Feige, U. A tight upper bound on the cover time for random walks on graphs. Random Structures and Algorithms 6, 1 (1995), 51--54.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....expected amount of time that simple random walk started at Q(v 1 ) takes before reaching Q(v 2 ) conditioned on Q(v 2 ) being the first distinguished vertex other than Q(v 1 ) to be visited. 34 Also define E = E n (T Q (v 1 , v 2 ) We can bound E because Q( w, w # ) Thus by [12] the covering time of . This implies . 11) Given t choose n to be the smallest integer such that n log t for all t. This implies that there exists C 14 such that E . Fix a good graph Q and v V (Q) If v is not a distinguished vertex let d 1 and d 2 be such that v Q(Q(d 1 ....

U. Feige. A tight upper bound on the cover time for random walks on graphs. Random Structures Algorithms (1995) 6, 51--54.

....random walk W on G starting at v, to visit every vertex of G. The cover time CG of G is de ned as CG = max v2V C v . The cover time of connected graphs has been extensively studied. It is a classic result of Aleliunas, Karp, Lipton, Lov asz and Racko [1] that CG 2m(n 1) It was shown by Feige [6], 7] that for any connected graph G (1 o(1) n log n CG (1 o(1) The lower bound is achieved by (for example) complete graph K n , whose cover time is determined by the Coupon Collector problem. In a previous paper [5] we studied the cover time of random graphs G n;p when np = c log ....

U. Feige, A tight upper bound for the cover time of random walks on graphs, Random Structures and Algorithms 6 (1995) 51-54.

....walk W on G starting at v, to visit every vertex of G. The cover time CG of G is de ned as CG = max v2V C v . The cover time of connected graphs has been extensively studied. It is a classic result of Aleliunas, Karp, Lipton, Lov asz and Racko [1] that CG 2m(n 1) It is also known (see Feige [6], 7] that for any connected graph G (1 o(1) n log n CG (1 o(1) 27 In this paper we study the cover time of the random graph, G 2 G n;p . It was shown by Jonasson [10] that whp (a) CG = 1 o(1) n log n if np log n 1. b) If c 1 is constant and np = c log n then CG (1 ....

U. Feige, A tight upper bound for the cover time of random walks on graphs, Random structures and algorithms 6 (1995) 51-54.

....walk W on G starting at v, to visit every vertex of G. The cover time CG of G is de ned as CG = max v2V C v . The cover time of connected graphs has been extensively studied. It is a classic result of Aleliunas, Karp, Lipton, Lov asz and Racko [1] that CG 2m(n 1) It is also known (see Feige [6], 7] that for any connected graph G (1 o(1) n log n CG (1 o(1) 27 In this paper we study the cover time of the random graph, G 2 G n;p . It was shown by Jonasson [10] that whp (a) CG = 1 o(1) n log n if np 1. b) If c 1 is constant and np = c log n then CG (1 )n log n ....

U. Feige, A tight upper bound for the cover time of random walks on graphs, Random structures and algorithms 6 (1995) 51-54.

....upper bound for the expected coverand return time. When the summation in the right hand side is minimized over all spanning trees, one gets the quantity R span and the upper bound 2jEjR span for the expected cover time that was first introduced in Chandra et al. 1989) and subsequently used by Feige (1995) and Coppersmith et al. 1996) to obtain maximal results for expected cover times. Notice that the factor 2 in (5) comes from the fact that every edge of the spanning tree is traversed twice. d) Take i 1 ; i N ; i N 1 = i 1 to be any tour that contains all vertices in V , and take all oe j ....

Feige, U. (1995) A tight upper bound on the cover time for random walks on graphs, Random Structures and Algorithms, 6, 51-54.

....or so, much work has been devoted to finding the cover time for different graphs and to giving general upper and lower bounds of the cover time. For an introduction, we refer the reader to the draft book by Aldous and Fill [2] in particular to Chapters 3, 5 and 6. It has been shown by Feige [9, 8] that (1 o(1) n log n E v C (1 o(1) 4 27 n 3 ; and these bounds are tight. In this paper, we show that for bounded degree planar graphs, one has better bounds, namely, Theorem 1.1 Let G = V; E) be a finite connected planar graph with n vertices and maximal degree M . Then for every ....

U. Feige, A Tight Upper Bound on the Cover Time for Random Walks on Graphs, Random Struct. Alg. 6 (1995), 51--54.

....11 (Brightwell Winkler [8] max max v;x E v T x is attained by the lollipop (Chapter 5 Example yyy) with m 1 = b(2n 1) 3c, taking x to be the leaf. Note that the implied asymptotic behavior is max max v;w E v Tw 4 27 n 3 : 15) Further asymptotic results are given by Theorem 12 (Feige [20, 18]) maxmax v E v C 4 27 n 3 (16) max min v E v C 3 27 n 3 (17) maxmin v E v C 2 27 n 3 (18) The value in (16) is asymptotically attained on the lollipop, as in Theorem 11. Note that (15) and (16) imply the same 4n 3 =27 behavior for intermediate quantities such as ....

