Beeping a Deterministic Time-Optimal Leader Election

The beeping model is an extremely restrictive broadcast communication model that relies only on carrier sensing. In this model, we solve the leader election problem with an asymptotically optimal round complexity of O(D + log n), for a network of unknown size n and unknown diameter D (but with unique identifiers). Contrary to the best previously known algorithms in the same setting, the proposed one is deterministic. The techniques we introduce give a new insight as to how local constraints on the exchangeable messages can result in efficient algorithms, when dealing with the beeping model. Using this deterministic leader election algorithm, we obtain a randomized leader election algorithm for anonymous networks with an asymptotically optimal round complexity of O(D + log n) w.h.p. In previous works this complexity was obtained in expectation only. Moreover, using deterministic leader election, we obtain efficient algorithms for symmetry-breaking and communication procedures: O(log n) time MIS and 5-coloring for tree networks (which is time-optimal), as well as k-source multi-broadcast for general graphs in O(min(k,log n) * D + k log{(n M)/k}) rounds (for messages in {1,..., M}). This latter result improves on previous solutions when the number of sources k is sublogarithmic (k = o(log n))

Sleeping is Superefficient: MIS in Exponentially Better Awake Complexity

Maximal Independent Set (MIS) is one of the central and most well-studied problems in distributed computing. Even after four decades of intensive research, the best-known (randomized) MIS algorithms take $O(\log{n})$ worst-case rounds on general graphs (where $n$ is the number of nodes), while the best-known lower bound is $\Omega\left(\sqrt{\frac{\log{n}}{\log{\log{n}}}}\right)$ rounds. Breaking past the $O(\log{n})$ worst-case bound or showing stronger lower bounds have been longstanding open problems. Our main contribution is that we show that MIS can be computed in (worst-case) awake complexity of $O(\log \log n)$ rounds that is (essentially) exponentially better compared to the (traditional) round complexity lower bound of $\Omega\left(\sqrt{\frac{\log{n}}{\log{\log{n}}}}\right)$. Specifically, we present the following results. (1) We present a randomized distributed (Monte Carlo) algorithm for MIS that with high probability computes an MIS and has $O(\log\log{n})$-rounds awake complexity. This algorithm has (traditional) {\em round complexity} that is $O(poly(n))$. Our bounds hold in the $CONGEST(O(polylog n))$ model where only $O(polylog n)$ (specifically $O(\log^3 n)$) bits are allowed to be sent per edge per round. (2) We also show that we can drastically reduce the round complexity at the cost of a slight increase in awake complexity by presenting a randomized MIS algorithm with $O(\log \log n \log^* n )$ awake complexity and $O(\log^3 n \log \log n \log^*n)$ round complexity in the $CONGEST(O(polylog n))$ model.Comment: Abstract shortened to fit arXiv constraint

Distributed MIS in O(log log n) Awake Complexity

Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best known (randomized) MIS algorithms have O(log n) round complexity on general graphs [Luby, STOC 1986] (where n is the number of nodes), while the best known lower bound is [EQUATION] [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the O(log n) round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Energy is a premium resource in various settings such as battery-powered wireless networks and sensor networks. The bulk of the energy is used by nodes when they are awake, i.e., when they are sending, receiving, and even just listening for messages. On the other hand, when a node is sleeping, it does not perform any communication and thus spends very little energy. Several recent works have addressed the problem of designing energy-efficient distributed algorithms for various fundamental problems. These algorithms operate by minimizing the number of rounds in which any node is awake, also called the (worst-case) awake complexity. An intriguing open question is whether one can design a distributed MIS algorithm that has significantly smaller awake complexity compared to existing algorithms. In particular, the question of obtaining a distributed MIS algorithm with o(log n) awake complexity was left open in [Chatterjee, Gmyr, Pandurangan, PODC 2020]. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of O(log n) and also bypassing its fundamental [EQUATION] round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in O(log log n) awake complexity with high probability.1 However, this algorithm has a round complexity that is O(poly(n)). We then show how to drastically improve the round complexity at the cost of a slight increase in awake complexity by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability computes an MIS in O((log log n) log* n) awake complexity and O((log3 n)(log log n) log* n) round complexity. Our algorithms work in the CONGEST model where messages of size O(log n) bits can be sent per edge per round

