J.D. Kahn, N. Linial, N. Nisan, and M.E. Saks. On the cover time of random walks on graphs. J. Theoretical Probab., 2:121--128, 1989. 23  Home/Search   Document Not in Database   Summary   Related Articles   Check

....all rooted spanning trees of G. The proof, given in section 2, is part of a known circle of ideas relating spanning trees to Markov chain stationary distributions. Clearly Proposition 1 yields an algorithm for generating a uniform spanning tree in time O(EC) It is known [4] that EC , and [13] that for regular graphs EC 8n . Andrei Broder [7] has independently noted Proposition 1 and discusses these algorithmic aspects at greater length. Our emphasis is rather on theoretical properties of uniform spanning trees, a topic on which there seems no literature at all. Proposition 1 ....

J.D. Kahn, N. Linial, N. Nisan, and M.E. Saks. On the cover time of random walks on graphs. J. Theoretical Probab., 2:121--128, 1989. 23

....Science and Engineering, FR 35, University of Washington, Seattle, Washington, U.S.A. 98195 1 Table 1 Best Known Bounds on Length of Universal Traversal Sequences Bound Relevant Range Source U (d; n) O(n 3 ) d = 2 Aleliunas [1] U (d; n) O(dn 3 log n) 3 d n=2 0 1 Kahn et al. [7] U (d; n) O(n 3 log n) n=2 0 1 d Chandra et al. 6] U (d; n) Omega# n 1:29 ) d = 2 This paper U (d; n) Omega# d 0:71 n 2:29 ) 3 d n log 6 1:5 This paper U (d; n) Omega# d 2 n 2 ) n log 6 1:5 d n=3 0 2 Borodin et al. 5] U (d; n) Omega# n 2 ) n=3 0 2 d ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks, On the cover time of random walks on graphs, Journal of Theoretical Probability, 2 (1989), pp. 121--128. 8

.... This bound is improved in Corollary 8 to U(2; n) Omega# n log 5 10 ) Omega# n 1:43 ) 4 Table 1: Best known bounds on length of universal traversal sequences Bound Relevant Range Source U(d; n) O(n 3 ) d = 2 Aleliunas [1] U(d; n) O(dn 3 log n) 3 d n=2 0 1 Kahn et al. [6] U(d; n) O(n 3 log n) n=2 0 1 d Chandra et al. 5] U(d; n) Omega# n 1:43 ) d = 2 This paper U(d; n) Omega# d 0:57 n 2:43 ) 3 d n log 10 2 This paper U(d; n) Omega# d 2 n 2 ) n log 10 2 d n=3 0 2 Borodin et al. 4] U(d; n) Omega# n 2 ) n=3 0 2 d Alon et ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. On the cover time of random walks on graphs. Journal of Theoretical Probability, 2(1):121--128, Jan. 1989.

....The Weizmann Institute, Rehovot, Israel. feige wisdom.weizmann.ac.il. Supported by a Koret Foundation fellowship. z IBM T.J. Watson Research Center, Yorktown Heights, New York. 1 Aleliunas et al. 2] showed that for any connected graph, EC[G] 2nm. This bound has been refined by Kahn et al. [14], who proved a bound of EC[G] 4n 2 d ave =d min , where d ave is the average degree of the graph, and d min is its minimum degree. This bound takes into account the structure of the graph: for regular graphs the bound is low, O(n 2 ) whereas for irregular graphs, those which have a high ....

....it is computable in deterministic logarithmic space) with loss of only a constant factor in the accuracy of the value computed for the cyclic cover time. A simple corollary of Theorem 1 is that for any connected graph, EC[G] 10 3 n 2 d ave (d Gamma1 ) ave . This is an improvement of the [14] bound on the cover time, since for every graph, d ave (d Gamma1 ) ave d ave =d min . However, the bounds that we obtain on the cover time are far from tight. Observe that the bounds that we derive are Omega Gamma n 2 ) whereas the 2 cover time of a graph may be as low as O(n log n) for ....

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J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. "On the cover time of random walks on graphs". Journal of Theoretical Probability, 2(1):121--128, January 1989.

....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, 14, 17]) that for connected graphs on n vertices, minG min v E v [G] 1 o(1) n ln n, where ln n denotes the natural logarithm of n, and o(1) denotes a (possibly negative) term that tends to 0 as n tends to 1. We prove this conjecture. This lower bound is best possible up to low order terms, as ....

