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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. For many classic global properties, like checking the acyclicity of the network, the optimal size of the certificates depends on the size of the network, n. In this paper, we focus on properties for which the size of the certificates does not depend on n but on other parameters.
We focus on three such important properties and prove tight bounds for all of them. Namely, we prove that the optimal certification size is: Θ(log k) for k-colorability (and even exactly ⌈ log k ⌉ bits in the anonymous model while previous works had only proved a 2-bit lower bound); (1/2)log t+o(log t) for dominating sets at distance t (an unexpected and tighter-than-usual bound) ; and Θ(log Δ) for perfect matching in graphs of maximum degree Δ (the first non-trivial bound parameterized by Δ). We also prove some surprising upper bounds, for example, certifying the existence of a perfect matching in a planar graph can be done with only two bits. In addition, we explore various specific cases for these properties, in particular improving our understanding of the trade-off between locality of the verification and certificate size.

Nicolas Bousquet, Laurent Feuilloley, and Sébastien Zeitoun. Local Certification of Local Properties: Tight Bounds, Trade-Offs and New Parameters. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bousquet_et_al:LIPIcs.STACS.2024.21, author = {Bousquet, Nicolas and Feuilloley, Laurent and Zeitoun, S\'{e}bastien}, title = {{Local Certification of Local Properties: Tight Bounds, Trade-Offs and New Parameters}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {21:1--21:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.21}, URN = {urn:nbn:de:0030-drops-197317}, doi = {10.4230/LIPIcs.STACS.2024.21}, annote = {Keywords: Local certification, local properties, proof-labeling schemes, locally checkable proofs, optimal certification size, colorability, dominating set, perfect matching, fault-tolerance, graph structure} }

Document

**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

Consider a dynamic network and a given distributed problem. At any point in time, there might exist several solutions that are equally good with respect to the problem specification, but that are different from an algorithmic perspective, because some could be easier to update than others when the network changes. In other words, one would prefer to have a solution that is more robust to topological changes in the network; and in this direction the best scenario would be that the solution remains correct despite the dynamic of the network.
In [Arnaud Casteigts et al., 2020], the authors introduced a very strong robustness criterion: they required that for any removal of edges that maintain the network connected, the solution remains valid. They focus on the maximal independent set problem, and their approach consists in characterizing the graphs in which there exists a robust solution (the existential problem), or even stronger, where any solution is robust (the universal problem). As the robustness criteria is very demanding, few graphs have a robust solution, and even fewer are such that all of their solutions are robust. In this paper, we ask the following question: Can we have robustness for a larger class of networks, if we bound the number k of edge removals allowed?
To answer this question, we consider three classic problems: maximal independent set, minimal dominating set and maximal matching. For the universal problem, the answers for the three cases are surprisingly different. For minimal dominating set, the class does not depend on the number of edges removed. For maximal matching, removing only one edge defines a robust class related to perfect matchings, but for all other bounds k, the class is the same as for an arbitrary number of edge removals. Finally, for maximal independent set, there is a strict hierarchy of classes: the class for the bound k is strictly larger than the class for bound k+1.
For the robustness notion of [Arnaud Casteigts et al., 2020], no characterization of the class for the existential problem is known, only a polynomial-time recognition algorithm. We show that the situation is even worse for bounded k: even for k = 1, it is NP-hard to decide whether a graph has a robust maximal independent set.

Swan Dubois, Laurent Feuilloley, Franck Petit, and Mikaël Rabie. When Should You Wait Before Updating? - Toward a Robustness Refinement. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dubois_et_al:LIPIcs.SAND.2023.7, author = {Dubois, Swan and Feuilloley, Laurent and Petit, Franck and Rabie, Mika\"{e}l}, title = {{When Should You Wait Before Updating? - Toward a Robustness Refinement}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.7}, URN = {urn:nbn:de:0030-drops-179435}, doi = {10.4230/LIPIcs.SAND.2023.7}, annote = {Keywords: Robustness, dynamic network, temporal graphs, edge removal, connectivity, footprint, packing/covering problems, maximal independent set, maximal matching, minimum dominating set, perfect matching, NP-hardness} }

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**Published in:** LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