U. Feige. A tight upper bound on the cover time for random walks on graphs. Random Struct. Alg., 6, 1995.

....independent of the starting distribution) Aldous [1] proved that this is true up to a constant factor if the starting point is drawn at random, from the stationary distribution. The asymptotically best possible upper and lower bounds on the cover time given in (b) are recent results of Feige [31,32]. For the case of regular graphs, a quadratic bound on the cover time was first obtained by Kahn, Linial, Nisan and Saks (1989) The bound given in (c) is due to Feige [33] Theorem 2.1. a) The access time between any two nodes of a graph on n nodes is at most (4=27)n 3 Gamma (1=9)n 2 ....

U. Feige, A Tight Upper Bound on the Cover Time for Random Walks on Graphs, Random Structures and Algorithms 6(1995), 51--54.

....reader is recommended to look into the draft book by Aldous and Fill [2] in particular Chapters 3, 5 and 6. For instance it is known that a general lower bound for min v E v C is (1 o(1) n log n and that a general upper bound for max v E v C is (1 o(1) 4n 3 =27. For proofs see Feige [8] and [7]. These bounds are tight up to small order terms; the cover time on the complete graph on n vertices is readily seen to be n P n Gamma1 i=1 1=i n log n and the lollipop graph, L n , where a path of length n=3 extends from a clique of size 2n=3, has a cover time of (1 o(1) 4n 3 =27 provided ....

U. FEIGE, A Tight Upper Bound on the Cover Time for Random Walks on Graphs, Random Struct. Alg. 6 (1995), 51-54.

....the family of all connected n vertex graphs. For a graph G 2 G(n) let h(G) max u;v2V h(u; v) C(G) max u;v2V C(u; v) and C(G) max u2V C u . Furthermore, let C (G) max u2V C u where C u are the cover and return times; C u = E u minfk : X k = u and k T v for all vg. Feige [5] proves that maxH2G(n) C (H) 4=27 o(1) n 3 . Indeed he proves something stronger, namely that the above bound holds even for the so called cyclic cover time. This immediately gives the same upper bound for hitting, commute and cover times as h(G) C(G) C (G) and h(G) C(G) C ....

U. FEIGE, A Tight Upper Bound on the Cover Time for Random Walks on Graphs, Random Struct. Alg. 6 (1995), 51-54.

....random walk started at v 0 has not reached v 00 by time 2EC(G 0 ) is less than .5. This implies that the probability that simple random walk started at v 0 has not reached v 00 by time cEC(G 0 ) is less than 2 Gammac=2 . The cover time for a graph with N vertices is bounded by N 3 [10]. For any vertex v in the n grid, the bush attached to v has at most (2nd) d vertices. Thus the cover time for the bush attached to v is bounded by (2nd) 3d . Let X v be the number of steps that a simple random walk started at v takes before it hits one of the two neighboring vertices in the ....

U. Feige. A tight upper bound on the cover time for random walks on graphs. (1995) Random Structures Algorithms (1995) 6, 51--54.

....this relationship. The mean cover time E[ C(G) of a graph G represents the maximum expected number of steps required by a predator until each vertex has been visited at least once. For arbitrary graphs the mean cover time can be bracketed by (1 o(1) n log n E[ C(G) 4 27 n 3 o(n 3 ) [8, 9]. A cubic increase in the number of vertices is certainly prohibitive for practical use. But the cubic upper bound decreases to a quadratic one for vertex and edge symmetric graphs [10] More specifically, n Gamma 1) Hn E[ C(G) 2 (n Gamma 1) 2 where Hn is the nth Harmonic number. The ....

U. Feige. A tight upper bound on the cover time for random walks on graphs. Random Structures and Algorithms, 6(1):51--54, 1995.

....of the inverse of the degrees of the vertices. This measure obtains its minimum 1 on regular graphs, and its maximum Omega Gamma n) on highly irregular graphs that have a linear number of vertices of constant degree and a linear number of vertices of degree Omega Gamma n) It was argued in [10] that this measure is preferable to d ave =d min , since it is more robust. Introduction of even a single vertex of small degree can cause d ave =d min to increase by a multiplicative factor of Theta(n) whereas d ave (d Gamma1 ) ave would increase by at most a constant factor. Using our ....

....span , the weight of the spanning tree of minimum weight (resistance) By previous discussion it follows that the cyclic cover time of G is at most 2mR span . The approach of using R span in order to bound the cover time originates from [2, 14] Our current work was motivated by the conjecture in [10] that R span = Theta( P v2V 1=d v ) 4 The excess resistance lemma Definition: The excess resistance ffi [ u; v) of edge (u; v) 2 E is defined by: ffi[ u; v) R[u; v] Gamma ( 1 d u 1 1 d v 1 ) Observe that by Proposition 2, ffi [ u; v) 0, with equality if and only if d u = ....

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U. Feige. "A Tight Upper Bound on the Cover Time for Random Walks on Graphs". Random Structures and Algorithms, 6(1):51--54, 1995.