....in which a short path extends from a clique, and the random walk starts from the end point of the path. See [1, 3, 5] for more details. 1.1 Related work To the best of the author s knowledge, the lower bound conjecture was first formulated by Linial. The conjecture has been proven for trees [14] (see [10, 5] for improved bounds) The Department of Applied Math and Computer Science, The Weizmann Institute, Rehovot, Israel. feige wisdom.weizmann.ac.il. Research supported by a Koret Foundation fellowship. 1 conjecture has been proven, up to the leading constant factor, for graphs ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. "On the cover time of random walks on graphs". Journal of Theoretical Probability, 2(1):121--128, January 1989.

....including ST = O(m 3=2 n 1=2 ) Barnes and Feige conjectured that the MD algorithm achieves a time space tradeoff of ST = O(mn=d min ) where d min is the minimum degree in the graph. This conjectured tradeoff of O(mn=d min ) for MD is patterned after the similar bound of Kahn et al. [KLNS] on the cover time of graphs. In particular, its value is O(n 2 ) for regular graphs. 1.2 Our results We suggest a new landmark distribution scheme, ID (inverse distribution) ID distributes p=2 landmarks according to the stationary distribution (probability that v receives a landmark is ....

....to discover its N th distinct vertex is O(N 3 ) They in turn conjectured that this bound can be refined to O(N N 2 min[N ; d max ] d min ) and proved this bound up to a multiplicative logarithmic factor. The conjecture is known to hold if N = Omega Gamma n) by the results of Kahn et al. [KLNS]. We prove the conjecture for arbitrary graphs and arbitrary values of N . The proof combines ideas that appear in Section 2, some of which originate in the works of [aldous] and [BF] We remark that for the special case of regular graphs, 18 the proof can be simplified, and it can be obtained as ....

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J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. "On the cover time of random walks on graphs". Journal of Theoretical Probability, 2(1):121--128, January 1989.

....at the same time a spanning tree of G . By the above claim, it has weight 2m(n Gamma 1) n 3 . But the weight of MST(G 00 ) is at most that of T , implying E s [G] n 3 . In subsequent work, the existence of spanning trees of weight less than n 3 in G 00 was investigated. Kahn et al. [13] showed that for any connected regular graph G , MST [G 00 ] O(n 2 ) Coppersmith et al. 6] generalized this result to any connected graph, showing that MST [G 00 ] Theta(n 2 d ave ( 1 d ) ave ) where d ave is the average degree in G , and ( 1 d ) ave is the average of the ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. "On the cover time of random walks on graphs". Journal of Theoretical Probability, 2(1):121--128, January 1989.

....a simple random walk on an undirected connected graph, G, traverses its M th distinct edge, M m This paper gives upper bounds on E[T (N ) and E[T (M) for arbitrary graphs. While a great deal was previously known about how quickly a random walk covers the entire graph (see, for example, [2, 4, 7, 9, 18, 22, 23]) little was known about the behavior of a random walk before the vertices are covered. These bounds help fill the gaps in our knowledge of random walks, giving a picture of the rate at which a random walk explores a finite or an infinite graph. Aleliunas et al. 4] show that the expected time ....

....= O(nM) Our theorems are the best possible in the sense that there exist graphs for which the bounds are tight up to constant factors (e.g. the n cycle for Theorem 1.2) However, these bounds can be refined if additional information regarding the structure of G is given. The work of Kahn et al. [18] indicates that dmin , the minimum degree of the vertices in the graph G, is a useful parameter to consider. They show that the expected cover time of any connected graph is O(mn=dmin ) implying a cover time of O(n 2 ) for regular graphs. This inverse dependency on dmin applies also to short ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks, On the cover time of random walks on graphs, Journal of Theoretical Probability, 2 (1989), pp. 121--128.

....to collect n coupons is Theta(n log n) This is essentially equivalent to showing that the expected cover time for a random walk on the complete graph is Theta(n log n) There are many other specific graphs for which the expected cover time has been computed. These include paths, cycles, trees [10], bar bell graphs 1 [14] and d dimensional cubes [4] For arbitrary connected graphs, Aleliunas et al. 6] showed a general upper bound E(C i ) O(jEj jV j) starting from any vertex i, where jEj is the number of edges and jV j is the number of vertices. A superficial examination of these ....

....problem for undirected graphs. Aleliunas et al. 6] have shown that there exist sequences of length O(d 2 n 3 log n) that are universal for all labeled graphs on n vertices with degree bounded by d. For d regular graphs the bound can be improved to O(dn 3 log n) via the results in [10]. Using the results of Lemmas 1 and 3, we show that there exist sequences of length O(dn 2 log n) that are universal for graphs with 2 bounded away from 1. We start by considering connected graphs that are not bipartite (and hence the Markov chain is aperiodic) and then generalize the result to ....

J.D. Kahn, N. Linial, N. Nisan, and M.E. Saks, "On the cover time of random walks in graphs," manuscript, 1988.

