R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems". In 20th Annual Symposium on Foundations of Computer Science, pages 218--223, San Juan, Puerto Rico, October 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Center, Yorktown Heights, New York. copper watson.ibm.com. y Department of Applied Math. 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 ....

....are assigned infinite weight . Our goal is to upper bound R 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 ....

[Article contains additional citation context not shown here]

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems". In 20th Annual Symposium on Foundations of Computer Science, pages 218--223, San Juan, Puerto Rico, October 1979.

....Fast Randomized LOGSPACE Algorithm for Graph Connectivity Uriel Feige August 5, 1996 Abstract We study the relationship between undirected graph reachability and graph connectivity, in the context of randomized LOGSPACE algorithms. Aleluinas et al. [2] show that graph reachability (checking whether there is a path connecting vertices S and T ) can be decided in logarithmic space and polynomial time, by starting a random walk at S, and checking whether T is hit within some time limit. The randomized algorithm has one sided error (with small ....

....or reachability, can be decided more quickly by LOGSPACE algorithms This is a hypothetical question, since we do not know that any of these problems can be decided in deterministic LOGSPACE. The above question becomes more concrete in the context of randomized algorithms. Aleluinas et al. [2] show that reachability can be decided in logarithmic space and polynomial time, by a randomized algorithm that starts a random walk at S, and checks whether T is hit within some time limit. This time limit is polynomially related to n, and is easily computable in LOGSPACE from the description of ....

[Article contains additional citation context not shown here]

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems". In 20th FOCS, 218--223, 1979.

.... 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 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 ....

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems". In 20th Annual Symposium on Foundations of Computer Science, pages 218--223, San Juan, Puerto Rico, October 1979.

.... Omega notation is used in order to suppress polylogarithmic terms. e.g. n log n is O(n) We also use the convention that suppressed terms are polylogarithmic in n, regardless of the explicit terms that appear in O( Hence log n is O(1) and p log n is O(p) Aleliunas et al.[AKLLR] showed that a randomized algorithm can solve USTCON in space O(log n) Their algorithm performs a random walk on the vertices of G, where at each time step the walk moves to a vertex chosen uniformly at random from the neighbors of the current vertex. Aleliunas et al. AKLLR] prove that if S and T ....

....O(p) Aleliunas et al. AKLLR] showed that a randomized algorithm can solve USTCON in space O(log n) Their algorithm performs a random walk on the vertices of G, where at each time step the walk moves to a vertex chosen uniformly at random from the neighbors of the current vertex. Aleliunas et al.[AKLLR] prove that if S and T are connected, then a random walk that starts at S is expected to reach T in mn steps. This leads to a randomized algorithm with one sided error: start a walk at S. If T is reached within 3mn steps, declare that S and T are connected. Otherwise not connected. The error ....

[Article contains additional citation context not shown here]

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems". In Proc. of 20th Annual Symposium on Foundations of Computer Science, 218--223, San Juan, Puerto Rico, October 1979.

....cover times. The expected running time of this algorithm is proportional to the actual cover time, while the accuracy of the approximation depends on the variance of the cover time. Since for random walks both the cover time and its variance are bounded by a polynomial in n = jV (G)j (see, e.g. [2]) the above algorithm does achieve the goal. One negative feature of this algorithm (from a purely theoretical point of view) is that it does not give much information about the relations between the global structure of the graph and its cover time. All this does not apply to arbitrary ....

....number of vertices of the graph, or on some property of the subclass from which G has been driven. These general bounds do not directly lead to any reasonable approximation of the cover time of an input graph with specified starting vertex. We survey some of the known results. Aleliunas et al. [2] were the first to give an n 3 upper bound on the cover time for random walks. Their approach can be described as follows. From graph G one can derive a digraph G 0 , where every pair of vertices (u; v) is connected by a (directed) edge of weight H [u; v] Let HPATH s (G 0 ) be the weight of ....

[Article contains additional citation context not shown here]

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems ". In 20th Annual Symposium on Foundations of Computer Science, pages 218--223, San Juan, Puerto Rico, October 1979.

....it takes a random walk that starts at v to visit all vertices of the graph. For a graph G(V; E) its hitting time H[G] commute time C[G] cover time EC[G] respectively) is defined as H[G] max u;v2V [H[u; v] C[G] max u;v2V [C[u; v] EC[G] max v2V [EC[v] respectively) Aleliunas et al. [2] have shown that for any connected graph, EC n 3 . It has been conjectured that for any graph, EC 4 27 n 3 o(n 3 ) see [1] This bound is best possible (up to low order terms) the hitting time for the lollipop graph (a path of length n=3 connected to a clique of size 2n=3) is 4 ....

