What can be decided locally without identifiers?

Structural Information and Communication Complexity

Hirvonen

Suomela

2015

Self Cite

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.

Section: Our Resultsmentioning

confidence: 98%

“…The fact that we consider only computable algorithms is crucial here-without this restriction we would have LDO = LD [13].…”

Section: Local Decision Tasksmentioning

confidence: 99%

See 1 more Smart Citation

Node Labels in Local Decision

Structural Information and Communication Complexity

Hirvonen

Suomela

2015

Self Cite

“…Several aspects of network computing play a role, including the presence or absence of identities [12,17], the ability to output values of unbounded size [18], and the presence or absence of a priori knowledge about the network [21]. A crucial step was made in [28] which essentially shows that randomization does not help for local computing as long as one aims at solving problems whose solutions can be checked locally.…”

Section: Related Workmentioning

confidence: 99%

Randomized Local Network Computing

Feuilloley

Proceedings of the 27th ACM Symposium on Parallelism in Algorithms and Architectures

2015

Self Cite

International audienceIn this paper, we carry on investigating the line of research questioning the power of randomization for the design of distributed algorithms. In their seminal paper, Naor and Stockmeyer [STOC 1993] established that, in the context of network computing, in which all nodes execute the same algorithm in parallel, any construction task that can be solved locally by a randomized Monte-Carlo algorithm can also be solved locally by a deterministic algorithm. This result however holds in a specific context. In particular, it holds only for distributed tasks whose solutions that can be locally checked by a deterministic algorithm. In this paper, we extend the result of Naor and Stockmeyer to a wider class of tasks. Specifically, we prove that the same derandomization result holds for every task whose solutions can be locally checked using a 2-sided error randomized Monte-Carlo algorithm. This extension finds applications to, e.g., the design of lower bounds for construction tasks which tolerate that some nodes compute incorrect values. In a nutshell, we show that randomization does not help for solving such resilient tasks

“…Paper [12] generated several following up contributions, including, e.g., studies on the impact of randomization [11], studies on the impact of node identifiers [10], studies on verification tasks where certificates include node IDs [17], etc. See also [16] for other forms of local checking, and for their impact on distributed graph-optimization problems.…”

Section: Related Workmentioning

confidence: 99%

Distributedly Testing Cycle-Freeness

Arfaoui

Graph-Theoretic Concepts in Computer Science

Ilcinkas

et al. 2014

Self Cite

Abstract. We tackle local distributed testing of graph properties. This framework is well suited to contexts in which data dispersed among the nodes of a network can be collected by some central authority (like in, e.g., sensor networks). In local distributed testing, each node can provide the central authority with just a few information about what it perceives from its neighboring environment, and, based on the collected information, the central authority is aiming at deciding whether or not the network satisfies some property. We analyze in depth the prominent example of checking cycle-freeness, and establish tight bounds on the amount of information to be transferred by each node to the central authority for deciding cycle-freeness. In particular, we show that distributedly testing cycle-freeness requires at least ⌈log d⌉ − 1 bits of information per node in graphs with maximum degree d, even for connected graphs. Our proof is based on a novel version of the seminal result by Naor and Stockmeyer (1995) enabling to reduce the study of certain kinds of algorithms to order-invariant algorithms, and on an appropriate use of the known fact that every free group can be linearly ordered.