SIROCCO 2015 · 22nd International Colloquium on Structural Information and Communication Complexity, Montserrat, Spain, July 2015 · doi:10.1007/978-3-319-25258-2_3

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 (1) they cannot be solved without unique identifiers, and (2) unique 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 oracles* $f$ that, for any given $n$, provide a multiset $f(n)$ 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 (1) any large oracle is at least as powerful as the unique identifiers in the context of local decision problems, while (2) for any small oracle there are local decision problems that still benefit from unique identifiers.

Christian Scheideler (Ed.): *Structural Information and Communication Complexity, 22nd International Colloquium, SIROCCO 2015, Montserrat, Spain, July 14–16, 2015, Post-Proceedings*, volume 9439 of *Lecture Notes in Computer Science*, pages 31–45, Springer, Berlin, 2015

ISBN 978-3-319-25257-5

- Pierre Fraigniaud, Juho Hirvonen, and Jukka Suomela: Node labels in local decision ·
*Theoretical Computer Science*751:61–73, 2018