Topology recognition with advice

In topology recognition, each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a (possibly different) string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes.We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size n and diameter D ? α n , for any constant α < 1 . In most cases, our bounds are asymptotically tight.

[1]  Mikkel Thorup,et al.  Approximate distance oracles , 2001, JACM.

[2]  Haim Kaplan,et al.  Compact labeling schemes for ancestor queries , 2001, SODA '01.

[3]  Adrian Kosowski,et al.  Constructing a Map of an Anonymous Graph: Applications of Universal Sequences , 2010, OPODIS.

[4]  Andrzej Pelc,et al.  Drawing maps with advice , 2010, J. Parallel Distributed Comput..

[5]  Andrzej Pelc,et al.  Trade-offs Between the Size of Advice and Broadcasting Time in Trees , 2008, SPAA '08.

[6]  Andrzej Pelc,et al.  How Much Memory Is Needed for Leader Election , 2010, DISC.

[7]  David Peleg,et al.  Labeling schemes for flow and connectivity , 2002, SODA '02.

[8]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[9]  Sebastiano Vigna,et al.  Computing anonymously with arbitrary knowledge , 1999, PODC '99.

[10]  Andrzej Pelc,et al.  Distributed computing with advice: information sensitivity of graph coloring , 2007, Distributed Computing.

[11]  Nicolas Nisse,et al.  Graph searching with advice , 2007, Theor. Comput. Sci..

[12]  Andrzej Pelc,et al.  Communication algorithms with advice , 2010, J. Comput. Syst. Sci..

[13]  Masafumi Yamashita,et al.  Computing on Anonymous Networks: Part I-Characterizing the Solvable Cases , 1996, IEEE Trans. Parallel Distributed Syst..

[14]  Roger Wattenhofer,et al.  Anonymous networks: randomization = 2-hop coloring , 2014, PODC.

[15]  Reuven Cohen,et al.  Label-guided graph exploration by a finite automaton , 2005, TALG.

[16]  Roger Wattenhofer,et al.  Computability in Anonymous Networks: Revocable vs. Irrecovable Outputs , 2014, ICALP.

[17]  Ran Raz,et al.  Distance labeling in graphs , 2001, SODA '01.

[18]  Saverio Caminiti,et al.  Engineering Tree Labeling Schemes: A Case Study on Least Common Ancestors , 2008, ESA.

[19]  Dana Angluin,et al.  Local and global properties in networks of processors (Extended Abstract) , 1980, STOC '80.

[20]  Evangelos Kranakis,et al.  Computing Boolean Functions on Anonymous Networks , 1994, Inf. Comput..

[21]  Hagit Attiya,et al.  Renaming in an asynchronous environment , 1990, JACM.

[22]  S. Tani.  Compression of View on Anonymous Networks—Folded View— , 2012, IEEE Transactions on Parallel and Distributed Systems.

[23]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[24]  Pierre Fraigniaud,et al.  Local MST computation with short advice , 2007, SPAA.

[25]  Gary L. Peterson,et al.  An O(nlog n) Unidirectional Algorithm for the Circular Extrema Problem , 1982, TOPL.

[26]  Andrzej Pelc,et al.  Fast radio broadcasting with advice , 2008, Theor. Comput. Sci..

[27]  Daniel S. Hirschberg,et al.  Decentralized extrema-finding in circular configurations of processors , 1980, CACM.

[28]  Shay Kutten,et al.  Proof labeling schemes , 2005, PODC '05.

[29]  Baruch Awerbuch,et al.  Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems , 1987, STOC.

[30]  Pierre Fraigniaud,et al.  Online computation with advice , 2009, Theor. Comput. Sci..

[31]  Andrzej Pelc,et al.  Tree exploration with advice , 2008, Inf. Comput..