....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 bounds on connected graphs, Aleluinas et al. 4] prove that maxG max v E v [G] n(n Gamma 1) 2 . This upper bound was improved to (1 o(1) 4n 3 =27 [12], giving the best possible leading constant. As for min max bounds, minG max v E v [G] 1 o(1) n ln n follows from the current paper, and maxG min v E v [G] 1 o(1) 2n 3 =27 is shown in [13] 1.2 Useful technical background For a vertex u, d u denotes its degree (the number of vertices ....

U. Feige. "A Tight Upper Bound on the Cover Time for Random Walks on Graphs". To appear in Random Structures & Algorithms.

....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 inverse of the degrees. Feige [9] showed that MST [G 00 ] 4n 3 =27 up to low order terms, implying a similar upper bound on the cover time. This upper bound matches (up to low order terms) the cover time for the lollipop graph a clique with 2n=3 vertices connected to a path of length n=3 . A different approach, suitable ....

U. Feige. "A Tight Upper Bound on the Cover Time for Random Walks on Graphs". Random Structures & Algorithms, 6(1), 1995, 51--54.

.... particularly useful for graphs that have slow cover time, such as the lollipop graph (a clique of size 2n=3 connected to a path of size n=3) It was used successfully to provide a tight bound (upto low order terms) of 4n 3 =27 on the cover time of any n vertex Collection coupons on trees 3 graph [8]. In this paper we study cover problems where neither the coupon collector approach nor the spanning tree approach provide a satisfactory bound. Our research was motivated by a question of Aldous (private communication) namely, how high can maxG min u E u [G] be 1.1. Additional notation. The ....

....cover time is at most W [T ] 2. This follows from the fact that min u max v [D[u; v] 0 [6] In the terminology of [13] a vertex u for which D[u; v] 0 for all v is called a remote vertex. Since the minimum weight spanning tree of any connected graph satisfies W [T ] 1 o(1) 4n 3 =27 [8], we obtain: Corollary 1.4. For connected graphs on n vertices, maxG min u [E u [G] 1 o(1) 2n 3 =27. This answers an open question of Aldous. Using Theorem 1.3 we show: Corollary 1.5. For connected graphs on n vertices, maxG min u [E u [G] 1 o(1) n 3 =9. The bounds in ....

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U. Feige, A Tight Upper Bound on the Cover Time for Random Walks on Graphs. Random Structures and Algorithms, Vol. 6, No. 1 (1995) 51--54.

....[W [T ] and similarly, E u [G] min T [W [T ] The spanning tree approach is particularly useful for graphs that have slow cover time, such as the lollipop graph. It was used successfully to provide a tight bound (upto low order terms) of 4n 3 =27 on the cover time of any n vertex graph [6]. In this paper we study cover problems where neither the coupon collector approach nor the spanning tree approach provide a satisfactory bound. Our research was motivated by a question of Aldous, namely, how high can maxG min u E u [G] be [1] 1.1 Additional notation We use the notation H T ....

....Theorem 1 was to show that for any graph, there is a good starting point from which the expected cover time is at most W [T ] 2. This follows from the fact that min u max v [D[u; v] 0. Since the minimum weight spanning tree of any connected graph satisfies W [T ] 1 o(1) 4n 3 =27 [6], we obtain: Corollary 3 For connected graphs on n vertices, maxG min u [E u [G] 1 o(1) 2n 3 =27. This answers an open question of Aldous. Using Theorem 2 we show: Corollary 4 For connected graphs on n vertices, maxG min u [E u [G] 1 o(1) n 3 =9. For some families of graphs ....

[Article contains additional citation context not shown here]

U. Feige. "A Tight Upper Bound on the Cover Time for Random Walks on Graphs". Technical report CS93-08, the Weizmann Institute, 1993.

....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 on connected graphs, Aleluinas et al. 4] prove that maxG max v E v [G] n(n Gamma 1) 2 . This upper bound was improved to (1 o(1) 4n 3 =27 [12], giving the best possible leading constant. 1.2 Useful technical background For a vertex u, d u denotes its degree (the number of vertices adjacent to it) For two vertices u; v 2 G, the hitting time H [u; v] is the expected number of steps it takes a walk that starts at u to reach v, and the ....

U. Feige. "A Tight Upper Bound on the Cover Time for Random Walks on Graphs". Technical report CS93-08, the Weizmann Institute, Israel, 1993.

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Feige, U. A tight upper bound on the cover time for random walks on graphs. Random Structures and Algorithms 6, 1 (1995), 51--54.

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U. Feige, \A Tight Upper Bound on the Cover Time for Random Walks on Graphs," Random Structures and Algorithms, 6(4):51-54, 1995.

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U.Feige, A Tight Upper Bound on the Cover Time for Random Walks on Graphs, Random Structures and Algorithms 6 (1995) 51-54.

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U. Feige, A tight upper bound for the cover time of random walks on graphs, Random Structures and Algorithms 6 (1995) 51-54.

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