One of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions. First, the problem of recoloring, that is computing an efficient transformation between two given colorings (instead of computing a new coloring), has been considered, both to model radio network updates, and as a useful subroutine for coloring. Second, as it appears that general graphs and bounded-degree graphs do not model real networks very well (with, respectively, pathological worst-case topologies and too strong assumptions), coloring has been studied in more specific graph classes. In this paper, we study the intersection of these two directions: distributed recoloring in two relevant graph classes, interval and chordal graphs.
More formally, the question of recoloring a graph is as follows: we are given a network, an input coloring α and a target coloring β, and we want to find a schedule of colorings to reach β starting from α. In a distributed setting, the schedule needs to be found within the LOCAL model, where nodes communicate with their direct neighbors synchronously. The question we want to answer is: how many rounds of communication {are} needed to produce a schedule, and what is the length of this schedule?
In the case of interval and chordal graphs, we prove that, if we have less than 2ω colors, ω being the size of the largest clique, extra colors will be needed in the intermediate colorings. For interval graphs, we produce a schedule after O(poly(Δ)log*n) rounds of communication, and for chordal graphs, we need O(ω²Δ²log n) rounds to get one.
Our techniques also improve classic coloring algorithms. Namely, we get ω+1-colorings of interval graphs in O(ωlog*n) rounds and of chordal graphs in O(ωlog n) rounds, which improves on previous known algorithms that use ω+2 colors for the same running times.

Nicolas Bousquet, Laurent Feuilloley, Marc Heinrich, and Mikaël Rabie. Distributed Recoloring of Interval and Chordal Graphs. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bousquet_et_al:LIPIcs.OPODIS.2021.19, author = {Bousquet, Nicolas and Feuilloley, Laurent and Heinrich, Marc and Rabie, Mika\"{e}l}, title = {{Distributed Recoloring of Interval and Chordal Graphs}}, booktitle = {25th International Conference on Principles of Distributed Systems (OPODIS 2021)}, pages = {19:1--19:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-219-8}, ISSN = {1868-8969}, year = {2022}, volume = {217}, editor = {Bramas, Quentin and Gramoli, Vincent and Milani, Alessia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.19}, URN = {urn:nbn:de:0030-drops-157941}, doi = {10.4230/LIPIcs.OPODIS.2021.19}, annote = {Keywords: Distributed coloring, distributed recoloring, interval graphs, chordal graphs, intersection graphs} }

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**Published in:** LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class 𝒢. Such certifications with labels of size O(log n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors.
In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log n). We also show matching lower bounds using a new proof technique.

Nicolas Bousquet, Laurent Feuilloley, and Théo Pierron. Local Certification of Graph Decompositions and Applications to Minor-Free Classes. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bousquet_et_al:LIPIcs.OPODIS.2021.22, author = {Bousquet, Nicolas and Feuilloley, Laurent and Pierron, Th\'{e}o}, title = {{Local Certification of Graph Decompositions and Applications to Minor-Free Classes}}, booktitle = {25th International Conference on Principles of Distributed Systems (OPODIS 2021)}, pages = {22:1--22:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-219-8}, ISSN = {1868-8969}, year = {2022}, volume = {217}, editor = {Bramas, Quentin and Gramoli, Vincent and Milani, Alessia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.22}, URN = {urn:nbn:de:0030-drops-157970}, doi = {10.4230/LIPIcs.OPODIS.2021.22}, annote = {Keywords: Local certification, proof-labeling schemes, locally checkable proofs, graph decompositions, minor-free graphs} }

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**Published in:** LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

Given a boolean predicate Π on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for Π is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying Π. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size n of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of O(log log n) bits per node in any n-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use Ω(log log n)-bit per node registers in some n-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms.

Lélia Blin, Laurent Feuilloley, and Gabriel Le Bouder. Optimal Space Lower Bound for Deterministic Self-Stabilizing Leader Election Algorithms. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 24:1-24:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blin_et_al:LIPIcs.OPODIS.2021.24, author = {Blin, L\'{e}lia and Feuilloley, Laurent and Le Bouder, Gabriel}, title = {{Optimal Space Lower Bound for Deterministic Self-Stabilizing Leader Election Algorithms}}, booktitle = {25th International Conference on Principles of Distributed Systems (OPODIS 2021)}, pages = {24:1--24:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-219-8}, ISSN = {1868-8969}, year = {2022}, volume = {217}, editor = {Bramas, Quentin and Gramoli, Vincent and Milani, Alessia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.24}, URN = {urn:nbn:de:0030-drops-157997}, doi = {10.4230/LIPIcs.OPODIS.2021.24}, annote = {Keywords: Space lower bound, memory tight bound, self-stabilization, leader election, anonymous, identifiers, state model, ring topology} }

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Brief Announcement

**Published in:** LIPIcs, Volume 209, 35th International Symposium on Distributed Computing (DISC 2021)

Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class 𝒢. Such certifications with labels of size O(log n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors.
In this paper, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log n). We also show matching lower bounds with a new simple proof technique.