....into (3) max v E v C e 8 3 (d2m 3=2 e 2 max v E v C) 4 Now Theorem 1 says max v E v C m(n Gamma 1) mn Gamma 1, so max v E v C e 8 3 (2m 3=2 2mn) 16 3 m(m 1=2 n) 32 3 mn establishing the Proposition. 2 Another variant of Theorem 1, due to Kahn et al. [24] (whose proof we follow) uses a graph theoretical lemma to produce a good spanning tree in graphs of high degree. Theorem 4 Writing d = min v d v , max v E v C 6 dn 2 d (5) and so on a regular graph max v E v C 6n 2 : 6) To appreciate (6) consider Example 5 Take ....

J.D. Kahn, N. Linial, N. Nisan, and M.E. Saks. On the cover time of random walks on graphs. J. Theoretical Probab., 2:121--128, 1989.

.... is improved in Corollary 7 to U(2; n) Omega Gamma n log 5 10 ) Omega Gamma n 1:43 ) 2 Table 1: Best Known Bounds on Length of Universal Traversal Sequences Bound Relevant Range Source U(d; n) O(n 3 ) d = 2 Aleliunas [1] U(d; n) O(dn 3 log n) 3 d n=2 Gamma 1 Kahn et al. [6] U(d; n) O(n 3 log n) n=2 Gamma 1 d Chandra et al. 5] U(d; n) Omega Gamma n 1:43 ) d = 2 This paper U(d; n) Omega Gamma d 0:57 n 2:43 ) 3 d n log 10 2 This paper U(d; n) Omega Gamma d 2 n 2 ) n log 10 2 d n=3 Gamma 2 Borodin et al. 4] U(d; n) ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. On the cover time of random walks on graphs. Journal of Theoretical Probability, 2(1):121--128, Jan. 1989.

....d regular n vertex graphs, and at most n 4m labeled nonregular m edge, n vertex graphs. 2 We remark that Lemma 4 implies the same bounds for lengths of vertex universal traversal sequences, asymptotically matching the best known upper bounds for both regular (Aleliunas et al. 2] Kahn et al. [36]) and nonregular graphs. The main technical result of this section is the following lemma. For the purposes of this lemma, it is convenient to think of the JAG as a transducer, i.e. as a machine equipped with a one way, write only output tape, excluded from the space bound, on which it writes ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. On the cover time of random walks on graphs. Journal of Theoretical Probability, 2(1):121--128, Jan. 1989.

....a random walk will with high probability manage to explore the netlist structure in a small number of steps. Fact: The cover time of a random walk in a d regular graph of n nodes is O(n 2 ) and Omega Gamma n log n) and there and there exist examples which show that both bounds are tight [10]. The O(n 2 ) upper bound also applies to cover times for the class of d bounded graphs [5] which includes gatelevel netlists. Therefore, we may compute a single random walk of length Theta(n 2 ) and expect to sample the entire netlist graph. We propose a method for extracting clusters from ....

J. D. Kahn, N. Linial, N. Nisan and M. E. Saks, "On the Cover Time of Random Walks on Graphs", J. of Theoretical Probability 2(1) (1989), pp. 121-128.

....and Snell) reiterates the fact that the electrical properties of the network underlying a graph are innately tied to the random walk. Prior work in the study of the cover time of graphs has used techniques from Markov chain theory (Aleliunas et al. 1979, Gobel Jagers 1974) from combinatorics (Kahn et al. 1989) , from linear algebra (Broder Karlin 1989) and from graph theory (Jerrum Sinclair 1989) The electrical approach used here provides an intuitive basis for understanding a variety of phenomena about random walks that had hitherto seemed counterintuitive. As an example, a simple and plausible ....

....of n=2 cliques connected by a single edge provides a simple example: R span = O(1) hence cover time is O(n 2 ) which is better than the bounds given by Aleliunas et al. 1979) or (1.1) Section 2, Theorem 2.4. For d regular graphs, the Aleliunas et al. bound for cover time is O(dn 2 ) Kahn et al. 1989) improved this bound for d regular graphs to O(n 2 ) Reexamination of their proof reveals that it supports the stronger statement that R span = O(n=d) for any d regular graph, hence cover time is O(n 2 ) by (1.2) Kahn et al. 1989) also give examples, for any d bn=2c Gamma 1, of n vertex, ....

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J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks, On the cover time of random walks on graphs. J. Theoret. Probability 2(1) (1989), 121--128.

....part in establishing tight bounds for cov tends to be deciding the extra log n factor. The basic technique for showing upper bounds of O( is one based on spanning trees, first used in [AKLLR] to show an upper bound for all graphs of O(jV jjEj) even though can be Theta(jV jjEj) [KLNS] extend this technique to show the general upper bound O(jV jjEj=d min ) and [CRRST] mention another application of this technique. By contrast, a variety of techniques have been used to show the lower bound Omega Gamma log n) even though most of the work on lower bounds has been ....