Distributed MIS in O(log log n) Awake Complexity

Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best known (randomized) MIS algorithms have O(log n) round complexity on general graphs [Luby, STOC 1986] (where n is the number of nodes), while the best known lower bound is [EQUATION] [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the O(log n) round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Energy is a premium resource in various settings such as battery-powered wireless networks and sensor networks. The bulk of the energy is used by nodes when they are awake, i.e., when they are sending, receiving, and even just listening for messages. On the other hand, when a node is sleeping, it does not perform any communication and thus spends very little energy. Several recent works have addressed the problem of designing energy-efficient distributed algorithms for various fundamental problems. These algorithms operate by minimizing the number of rounds in which any node is awake, also called the (worst-case) awake complexity. An intriguing open question is whether one can design a distributed MIS algorithm that has significantly smaller awake complexity compared to existing algorithms. In particular, the question of obtaining a distributed MIS algorithm with o(log n) awake complexity was left open in [Chatterjee, Gmyr, Pandurangan, PODC 2020]. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of O(log n) and also bypassing its fundamental [EQUATION] round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in O(log log n) awake complexity with high probability.1 However, this algorithm has a round complexity that is O(poly(n)). We then show how to drastically improve the round complexity at the cost of a slight increase in awake complexity by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability computes an MIS in O((log log n) log* n) awake complexity and O((log3 n)(log log n) log* n) round complexity. Our algorithms work in the CONGEST model where messages of size O(log n) bits can be sent per edge per round

An Almost Singularly Optimal Asynchronous Distributed MST Algorithm

A singularly (near) optimal distributed algorithm is one that is (near) optimal in \emph{two} criteria, namely, its time and message complexities. For \emph{synchronous} CONGEST networks, such algorithms are known for fundamental distributed computing problems such as leader election [Kutten et al., JACM 2015] and Minimum Spanning Tree (MST) construction [Pandurangan et al., STOC 2017, Elkin, PODC 2017]. However, it is open whether a singularly (near) optimal bound can be obtained for the MST construction problem in general \emph{asynchronous} CONGEST networks. We present a randomized distributed MST algorithm that, with high probability, computes an MST in \emph{asynchronous} CONGEST networks and takes $\tilde{O}(D^{1+\epsilon} + \sqrt{n})$ time and $\tilde{O}(m)$ messages, where $n$ is the number of nodes, $m$ the number of edges, $D$ is the diameter of the network, and $\epsilon >0$ is an arbitrarily small constant (both time and message bounds hold with high probability). Our algorithm is message optimal (up to a polylog$(n)$ factor) and almost time optimal (except for a $D^{\epsilon}$ factor). Our result answers an open question raised in Mashregi and King [DISC 2019] by giving the first known asynchronous MST algorithm that has sublinear time (for all $D = O(n^{1-\epsilon})$) and uses $\tilde{O}(m)$ messages. Using a result of Mashregi and King [DISC 2019], this also yields the first asynchronous MST algorithm that is sublinear in both time and messages in the $KT_1$ CONGEST model. A key tool in our algorithm is the construction of a low diameter rooted spanning tree in asynchronous CONGEST that has depth $\tilde{O}(D^{1+\epsilon})$ (for an arbitrarily small constant $\epsilon > 0$) in $\tilde{O}(D^{1+\epsilon})$ time and $\tilde{O}(m)$ messages. To the best of our knowledge, this is the first such construction that is almost singularly optimal in the asynchronous setting.Comment: 27 pages, accepted to DISC 202

