A.Z. Broder, A.R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In ACM Symposium on Theory of Computing (STOC), pages 543--549, 1989.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....about global properties. This is very useful in computation, where limited resources are used to determine global information. For example, random walks have been used in many applications in computer science, including on line algorithms [9] space efficient algorithms for undirected connectivity [7], approximation algorithms [10, 14] generation of random spanning trees [2, 6, 16] assigning processes to nodes in networks [4] and token management schemes and self stabilizing in distributed computing [13, 18] Frequently, we ask for the cover time of a graph, i.e. the expected time for a ....

A.Z. Broder, A.R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In ACM Symposium on Theory of Computing (STOC), pages 543--549, 1989.

....works by repeatedly moving from a vertex to a new vertex, where the new vertex is chosen at random from the neighbors of the old vertex. Aleliunas et al. show that if s is connected to t, then such a walk of length 2m(n 0 1) taken from s will visit t with probability at least 1=2. Broder et al. BKRU89] show a nearly smooth time space tradeoff between the space optimal random walk algorithm and the time optimal depth and breadthfirst search algorithms. Note that both algorithms have a time space product of 2(nm log n) For space O(s) log n s n log n, Broder et al. s algorithm uses 2(m 2 ....

....landmark to vg generate bbfs(v; b) if v s neighborhood is not full then return ( small ) for all landmarks, l, in order do begin generate bbfs(l; b) if v s neighborhood overlaps l s then return (l) end; end cl. Figure 2. 2: The cl function of the simple algorithm see Broder et al. BKRU89] and a factor of n when the space approaches its upper bound (O(n log n) In summary, we have shown the following. Theorem 2.2 For any n 1=2 log n s n log n, the simple algorithm, presented above, solves ustcon for arbitrary n vertex, m edge graphs in space O(s) and time O( m n)n 2 ....

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A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, pages 543--549, Seattle, WA, May 1989.

....achieve a time space tradeoff of ST = O(mn) where T denotes time and S denotes space) Is there a randomized algorithm that for any space bound log n S n, requires time T = O(mn=S) In this paper we answer this question in the affirmative. 1. 1 Previous related work Broder et al. [BKRU] were the first to develop an algorithm that achieves a nontrivial tradeoff. Their algorithm is based on the following observation: the fraction of wasted steps of the random walk (steps in which the walk revisits previously visited vertices) increases as the walk grows longer. Hence many short ....

....previously visited vertices) increases as the walk grows longer. Hence many short random walks are more efficient in exploring a graph than one long random walk. We now give a high level overview of the algorithm of Broder et al. Some implementation details are left out, and can be found in [BKRU]. The algorithm that is developed in the current paper is obtained via simple modifications of [BKRU] s algorithm. The overview is presented in a way that will simplify subsequent presentation of modifications to the algorithm. Input to the algorithm: A graph G with n vertices and m edges, two ....

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A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal. "Trading space for time in undirected s-t connectivity". SIAM J. on Computing, Vol 23, 1994, 324--334.

....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 algorithms [6, 12, 17] efficient constructions in cryptography [14] and self stabilizing distributed computing [11, 16] Frequently (see, for example, Karger et al. 19] and Nisan et al. 20] we are interested in E[T (N ....

....Karger et al. 19] and Nisan et al. 20] we are interested in E[T (N ) the expected time before a simple random walk on an undirected connected graph, G, visits its N th distinct vertex, N n. The corresponding question for edges is also interesting, and arises in the work of Broder et al. [8]: how large is E[T (M) the expected time before 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 ....

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A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal, Trading space for time in undirected s-t connectivity, in Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing, Seattle, WA, May 1989, pp. 543--549.

....to a neighbour chosen with probability proportional to the weight of traversed edge. Random walks have been studied extensively, and have numerous applications, including generation of random spanning trees [1, 6, 16, 18] online algorithms[8] space efficient algorithms for undirected connectivity[7], approximation algorithms [9, 14] assigning processes to nodes in networks [4] and token management schemes and self stabilizing in distributed computing [13, 17] The cover time of a random walk on a graph is the number of steps required for all vertices to be visited. The hitting time between ....

A.Z. Broder, A.R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In ACM Symposium on Theory of Computing (STOC), pages 543--549, 1989.

....about global properties. This is very useful in computation, where limited resources are used to determine global information. For example, random walks have been used in many applications in computer science, including on line algorithms [9] space efficient algorithms for undirected connectivity [7], approximation algorithms [10, 14] generation of random spanning trees [2, 6, 16] assigning processes to nodes in networks [4] and token management schemes and self stabilizing in distributed computing [13, 18] Frequently, we ask for the cover time of a graph, i.e. the expected time for a ....

