The Next 700 Impossibility Results in Time-Varying Graphs

Braud-Santoni, Nicolas; Dubois, Swan; Kaaouachi, Mohamed Hamza; Petit, Franck

doi:10.15803/ijnc.6.1_27

Cited by 17 publications

(30 citation statements)

References 12 publications

(21 reference statements)

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“…The proof of Theorem 1 relies on a generic framework introduced by Braud-Santoni et al [5]. Note that even though this generic framework is designed for another model (namely, the classical message passing model), it is straightforward to borrow it for our current model.…”

Section: Impossibility Resultsmentioning

confidence: 99%

“…We present briefly this framework here. The interested reader is referred to the original work [5] for more details.…”

Section: Impossibility Resultsmentioning

confidence: 99%

“…Many recent works focus on defining pertinent assumptions for capturing the dynamics of those systems [7,16,21]. When these assumptions become very weak, that is, when the system becomes highly dynamic, a somewhat frustrating but not very surprising conclusion emerge: many fundamental distributed problems are impossible at least, in their classical form [2,5,6].…”

Section: Introductionmentioning

confidence: 99%

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Gracefully Degrading Gathering in Dynamic Rings

Bournat¹,

Dubois²,

Petit³

2018

Lecture Notes in Computer Science

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Gracefully degrading algorithms [Biely et al., TCS 2018] are designed to circumvent impossibility results in dynamic systems by adapting themselves to the dynamics. Indeed, such an algorithm solves a given problem under some dynamics and, moreover, guarantees that a weaker (but related) problem is solved under a higher dynamics under which the original problem is impossible to solve. The underlying intuition is to solve the problem whenever possible but to provide some kind of quality of service if the dynamics become (unpredictably) higher.In this paper, we apply for the first time this approach to robot networks. We focus on the fundamental problem of gathering a squad of autonomous robots on an unknown location of a dynamic ring. In this goal, we introduce a set of weaker variants of this problem. Motivated by a set of impossibility results related to the dynamics of the ring, we propose a gracefully degrading gathering algorithm. arXiv:1805.05137v2 [cs.DC] 2 Aug 2018 Rules for Phase K K 1 :: AllButT woW aitingW alker() −→ InitiateWalk() K 2 :: W aitingW alker() −→ StopMoving() K 3 :: ∃r ∈ N odeM ate(), P otentialM inOrSearcherW ithM inW aiting(r ) −→ BecomeWaitingWalker(r') K 4 :: ∃r ∈ N odeM ate(), RighterW ithM inW aiting(r ) ∧ ExistsEdge(right, current) −→ BecomeAwareSearcher(r') Rules for Phase M M 1 :: P otentialM inOrRighter() ∧ M inDiscovery() −→ BecomeMinWaitingWalker(r) M 2 :: ∃r ∈ N odeM ate(), N otW alkerW ithHeadW alker(r ) ∧ ExistsEdge(right, current) −→ BecomeAwareSearcher(r') M 3 :: ∃r ∈ N odeM ate(), N otW alkerW ithHeadW alker(r ) −→ BecomeAwareSearcher(r'); StopMoving() M 4 :: ∃r ∈ N odeM ate(), N otW alkerW ithT ailW alker(r ) −→ BecomeTailWalker(r'); Walk() M 5 :: ∃r ∈ N odeM ate(), P otentialM inW ithAwareSearcher(r ) −→ BecomeAwareSearcher(r'); Search() M 6 :: AllButOneRighter() −→ InitiateSearch() M 7 :: ∃r ∈ N odeM ate(), RighterW ithSearcher(r ) −→ BecomeAwareSearcher(r'); Search() M 8 :: P otentialM inOrRighter() −→ MoveRight() M 9 :: DumbSearcherM inRevelation() −→ BecomeAwareSearcher(r); Search() M 10 :: ∃r ∈ N odeM ate(), DumbSearcherW ithAwareSearcher(r ) −→ BecomeAwareSearcher(r'); Search() M 11 :: Searcher() −→ Search()

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Section: Impossibility Resultsmentioning

confidence: 99%

“…We present briefly this framework here. The interested reader is referred to the original work [5] for more details.…”

