G. Barnes and U. Feige. Short random walks on graphs. In Proceedings of the 25rd Annual ACM Symposium on Theory of Computing, San Diego, California, pages 728-737, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... steps. Proof. If the pursuer reaches a mote 1 , then there exists a path between the pursuer and the evader that is at most of length . This distance does not increase in the following program steps (due to maximal parallel execution semantics and the program actions) In [6], it is proven that during a random walk on a graph the expected time to find distinct vertices is . However, a recent result [12] shows that by using a local topology information (i.e. degree information of neighbor vertices) it is possible to achieve the cover time . ....

G. Barnes and U. Feige. Short random walks on graphs. SIAM Journal on Discrete Mathematics, 9(1):19--28, 1996.

....undirected graph in O(m n) time, but requires Omega# n) space. Alternatively, a random walk can traverse an undirected graph using only O(log n) space, but requires 2(mn) expected time (Aleliunas et al. 2] In fact, Feige [23] based on earlier work of Broder et al. 18] and Barnes and Feige [7], has shown that there is a spectrum of compromises between time and space for this problem: any graph can be traversed in space S and expected time T , where ST mn(log n) This raises the intriguing prospect of proving that logarithmic space and linear time are not simultaneously achievable ....

G. Barnes and U. Feige. Short random walks on graphs. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pages 728--737, San Diego, CA, May 1993.

....vertex v of the graph a large virtual neighbourhood. Karger, Nisan and Parnas [KNP92] construct the virtual neighbourhoods using relatively short random walks. A similar approach was used by Aleliunas et al. AKL 79] and recently by Nisan, Szemeredi and Wigderson [NSW92] and Barnes and Feige [BF93] to obtain space efficient algorithms and time space tradeoffs for undirected s t connectivity. Barnes and Feige [BF93] proving a conjecture of Linial, showed that a random walk of length s in an undirected connected graph is likely to visit at least Omega Gamma s 1=3 ) vertices (or all the ....

....using relatively short random walks. A similar approach was used by Aleliunas et al. AKL 79] and recently by Nisan, Szemeredi and Wigderson [NSW92] and Barnes and Feige [BF93] to obtain space efficient algorithms and time space tradeoffs for undirected s t connectivity. Barnes and Feige [BF93] proving a conjecture of Linial, showed that a random walk of length s in an undirected connected graph is likely to visit at least Omega Gamma s 1=3 ) vertices (or all the vertices of the graph) A slightly weeker bound of Omega Gamma s 1=4 ) appears in Karger et al. KNP92] The ....

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G. Barnes and U. Feige. Short random walks on graphs. In Proceedings of the 25rd Annual ACM Symposium on Theory of Computing, San Diego, California, pages 728--737, 1993.

....as f C (t) 2 3=2 Gamma1=2 t Gamma3=2 1 X m=1 ( Gamma1) m Gamma1 m 2 exp( Gamma m 2 2t ) Sbihi [28] gives a direct derivation of a different representation of f C . Section 4. Use of Lemma 10 in the random walk context goes back at least to Flatto et al. [21] Barnes and Feige [5] give a more extensive treatment of short time bounds in the irregular setting, and their applications to covering with multiple walks (cf. Proposition 17 and section 8.2) They also give bounds on the mean time taken to cover different edges or different vertices their bound for the latter ....

G. Barnes and U. Feige. Short random walks on graphs. 1992.

....receives a graph G and a number m as inputs. Since it is a matrix algorithm, it receives two more indices, s; u, and it should output the value of the (s; u)th entry of the output matrix. Let us denote k = 2 m as was used in section 3. We start with the following Lemma of Barnes and Feige [BF93] Lemma 7.1 Suppose G is a connected graph and s is an arbitrary vertex in G. Let k be an integer. Then with probability at least 3=4, a random walk of length l = l(k) O(k 3 ) on G from s visits at least k distinct vertices or the whole connected component of s. We could equally use a ....

Greg Barnes and Uriel Feige. Short random walks on graphs. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pages 728--737. ACM, 1993.

....is a drawback of DFS. Using randomness in resolving ties, the average cover time by DFS can be reduced to 1:5m [GA90] Another, memoryless, method is the random walk just choose a random neighbor of the current vertex and go there. Clearly, covering by a random walk is rather slow; in [AKLLR79] [BF93] it was shown to cover a graph within expected time O(mn) where m is the number of edges and n the number of vertices. It was also shown that, under a proper initial distribution of k agents, their simultaneous random walk covers the graph in expected time O(mn=k) BKRU94] More sophisticated ....

G. Barnes, U. Feige, "Short Random Walks on Graphs," SIAM J. Disc. Math., 9(1) (1996), pp. 19-28.

....graph in O(m n) time, but requires Omega Gamma n) space. Alternatively, a random walk can traverse an undirected graph using only O(log n) space, but requires Theta(mn) expected time (Aleliunas et al. 2] In fact, Feige [28] based on earlier work of Broder et al. 20] and Barnes and Feige [7], has shown that there is a spectrum of compromises between time and space for this problem: any graph can be traversed in space S and expected time T , where ST mn(log n) O(1) d min and d min is the minimum degree of any vertex. This raises the intriguing prospect of proving that logarithmic ....