A.Z. Broder, A.R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In ACM Symposium on Theory of Computing (STOC), pages 543--549, 1989.

.... there exists at most one path joining any two nodes, have many applications too [AS] OSW] Ya78] and have been studied a lot [Br] KP] EP] Ke] MM] Pro] Problems similar to graph searching have been studied too, like the s t connectivity, that decides if a node t is accessible from a node s [BKRU], random walks [GJ] traversal sequences [AKLLR] or parallel searching [KUW] In undirected graphs, when we count the number of edges used, we count the edges we use in both directions twice. The total number of edges counted may be up to twice the number of existing edges, but the same ....

....will fail only with probability O(n Gamma2 ) AV] ffl The algorithms of Dyer and Frieze that address the NP complete problems of graph coloring, minimum cut, and graph partitioning [DF] ffl The algorithm of Broder, Karlin, Raghavan and Upfal that detects s t connectivity in limited space. [BKRU]. ffl The algorithm of Cherigan and Hagerup that finds the maximum flow in O(nm) average time [CH] 17 Some algorithms use properties of random graphs and produce results that are valid for almost all graphs, and these algorithms run much faster than their deterministic counterparts [Ti] ....

A.Z. Broder, A.R. Karlin Prabhakar Raghavan, and Eli Upfal. Trading space for time in undirected s-t connectivity, Proceedings of the 21st Annual ACM Symposium on the Theory of Computing, 1989, 543-549.

.... is allowed, these bounds can be improved to O (log n) space and O Gamma n 4 log n Delta time [CRRST89, KLNS89] If probabilism is allowed, random walks can traverse any component of the graph using O (log 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. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, pages 543--549, Seattle, WA, May 1989.

....slow algorithm for ustcon. At the other extreme, well known deterministic methods like depth and breadth first search provide time optimal, but space intensive solutions. Both extremes exhibit a time space product of O(mn log n) The probabilistic algorithm of Broder, Karlin, Raghavan, and Upfal [10] provides a spectrum of compromises roughly between these two extremes: time O(m 2 log 5 n=s) with space s. For deterministic algorithms, less is known. For directed graphs, depthand breadth first search are still applicable, still time optimal, and still require space 2(n log n) in the worst ....

.... there is one algorithm solving a restricted case of ustcon in O(log 2 n= log log n) space, namely the case of undirected graphs of (log n) bounded genus, solved by Kriegel [17] None of these results provides the sort of spectrum of compromises between time and space that Broder, et al. [10] give. The main results of our paper are three new deterministic algorithms for undirected s t connectivity that achieve sublinear space and polynomial time simultaneously. The algorithms provide a nearly smooth tradeoff for ustcon between time efficient, space intensive algorithms, such as ....

[Article contains additional citation context not shown here]

A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, pages 543-- 549, Seattle, WA, May 1989.

....n node, m edge graph. However, they require Omega (n) space on a RAM. Alternatively, this problem can be solved probabilistically using random walks. The expected time to traverse any component of the graph is only 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 ....

A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, pages 543--549, Seattle, WA, May 1989.

....any n vertex, m edge 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 ....

A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. SIAM Journal on Computing, 23(2):324--334, Apr. 1994.

....vertex, m edge undirected 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 ....

....lower bounds for variants 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 ....

[Article contains additional citation context not shown here]

A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. SIAM Journal on Computing, 23(2):324--334, Apr. 1994.

....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 traced the origins of the topic back into ....

A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal, Trading space for time in undirected s-t connectivity. SIAM J. Comput. 23(2) (1994), 324--334.

....as follows. Depth first or breadth first search can traverse any n vertex, m edge undirected graph in O(m n) time, but requires Omega0 n) space. Alternatively, a random walk can traverse an undirected graph using only O(log n) space, but requires 2(mn) expected time [2] In fact, Broder et al. [16] have 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 = O(m 2 log 5 n) This raises the intriguing prospect of proving that logarithmic space and linear time are not simultaneously ....

....perhaps 2(mn) The main results of this paper are lower bounds for variants 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 = O(m 2 log 5 n) by Broder et al. [16] is established on a model that is actually a restricted variant of the JAG. In their algorithm, the JAG initially drops P 0 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 landmarks. In ....

[Article contains additional citation context not shown here]

A. Z. Broder, A. R. Karlin, P. Raghavan, and E. Upfal. Trading space for time in undirected s-t connectivity. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, pages 543--549, Seattle, WA, May 1989.

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