experimental means.

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 present a new finger search tree with
O(
log log
d)
expected search time
in the Random Access Machine (RAM) model of computation for a large class of
input distributions. The parameter
d
represents the number of elements (distance) be-
tween the search element and an element pointed to by a finger, in a finger search tree
that stores
n
elements. Our data structure improves upon a previous result by Andersson and Mattsson that exhibits expected
O(
log log
n)
search time by incorporating the
distance
d
into the search time complexity, and thus removing the dependence on
n
.
We are also able to show that the search time is
O(
log log
d
+
φ(n))
with high prob-
ability, where
φ(n)
is
any
slowly growing function of
n
. For the need of the analysis
we model the updates by a “balls and bins” combinatorial game that is interesting in
its own right as it involves insertions and deletions of balls according to an unknown
distribution.

Abstract: We present a new finger search tree with O(1) worst-case
update time and O(log log d) expected search time with high probability
in the Random Access Machine (RAM) model of computation for a large
class of input distributions. The parameter d represents the number of
elements (distance) between the search element and an element pointed
to by a finger, in a finger search tree that stores n elements. For the need
of the analysis we model the updates by a "balls and bins" combinatorial
game that is interesting in its own right as it involves insertions and
deletions of balls according to an unknown distribution.

Abstract: We present a new finger search tree with O(log log d) expected search time in the Random
Access Machine (RAM) model of computation for a large class of input distributions. The
parameter d represents the number of elements (distance) between the search element and an
element pointed to by a finger, in a finger search tree that stores n elements. Our data structure
improves upon a previous result by Andersson and Mattsson that exhibits expected O(log log n)
search time by incorporating the distance d into the search time complexity, and thus removing
the dependence on n. We are also able to show that the search time is O(log log d + φ(n)) with
high probability, where φ(n) is any slowly growing function of n. For the need of the analysis
we model the updates by a “balls and bins” combinatorial game that is interesting in its own
right as it involves insertions and deletions of balls according to an unknown distribution.

Abstract: An ad-hoc mobile network is a collection of mobile hosts, with
wireless communication capabilities, forming a temporary network
without the aid of any established fixed infrastructure.
In such networks, topological connectivity is subject to frequent,
unpredictable change. Our work focuses on networks with high
rate of such changes to connectivity. For such dynamic changing
networks we propose protocols which exploit the co-ordinated
(by the protocol) motion of a small part of the network.
We show that such protocols can be designed to work
correctly and efficiently even in the case of arbitrary (but not
malicious) movements of the hosts not affected by the protocol.
We also propose a methodology for the analysis of the expected
behaviour of protocols for such networks, based on the assumption that mobile hosts (whose motion is not guided by
the protocol) conduct concurrent random walks in their
motion space.
Our work examines some fundamental problems such as pair-wise
communication, election of a leader and counting, and proposes
distributed algorithms for each of them. We provide their
proofs of correctness, and also give rigorous analysis by
combinatorial tools and also via experiments.

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.