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems". In 20th Annual Symposium on Foundations of Computer Science, pages 218--223, San Juan, Puerto Rico, October 1979.

....state i, and the cover time C i is the first time every state in S is visited at least once starting from state i. Computer scientists originally became interested in analyzing the expected cover times for graphs in an attempt to obtain bounds on the space complexity of undirected st connectivity [6] (Given an undirected graph G and two specified vertices s and t in G, determine if there is a path connecting s and t) There are several other topics in computer science and graph theory that motivate the investigation of cover times: ffl The study of the relations between combinatorial ....

....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 problems might lead one to conjecture that adding more edges to the graph would reduce the expected cover time. ....

[Article contains additional citation context not shown here]

R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz and C. Rackoff, "Random walks, universal traversal sequences, and the complexity of maze problems," Proceedings of the 20th IEEE Symposium on Foundations of Computer Science, 1979, 218--233.

....problem becomes much harder. However, the additional complications that arise from the extra states do not have so much to do with the checking problem itself, but instead are due to the well known difficulties of traversing unknown graphs, i.e. the universal traversal problem for directed graphs [AKLLR]. Suppose that A is a specification FSM with n states and p inputs and that B is an implementation machine with at most n D states. Suppose that B is identical to A except for the D extra states that hang off from a state of B where there happens to be an incorrect transition. Then it is like ....

....techniques to the composite machine. Again, we may run into the state explosion problem. There are heuristic procedures for test generation for CFSM s such as random walk; we select the next input at random and on line. It is well known that we may be trapped in one small portion of the system [AKLLR]. To cope with this, guided random walk was suggested. Specifically, we only want to test transitions of each component machine and that often provides a reasonable fault coverage. On the other hand, we can afford to keep track of each component machine instead of the composite machine. For this ....

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lovasz, and C. Rackoff, "Random walks, universal traversal sequences, and the complexity of maze problems," in Proc. 20th Ann. Symp. on Foundations of Computer Science, pp. 218-223, 1979.

....failures 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 ....

....on cover times for a sample of graph covering algorithms, assuming the graph to be static and to have n vertices, m edges, diameter d and maximum vertex degree Delta. For randomized algorithms the expected bound is given. method reference cover time depth first search [T95] 2m random walk [AKLLR79] O(mn) exp. k random walkers [BKRU94] O(mn=k) exp. semi random [GA90] 1:5m(exp. learning real time A [K90] O(mn) counter based [T92] O(n 2 d) nearest neighbor [KS96] O(m log n) k edge ant walkers [WLB96a] O( Deltan 2 =k) vertex ant walk [WLB98] O(nd) All the algorithms and cover ....

R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz, C. Rakoff, "Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems," Proc. FOCS'79, (1979) pp. 218-223.

....to this algorithm as the expected hitting time routing algorithm, and investigate its performance on general networks as well as on specific networks, particularly, trees and cycles. 4.6. 1 General Case Analysis The hitting time of random walks on general graphs was analyzed by Aleliunas et al. [2]. One of their results is a bound for the expected hitting time in general graphs, H max 2mD, where m is the number of edges in the graph, and D is its diameter. An improved bound for H max was presented by Chandra et al. 10] Their work links the commute time between two nodes to the effective ....

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lovasz, C. Rackoff. "Random walks, universal traversal sequences and the complexity of maze problems ". 20th Annual Symposium on Foundations of Computer Science, pp. 218-223, San Juan, Puerto Rico, October 1979.

....work) is different, however, because the probability 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 ....

....UTSA In this section we shall show that if F is the set of all n size unit grid polygons, i.e. polygons made of n attached 1 Theta 1 squares) then such a sequence exists and has a cover time polynomial in n 10 . For this purpose we follow the probabilistic method invented by Erdos and used in [Aleliunas, Karp, Lipton, Lovasz Rakoff 1979] to prove that a sequence of length O(n 4 log n) exists that covers any edge labelled k regular graph 11 with n vertices. Theorem 4 There exists a sequence of angles that covers, within time 4n 4 log n, any rectilinear gridded polygon of size n. Proof: First let us observe that if F is ....

R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz, C. Rakoff, "Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems," in 20'th Annual Symposium on Foundations of Computer Science, p. 218-223, San Juan, Puerto Rico, October 1979.

....collector argument was generalized by Matthews to give the upper bound E u [G] max x;y [H [x; y] ln n [8] This bound is particularly useful for graphs that have very fast cover times, such as expander graphs. A different approach for bounding the cover time was introduced by Aleluinas et al. [2]. In this approach, the coupons are to be collected in a particular favorable order. A convenient order to consider is one that agrees with Deapth First Search of a minimum weight spanning tree of the graph. Let T be a tree defined on the vertices of G (though tree edges need not correspond to ....

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems". In 20th Annual Symposium on Foundations of Computer Science, pages 218--223, San Juan, Puerto Rico, October 1979.

....investigating the recuerrence properties of random walks on 1, 2 and 3 dimensional grids. The rate of coverage of graphs by a random walk has been studied intensively. Two 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 [2], and O(mae log n) where ae is the resistance of the graph, assuming all edges to be 1 Ohm resistors [11] In [10] it was shown that several random walkers, if properly distributed in the graph, can bring a significant speed up to the process of covering. On the other hand, Coverage of continuous ....

....point. In this section we shall show that if F is the set of all n size unit grid polygons, i.e. polygons made of n attached 1 Theta 1 squares) then such a sequence exists and has a length polynomial in n. For this purpose we follow the probabilistic method invented by Erdos and used in [2] to prove that a sequence of length O(n 3 log n) exists that covers any edge labelled k regular graph 8 with n vertices. 6 It is of interest to mention a lumped circuit analogy: a square m Theta m mesh of 1 Ohm resistors is known [11] to have resistance Theta(log n) 7 This value of the ....

R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz, C. Rakoff,"Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems,"in 20'th Annual Symposium on Foundations of Computer Science, p. 218-223,San Juan, Puerto Rico, October 1979.

....and 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 ....

R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz, C. Rakoff, "Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems," in 20'th Annual Symposium on Foundations of Computer Science, p. 218-223, San Juan, Puerto Rico, October 1979.

....that this happens is at least 1=n, thus in expected n attempts it will indeed occur. Now it follows that the expected time for a target trail of initial length l to become a single edge is E[ l Delta n Delta E[ r ] n 2 E[ r ] 2) For the return time r , combining arguments of [3] and averaging principles (about odd and even cycles) it can be shown that E[ r ] O(m 2 ) 3) Now (1) 2) and (3) complete the proof of the Main Theorem 3.1. 4 More General Protocols and Cover Times Three main restrictions define the class of SDFF s: symmetry, two states per enity, and ....

....to its starting combined state. Figure 3: Assymmetric counter example to rapid mixing. For a random walk on a graph, the cover time is the time by which all vertices have been visited at least once. All symmetric (undirected) graphs are known to have cover times at most cubic in their sizes [3], and most graphs have cover times slightly bigger than linear. Thus all symmetric protocols posses the small cover time property, and, in the sense of exhaustive search of the reachable state space, are amenable to effective testing by random walk. 0 1 carry b 1 a 0 b 0 a 1 a 0 b 0 a 1 b 1 ....

Aleliunas, R., Karp, R.M., Lipton, J.R., Lovasz, L., and Rackoff, C., "Random walks, universal traversal sequences, and the complexity of maze problems", Proc. 20th IEEE Symp. on Foundations of Computer Science, 1979, 218-233.

.... 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 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, ....

R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lov'asz, and C. Rackoff. "Random walks, universal traversal sequences, and the complexity of maze problems". In 20th Annual Symposium on Foundations of Computer Science, pages 218--223, San Juan, Puerto Rico, October 1979.

....more than once in T units 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 ....

R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz, C. Rakoff, "Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems," in 20'th Annual Symposium on Foundations of Computer Science, p. 218-223, San Juan, Puerto Rico, October 1979.

No context found.

R. Aleliunas, R.M. Karp, R.J. Lipton, L. Lovasz, and C. Rackoff, "Random Walks, Universal Traversal Sequences, and the Complexity of the Maze Problem, " Proceedings of the 20th annual IEEE Found. of Comp. Sci. Symp. , pages 218-223, October 1979.

No context found.

R. Aleliunas, R. Karp, R. Lipton, L. Lovasz, and C. Rackoff, "Random Walks, Universal Traversal Sequences, and the Complexity of the Maze Problem, " Proceedings of the 20th annual IEEE Found. of Comp. Sci. Symp. (1979), 218-223.

No context found.

Aleliunas, Karp, Lipton, Lov'az, and Rackoff, "Random Walks, Universal Traversal Sequences, and the Complexity of Maze Problems," 20th IEEE FOCS Symp., 1979, (218-233).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC