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On the Impact of Identifiers on Local Decision
"... The issue of identifiers is crucial in distributed computing. Informally, identities are used for tackling two of the fundamental difficulties that are inherent to deterministic distributed computing, namely: (1) symmetry breaking, and (2) topological information gathering. In the context of local c ..."
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The issue of identifiers is crucial in distributed computing. Informally, identities are used for tackling two of the fundamental difficulties that are inherent to deterministic distributed computing, namely: (1) symmetry breaking, and (2) topological information gathering. In the context of local computation, i.e., when nodes can gather information only from nodes at bounded distances, some insight regarding the role of identities has been established. For instance, it was shown that, for large classes of construction problems, the role of the identities can be rather small. However, for the identities to play no role, some other kinds of mechanisms for breaking symmetry must be employed, such as edgelabeling or sense of direction. When it comes to local distributed decision problems, the specification of the decision task does not seem to involve symmetry breaking. Therefore, it is expected that, assuming nodes can gather sufficient information about their neighborhood, one could get rid of the identities, without employing extra mechanisms for breaking symmetry.
Deterministic local algorithms, unique identifiers, and fractional graph colouring
 In Proc. 19th Colloquium on Structural Information and Communication Complexity (SIROCCO 2012), volume 7355 of LNCS
, 2012
"... Abstract. We show that for any α> 1 there exists a deterministic distributed algorithm that finds a fractional graph colouring of length at most α(∆+ 1) in any graph in one synchronous communication round; here ∆ is the maximum degree of the graph. The result is neartight, as there are graphs in ..."
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Abstract. We show that for any α> 1 there exists a deterministic distributed algorithm that finds a fractional graph colouring of length at most α(∆+ 1) in any graph in one synchronous communication round; here ∆ is the maximum degree of the graph. The result is neartight, as there are graphs in which the optimal solution has length ∆+ 1. The result is, of course, too good to be true. The usual definitions of scheduling problems (fractional graph colouring, fractional domatic partition, etc.) in a distributed setting leave a loophole that can be exploited in the design of distributed algorithms: the size of the local output is not bounded. Our algorithm produces an output that seems to be perfectly good by the usual standards but it is impractical, as the schedule of each node consists of a very large number of short periods of activity. More generally, the algorithm shows that when we study distributed algorithms for scheduling problems, we can choose virtually any tradeoff between the following three parameters: T, the running time of the algorithm, `, the length of the schedule, and κ, the maximum number of periods of activity for a any single node. Here ` is the objective function of the optimisation problem, while κ captures the “subjective ” quality of the solution. If we study, for example, boundeddegree graphs, we can trivially keep T and κ constant, at the cost of a large `, or we can keep κ and ` constant, at the cost of a large T. Our algorithm shows that yet another tradeoff is possible: we can keep T and ` constant at the cost of a large κ. 1
Distributed maximal matching: greedy is optimal
 Manuscript
, 2011
"... We study distributed algorithms that find a maximal matching in an anonymous, edgecoloured graph. If the edges are properly coloured with k colours, there is a trivial greedy algorithm that finds a maximal matching in k − 1 synchronous communication rounds. The present work shows that the greedy ..."
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We study distributed algorithms that find a maximal matching in an anonymous, edgecoloured graph. If the edges are properly coloured with k colours, there is a trivial greedy algorithm that finds a maximal matching in k − 1 synchronous communication rounds. The present work shows that the greedy algorithm is optimal in the general case: if A is a deterministic distributed algorithm that finds a maximal matching in anonymous, kedgecoloured graphs, then there is a worstcase input in which the running time of A is at least k − 1 rounds. If we focus on graphs of maximum degree ∆, it is known that a maximal matching can be found in O( ∆ + log ∗ k) rounds, and prior work implies a lower bound of Ω(polylog(∆) + log ∗ k) rounds. Our work closes the gap between upper and lower bounds: the complexity is Θ( ∆ + log ∗ k) rounds. To our knowledge, this is the first linearin ∆ lower bound for the distributed complexity of a classical graph problem.
What can be decided locally without identifiers
 In Proc. 32nd ACM Symp. on Principles of Distributed Computing
, 2013
"... Abstract. Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G ∈ P by having each node run a constanttime distributed decision algorithm. If G ∈ P, ..."
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Abstract. Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G ∈ P by having each node run a constanttime distributed decision algorithm. If G ∈ P, all the nodes should output yes; if G / ∈ P, at least one node should output no. A recent work (Fraigniaud et al., OPODIS 2012) studied the role of identifiers in local decision and gave several conditions under which identifiers are not needed. In this article, we answer their original question. More than that, we do so under all combinations of the following two critical variations on the underlying model of distributed computing: − (B): the size of the identifiers is bounded by a function of the size of the input network; as opposed to (¬B): the identifiers are unbounded. − (C): the nodes run a computable algorithm; as opposed to (¬C): the nodes can compute any, possibly uncomputable function. While it is easy to see that under (¬B,¬C) identifiers are not needed, we show that under all other combinations there are properties that can be decided locally if and only if identifiers are present. Our constructions use ideas from classical computability theory.
