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.

Abstract A random walk on a finite graph can be used to construct a uniformrandom spanning tree. We show how random walk techniques can be applied to the study of several properties of the uniform randomspanning tree: the proportion of leaves, the distribution of degrees, and the diameter.

The complexity of next-generation VLSI systems will exceed the capabilities of top-down layout synthesis algorithms, particularly in netlist partitioning and module placement. Bottom-up clustering is needed to "condense" the netlist so that the problem size becomes tractable to existing optimization methods. In this paper, we establish the DS quality measure, the first general metric for evaluation of clustering algorithms. The DS metric in turn motivates our RWST algorithm, a new self-tuning clustering method based on random walks in the circuit netlist. RWST efficiently captures a globally good circuit clustering. When incorporated within a two-phase iterative Fiduccia-Mattheyses partitioning strategy, the RW-ST clustering method improves bisection width by an average of 17% over previous matching-based methods. 1 Introduction Top-down approaches are widely used to cope with increasing problem complexity in layout synthesis. Recursive calls to a partitioning algorithm generate a ci...

We prove that the expected time for a random walk to visit all n vertices of a connected graph is at most 4 27 n 3 + o(n 3 ). 1 Introduction Let G = G(V; E) be a simple connected undirected graph with n vertices and m edges. We consider simple 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 u and v denote two vertices. The hitting time H[u; v] is the expected number of steps it takes a walk that starts at u to reach v. The commute time C[u; v] is the expected number of steps that it takes a walk to go from u to v and back to u (that is, C[u; v] = H[u; v] +H[v;u]). The cover time EC[v] is the expected number of steps it takes a random walk that starts at v to visit all vertices of the graph. For a graph G(V; E) its hitting time H[G] (commute time C[G], cover time EC[G], respectively) is defined as H[G] = max u;v2V [H[u; v]] (C[G] = max u;v2V [C[u; v]] , EC[G] = max v...

In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.

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 ...

. The short term behavior of random walks on graphs is studied, in particular, the rate at which a random walk discovers new vertices and edges. A conjecture by Linial that the expected time to find N distinct vertices is O(N 3 ) is proved. In addition, an upper bound of O(M 2 ) on the expected time to traverse M edges, and O(MN ) on the expected time to either visit N vertices or traverse M edges (whichever comes first) is proved. Key words. random walk, graph, Markov chain AMS subject classification. 60J15 1. Introduction. Consider a simple random walk on G, an undirected graph with n vertices and m edges. At each time step, if the walk is at vertex v, it moves to a vertex chosen uniformly at random from the neighbors of v. Random walks have been studied extensively, and have numerous applications in theoretical computer science, including space-efficient algorithms for undirected connectivity [4, 8], derandomization [1], recycling of random bits [10, 15], approximation algori...

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...

For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list, until it completes a full cycle. The main result of this paper is a characterization of the cyclic cover time in terms of simple and easy to compute graph properties. Namely, for any connected graph, the cyclic cover time is \Theta(n 2 d ave (d \Gamma1 ) ave ), where n is the number of vertices in the graph, d ave is the average degree of its vertices, and (d \Gamma1 ) ave is the average of the inverse of the degree of its vertices. Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of graphs, and a simplified proof that the maximum commute time in any connected graph is at most 4n 3 =27 + o(n 3 ). Key words: Random walks, graphs, electrical resist...

In this paper we introduce a new notion of traversal sequences that we call exploration sequences. Exploration sequences share many properties with the traversal sequences defined in [AKL+], but they also exhibit some new properties. In particular, they have an ability to backtrack, and their random properties are robust under choice of the probability distribution on labels. Further, we present extremely simple constructions of polynomial length universal exploration sequences for some previously studied classes of graphs (e.g., 2-regular graphs, cliques, expanders), and we also present universal exploration sequences for trees. Our constructions beat previously known lower-bounds on the length of universal traversal sequences. 1