arXiv subjects

Laurent Feuilloley

Publications and source records attributed to Laurent Feuilloley.

At least 37 records · Page 2Linked to original sources

arXiv2021

Classifying grounded intersection graphs via ordered forbidden patterns

It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered subgraphs, called patterns. Very recently, all the classes corresponding to patterns on three vertices (including the ones mentioned above) have been listed, and proved to be efficiently recognizable. In contrast, very little is known about patterns on four vertices. One of the few graph classes characterized by a pattern on four vertices is the class of intersection graphs of rectangles that are said to be grounded on a line. This class appears naturally in the study of intersection graphs, and similar grounded classes have recently attracted a lot of attention. This paper contains three parts. First, we make a survey of grounded intersection graph classes, summarizing all the known inclusions between these various classes. Second, we show that the correspondence between a pattern on four vertices and grounded rectangle graphs is not an isolated phenomenon. We establish several other pattern characterizations for geometric classes, and show that the hierarchy of grounded intersection graph classes is tightly interleaved with the classes defined patterns on four vertices. We claim that forbidden patterns are a useful tool to classify grounded intersection graphs. Finally, we give an overview of the complexity of the recognition of classes defined by forbidden patterns on four vertices and list several interesting open problems.

cs.DM↗

arXiv2021

Local certification of MSO properties for bounded treedepth graphs

The graph model checking problem consists in testing whether an input graph satisfies a given logical formula. In this paper, we study this problem in a distributed setting, namely local certification. The goal is to assign 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. We first investigate which properties can be locally certified with small certificates. Not surprisingly, this is almost never the case, except for not very expressive logic fragments. Following the steps of Courcelle-Grohe, we then look for meta-theorems explaining what happens when we parameterize the problem by some standard measures of how simple the graph classes are. In that direction, our main result states that any MSO formula can be locally certified on graphs with bounded treedepth with a logarithmic number of bits per node, which is the golden standard in certification.

cs.DC↗

arXiv2021

Introduction to local certification

A distributed graph algorithm is basically an algorithm where every node of a graph can look at its neighborhood at some distance in the graph and chose its output. As distributed environment are subject to faults, an important issue is to be able to check that the output is correct, or in general that the network is in proper configuration with respect to some predicate. One would like this checking to be very local, to avoid using too much resources. Unfortunately most predicates cannot be checked this way, and that is where certification comes into play. Local certification (also known as proof-labeling schemes, locally checkable proofs or distributed verification) consists in assigning labels to the nodes, that certify that the configuration is correct. There are several point of view on this topic: it can be seen as a part of self-stabilizing algorithms, as labeling problem, or as a non-deterministic distributed decision. This paper is an introduction to the domain of local certification, giving an overview of the history, the techniques and the current research directions.

cs.DC↗

arXiv2021

Local certification of graph decompositions and applications to minor-free classes

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~$\mathcal{G}$. 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 with a simple new proof technique.

cs.DC↗

arXiv2021

Distributed recoloring of interval and chordal graphs

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.

cs.DC↗

arXiv2020

The Secretary Problem with Independent Sampling

In the secretary problem we are faced with an online sequence of elements with values. Upon seeing an element we have to make an irrevocable take-it-or-leave-it decision. The goal is to maximize the probability of picking the element of maximum value. The most classic version of the problem is that in which the elements arrive in random order and their values are arbitrary. However, by varying the available information, new interesting problems arise. Also the case in which the arrival order is adversarial instead of random leads to interesting variants that have been considered in the literature. In this paper we study both the random order and adversarial order secretary problems with an additional twist. The values are arbitrary, but before starting the online sequence we independently sample each element with a fixed probability $p$. The sampled elements become our information or history set and the game is played over the remaining elements. We call these problems the random order secretary problem with $p$-sampling (ROS$p$ for short) and the adversarial order secretary problem with $p$-sampling (AOS$p$ for short). Our main result is to obtain best possible algorithms for both problems and all values of $p$. As $p$ grows to 1 the obtained guarantees converge to the optimal guarantees in the full information case. In the adversarial order setting, the best possible algorithm turns out to be a simple fixed threshold algorithm in which the optimal threshold is a function of $p$ only. In the random order setting we prove that the best possible algorithm is characterized by a fixed sequence of time thresholds, dictating at which point in time we should start accepting a value that is both a maximum of the online sequence and has a given ranking within the sampled elements.

