Compact Routing on Internet-Like Graphs

The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing scheme delivering a nearly optimal local memory upper bound. Using both direct analysis and simulation, we calculate the stretch distribution of this routing scheme on random graphs with power-law node degree distributions, $P_k \sim k^{-\gamma}$. We find that the average stretch is very low and virtually independent of $\gamma$. In particular, for the Internet interdomain graph, $\gamma \sim 2.1$, the average stretch is around 1.1, with up to 70% of paths being shortest. As the network grows, the average stretch slowly decreases. The routing table is very small, too. It is well below its upper bounds, and its size is around 50 records for $10^4$-node networks. Furthermore, we find that both the average shortest path length (i.e. distance) $\bar{d}$ and width of the distance distribution $\sigma$ observed in the real Internet inter-AS graph have values that are very close to the minimums of the average stretch in the $\bar{d}$- and $\sigma$-directions. This leads us to the discovery of a unique critical quasi-stationary point of the average TZ stretch as a function of $\bar{d}$ and $\sigma$. The Internet distance distribution is located in a close neighborhood of this point. This observation suggests the analytical structure of the average stretch function may be an indirect indicator of some hidden optimization criteria influencing the Internet's interdomain topology evolution.Comment: 29 pages, 16 figure

Distributed Computation in Dynamic Networks

In this report we investigate distributed computation in dynamic networks in which the network topology changes from round to round. We consider a worst-case model in which the communication links for each round are chosen by an adversary, and nodes do not know who their neighbors for the current round are before they broadcast their messages. The model is intended to capture mobile networks and wireless networks, in which mobility and interference render communication unpredictable. The model allows the study of the fundamental computation power of dynamic networks. In particular, it captures mobile networks and wireless networks, in which mobility and interference render communication unpredictable. In contrast to much of the existing work on dynamic networks, we do not assume that the network eventually stops changing; we require correctness and termination even in networks that change continually. We introduce a stability property called T-interval connectivity (for T >= 1), which stipulates that for every T consecutive rounds there exists a stable connected spanning subgraph. For T = 1 this means that the graph is connected in every round, but changes arbitrarily between rounds. Algorithms for the dynamic graph model must cope with these unceasing changes. We show that in 1-interval connected graphs it is possible for nodes to determine the size of the network and compute any computable function of their initial inputs in O(n^2) rounds using messages of size O(log n + d), where d is the size of the input to a single node. Further, if the graph is T-interval connected for T > 1, the computation can be sped up by a factor of T, and any function can be computed in O(n + n^2 / T) rounds using messages of size O(log n + d). We also give two lower bounds on the gossip problem, which requires the nodes to disseminate k pieces of information to all the nodes in the network. We show an Omega(n log k) bound on gossip in 1-interval connected graphs against centralized algorithms, and an Omega(n + nk / T) bound on exchanging k pieces of information in T-interval connected graphs for a restricted class of randomized distributed algorithms. The T-interval connected dynamic graph model is a novel model, which we believe opens new avenues for research in the theory of distributed computing in wireless, mobile and dynamic networks

Distributed computation in wireless and dynamic networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 211-221) and index.Today's wireless networks tend to be centralized: they are organized around a fixed central backbone such as a network of cellular towers or wireless access points. However, as mobile computing devices continue to shrink in size and in cost, we are reaching the point where large-scale ad-hoc wireless networks, composed of swarms of cheap devices or sensors, are becoming feasible. In this thesis we study the theoretical computation power of such networks, and ask what tasks are they capable of carrying out. how long does solving particular tasks take. and what is the effect of the unpredictable network topology on the network's computation power. In the first part of the thesis we introduce an abstract model for dynamic networks. In contrast to much of the literature on mobile and ad-hoc networks, our model makes fairly minimalistic assumptions; it allows the network topology to change arbitrarily from round to round, as long as in each round the communication graph is connected. We show that even in this weak model, global computation is still possible, and any function of the nodes' initial inputs can be computed efficiently. Also, using tools from the field of epistemic logic, we analyze information flow in dynamic networks, and study the time required to achieve various notions of coordination. In the second part of the thesis we restrict attention to static networks, which retain an important feature of wireless networks: they are potentially (symmetric. We show that in this setting. classical data aggregation tasks become much harder. and we develop both upper and lower bounds on computing various classes of functions. Our main tool in this part of the thesis is communication complexity: we use existing lower bounds in two-player communication complexity, and also introduce a new problem, task allocation, and study its communication complexity in the two-player and multi-player settings.by Rotei Oshman.Ph.D

Compact Routing Schemes for Dynamic Ring Networks

We consider the problem of routing in an asynchronous dynamically changing ring of processors using schemes that minimize the storage space for the routing information. In general, applying static techniques to a dynamic network would require significant re-computation. Moreover, the known dynamic techniques applied to the ring lead to inefficient schemes. In this paper we introduce a new technique, Dynamic Interval Routing, and we show tradeoffs between the stretch factor, the adaptation cost, and the size of the update messages used by routing schemes based upon it. We give three algorithms for rings of maximum size N: the first two are deterministic, one with adaptation cost zero but worst case stretch factor ⌊N/2⌋, the other with worst case adaptation cost O(N) update messages of O(log N) bits and stretch factor 1. The third algorithm is randomized, uses update messages of size O(k log N), has adaptation cost O(k), and expected stretch factor 1 + 1/k, for any integer k ≥ 3. All schemes require O(log N) bits per node for the routing information and all messages headers are of O(log N) bits