....lower bound of Omega Gamma n log n) for all graphs, regardless of the start vertex. Aldous [A3] has proved this bound if the walk starts from the stationary distribution. Examples of the different techniques are an inductive argument to show the Omega Gamma n log n) conjecture for trees [KLNS], a coupon collector type argument to show the Omega Gamma n log n) conjecture for rapidly mixing walks [BK] and use of the Theta Gamma p n Delta standard deviation law to show an Omega Gamma n log 2 n= log 2 dmax ) lower bound for trees with small degree [Z] In this paper, we ....

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J.D. Kahn, N. Linial, N. Nisan, and M.E. Saks, On the cover time of random walks in graphs, J. Theoretical Probab., 2 (1989), pp. 121-28.

.... Edges of a Graph David Zuckerman Division of Computer Science University of California Berkeley, CA 94720 November 26, 1997 Abstract The expected time for a random walk on an undirected graph G = V; E) to visit all the vertices is O(jV jjEj) AKLLR] and is O(jV j 2 ) for regular graphs [KLNS]. Here we show that both bounds hold even if we are required to traverse all the edges, although our bound for regular graphs requires that the degree be jV j ffi for some ffi 1. Keywords: random walk, cover time, combinatorial problems 1 Introduction A random walk on an undirected graph is ....

....expected time to traverse all the directed edges, where a random walk traverses directed edge (v; w) if it visits v and w consecutively. The most natural method gives a bound of O(jV jjEj log jV j) In this note we obtain a bound of O(jV jjEj) for this problem. For regular graphs, Kahn et al. [KLNS] have improved the bound on visiting all vertices to O(jV j 2 ) Here again the most natural way to prove a similar result about traversing all directed edges adds a log jV j factor, giving O(jV j 2 log jV j) We obtain a bound of O(jV j 2 ) if the degree is jV j ffi for some ffi 1. We ....

J.D. Kahn, N. Linial, N. Nisan, and M.E. Saks, On the cover time of random walks in graphs, J. Theoretical Probab., 2 (1989), pp. 121-28.

....traversal sequence (UTS) for G(d,n) if U traverses every G G(d,n) starting at any vertex in G. Let U(d,n) denote the length of a shortest UTS for non empty G(d,n) and define U(d,n) U(d,n 1) in case G(d,n) is empty. The lower and upper bounds on U(d,n) for various ranges of d were studied in [1, 2, 3, 5, 8, 10, 12]. Prior to the current work, the best lower bounds on U(d,n) for d=2 and for 3 d n 17 1 were U(2,n) W(n log 5 10 ) and U(d,n) W(d 2 log 5 10 n 1 log 5 10 ) due to a personal communication [6] These lower bounds are improved in this paper to U(2,n) W(n log 7 17.82 ) and U(d,n) ....

J.D. Kahn, N. Linial, N. Nisan, and M.E. Saks. On the cover time of random walks on graphs. Journal of Theoretical Probability, 2(1):121--128, 1989.

....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. ....

....of Applied Math. The Weizmann Institute, Rehovot, Israel. feige wisdom.weizmann.ac.il. Supported by a Koret Foundation fellowship. 1.1 Related work To the best of the author s knowledge, the lower bound conjecture was first formulated by Linial. The conjecture has been proven for trees [13] (see [10, 5] for improved bounds) The conjecture has been proven, up to the leading constant factor, for graphs satisfying certain conditions on their eigenvalues [6, 16] and for regular graphs [3] More results on special types of graphs can be found by tracing the references in [3] On ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. "On the cover time of random walks on graphs". Journal of Theoretical Probability, 2(1):121--128, January 1989.

....d regular n vertex graphs, and at most n 4m labeled nonregular m edge, n vertex graphs. 2 We remark that Lemma 4 implies the same bounds for lengths of vertex universal traversal sequences, asymptotically matching the best known upper bounds for both regular (Aleliunas et al. 2] Kahn et al. [27]) and nonregular graphs. The main technical result of this section is the following lemma. For the purposes of this lemma, it is convenient to think of the JAG as a transducer, i.e. as a machine equipped with a one way, write only output tape on which it writes the string encoding the graph ....

J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. On the cover time of random walks on graphs. Journal of Theoretical Probability, 2(1):121--128, Jan. 1989.

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J. D. Kahn, N. Linial, N. Nisan, and M. E. Saks. On the cover time of random walks on graphs. Journal of Theoretical Probability, 2(1):121--128, January 1989.

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