cs.GT↗

arXiv2020

Silent MST approximation for tiny memory

In this paper we show that approximation can help reduce the space used for self-stabilization. In the classic \emph{state model}, where the nodes of a network communicate by reading the states of their neighbors, an important measure of efficiency is the space: the number of bits used at each node to encode the state. In this model, a classic requirement is that the algorithm has to be \emph{silent}, that is, after stabilization the states should not change anymore. We design a silent self-stabilizing algorithm for the problem of minimum spanning tree, that has a trade-off between the quality of the solution and the space needed to compute it.

cs.DC↗

arXiv2020

Redundancy in Distributed Proofs

Distributed proofs are mechanisms enabling the nodes of a network to collectivity and efficiently check the correctness of Boolean predicates on the structure of the network, or on data-structures distributed over the nodes (e.g., spanning trees or routing tables). We consider mechanisms consisting of two components: a \emph{prover} assigning a \emph{certificate} to each node, and a distributed algorithm called \emph{verifier} that is in charge of verifying the distributed proof formed by the collection of all certificates. In this paper, 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 for establishing perfect tradeoffs between the \emph{size of the certificate} stored at every node, and the \emph{number of rounds} of the verification protocol. If we allow every node to communicate to distance at most $t$, one might expect that the certificate sizes can be reduced by a multiplicative factor of at least~$t$. In trees, cycles and grids, we show that such tradeoffs can be established for \emph{all} network predicates, i.e., it is always possible to linearly decrease the certificate size. In arbitrary graphs, we show that any part of the certificates common to all nodes can be evenly redistributed among these nodes, achieving even a better tradeoff: this common part of the certificate can be reduced by the size of a smallest ball of radius $t$ in the network. In addition to these general results, we establish several upper and lower bounds on the certificate sizes used for distributed proofs for spanning trees, minimum-weight spanning trees, diameter, additive and multiplicative spanners, and more, improving and generalizing previous results from the literature.

cs.DC↗

arXiv2020

Graph classes and forbidden patterns on three vertices

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.

cs.DS↗

arXiv2020

Local Certification of Graphs with Bounded Genus

Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class can be translated into a $\mathsf{dMAM}(O(\log n))$ protocol for this class, that is, a distributed interactive protocol with $O(\log n)$-bit proof size in $n$-node graphs, and three interactions between the (centralizer) computationally-unbounded but non-trustable prover Merlin, and the (decentralized) randomized computationally-limited verifier Arthur. As a corollary, there is a $\mathsf{dMAM}(O(\log n))$ protocol for the class of planar graphs, as well as for the class of graphs with bounded genus. We show that there exists a distributed interactive protocol for the class of graphs with bounded genus performing just a single interaction, from the prover to the verifier, yet preserving proof size of $O(\log n)$ bits. This result also holds for the class of graphs with bounded demi-genus, that is, graphs that can be embedded on a non-orientable surface of bounded genus. The interactive protocols described in this paper are actually proof-labeling schemes, i.e., a subclass of interactive protocols, previously introduced by Korman, Kutten, and Peleg [PODC 2005]. In particular, these schemes do not require any randomization from the verifier, and the proofs may often be computed a priori, at low cost, by the nodes themselves. Our results thus extend the recent proof-labeling scheme for planar graphs by Feuilloley et al. [PODC 2020], to graphs of bounded genus, and to graphs of bounded demigenus.

cs.DC↗

arXiv2020

Compact Distributed Certification of Planar Graphs

Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a \emph{distributed interactive proof} for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a \dMAM\/ protocol), and uses small certificates, on $O(\log n)$ bits in $n$-node networks. We show that a single interaction from the prover suffices, and randomization is unecessary, by providing an explicit description of a \emph{proof-labeling scheme} for planarity, still using certificates on just $O(\log n)$ bits. We also show that there are no proof-labeling schemes -- in fact, even no \emph{locally checkable proofs} -- for planarity using certificates on $o(\log n)$ bits.

cs.DC↗

arXiv2020

Note on distributed certification of minimum spanning trees

A distributed proof (also known as local certification, or proof-labeling scheme) is a mechanism to certify that the solution to a graph problem is correct. It takes the form of an assignment of labels to the nodes, that can be checked locally. There exists such a proof for the minimum spanning tree problem, using $O(\log n \log W)$ bit labels (where $n$ is the number of nodes in the graph, and $W$ is the largest weight of an edge). This is due to Korman and Kutten who describe it in concise and formal manner in [Korman and Kutten 07]. In this note, we propose a more intuitive description of the result, as well as a gentle introduction to the problem.

