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International audience; Motivated by crowdsourced computation, peergrading, and recommendation systems, Braverman, Mao and Weinberg [7] studied the query and round complexity of fundamental problems such as finding the maximum (max), finding all elements above a certain value (threshold-ν) or computing the top−k elements (Top-k) in a noisy environment.For example, consider the task of selecting papers for a conference. This task is challenging due to the crowdsourcing nature of peer reviews: the results of reviews are noisy and it is necessary to parallelize the review process as much as possible. We study the noisy value model and the noisy comparison model: In the noisy value model, a reviewer is asked to evaluate a single element: “What is the value of paper i?” (e.g., accept). In the noisy comparison model (introduced in the seminal work of Feige, Peleg, Raghavan and Upfal [17]) a reviewer is asked to do a pairwise comparison: “Is paper i better than paper j?”In this paper, we introduce new lower bound techniques for these classic problems. In comparison to previous work, our lower bounds are much more fine-grained: they focus on the interplay between round and query complexity and the dependency on the output size. In the setting of conference papers, this translates into a trade-off between number of reviews per paper and the number of review rounds necessary in order find the best 100 papers for the conference.We complement these results with simple algorithms which show that our lower bounds are almost tight.We then go beyond the worst-case and address the question of the importance of knowledge of the instance by providing, for a large range of parameters, instance-optimal algorithms with respect to the query complexity. We complement these results by showing that for some family of instances, no instance-optimal algorithm can exist.

International audience

International audience; Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, Dasgupta framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a “good” hierarchical clustering is one that minimizes a particular cost function [23]. He showed that this cost function has certain desirable properties: To achieve optimal cost, disconnected components (namely, dissimilar elements) must be separated at higher levels of the hierarchy, and when the similarity between data elements is identical, all clusterings achieve the same cost.We take an axiomatic approach to defining “good” objective functions for both similarity- and dissimilarity-based hierarchical clustering. We characterize a set of admissible objective functions having the property that when the input admits a “natural” ground-truth hierarchical clustering, the ground-truth clustering has an optimal value. We show that this set includes the objective function introduced by Dasgupta.Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better and faster algorithms for hierarchical clustering. We also initiate a beyond worst-case analysis of the complexity of the problem and design algorithms for this scenario.

A fundamental problem in distributed computing is the distribution of requests to a set of uniform servers without a centralized controller. Classically, such problems are modeled as static balls into bins processes, where m balls (tasks) are to be distributed among n bins (servers). In a seminal work, Azar et al. (SIAM J Comput 29(1):180–200, 1999. https://doi.org/10.1137/S0097539795288490) proposed the sequential strategy GREEDY[d] for n=m . Each ball queries the load of d random bins and is allocated to a least loaded of them. Azar et al. (1999) showed that d=2 yields an exponential improvement compared to d=1 . Berenbrink et al. (SIAM J Comput 35(6):1350–1385, 2006. https://doi.org/10.1137/S009753970444435X) extended this to m≫n , showing that for d=2 the maximal load difference is independent of m (in contrast to the d=1 case). We propose a new variant of an infinite balls-into-bins process. In each round an expected number of λn new balls arrive and are distributed (in parallel) to the bins. Subsequently, each non-empty bin deletes one of its balls. This setting models a set of servers processing incoming requests, where clients can query a server’s current load but receive no information about parallel requests. We study the GREEDY[d] distribution scheme in this setting and show a strong self-stabilizing property: for any arrival rate λ=λ(n)

This thesis is devoted to the study of stochastic decentralized processes. Typical examples in the real world include the dynamics of weather and temperature, of traffic, the way we meet our friends, etc. We take the rich tool set from probability theoryfor the analysis of Markov Chains and employ it to study a wide range of such distributed processes: Forest Fire Model (social networks), Balls-into-Bins with Deleting Bins, and fundamental consensus dynamics and protocols such as the Voter Model, 2-Choices, and 3-Majority., Cette thèse est consacrée à l'étude des processus stochastiques décentralisés. Parmi les exemples typiques de ces processus figurent la dynamique météorologique, la circulation automobile, la façon dont nous rencontrons nos amis, etc. Dans cette thèse, nous exploitons une large palette d'outils probabilistes permettant d'analyser des chaînes de Markov afin d'étudier un large éventail de ces processus distribués : modèle des feux de forêt (réseaux sociaux), balls-into-bins avec suppression, et des dynamiques et protocoles de consensus fondamentaux tels que Voter Model, 2-Choices, et 3-Majority.

