Node Labels in Local Decision

The role of unique node identifiers in network computing is well understood as far as symmetry breaking is concerned. However, the unique identifiers also leak information about the computing environment--in particular, they provide some nodes with information related to the size of the network. It was recently proved that in the context of local decision, there are some decision problems such that 1i?źthey cannot be solved without unique identifiers, and 2i?źunique node identifiers leak a sufficient amount of information such that the problem becomes solvable PODC 2013. In this work we study what is the minimal amount of information that we need to leak from the environment to the nodes in order to solve local decision problems. Our key results are related to scalar oraclesf that, for any given n, provide a multiset fn of n labels; then the adversary assigns the labels to the n nodes in the network. This is a direct generalisation of the usual assumption of unique node identifiers. We give a complete characterisation of the weakest oracle that leaks at least as much information as the unique identifiers. Our main result is the following dichotomy: we classify scalar oracles as large and small, depending on their asymptotic behaviour, and show that 1i?źany large oracle is at least as powerful as the unique identifiers in the context of local decision problems, while 2i?źfor any small oracle there are local decision problems that still benefit from unique identifiers.

[1]  Jukka Suomela,et al.  Deterministic local algorithms, unique identifiers, and fractional graph colouring , 2012, Theor. Comput. Sci..

[2]  Sebastiano Vigna,et al.  An Effective Characterization of Computability in Anonymous Networks , 2001, DISC.

[3]  Pierre Fraigniaud,et al.  On the Impact of Identifiers on Local Decision , 2012, OPODIS.

[4]  E. Kranakis.  Symmetry and computability in anonymous networks: a brief survey , 1996 .

[5]  Nancy Norris.  Classifying Anonymous Networks: When Can Two Networks Compute the Same Vector-Valued Functions? , 1994, SIROCCO.

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

[7]  Pierre Fraigniaud,et al.  What can be decided locally without identifiers? , 2013, PODC '13.

[8]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[9]  Masafumi Yamashita,et al.  Leader Election Problem on Networks in which Processor Identity Numbers Are Not Distinct , 1999, IEEE Trans. Parallel Distributed Syst..

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

[11]  Stéphane Pérennes,et al.  Assigning labels in an unknown anonymous network with a leader , 2001, Distributed Computing.

[12]  Jukka Suomela,et al.  Survey of local algorithms , 2013, CSUR.

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

[14]  Moni Naor,et al.  Local computations on static and dynamic graphs , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[15]  Pierre Fraigniaud,et al.  Local Distributed Decision , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

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

[17]  Jukka Suomela,et al.  Lower bounds for local approximation , 2012, PODC '12.

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

[19]  M. Kaufmann.  What Can Be Computed Locally ? , 2003 .

[20]  Krzysztof Diks,et al.  Anonymous Wireless Rings , 1995, Theor. Comput. Sci..

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

[22]  Nicola Santoro,et al.  Groupings and Pairings in Anonymous Networks , 2006, DISC.

[23]  Faith Ellen,et al.  Hundreds of impossibility results for distributed computing , 2003, Distributed Computing.

[24]  Christoph Lenzen,et al.  Leveraging Linial's Locality Limit , 2008, DISC.

[25]  David Peleg,et al.  Compact and localized distributed data structures , 2003, Distributed Computing.

[26]  Andrzej Czygrinow,et al.  Fast Distributed Approximations in Planar Graphs , 2008, DISC.