....of the JAG that provide progress toward proving this conjecture and, in fact, establish such a lower bound for one variant. These results are outlined below. The upper bound of ST mn(log n) O(1) d min by Feige [28] and the preceding upper bounds of Broder et al. 20] and Barnes and Feige [7], are established on a model that is actually a restricted variant of the JAG. In their algorithms, the JAG initially drops P Gamma 1 pebbles on random vertices, after which they are never moved. It then uses its last pebble to explore the graph (probabilistically) with the others as fixed ....

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G. Barnes and U. Feige. Short random walks on graphs. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pages 728--737, San Diego, CA, May 1993.

....to an adjacent vertex chosen uniformly at random. The study of random walks in graphs has many applications in the design of algorithms in the study of distributed computation (Broder Karlin 1989) space bounded computation (Aleliunas et al. 1979, Borodin et al. 1989) time space tradeoffs (Barnes Feige 1996, Broder et al. 2 Chandra, et al. 1994, Feige 1993) and in the design of approximation algorithms for some hard combinatorial problems (Dyer et al. 1991, Jerrum Sinclair 1989) Doyle Snell (1984) exposed many interesting connections between random walks and electrical network theory, and ....

G. Barnes and U. Feige, Short random walks on graphs. SIAM J. Disc. Math. 9(1) (1996), 19--28.

....of a point to be chosen as the next location of the robot depends on both the previous location and the shape of the region being explored. ffl Coverage Processes: The rate of coverage of graphs by random walk has been studied intensively (e.g. Aleliunas, Karp, Lipton, Lovasz Rakoff 1979] [Barnes Feige 1993], Broder, Karlin, Raghavan, Upfal 1994] Representative results in this context are the upper bounds of O(mn) on the cover time of a graph with m edges and n vertices, and O(mR log n) where R is the resistance of the graph, assuming all edges to be 1 Ohm resistors. On the other hand, Coverage of ....

G. Barnes, U. Feige, "Short Random Walks on Graphs," in Proc. of the 25'th ACM STOC, 1993.

.... following chain of complexity classes NC 1 L NL SAC 1 = LOGCFL AC 1 NC 2 : The research on graph connectivity is voluminous and even since Wigderson s excellent survey of the state of the art in 1992 [17] there have been significant new developments in connectivity algorithms [4, 12] and lower bounds on restricted models of computation [11, 3, 10, 19] The key tool in showing that every problem in NL may be solved with circuits of relatively small depth is the Repeated Squaring or Pointer Doubling algorithm for transitive closure. Another way of phrasing some of these ....

Greg Barnes and Uri Feige. Short random walks on graphs. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pages 728--737, San Diego, CA, May 1993.

....The algorithm we describe uses short random walks, as suggested by Karger, Nisan and Parnas [KNP92] Before describing our algorithm, we shortly review the algorithm of Karger et al. and sketch the changes that we make to it. 5. 1 The algorithm of Karger, Nisan and Parnas Barnes and Feige [BF93] showed that a random walk of length s in a connected graph is likely to visit at least Omega Gamma s 1=3 ) vertices (or all the vertices of the graph) More precisely, for some constant c0 0, the probability that such a random walk visits at least c0 Delta s 1=3 vertices is at least 3=4. ....

G. Barnes and U. Feige. Short random walks on graphs. In Proceedings of the 25rd Annual ACM Symposium on Theory of Computing, San Diego, California, pages 728--737, 1993.

....n) space and only Theta (mn) time [AKLLR79] More generally, Broder et al. BKRU89] have exhibited a family of probabilistic algorithms that achieves a tradeoff between the time and the space of S Delta T 2 m 2 log O(1) n. This has been improved to S Delta T 2 m 1:5 n :5 log O(1) n [BF93]. A long term goal is to prove a matching lower bound. Proving lower bounds for a general model of computation, such as a Turing machine, is beyond the reach of the current techniques. Thus it is natural to consider a structured model [Bo82] whose basic operations are based on the structure of ....

G. Barnes, and U. Feige. Short random walks on graphs. In Proceedings of the Twenty Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, May 1993.

....and computing the average take space no more than O(log p log n) O(a log m log n) So the total space needed is O(a log n) as desired. 6 Pseudo random Walks In this section we present algorithm WALK and prove Lemma 3.3. We start with the following Lemma whose proof is shown in [BF93] Lemma 6.1 Suppose G is a connected graph and i is an arbitrary vertex in G. Let p be an integer. Then with probability at least 3=4, a random walk of length l = l (p) O(p 3 ) on G from i visits at least p distinct vertices. Since to proceed a random walk on a graph takes space ....

G. Barnes and U. Feige. Short random walks on graphs. In Proc. 25th ACM Symposium on Theory of Computing (STOC), pages 728--737, 1993.