We consider plurality consensus in networks of n nodes. Initially, each node has one of k opinions. The nodes execute a (randomized) distributed protocol to agree on the plurality opinion (the opinion initially supported by the most nodes). In certain types of networks the nodes can be quite cheap and simple, and hence one seeks protocols that are not only time efficient but also simple and space efficient. Typically, protocols depend heavily on the employed communication mechanism, which ranges from sequential (only one pair of nodes communicates at any time) to fully parallel (all nodes communicate with all their neighbors at once) and everything in-between. We propose a framework to design protocols for a multitude of communication mechanisms. We introduce protocols that solve the plurality consensus problem and are, with probability 1-o(1), both time and space efficient. Our protocols are based on an interesting relationship between plurality consensus and distributed load balancing. This relationship allows us to design protocols that generalize the state of the art for a large range of problem parameters.

In the deterministic binary majority process we are given a simple graph where each node has one out of two initial opinions. In every round, every node adopts the majority opinion among its neighbors. By using a potential argument first discovered by Goles and Olivos (1980), it is known that this process always converges in $O(|E|)$ rounds to a two-periodic state in which every node either keeps its opinion or changes it in every round. It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the $O(|E|)$ bound on the convergence time of the deterministic binary majority process is indeed tight even for dense graphs. However, in many graphs such as the complete graph, from any initial opinion assignment, the process converges in just a constant number of rounds. By carefully exploiting the structure of the potential function by Goles and Olivos (1980), we derive a new upper bound on the convergence time of the deterministic binary majority process that accounts for such exceptional cases. We show that it is possible to identify certain modules of a graph $G$ in order to obtain a new graph $G^\Delta$ with the property that the worst-case convergence time of $G^\Delta$ is an upper bound on that of $G$. Moreover, even though our upper bound can be computed in linear time, we show that, given an integer $k$, it is NP-hard to decide whether there exists an initial opinion assignment for which it takes more than $k$ rounds to converge to the two-periodic state., Comment: full version of brief announcement accepted at DISC'15

A fundamental problem in distributed computing is the distribution of requests to a set of uniform servers without a centralized controller. Classically, such problems are modelled as static balls into bins processes, where m balls (tasks) are to be distributed to n bins (servers). In a seminal work, [Azar et al.; JoC'99] proposed the sequential strategy Greedy[d] for n = m. When thrown, a ball queries the load of d random bins and is allocated to a least loaded of these. [Azar et al.; JoC'99] showed that d=2 yields an exponential improvement compared to d=1. [Berenbrink et al.; JoC'06] extended this to m ⇒ n, showing that the maximal load difference is independent of m for d=2 (in contrast to d=1). We propose a new variant of an infinite balls into bins process. In each round an expected number of λ n new balls arrive and are distributed (in parallel) to the bins and each non-empty bin deletes one of its balls. This setting models a set of servers processing incoming requests, where clients can query a server's current load but receive no information about parallel requests. We study the Greedy[d] distribution scheme in this setting and show a strong self-stabilizing property: For any arrival rate λ=λ(n) < 1, the system load is time-invariant. Moreover, for any (even super-exponential) round t, the maximum system load is (w.h.p.) O(1 over 1-λ•logn over 1-λ) for d=1 and O(log n over 1-λ) for d=2. In particular, Greedy[2] has an exponentially smaller system load for high arrival rates.

International audience; We study the deterministic binary majority process which is defined as fol- lows. We are given a graph G = (V,E) where each node has one out of two opinions. The process runs in discrete rounds where in every round each node computes and adopts the majority opinion among all of its neighbors.

In the voting model each node has an opinion and in every time step each node adopts the opinion of a random neighbour. We show that the consensus time, i.e., expected time required until one opinion prevails, is bounded by O(n / phi) for regular graphs, where phi is the conductance of the graph. This bound is tight for regular expanders. For non-regular graphs we obtain a bound of O( (davg / dmin)(n / phi) ), where davg is the average degree and dmin is the minimum degree. Additionally, we consider a generalization of the model where every opinion has a popularity. Here, every node chooses a neighbour and adopts its opinion with some probability which depends on the popularity of the neighbour's opinion. We show that the expected consensus time for a regular graph having initially two distinct opinions is bounded by O( logn / phi).

