Random walks which prefer unvisited edges : exploring high girth even degree expanders in linear time.

Let G = (V,E) be a connected graph with |V | = n vertices. A simple random walk on the vertex set of G is a process, which at each step moves from its current vertex position to a neighbouring vertex chosen uniformly at random. We consider a modified walk which, whenever possible, chooses an unvisited edge for the next transition; and makes a simple random walk otherwise. We call such a walk an edge-process (or E -process). The rule used to choose among unvisited edges at any step has no effect on our analysis. One possible method is to choose an unvisited edge uniformly at random, but we impose no such restriction. For the class of connected even degree graphs of constant maximum degree, we bound the vertex cover time of the E -process in terms of the edge expansion rate of the graph G, as measured by eigenvalue gap 1 -λmax of the transition matrix of a simple random walk on G. A vertex v is ℓ -good, if any even degree subgraph containing all edges incident with v contains at least ℓ vertices. A graph G is ℓ -good, if every vertex has the ℓ -good property. Let G be an even degree ℓ -good expander of bounded maximum degree. Any E -process on G has vertex cover time equation image This is to be compared with the Ω(nlog n) lower bound on the cover time of any connected graph by a weighted random walk. Our result is independent of the rule used to select the order of the unvisited edges, which could, for example, be chosen on-line by an adversary. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2013 As no walk based process can cover an n vertex graph in less than n - 1 steps, the cover time of the E -process is of optimal order when ℓ =Θ (log n). With high probability random r -regular graphs, r ≥ 4 even, have ℓ =Ω (log n). Thus the vertex cover time of the E -process on such graphs is Θ(n)

Multiple random walks on paths and grids

We derive several new results on multiple random walks on "low dimensional" graphs. First, inspired by an example of a weighted random walk on a path of three vertices given by Efremenko and Reingold, we prove the following dichotomy: as the path length n tends to infinity, we have a super-linear speed-up w.r.t. the cover time if and only if the number of walks k is equal to 2. An important ingredient of our proofs is the use of a continuous-time analogue of multiple random walks, which might be of independent interest. Finally, we also present the first tight bounds on the speed-up of the cover time for any d-dimensional grid with d >= 2 being an arbitrary constant, and reveal a sharp transition between linear and logarithmic speed-up

Euler Tour Lock-in Problem in the Rotor-Router Model

International audienceThe rotor-router model, also called the Propp machine, was first considered as a deterministic alternative to the random walk. It is known that the route in an undirected graph G=(V,E), where |V|=n and |E|=m, adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In recent work [Yan03] Yanovski et al. proved that independently of the initial configuration of the rotor-router mechanism in G the agent locks-in in time bounded by 2m diam, where diam is the diameter of G. This upper bound can be matched asymptotically in lollipop graphs. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. The case study is performed in the form of a game between a player pl intending to lock-in the agent in an Euler tour as quickly as possible and its adversary ad with the counter objective. First, we observe that in certain (easy) cases the lock-in can be achieved in time O(m). On the other hand we show that if adversary ad is solely responsible for the assignment of ports and pointers, the lock-in time Omega(m diam) can be enforced in any graph with m edges and diameter diam. Furthermore, we show that if ad provides its own port numbering after the initial setup of pointers by pl, the complexity of the lock-in problem is bounded by O(m min{log m,diam}). We also propose a class of graphs in which the lock-in requires time Omega(m log m). In the remaining two cases we show that the lock-in requires time Omega(m diam) in graphs with the worst-case topology. In addition, however, we present non-trivial classes of graphs with a large diameter in which the lock-in time is O(m)

Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading

We study gossip algorithms for the rumor spreading problem which asks one node to deliver a rumor to all nodes in an unknown network, and every node is only allowed to call one neighbor in each round. In this work we introduce two fundamentally new techniques in studying the rumor spreading problem: First, we establish a new connection between the rumor spreading process in an arbitrary graph and certain Markov chains. While most previous work analyzed the rumor spreading time in general graphs by studying the rate of the number of (un-)informed nodes after every round, we show that the mixing time of a certain Markov chain suffices to bound the rumor spreading time in an arbitrary graph. Second, we construct a reduction from rumor spreading processes to branching programs. This reduction gives us a general framework to derandomize the rumor spreading and other gossip processes. In particular, we show that, for any n-vertex expander graph, there is a protocol which informs every node in O(log n) rounds with high probability, and uses O (log n · log log n) random bits in total. The runtime of our protocol is tight, and the randomness requirement of O (log n· log log n) random bits almost matches the lower bound of Ω(log n) random bits. We further show that, for many graph families (defined with respect to the expansion and the degree), O (poly log n) random bits in total suffice for fast rumor spreading. These results give us an almost complete understanding of the role of randomness in the rumor spreading process, which was extensively studied over the past years

The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks

International audienceThe rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. The behavior of the rotor-router is fully deterministic but its performance characteristics (cover time, return time) closely resemble the expected values of the corresponding parameters of the random walk. In this work Research partially supported by the ANR Project DISPLEXITY (ANR-11-BS02-014). This study has been carried out in the frame of the Investments for the future Programme IdEx Bordeaux-CP

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum