Bounds on the cover time of parallel rotor walks

We consider the cover time of multi-agent rotor-router.Rotor-router is a deterministic process in which the environment determines moves of the agents.Rotor-router is a form of derandomization of the random walk.We show that k agents explore any graph between ? ( log ? k ) and ? ( k ) faster than single agent.It is conjectured that the speedup for multiple random walks is also between ? ( log ? k ) and ? ( k ) . The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of k identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node successively propagates walkers visiting it along its outgoing arcs in round-robin fashion, according to a fixed ordering. We consider the cover time of such a system, i.e., the number of steps after which each node has been visited by at least one walk, regardless of the initialization of the walks. We show that for any graph with m edges and diameter D, this cover time is at most ? ( m D / log ? k ) and at least ? ( m D / k ) , which corresponds to a speedup of between ? ( log ? k ) and ? ( k ) with respect to the cover time of a single walk.

[1]  Israel A. Wagner,et al.  A Distributed Ant Algorithm for\protect Efficiently Patrolling a Network , 2003, Algorithmica.

[2]  Y. Peres,et al.  Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile , 2007, 0704.0688.

[3]  Thomas Sauerwald,et al.  The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks , 2013, PODC '13.

[4]  Joshua N. Cooper,et al.  Simulating a Random Walk with Constant Error , 2004, Combinatorics, Probability and Computing.

[5]  Y. Peres,et al.  Spherical asymptotics for the rotor-router model in Zd , 2005, math/0503251.

[6]  Yukiko Yamauchi,et al.  Deterministic Random Walks for Rapidly Mixing Chains , 2018, SIAM J. Discret. Math..

[7]  Noga Alon,et al.  Many random walks are faster than one , 2007, SPAA '08.

[8]  Adrian Kosowski,et al.  Euler Tour Lock-In Problem in the Rotor-Router Model , 2009, DISC.

[9]  David S. Greenberg,et al.  Traversing Directed Eulerian Mazes , 2000, WG.

[10]  Thomas Sauerwald,et al.  Tight bounds for the cover time of multiple random walks , 2011, Theor. Comput. Sci..

[11]  Tobias Friedrich,et al.  Deterministic Random Walks on the Two-Dimensional Grid , 2009, Comb. Probab. Comput..

[12]  Uriel Feige.  A Spectrum of Time-Space Trade-Offs for Undirected s-t Connectivity , 1997, J. Comput. Syst. Sci..

[13]  Adrian Kosowski,et al.  Robustness of the Rotor-router Mechanism , 2009, OPODIS.

[14]  Adrian Kosowski,et al.  Does Adding More Agents Make a Difference? A Case Study of Cover Time for the Rotor-Router , 2014, ICALP.

[15]  Noga Alon,et al.  Many Random Walks Are Faster Than One , 2011, Comb. Probab. Comput..

[16]  Hoda Akbari,et al.  Parallel rotor walks on finite graphs and applications in discrete load balancing , 2013, SPAA.

[17]  Colin Cooper,et al.  Derandomizing random walks in undirected graphs using locally fair exploration strategies , 2011, Distributed Computing.

[18]  Dhar,et al.  Eulerian Walkers as a Model of Self-Organized Criticality. , 1996, Physical review letters.

[19]  Adrian Kosowski,et al.  Bounds on the Cover Time of Parallel Rotor Walks , 2014, STACS.

[20]  Eli Upfal,et al.  Trading Space for Time in Undirected s-t Connectivity , 1994, SIAM J. Comput..

[21]  Reid G. Simmons,et al.  Easy and Hard Testbeds for Real-Time Search Algorithms , 1996, AAAI/IAAI, Vol. 1.

[22]  Thomas Sauerwald,et al.  The Cover Time of Deterministic Random Walks , 2010, Electron. J. Comb..

[23]  Y. Peres,et al.  The rotor-router shape is spherical , 2005 .

[24]  Thomas Sauerwald.  Expansion and the cover time of parallel random walks , 2010, PODC '10.

[25]  Omer Reingold,et al.  How Well Do Random Walks Parallelize? , 2009, APPROX-RANDOM.

[26]  J. Propp,et al.  Rotor Walks and Markov Chains , 2009, 0904.4507.

[27]  Israel A. Wagner,et al.  Distributed covering by ant-robots using evaporating traces , 1999, IEEE Trans. Robotics Autom..