In the Palindrome Problem one tries to find all palindromes (palindromic substrings) in a given string. A palindrome is defined as a string which reads forwards the same as backwards, e.g., the string "racecar". A related problem is the Longest Palindromic Substring Problem in which finding an arbitrary one of the longest palindromes in the given string suffices. We regard the streaming version of both problems. In the streaming model the input arrives over time and at every point in time we are only allowed to use sublinear space. The main algorithms in this paper are the following: The first one is a one-pass randomized algorithm that solves the Palindrome Problem. It has an additive error and uses $O(\sqrt n$) space. The second algorithm is a two-pass algorithm which determines the exact locations of all longest palindromes. It uses the first algorithm as the first pass. The third algorithm is again a one-pass randomized algorithm, which solves the Longest Palindromic Substring Problem. It has a multiplicative error using only $O(\log(n))$ space. We also give two variants of the first algorithm which solve other related practical problems.

A fundamental problem in distributed computing is the distribution of requests to a set of uniform servers without a centralized controller. Classically, such problems are modeled as static balls into bins processes, where $m$ balls (tasks) are to be distributed to $n$ bins (servers). In a seminal work, Azar et al. proposed the sequential strategy \greedy{d} for $n=m$. When thrown, a ball queries the load of $d$ random bins and is allocated to a least loaded of these. Azar et al. showed that $d=2$ yields an exponential improvement compared to $d=1$. Berenbrink et al. extended this to $m\gg n$, showing that the maximal load difference is independent of $m$ for $d=2$ (in contrast to $d=1$). We propose a new variant of an \emph{infinite} balls into bins process. Each round an expected number of $\lambda n$ new balls arrive and are distributed (in parallel) to the bins. Each non-empty bin deletes one of its balls. This setting models a set of servers processing incoming requests, where clients can query a server's current load but receive no information about parallel requests. We study the \greedy{d} distribution scheme in this setting and show a strong self-stabilizing property: For \emph{any} arrival rate $\lambda=\lambda(n)

Coalescing random walks is a fundamental distributed process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress such as that of Cooper et al., the coalescence time for graphs, such as binary trees, d -dimensional tori, hypercubes, and, more generally, vertex-transitive graphs, remains unresolved. We provide a powerful toolkit that results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log 2 n ), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. Finally, we prove a tight worst-case bound for the coalescence time of O(n 3 ) . By duality, our results yield identical bounds on the voter model. Our techniques also yield a new bound on the hitting time and cover time of regular graphs, improving and tightening previous results by Broder and Karlin, as well as those by Aldous and Fill.

Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, Dasgupta (2016) framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a `good' hierarchical clustering is one that minimizes some cost function. He showed that this cost function has certain desirable properties, such as in order to achieve optimal cost disconnected components must be separated first and that in `structureless' graphs, i.e., cliques, all clusterings achieve the same cost. We take an axiomatic approach to defining `good' objective functions for both similarity and dissimilarity-based hierarchical clustering. We characterize a set of admissible objective functions (that includes the one introduced by Dasgupta) that have the property that when the input admits a `natural' ground-truth hierarchical clustering, the ground-truth clustering has an optimal value. Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better and faster algorithms for hierarchical clustering. For similarity-based hierarchical clustering, Dasgupta (2016) showed that a simple recursive sparsest-cut based approach achieves an O(log^3/2 n)-approximation on worst-case inputs. We give a more refined analysis of the algorithm and show that it in fact achieves an O(√log n)-approximation. This improves upon the LP-based O(log n)-approximation of Roy and Pokutta (2016). For dissimilarity-based hierarchical clustering, we show that the classic average-linkage algorithm gives a factor 2 approximation, and provide a simple and better algorithm that gives a factor 3/2 approximation. This aims at explaining the success of this heuristics in practice. Finally, we consider `beyond-worst-case' scenario through a generalisation of the stochastic block model for hierarchical clustering. We show that Dasgupta's cost function also has desirable properties for these inputs and we provide a simple algorithm that for graphs generated according to this model yields a 1 + o(1) factor approximation.