Random walks, universal traversal sequences, and the complexity of maze problems

Aleliunas, Romas; Karp, Richard M.; Lipton, Richard J.; Lovász, László; Rackoff, Charles

doi:10.1109/sfcs.1979.34

Cited by 563 publications

(540 citation statements)

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“…17 An important step towards the solution was made by Aleliunas, Karp, Lipton, Lovász, and Rackoff [AKL79]. They showed that this problem can be solved by a probabilistic logspace algorithm.…”

Section: Introduction As In Earlier Sections We Consider An (N D)-gmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,509

1,355

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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Section: Introduction As In Earlier Sections We Consider An (N D)-gmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,509

1,355

View full text Add to dashboard Cite

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“…The mixing-rate of a non-backtracking random walk on a regular graph, in terms of its eigenvalues, was computed in [2], using some properties of Chebyshev polynomials of the second kind. It is shown in [2] that this rate is always better than that of the simple random walk, provided that d = n o (1) . In fact, the mixing rate of the non-backtracking random walk may be up to twice faster, and the closer the graph is to being a Ramanujan graph (that is, a graph satisfying λ ≤ 2 √ d − 1), the closer the ratio between the two mixing-rates is to 2(d − 1)/d.…”

Section: Background and Definitionsmentioning

confidence: 97%

“…This fact was exploited for example in [1], [5] and [13], in the study of space efficient algorithms for S − T connectivity in undirected graphs. Another well known example is the conservation of random bits in the amplification of randomized algorithms (we will elaborate on this point later on).…”

Section: Background and Definitionsmentioning

confidence: 99%

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Poisson approximation for non-backtracking random walks

Alon

Lubetzky

2009

Isr. J. Math.

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Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. The authors of [2] studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length n on a high-girth n-vertex regular expander, is typically (1 + o(1)) log n log log n , as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes.In this work, we answer the above question by combining a generalized form of Brun's sieve with some extensions of the ideas in [2]. Let N t denote the number of vertices visited precisely t times by a non-backtracking random walk of length n on a regular n-vertex expander of fixed degree and girth g. We prove that if g = ω(1), then for any fixed t, N t /n is typically 1 et! + o(1). Furthermore, if g = Ω(log log n), then N t /n is typically 1+o (1) et! uniformly on all t ≤ (1 − o(1)) log n log log n and 0 for all t ≥ (1 + o(1)) log n log log n . In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.

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“…For k ≥ 2, it is possible to identify the k-classes and the corresponding distinguishing paths (from knowledge of the k − 1 classes) using the properties below. There exists a UXS for graphs of size at most N and maximum degree at most d, that is of length O(N 3 d 2 log N ) [1]. Using such a sequence for the traversal gives us an algorithm of move complexity O(N 3 n 3 d 3 log N ) for solving rendezvous.…”

Section: Agents Having Unbounded Memorymentioning

confidence: 99%

Deterministic Symmetric Rendezvous in Arbitrary Graphs: Overcoming Anonymity, Failures and Uncertainty

2013

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We consider the rendezvous problem of gathering two or more identical agents that are initially scattered among the nodes of an unknown graph. We discuss some of the recent results for this problem focusing only on deterministic algorithms for the general case when the graph topology is unknown, the nodes of the graph may not be uniquely labeled and the agents may not be synchronized with each other. In this scenario, the objective is to solve rendezvous whenever deterministically feasible, while optimizing on the amount of movement by the agents or the memory required (for the nodes or the agents) in the worst case. Further we also investigate some special scenarios such as (i) when the graph contains some dangerous nodes or, (ii) when there is no consistent ordering on the edges of a node. We present positive results, complexity analysis and some general techniques for dealing with such worst case scenarios for the symmetric rendezvous problem.

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Random walks, universal traversal sequences, and the complexity of maze problems

Cited by 563 publications

References 5 publications

Expander graphs and their applications

Expander graphs and their applications

Poisson approximation for non-backtracking random walks

Deterministic Symmetric Rendezvous in Arbitrary Graphs: Overcoming Anonymity, Failures and Uncertainty

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