Weak models of distributed computing, with connections to modal logic
 CoRR
"... This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and we study models of computing that are weaker versions of the widelystudied portnumbering model. In the portnumbering model, a node of degree d receives messages throu ..."
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This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and we study models of computing that are weaker versions of the widelystudied portnumbering model. In the portnumbering model, a node of degree d receives messages through d input ports and it sends messages through d output ports, both numbered with 1, 2,..., d. In this work, VVc is the class of all graph problems that can be solved in the standard portnumbering model. We study the following subclasses of VVc: VV: Input port i and output port i are not necessarily connected to the same neighbour. MV: Input ports are not numbered; algorithms receive a multiset of messages. SV: Input ports are not numbered; algorithms receive a set of messages. VB: Output ports are not numbered; algorithms send the same message to all output ports. MB: Combination of MV and VB. SB: Combination of SV and VB. Now we have many trivial containment relations, such as SB ⊆ MB ⊆ VB ⊆ VV ⊆ VVc, but it is not obvious if, e.g., either of VB ⊆ SV or SV ⊆ VB should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that SB � MB = VB � SV = MV = VV � VVc. The same holds for the constanttime versions of these classes. We also show that the constanttime variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, we can use tools from modal logic to study these classes.
No SublogarithmicTime Approximation Scheme for Bipartite Vertex Cover
, 2013
"... Abstract König’s theorem states that on bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover. It is known from prior work that for every ε> 0 there exists a constanttime distributed algorithm that finds a (1+ ε)approximation of a maximum matching on bounde ..."
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Abstract König’s theorem states that on bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover. It is known from prior work that for every ε> 0 there exists a constanttime distributed algorithm that finds a (1+ ε)approximation of a maximum matching on boundeddegree graphs. In this work, we show—somewhat surprisingly—that no sublogarithmictime approximation scheme exists for the dual problem: there is a constant δ> 0 so that no randomised distributed algorithm with running time o(logn) can find a (1+ δ)approximation of a minimum vertex cover on 2coloured graphs of maximum degree 3. In fact, a simple application of the Linial–Saks (1993) decomposition demonstrates that this runtime lower bound is tight. Our lowerbound construction is simple and, to some extent, independent of previous techniques. Along the way we prove that a certain cut minimisation problem, which might be of independent interest, is hard to approximate locally on expander graphs.
Node Labels in Local Decision
"... 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 net ..."
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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. ar X iv
Linearin ∆ Lower Bounds in the LOCAL Model
"... By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in O(∆) rounds, independently of n; here ∆ is the maximum degree of the graph and n is the number of nodes in the graph. We show that this is optimal: there is no distributed algori ..."
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By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in O(∆) rounds, independently of n; here ∆ is the maximum degree of the graph and n is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in o(∆) rounds, independently of n. Our work gives the first linearin ∆ lower bound for a natural graph problem in the standard LOCAL model of distributed computing—prior lower bounds for a wide range of graph problems have been at best logarithmic in ∆.
Local Coordination and Symmetry Breaking
"... This article gives a short survey of recent lower bounds for distributed graph algorithms. There are many classical graph problems (e.g., maximal matching) that can be solved in O( ∆ + log ∗ n) or O(∆) communication rounds, where n is the number of nodes and ∆ is the maximum degree of the graph. In ..."
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This article gives a short survey of recent lower bounds for distributed graph algorithms. There are many classical graph problems (e.g., maximal matching) that can be solved in O( ∆ + log ∗ n) or O(∆) communication rounds, where n is the number of nodes and ∆ is the maximum degree of the graph. In these algorithms, the key bottleneck seems to be a form of local coordination, which gives rise to the linearin ∆ term in the running time. Previously it has not been known if this linear dependence is necessary, but now we can prove that there are graph problems that can be solved in time O(∆) independently of n, and cannot be solved in time o(∆) independently of n. We will give an informal overview of the techniques that can be used to prove such lower bounds, and we will also propose a roadmap for future research, with the aim of resolving some of the major open questions of the field. 1
liafa.univparisdiderot.fr
"... Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G has property P by having each node run a constanttime distributed decision algorithm. In a yes ..."
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Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G has property P by having each node run a constanttime distributed decision algorithm. In a yesinstance all nodes should output yes, while in a noinstance at least one node should output no. Recently, Fraigniaud et al. (OPODIS 2012) gave several conditions under which identifiers are not needed, and they conjectured that identifiers are not needed in any decision problem. In the present work, we disprove the conjecture. More than that, we analyse two critical variations of the underlying model of distributed computing: (B): the size of the identifiers is bounded by a function of the size of the input network, (¬B): the identifiers are unbounded, (C): the nodes run a computable algorithm, (¬C): the nodes can compute any, possibly uncomputable function. While it is easy to see that under (¬B,¬C) identifiers are not needed, we show that under all other combinations there are properties that can be decided locally if and only if identifiers are present.