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View an n-vertex, m-edge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with the lengths of random walks on the graph. For example, the commute time between two vertices s and t (the expected length of a random walk from s to t and back) is precisely characterized by the e ective resistance Rst between s and t: commute time = 2mRst. As a corollary, the cover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistance R in the graph to within a factor of log n: mR cover time O(mR log n). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, we improve known bounds on cover times for high-degree graphs and expanders, and give new proofs of known results for multidimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.

We prove that the expected time for a random walk to cover all n vertices of a graph is at least (1 + o(1))n ln n. 1 Introduction Let G be a simple connected undirected graph on n vertices and m edges. We consider random walks on G, where at each step the random walk moves to a vertex chosen at random with uniform probability from the neighbors of the current vertex. Let E v [G] denote the cover time, the expected number of steps that it takes a walk that starts at v to visit all vertices of G. It is a well known conjecture (see for example [1, 6, 13, 16]) that for connected graphs on n vertices, minG min v E v [G] (1 + o(1))n ln n, where o(1) denotes a (possibly negative) term that tends to 0 an n tends to 1. We prove this conjecture. This lower bound is best possible up to low order terms, as demonstrated by the complete graph on n vertices. The complete graph is not extremal with respect to min v E v [G], and faster cover times are possible by considering graphs in which a short ...

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. In this paper, 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 e#cient in number of messages and su#cient 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.

We give a general technique for proving lower bounds on expected covering times of random walks on graphs in terms of expected hitting times between vertices. We use this technique to prove: i) A tight bound of \Omega\Gamma jV j log 2 jV j) for the 2-dimensional torus. ii) A tight bound of \Omega\Gamma jV j log 2 jV j= log dmax ) for trees with maximum degree dmax . iii) Tight bounds of \Omega\Gamma ¯ + log jV j) for rapidly mixing walks on vertex transitive graphs, where ¯ + denotes the maximum expected hitting time between vertices. In addition to these new results, our technique allows us to systematically prove several known lower bounds on cover times, often in a much simpler way. Finally, we use a different technique to prove an\Omega \Gamma 1=(1 \Gamma 2 ) \Delta lower bound on the cover time, where 2 is the second largest eigenvalue of the transition matrix. This was previously known only in the case where the walk starts in the stationary distribution [BK]. * Thi...

The cover time and mixing time of graphs has much relevance to algorithmic appli-cations and has been extensively investigated. Recently, with the advent of ad-hoc and sensor networks, an interesting class of random graphs, namely random geo-metric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r) is obtained by placing n points uni-formly 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 r ≥ ropt G(n, r) 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 in-finite cover time, there is a phase transition and the corresponding threshold width is O(rcon). On the other hand, the radius required for rapid mixing rrapid = ω(rcon), and, in particular, rrapid = Θ(1/poly(log n)). We are able to draw our results by giv-ing a tight bound on the electrical resistance and conductance of G(n, r) via certain constructed flows.

Consider a particle moving on the surface of the unit sphere in R 3 and heading towards a specific destination with a constant average speed, but subject to random deviations. The motion is modeled as a diffusion with drift restricted to the surface of the sphere. Expressions are set down for various characteristics of the process including expected travel time to a cap, the limiting distribution, the likelihood ratio and some estimates for parameters appearing in the model. KEY WORDS: Drift; great circle path; likelihood ratio; pole-seeking; skew product; spherical Brownian motion; stochastic differential equation; travel time. 1.

Let T (x; ") denote the rst hitting time of the disc of radius " centered at x for Brownian motion on the two dimensional torus T 2 We prove that sup x2T 2 T (x; ")=j log "j

Abstract. 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 G(n, r) 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 r ≥ ropt G(n, r) 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 G(n, r) via the power of certain constructed flows. 1

Three methods are described for exploring a continuous unknown planar region by a group of robots having limited sensors and no explicit communication. We formalize the problem, prove that its off-line version is NP-hard, and show a lower bound on the length of any solution. Then a deterministic mark and cover (MAC) algorithm is described for the on-line problem using short-lived navigational markers as means of navigation and indirect communication. The convergence of the algorithm is proved, and its cover time is shown to be the asymptotically optimal O(A/a), where A is the total area and a is the area covered by the robot in a single step. TheMAC algorithm is tested against an alternative randomized probabilistic covering (PC) method, which does not rely on sensors but is still able to cover an unknown region in an expected time that depends polynomially on the dimensions of the region. Both algorithms enable cooperation of several robots to achieve faster coverage. Finally, we show...