....be too big and they do not take part in any of the subsequent operations. It is easy to identify, in O(log s i ) time, all the trees that are small enough (i.e. not too big) This enables WALK HOOK to run in O(log s i log log n) time. It is shown in [HZ94] using a result of Barnes and Feige [BF93] that with v.h.p. each act vertex v has a cell [v; t] whose tree T [v;t] is not too big, and also satisfies at least one of the following three conditions: i) T [v;t] contains cells belonging to at least s 1=3 i different vertices; ii) T [v;t] contains a cell belonging to an ina vertex; ....

G. Barnes and U. Feige. Short random walks on graphs. In Proceedings of the 25rd Annual ACM Symposium on Theory of Computing, San Diego, California, pages 728--737, 1993.

....many other papers. In [49] an initial step is done towards developing an analytical approach to a cooperative cleaning method where the dirt on the floor is used as a marking. b) Computing Graph search is an old problem; several methods exist for deterministic (e.g. 48] 24] random (e.g. 2] [11], 13] and semi random ( 25] covering, but a lot more needs to be done in order to make the theory useful in the context of robotic covering problems. A step towards a trace oriented theory of search was done in [10] and [12] where pebbles are used to assist the search. Pebbles are tokens ....

G. Barnes, U. Feige, "Short Random Walks on Graphs," in Proc. of the 25'th ACM STOC, 1993.

....of time, and T dn, then our method guarantees that no change will be missed. Related work: Graph search is an old problem; several methods exist for deterministic (e.g. Even 1979] Fraenkel 1970] Hopcroft Tarjan 1973] Tarry 1895] Tarjan 1972] random (e.g. Aleliunas et al. 1979] [Barnes Feige 1993], Broder et al. 1994] and semirandom ( Gal Anderson 1990] covering. A step towards a trace oriented theory of search was done in [Blum Sakoda 1977] and [Blum Kozen 1978] where pebbles are used to assist the search. Pebbles are tokens that can be placed on the floor and later be removed. ....

G. Barnes, U. Feige, "Short Random Walks on Graphs," in Proc. of the 25'th ACM STOC, 1993.

.... Theta (mn) and uses only O (log n) space [AKLLR79] More generally, Broder et al. BKRU89] have exhibited a family of probabilistic algorithms that achieves a tradeoff of S Delta T 2 m 2 log O(1) n between space and time. This has been improved to S Delta T 2 m 1:5 n :5 log O(1) n [BF93]. A long term goal is to prove a matching lower bound. Deterministic non uniform algorithms for st connectivity can be constructed using Universal traversal sequence. The algorithm uses a single pebble to traverse the connected components of the graph. Sequences of length O Gamma n 4 log ....

G. Barnes, and U. Feige. Short random walks on graphs. In Proceedings of the Twenty Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, May 1993.

....Start a random walk of length t = O(s 2 ) at an end point of e. If for some edge e 0 along the walk, h(e 0 ) r, output large. Else output small. Clearly, if the component is small, then the probability of outputting small is at least 1=2. If the component is large, we may use the result of [3] to show that with high probability, the random walk visits at least 8e edges (regardless of the size and shape of the connected component) If h is chosen from a universal family of hash functions [5, 13] then it is likely that the hashed value of some visited vertex agrees with r. A different ....

....procedure for estimating the size of connected components. The tricky issue is to have this running time depend on R, which is a global property of the whole graph, and may not be reflected in local regions of the graph (a single connected component) To overcome this problem, we use an idea of [3, 9]. We artificially increase the number of edges in G by a constant factor, by placing self loops on vertices of G. This is done in such a way that guarantees quick estimation of sizes of connected components, where the size of a connected component now also takes into account its self loops. ....

G. Barnes and U. Feige. "Short Random Walks on Graphs". To appear in SIAM Journal on Discrete Mathematics. Preliminary version appeared in Proc. of 25 th STOC, 1993, pp. 728-737.

....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 random walks. Preliminary results in this direction (tight up to a logarithmic factor) were presented in an earlier version of this paper [5]. The superfluous logarithmic factor in these results was subsequently removed by Feige [13] building upon proof techniques that were developed by Aldous [3] Aldous is writing a textbook giving SHORT RANDOM WALKS 3 a systematic account of random walks on graphs and reversible Markov chains. The ....

....al. 8] One key property of Broder et al. s algorithm is that a short random walk from a given edge traverses many edges. Improved bounds on E[T (M) then, would seem to provide an improvement to their tradeoff. Partial results in this direction were presented in an earlier version of our paper [5], and further improvements are presented by Feige [13] 2. Proofs of theorems. The proofs of the theorems are best read in order. The proof of Theorem 1.1 introduces the proof techniques that are used in all subsequent proofs. The proof of Theorem 1.2 is a simple modification of this proof ....

G. Barnes and U. Feige, Short random walks on graphs, in Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, May 1993, pp. 728-- 737.

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G. Barnes and U. Feige. Short random walks on graphs. In Proceedings of the 25rd Annual ACM Symposium on Theory of Computing, San Diego, California, pages 728-737, 1993.

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G. Barnes and U. Feige. Short random walks on graphs. SIAM Journal on Discrete Mathematics, 9:19--28, 1996.

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