We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

Information flow in a telecommunication network is accomplished through the interaction of mechanisms at various design layers with the end goal of supporting the information exchange needs of the applications. In wireless networks in particular, the different layers interact in a nontrivial manner in order to support information transfer. In this text we will present abstract models that capture the cross-layer interaction from the physical to transport layer in wireless network architectures including cellular, ad-hoc and sensor networks as well as hybrid wireless-wireline. The model allows for arbitrary network topologies as well as traffic forwarding modes, including datagrams and virtual circuits. Furthermore the time varying nature of a wireless network, due either to fading channels or to changing connectivity due to mobility, is adequately captured in our model to allow for state dependent network control policies. Quantitative performance measures that capture the quality of service requirements in these systems depending on the supported applications are discussed, including throughput maximization, energy consumption minimization, rate utility function maximization as well as general performance functionals. Cross-layer control algorithms with optimal or suboptimal performance with respect to the above measures are presented and analyzed. A detailed exposition of the related analysis and design techniques is provided.

/pdf/resource-allocation-and-cross-layer-control-in-wireless-1bikwdtjww.pdf

Resource Allocation and Cross Layer Control in Wireless Networks

We present a simple dictionary with worst case constant lookup time, equaling the theoretical performance of the classic dynamic perfect hashing scheme of Dietzfelbinger et al. [SIAM J. Comput. 23 (4) (1994) 738-761]. The space usage is similar to that of binary search trees. Besides being conceptually much simpler than previous dynamic dictionaries with worst case constant lookup time, our data structure is interesting in that it does not use perfect hashing, but rather a variant of open addressing where keys can be moved back in their probe sequences. An implementation inspired by our algorithm, but using weaker hash functions, is found to be quite practical. It is competitive with the best known dictionaries having an average case (but no nontrivial worst case) guarantee on lookup time.

Cuckoo hashing

In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

We investigate balls-into-bins processes allocating m balls into n bins based on the multiple-choice paradigm. In the classical single-choice variant each ball is placed into a bin selected uniformly at random. In a multiple-choice process each ball can be placed into one out of $d \ge 2$ randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when $m \approx n$. In this paper we present the first tight analysis in the heavily loaded case, that is, when $m \gg n$ rather than $m \approx n$.The best previously known results for the multiple-choice processes in the heavily loaded case were obtained using majorization by the single-choice process. This yields an upper bound of the maximum load of bins of $m/n + {\mbox{$\cal O$}}(\sqrt{m \ln n \,/\, n})$ with high probability. We show, however, that the multiple-choice...

Balanced allocations: the heavily loaded case

The (asymptotic) degree distributions of the best known "scale free" network models are all similar and are independent of the seed graph used Hence it has been tempting to assume that networks generated by these models are similar in general In this paper we observe that several key topological features of such networks depend heavily on the specific model and the seed graph used Furthermore, we show that starting with the "right" seed graph, the duplication model captures many topological features of publicly available PPI networks very well

/pdf/not-all-scale-free-networks-are-born-equal-the-role-of-the-48yufzwog9.pdf

Not all scale free networks are Born equal: the role of the seed graph in PPI network emulation

Balanced Allocations: The Heavily Loaded Case

Suppose that a set of $m$ tasks are to be shared as equally as possible among a set of $n$ resources. A game-theoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent” and require each agent to select a resource, with the cost of a resource being the number of agents that select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced. Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For $m \gg n$, the system becomes approximately balanced (an $\epsilon$-Nash equilibrium) in expected time $O(\log \log m)$. We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time $O(\log \log m + n^4)$. We also give a lower bound of $\Omega(\max\{\log \log m, n\})$ for the convergence time.

/pdf/distributed-selfish-load-balancing-5ahv34g05t.pdf

Distributed Selfish Load Balancing

In this paper we analyse a very simple dynamic work-stealing algorithm. In the work-generation model, there are n generators which are arbitrarily distributed among a set of n processors. During each time-step, with probability /spl lambda/, each generator generates a unit-time task which it inserts into the queue of its host processor. After the new tasks are generated, each processor removes one task from its queue and services it. Clearly, the work-generation model allows the load to grow more and more imbalanced, so, even when /spl lambda/<1, the system load can be unbounded. The natural work-stealing algorithm that we analyse works as follows. During each time step, each empty processor sends a request to a randomly selected other processor. Any non-empty processor having received at least one such request in turn decides (again randomly) in favour of one of the requests. The number of tasks which are transferred from the non-empty processor to the empty one is determined by the so-called work-stealing function f. We analyse the long-term behaviour of the system as a function of /spl lambda/ and f. We show that the system is stable for any constant generation rate /spl lambda/<1 and for a wide class of functions f. We give a quantitative description of the functions f which lead to stable systems. Furthermore, we give upper bounds on the average system load (as a function of f and n).

/pdf/the-natural-work-stealing-algorithm-is-stable-5bfdfbx9xr.pdf

Petra Berenbrink

Papers

Balanced allocations: the heavily loaded case

Not all scale free networks are Born equal: the role of the seed graph in PPI network emulation

Balanced Allocations: The Heavily Loaded Case

Distributed Selfish Load Balancing

The natural work-stealing algorithm is stable