cs.DC↗

arXiv2018

Lower bounds for text indexing with mismatches and differences

In this paper we study lower bounds for the fundamental problem of text indexing with mismatches and differences. In this problem we are given a long string of length $n$, the "text", and the task is to preprocess it into a data structure such that given a query string $Q$, one can quickly identify substrings that are within Hamming or edit distance at most $k$ from $Q$. This problem is at the core of various problems arising in biology and text processing. While exact text indexing allows linear-size data structures with linear query time, text indexing with $k$ mismatches (or $k$ differences) seems to be much harder: All known data structures have exponential dependency on $k$ either in the space, or in the time bound. We provide conditional and pointer-machine lower bounds that make a step toward explaining this phenomenon. We start by demonstrating lower bounds for $k = Θ(\log n)$. We show that assuming the Strong Exponential Time Hypothesis, any data structure for text indexing that can be constructed in polynomial time cannot have $\mathcal{O}(n^{1-δ})$ query time, for any $δ>0$. This bound also extends to the setting where we only ask for $(1+\varepsilon)$-approximate solutions for text indexing. However, in many applications the value of $k$ is rather small, and one might hope that for small~$k$ we can develop more efficient solutions. We show that this would require a radically new approach as using the current methods one cannot avoid exponential dependency on $k$ either in the space, or in the time bound for all even $\frac{8}{\sqrt{3}} \sqrt{\log n} \le k = o(\log n)$. Our lower bounds also apply to the dictionary look-up problem, where instead of a text one is given a set of strings.

cs.DS↗

arXiv2018

Local verification of global proofs

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.

cs.DC↗

arXiv2017

How Long It Takes for an Ordinary Node with an Ordinary ID to Output?

In the context of distributed synchronous computing, processors perform in rounds, and the time-complexity of a distributed algorithm is classically defined as the number of rounds before all computing nodes have output. Hence, this complexity measure captures the running time of the slowest node(s). In this paper, we are interested in the running time of the ordinary nodes, to be compared with the running time of the slowest nodes. The node-averaged time-complexity of a distributed algorithm on a given instance is defined as the average, taken over every node of the instance, of the number of rounds before that node output. We compare the node-averaged time-complexity with the classical one in the standard LOCAL model for distributed network computing. We show that there can be an exponential gap between the node-averaged time-complexity and the classical time-complexity, as witnessed by, e.g., leader election. Our first main result is a positive one, stating that, in fact, the two time-complexities behave the same for a large class of problems on very sparse graphs. In particular, we show that, for LCL problems on cycles, the node-averaged time complexity is of the same order of magnitude as the slowest node time-complexity. In addition, in the LOCAL model, the time-complexity is computed as a worst case over all possible identity assignments to the nodes of the network. In this paper, we also investigate the ID-averaged time-complexity, when the number of rounds is averaged over all possible identity assignments. Our second main result is that the ID-averaged time-complexity is essentially the same as the expected time-complexity of randomized algorithms (where the expectation is taken over all possible random bits used by the nodes, and the number of rounds is measured for the worst-case identity assignment). Finally, we study the node-averaged ID-averaged time-complexity.

cs.DC↗

arXiv2017

Survey of Distributed Decision

We survey the recent distributed computing literature on checking whether a given distributed system configuration satisfies a given boolean predicate, i.e., whether the configuration is legal or illegal w.r.t. that predicate. We consider classical distributed computing environments, including mostly synchronous fault-free network computing (LOCAL and CONGEST models), but also asynchronous crash-prone shared-memory computing (WAIT-FREE model), and mobile computing (FSYNC model).

cs.DC↗

arXiv2016

A hierarchy of local decision

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 $Ω(\log^2 n)$-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 $Ω(n^2)$ 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.

cs.DC↗

arXiv2015

Brief Announcement : Average Complexity for the LOCAL Model

A standard model in network synchronised distributed computing is the LOCAL model. In this model, the processors work in rounds and, in the classic setting, they know the number of vertices of the network, $n$. Using $n$, they can compute the number of rounds after which they must all stop and output. It has been shown recently that for many problems, one can basically remove the assumption about the knowledge of $n$, without increasing the asymptotic running time. In this case, it is assumed that different vertices can choose their final output at different rounds, but continue to transmit messages. In both models, the measure of the running time is the number of rounds before the last node outputs. In this brief announcement, the vertices do not have the knowledge of $n$, and we consider an alternative measure: the average, over the nodes, of the number of rounds before they output. We prove that the complexity of a problem can be exponentially smaller with the new measure, but that Linial's lower bound for colouring still holds.

cs.DC↗