Zuckerman, D. A technique for lower bounding the cover time. In Proc. of the twenty-second annual ACM symposium on Theory of computing (1990), ACM Press, pp. 254--259.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 ....

....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 satisfying certain conditions on their eigenvalues [6, 17], and for regular graphs [3] More results on special types of graphs can be found by tracing the references in [3] On arbitrary graphs, Aldous [2, 3] proves an Omega Gamma n ln n) lower bound for walks that start at the stationary distribution (rather than any vertex) In terms of upper ....

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D. Zuckerman. "A Technique for Lower Bounding the Cover Time". SIAM J. Disc. Math., 5:81-87, 1992. 7

....Matthews [16] establishes also a lower bound: E s [G] min u;v H [u; v] ln n . While sometimes this bound is far from optimal (e.g. it may happen that min u;v H [u; v] 1) the method by which it was obtained often proves useful. Refinements of this method were employed by Zuckerman [21] in order to show that the cover time of the p n Theta p n mesh is Theta(n(log n) 2 ) and also by Feige [10] who has established a general lower bound E s [G] n ln n (up to low order terms) for arbitrary graphs. Much information on a Markov chain can be obtained also by considering ....

D. Zuckerman. "A Technique for Lower Bounding the Cover Time". SIAM J. Disc. Math., 5:81-87, 1992.

....a result of Zuckerman [16] showing that min v E v C cn(log n) 2 for bounded degree trees on n vertices. If G = Z d [ Gammam; m] d , a finite portion of the d dimensional integer lattice, then E v C is Theta(n 2 ) for d = 1, Theta(n(log n) 2 ) for d = 2 and Theta(n log n) for d 3 [1,17]. Here, n = 2m 1) d = jV j. The cases d = 1 and d = 2 show that Theorem 1.1 is tight (up to the constants) The case d = 3 shows that the planarity assumption is necessary. The upper bound in Theorem 1.1 is quite easy. The lower bound will be based on Koebe s [11] Circle Packing Theorem ....

D. Zuckerman, A technique for lower bounding the cover time, SIAM J.

....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 ....

D. I. Zuckerman, A technique for lower bounding the cover time, in Proceedings of the TwentySecond Annual ACM Symposium on Theory of Computing, Baltimore, MD, May 1990, SHORT RANDOM WALKS 11 pp. 254--259.

....6 Attributions for what I regard as the main ideas were given in the text. The literature contains a number of corollaries or variations of these ideas, some of which I ve used without attribution, and many of which I haven t mentioned at all. A number of these ideas can be found in Zuckerman [29, 31], Palacios [27, 26] and the Ph.D. thesis of Sbihi [28] as well as papers cited elsewhere. Section 1. The conference proceedings paper [3] proving Theorem 1 was not widely known, or at least its implications not realized, for some years. Several papers subsequently appeared proving results which ....

....starting from the unattached end of the necklace the mean cover time is Theta(m 1 log m 1 m 2 2 ) Taking m 1 log m 1 m 2 2 ) o(m 1 m 2 ) gives the desired example. Section 6. The subset version of Matthews lower bound (Theorem 26) and its application to trees were noted by Zuckerman [31], Sbihi [28] and others. As well as giving a lower bound for balanced trees, these authors give several lower bounds for more general trees satisfying various constraints (cf. the unconstrained result, Proposition 7) As an illustration, Devroye Sbihi [16] show that on a tree min v E v C (1 ....

D. Zuckerman. A technique for lower bounding the cover time. SIAM J. Disc.Math., 5:81--87, 1992. 33

.... Gamma 1 E i T j : Section 1.5. Before Matthews method was available, a result like Corollary 5 (c) required a lot of work see Aldous [1] for a result in the setting of non reversible random flight on a group. The present version of Corollary 5 (c) is a slight polishing of ideas in Zuckerman [38] section 6. The fact that (17) implies (15) is a slight variation of the usual textbook forms of the continuity theorem ( 17] 2.3.4 and 2.3.11) for Fourier and Laplace transforms. By the same argument as therein, it is enough for the limit transform to be continuous at = 0, which holds in our ....

....is not necessarily arc transitive [20] Gobel and Jagers [19] observed that the property E v Tw EwT v = 2(n Gamma 1) for all edges (v; w) equivalently: the effective resistance across each edge is constant) holds for arc transitive graphs and for trees. Section 2.2. Sbihi [33] and Zuckerman [38] noted that the subset version of Matthews method could be applied to the d torus to give Corollaries 24 and 25. The related topic of the time taken by random walk on the infinite lattice Z d to cover a ball centered at the origin has been studied independently see Revesz [32] Chapter 22 and ....

