Chen Avin

Efficient and robust query processing in dynamic environments using random walk techniques

Many existing systems for sensor networks rely on state information stored in the nodes for proper operation (e.g., pointers to parent in a spanning tree, routing information, etc). In dynamic environments, such systems must adopt failure recovery mechanisms, which significantly increase the complexity and impact the overall performance. We investigate alternative schemes for query processing based on random walk techniques. The robustness of this approach under dynamics follows from the simplicity of the process, which only requires the connectivity of the neighborhood to keep moving. In addition we show that visiting a constant fraction of sensor network, say 80%, using a random walk is efficient in number of messages and sufficient for answering many interesting queries with high quality. Finally, the natural behavior of a random walk, also provide the important properties of load-balancing and scalability.

How to Explore a Fast-Changing World (Cover Time of a Simple Random Walk on Evolving Graphs)

Motivated by real world networks and use of algorithms based on random walks on these networks we study the simple random walks on dynamicundirected graphs with fixed underlying vertex set, i.e., graphs which are modified by inserting or deleting edges at every step of the walk. We are interested in the expected time needed to visit all the vertices of such a dynamic graph, the cover time, under the assumption that the graph is being modified by an oblivious adversary. It is well known that on connected staticundirected graphs the cover time is polynomial in the size of the graph. On the contrary and somewhat counter-intuitively, we show that there are adversary strategies which force the expected cover time of a simple random walk on connected dynamic graphs to be exponential. We relate this result to the cover time of static directed graphs. In addition we provide a simple strategy, the lazyrandom walk, that guarantees polynomial cover time regardless of the changes made by the adversary.

Many random walks are faster than one

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time - the expected time required to visit every node in a graph at least once - and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probablistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t-connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

On the cover time of random geometric graphs

The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad-hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph

Tight bounds for algebraic gossip on graphs

{\mathcal G}(n,r)

SINR diagrams: towards algorithmically usable SINR models of wireless networks

is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius ropt such that for any

Fast randomized algorithm for hierarchical clustering in Vehicular Ad-Hoc Networks

r \geq r_{\rm opt} {\mathcal G}(n,r)

The power of choice in random walks: an empirical study

has optimal cover time of Θ(n log n) with high probability, and, importantly, ropt=Θ(rcon) where rcon denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(rcon). We are able to draw our results by giving a tight bound on the electrical resistance of

Probabilistic quorum systems in wireless Ad Hoc networks

{\mathcal G}(n,r)

A Note on Uniform Power Connectivity in the SINR Model

via the power of certain constructed flows.