Node Labels in Local Decision

Abstract

Abstract. 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 give 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.

Deciding and verifying network properties locally with few output bits

“…The same paper also defines the class BPLD, the probabilistic version of LD. Interestingly, [15] [17,19], which demonstrate that, perhaps surprisingly, identifiers help local decision even if such tasks do not require symmetry breaking. Randomized distributed decision was extensively studied in [20], where it is shown that, as opposed to the sequential setting, the probability of success for the randomized decision algorithm cannot be boosted.…”

A hierarchy of local decision

“…For instance, after noticing that the identities are leaking information to the nodes about the size of the network (e.g., at least one node has an ID at least n − 1 in n-node network), it was recently proved that restricting the algorithms to be identity-oblivious reduces the ability to decide languages locally in Σ 0 (see [10]), while this is not the case for Σ 1 (see [11]). Recently, [12] characterized the "power of the IDs" in local distributed decision. In [5], the authors discussed what can be computed in an anonymous networks, and showed that the answer to this question varies a lot depending on the commitment of the nodes to their first computed output value, i.e., whether it is revocable or not.…”

Survey of Distributed Decision

“…In [36,37], the power of IDs in local decision is characterized using oracles. An oracle is a trustable party with full knowledge of the input, who can provide nodes with information about this input.…”

“…, f n ) of n values that are assigned arbitrarily to the n nodes in a one-to-one manner. A scalar oracle f is large if, for any set of k nodes, the largest value provided by f to the nodes in this set grows with k. [37] proved that, for any computable scalar oracle f , we have LDO f = LD f if and only if f is large, where LD f (resp., LDO f ) is the class of languages that can be locally decided in O(1) rounds in the LOCAL model by an algorithm (resp., by an ID-oblivious algorithm) which uses the information provided by f available at the nodes.…”