D. Zuckerman. A technique for lower bounding the cover time. SIAM J. Disc.Math., 5:81--87, 1992. 33

....[ f(i Gamma 1; i) i = 2n=3 1; ng. This graph will be mentioned again later on. Let Z n be the n path, i.e. the graph on vertex set f1; ng with edge set f(i; i 1) i = 1; n Gamma 1g, and consider the product graph Z k n , k = 1; 2; It is known (see [1] and [12]) that for k = 1, E v C = Theta(n 2 ) and for k = 2, E v C = Theta(n 2 (log n) 2 ) whereas for k 3, E v C = Theta(n k log n) Thus, in terms of the number of vertices, Z 2 n is covered faster than Z n and Z k n , k 3, is even faster. The question 2 JOHAN JONASSON we address in ....

....1 2 1 2 1 3 1 4 1 6 1 6 1 12 1 12 1 6 1 6 1 4 1 3 1 2 Figure 2: Z 2 4 and the unit flow j. 3 Proofs The following result for Z k n will provide a useful tool. It should be noted that this result is not an original result of the present paper and that the proof is essentailly covered by [6] and [12]. Lemma 3.1 (a) Let u and v be any two vertices of Z 2 n . Then R(u; v) 8h n : On the other hand one can find a subset V 0 of vertices of Z 2 n such that jV 0 j = n and such that R(u; v) h n =16 for any u; v 2 V 0 . b) Consider Z k n for some k 3. There exists a finite constant K k ....

D. ZUCKERMAN, A technique for lower bounding the cover time, SIAM J. Disc. Math. 5 (1992), 81-87.

....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 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 arbitrary graphs, Aldous [2, 3] proves an Omega Gamma n log n) lower bound for walks that start at the stationary distribution (rather than any vertex) In terms of upper bounds ....

[Article contains additional citation context not shown here]

D. Zuckerman. "A Technique for Lower Bounding the Cover Time". SIAM J. Disc. Math., 5:81-87, 1992.

....obvious upper bound on c e , we note that for any two edges e 1 and e 2 , E e 1 T e 2 t max m. Thus the results of e.g. Ma] imply that c e = O(t max log m m log m) For those graphs having c v = Omega Gamma t max log n) such as the hypercube and k dimensional toruses for k 1 (see [A1] and [Z]) this is tight to within a constant factor. This is because c e c v , c v = O(tmax log n) see e.g. Ma] and that c e = Omega Gamma m log m) see [A2] For general graphs, however, it gives a bound of c e = O(mn log n) To improve this to O(mn) we prove the following lemma: Lemma 2 For ....

D. Zuckerman, A technique for lower bounding the cover time, STOC, 1990, to appear.

.... is C = O(H log n) 1, 18] We strengthen this by showing that in fact B = O(H log n) This establishes our conjecture in the very common case where C = Theta(H log n) This is the case for almost all graphs, as well as most natural graphs: the hypercube [1] k dimensional lattices for k 2 [1, 24], balanced k ary trees [23, 11, 24] and expanders [7, 20, 6] The classic examples of graphs for which C = Theta(H ) are paths and cycles. In our second result, we show that for these graphs our conjecture also holds. Aldous [3] has subsequently shown that this second theorem also follows from ....

.... strengthen this by showing that in fact B = O(H log n) This establishes our conjecture in the very common case where C = Theta(H log n) This is the case for almost all graphs, as well as most natural graphs: the hypercube [1] k dimensional lattices for k 2 [1, 24] balanced k ary trees [23, 11, 24], and expanders [7, 20, 6] The classic examples of graphs for which C = Theta(H ) are paths and cycles. In our second result, we show that for these graphs our conjecture also holds. Aldous [3] has subsequently shown that this second theorem also follows from deep but well known facts about ....

D. Zuckerman, A technique for lower bounding the cover time, SIAM Journal on Discrete Mathematics 5 (1992), 81-87

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Zuckerman, D. A technique for lower bounding the cover time. In Proc. of the twenty-second annual ACM symposium on Theory of computing (1990), ACM Press, pp. 254--259.

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D. Zuckerman. A technique for lower bounding the cover time. In Proc. of the twenty-second annual ACM symposium on Theory of computing, pages 254--259. ACM Press, 1990.

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