Nicolas Bousquet, Laurent Feuilloley, and Théo Pierron. Brief Announcement: Local Certification of Graph Decompositions and Applications to Minor-Free Classes. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 49:1-49:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bousquet_et_al:LIPIcs.DISC.2021.49, author = {Bousquet, Nicolas and Feuilloley, Laurent and Pierron, Th\'{e}o}, title = {{Brief Announcement: Local Certification of Graph Decompositions and Applications to Minor-Free Classes}}, booktitle = {35th International Symposium on Distributed Computing (DISC 2021)}, pages = {49:1--49:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-210-5}, ISSN = {1868-8969}, year = {2021}, volume = {209}, editor = {Gilbert, Seth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2021.49}, URN = {urn:nbn:de:0030-drops-148515}, doi = {10.4230/LIPIcs.DISC.2021.49}, annote = {Keywords: Local certification, proof-labeling schemes, locally checkable proofs, graph decompositions, minor-free graphs} }

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Brief Announcement

**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

In the context of self-stabilization, a silent algorithm guarantees that the communication registers (a.k.a register) of every node do not change once the algorithm has stabilized. At the end of the 90’s, Dolev et al. [Acta Inf. '99] showed that, for finding the centers of a graph, for electing a leader, or for constructing a spanning tree, every silent deterministic algorithm must use a memory of Omega(log n) bits per register in n-node networks. Similarly, Korman et al. [Dist. Comp. '07] proved, using the notion of proof-labeling-scheme, that, for constructing a minimum-weight spanning tree (MST), every silent algorithm must use a memory of Omega(log^2n) bits per register. It follows that requiring the algorithm to be silent has a cost in terms of memory space, while, in the context of self-stabilization, where every node constantly checks the states of its neighbors, the silence property can be of limited practical interest. In fact, it is known that relaxing this requirement results in algorithms with smaller space-complexity.
In this paper, we are aiming at measuring how much gain in terms of memory can be expected by using arbitrary deterministic self-stabilizing algorithms, not necessarily silent. To our knowledge, the only known lower bound on the memory requirement for deterministic general algorithms, also established at the end of the 90’s, is due to Beauquier et al. [PODC '99] who proved that registers of constant size are not sufficient for leader election algorithms. We improve this result by establishing the lower bound Omega(log log n) bits per register for deterministic self-stabilizing algorithms solving (Delta+1)-coloring, leader election or constructing a spanning tree in networks of maximum degree Delta.

Lélia Blin, Laurent Feuilloley, and Gabriel Le Bouder. Brief Announcement: Memory Lower Bounds for Self-Stabilization. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 37:1-37:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{blin_et_al:LIPIcs.DISC.2019.37, author = {Blin, L\'{e}lia and Feuilloley, Laurent and Le Bouder, Gabriel}, title = {{Brief Announcement: Memory Lower Bounds for Self-Stabilization}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {37:1--37:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.37}, URN = {urn:nbn:de:0030-drops-113442}, doi = {10.4230/LIPIcs.DISC.2019.37}, annote = {Keywords: Space lower bound, memory tight bound, deterministic self-stabilization, leader election, anonymous, identifiers, state model, ring} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

Distributed proofs are mechanisms enabling the nodes of a network to collectively and efficiently check the correctness of Boolean predicates on the structure of the network (e.g. having a specific diameter), or on data structures distributed over the nodes (e.g. a spanning tree). We consider well known mechanisms consisting of two components: a prover that assigns a certificate to each node, and a distributed algorithm called verifier that is in charge of verifying the distributed proof formed by the collection of all certificates. We show that many network predicates have distributed proofs offering a high level of redundancy, explicitly or implicitly. We use this remarkable property of distributed proofs to establish perfect tradeoffs between the size of the certificate stored at every node, and the number of rounds of the verification protocol.

Laurent Feuilloley, Pierre Fraigniaud, Juho Hirvonen, Ami Paz, and Mor Perry. Redundancy in Distributed Proofs. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{feuilloley_et_al:LIPIcs.DISC.2018.24, author = {Feuilloley, Laurent and Fraigniaud, Pierre and Hirvonen, Juho and Paz, Ami and Perry, Mor}, title = {{Redundancy in Distributed Proofs}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {24:1--24:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.24}, URN = {urn:nbn:de:0030-drops-98139}, doi = {10.4230/LIPIcs.DISC.2018.24}, annote = {Keywords: Distributed verification, Distributed graph algorithms, Proof-labeling schemes, Space-time tradeoffs, Non-determinism} }

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**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

In this work we study the cost of local and global proofs on distributed verification. In this setting the nodes of a distributed system are provided with a nondeterministic proof for the correctness of the state of the system, and the nodes need to verify this proof by looking at only their local neighborhood in the system.
Previous works have studied the model where each node is given its own, possibly unique, part of the proof as input. The cost of a proof is the maximum size of an individual label. We compare this model to a model where each node has access to the same global proof, and the cost is the size of this global proof.
It is easy to see that a global proof can always include all of the local proofs, and every local proof can be a copy of the global proof. We show that there exists properties that exhibit these relative proof sizes, and also properties that are somewhere in between. In addition, we introduce a new lower bound technique and use it to prove a tight lower bound on the complexity of reversing distributed decision and establish a link between communication complexity and distributed proof complexity.

Laurent Feuilloley and Juho Hirvonen. Local Verification of Global Proofs. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{feuilloley_et_al:LIPIcs.DISC.2018.25, author = {Feuilloley, Laurent and Hirvonen, Juho}, title = {{Local Verification of Global Proofs}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.25}, URN = {urn:nbn:de:0030-drops-98146}, doi = {10.4230/LIPIcs.DISC.2018.25}, annote = {Keywords: Proof-labeling schemes, distributed verification, non-determinism, local proofs} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

Proof-labeling schemes are known mechanisms providing nodes of networks with certificates that can be verified locally by distributed algorithms. Given a boolean predicate on network states, such schemes enable to check whether the predicate is satisfied by the actual state of the network, by having nodes interacting with their neighbors only. Proof-labeling schemes are typically designed for enforcing fault-tolerance, by making sure that if the current state of the network is illegal with respect to some given predicate, then at least one node will detect it. Such a node can raise an alarm, or launch a recovery procedure enabling the system to return to a legal state. In this paper, we introduce error-sensitive proof-labeling schemes. These are proof-labeling schemes which guarantee that the number of nodes detecting illegal states is linearly proportional to the edit-distance between the current state and the set of legal states. By using error-sensitive proof-labeling schemes, states which are far from satisfying the predicate will be detected by many nodes, enabling fast return to legality. We provide a structural characterization of the set of boolean predicates on network states for which there exist error-sensitive proof-labeling schemes. This characterization allows us to show that classical predicates such as, e.g., acyclicity, and leader admit error-sensitive proof-labeling schemes, while others like regular subgraphs don't. We also focus on compact error-sensitive proof-labeling schemes. In particular, we show that the known proof-labeling schemes for spanning tree and minimum spanning tree, using certificates on O(log n) bits, and on O(log^2 n) bits, respectively, are error-sensitive, as long as the trees are locally represented by adjacency lists, and not just by parent pointers.

Laurent Feuilloley and Pierre Fraigniaud. Error-Sensitive Proof-Labeling Schemes. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{feuilloley_et_al:LIPIcs.DISC.2017.16, author = {Feuilloley, Laurent and Fraigniaud, Pierre}, title = {{Error-Sensitive Proof-Labeling Schemes}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {16:1--16:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.16}, URN = {urn:nbn:de:0030-drops-80180}, doi = {10.4230/LIPIcs.DISC.2017.16}, annote = {Keywords: Fault-tolerance, distributed decision, distributed property testing} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We extend the notion of distributed decision in the framework of distributed network computing, inspired by recent results on so-called distributed graph automata. We show that, by using distributed decision mechanisms based on the interaction between a prover and a disprover, the size of the certificates distributed to the nodes for certifying a given network property can be drastically reduced. For instance, we prove that minimum spanning tree can be certified with O(log(n))-bit certificates in n-node graphs, with just one interaction between the prover and the disprover, while it is known that certifying MST requires Omega(log^2(n))-bit certificates if only the prover can act. The improvement can even be exponential for some simple graph properties.
For instance, it is known that certifying the existence of a nontrivial automorphism requires Omega(n^2) bits if only the prover can act. We show that there is a protocol with two interactions between the prover and the disprover enabling to certify nontrivial automorphism with O(log(n))- bit certificates. These results are achieved by defining and analysing a local hierarchy of decision which generalizes the classical notions of proof-labelling schemes and locally checkable proofs.

Laurent Feuilloley, Pierre Fraigniaud, and Juho Hirvonen. A Hierarchy of Local Decision. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 118:1-118:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{feuilloley_et_al:LIPIcs.ICALP.2016.118, author = {Feuilloley, Laurent and Fraigniaud, Pierre and Hirvonen, Juho}, title = {{A Hierarchy of Local Decision}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {118:1--118:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.118}, URN = {urn:nbn:de:0030-drops-62536}, doi = {10.4230/LIPIcs.ICALP.2016.118}, annote = {Keywords: Distributed Network Computing, Distributed Algorithm, Distributed Decision, Locality} }

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