Section: Impossibility Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gracefully Degrading Gathering in Dynamic Rings

Bournat¹,

Dubois²,

Petit³

2018

Lecture Notes in Computer Science

Self Cite

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“…In [11], the temporal definition above is extended to infinite lifetime networks, by requiring that the covering relation (domination or else) be satisfied infinitely often. The authors observe that whenever the network is temporally connected in a recurrent way, which corresponds to Class T C R in [7] (and Class 5 in [8]), one has the guarantee that among all the edges that appear at some point, at least a connected spanning subset of them must reappear forever [5]. In other words, an equivalent characterization of T C R is that the eventual footprint of the network (i.e., the union of those edges which reappear forever) is connected.…”

Section: Further Discussion On the Temporal Interpretation Of Robustnessmentioning

confidence: 99%

Robustness: A new form of heredity motivated by dynamic networks

Casteigts

Dubois

Petit

et al. 2020

Theoretical Computer Science

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We investigate a special case of hereditary property in graphs, referred to as robustness. A property (or structure) is called robust in a graph G if it is inherited by all the connected spanning subgraphs of G. We motivate this definition using two different settings of dynamic networks. The first corresponds to networks of low dynamicity, where some links may be permanently removed so long as the network remains connected. The second corresponds to highly-dynamic networks, where communication links appear and disappear arbitrarily often, subject only to the requirement that the entities are temporally connected in a recurrent fashion (i.e. they can always reach each other through temporal paths). Each context induces a different interpretation of the notion of robustness.We start by motivating the definition and discussing the two interpretations, after what we consider the notion independently from its interpretation, taking as our focus the robustness of maximal independent sets (MIS). A graph may or may not admit a robust MIS. We characterize the set of graphs RMIS ∀ in which all MISs are robust. Then, we turn our attention to the graphs that admit a robust MIS (RMIS ∃ ). This class has a more complex structure; we give a partial characterization in terms of elementary graph properties, then a complete characterization by means of a (polynomial time) decision algorithm that accepts if and only if a robust MIS exists. This algorithm can be adapted to construct such a solution if one exists.

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“…Applying the theorem of [5], we obtain that, until time t i , the execution of A on G ω is identical to the one on G i . This implies that, executing A on G ω (of size strictly greater than 3), r 1 and r 2 only visit the nodes u, v, and w. This is contradictory with the fact that A satisfies the perpetual exploration specification on connected over time rings of size strictly greater than 3 using two robots.…”

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confidence: 96%

Computability of Perpetual Exploration in Highly Dynamic Rings

Bournat¹,

Dubois²,

Petit³

2017

2017 IEEE 37th International Conference on Distributed Computing Systems (ICDCS)

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We consider systems made of autonomous mobile robots evolving in highly dynamic discrete environment i.e., graphs where edges may appear and disappear unpredictably without any recurrence, stability, nor periodicity assumption. Robots are uniform (they execute the same algorithm), they are anonymous (they are devoid of any observable ID), they have no means allowing them to communicate together, they share no common sense of direction, and they have no global knowledge related to the size of the environment. However, each of them is endowed with persistent memory and is able to detect whether it stands alone at its current location. A highly dynamic environment is modeled by a graph such that its topology keeps continuously changing over time. In this paper, we consider only dynamic graphs in which nodes are anonymous, each of them is infinitely often reachable from any other one, and such that its underlying graph (i.e., the static graph made of the same set of nodes and that includes all edges that are present at least once over time) forms a ring of arbitrary size.In this context, we consider the fundamental problem of perpetual exploration: each node is required to be infinitely often visited by a robot. This paper analyzes the computability of this problem in (fully) synchronous settings, i.e., we study the deterministic solvability of the problem with respect to the number of robots. We provide three algorithms and two impossibility results that characterize, for any ring size, the necessary and sufficient number of robots to perform perpetual exploration of highly dynamic rings.

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The Next 700 Impossibility Results in Time-Varying Graphs

Cited by 17 publications

References 12 publications

Gracefully Degrading Gathering in Dynamic Rings

Gracefully Degrading Gathering in Dynamic Rings

Robustness: A new form of heredity motivated by dynamic networks

Computability of Perpetual Exploration in Highly Dynamic Rings

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