Surmonter les interférences dans le modèle de communication par bips

Small inexpensive inter-communicating electronic devices have become widely available. Although the individual device has severely limited capabilities (e.g., basic communication, constant-size memory or limited mobility), multitudes of such weak devices communicating together are able to form a low-cost, easily deployable, yet highly performant network. Such distributed systems present significant challenges however when it comes to the design of efficient, scalable and simple algorithms. In this thesis, we are interested in studying such systems composed of devices with severely limited communication capabilities - using only simple bursts of energy. These distributed systems may be modeled using the beeping model, in which nodes communicate by beeping or listening to their neighbors (according to some undirected communication graph). Simultaneous communications (i.e., collisions) result in non-destructive interference: a node with two or more neighbors beeping simultaneously detects a beep. Its simple, general and energy efficient communication mechanism makes the beeping model widely applicable. However, that simplicity comes at a cost. Due to the poor expressiveness of beeps and the interference caused by simultaneous communications, algorithm design is challenging. Throughout this work, we overcome both difficulties in order to provide efficient communication primitives. A particular focus of the thesis is on deterministic and time-efficient solutions independent of the communication graph's parameters (i.e., uniform). The first part of the thesis considers a setting in which nodes wake up at the same time (i.e., the network has been set up a priori). To obtain efficient solutions to fundamental distributed communication problems, we first focus on efficiently solving problems for local symmetry-breaking: (Δ+1)-vertex coloring and maximal independent set (where Δ is the maximum degree of the communication graph). The solutions we devise are particularly efficient when the communication graph is sparse. They are then used to solve the 2-hop variants of these problems and to simulate message-passing. Finally, combining this simulation with existing results, which assume message-passing, gives the first vertex coloring algorithm using less than Δ+1 colors in the beeping model. Then, we study problems defined on a global scale, such as leader election and multi-broadcast (i.e., information dissemination). Leader election is a crucial building block in the design of distributed algorithms. We give the first two time-optimal leader election algorithms for the beeping model. One is deterministic, but requires unique identifiers. The second one does not need identifiers (useful for security and privacy reasons), but is randomized. Building upon the time-optimal leader election solution, computationally efficient and time-optimal algorithms for multi-broadcast are presented. Although a previous time-optimal solution was available, it required computationally expensive methods. The second part of the thesis considers a more difficult but more general setting, in which nodes wake up at some arbitrary time rounds. We focus on the desynchronization problem, and more precisely on its 2-hop variant, which can be used as medium access control method. We show that it is possible for nodes to communicate in a coherent manner beyond their 1-hop neighborhood. More concretely, a primitive allowing nodes to simulate communication on the square of the communication graph is presented. This primitive is a centerpiece in the design of the 2-hop desynchronization algorithm. Finally, by leveraging this solution, we show that higher-level primitives for sending and receiving messages can be obtained in this difficult setting.Les petits appareils électroniques peu coûteux et à communication sans fil sont devenus largement disponibles. Bien que chaque entité ait des capacités limitées (par exemple, communication basique ou mémoire de taille constante), un déploiement à grande échelle de telles entités communiquantes constitue un réseau performant, en plus d’être peu coûteux. De tels systèmes distribués présentent toutefois des défis importants en ce qui concerne la conception d'algorithmes simples, efficaces et évolutifs. Dans cette thèse, nous nous intéressons à l’étude de ces systèmes, composés d’appareils dotés de capacités de communication très limitées, à base de simples impulsions d’énergie. Ces systèmes distribués peuvent être modélisés à l'aide du modèle de bips, dans lequel les nœuds communiquent en émettant un bip, un simple signal indifférencié, ou en écoutant leurs voisins (selon un graphe de communication non orienté). Les communications simultanées (c'est-à-dire les collisions) entraînent des interférences non destructives : un nœud, dont deux voisins ou plus émettent simultanément un bip, détecte seulement un bip. Ce mécanisme de communication simple, général et économe en énergie rend les résultats obtenus dans le modèle de bips applicables à de nombreuses situations différentes, avec cependant un challenge. En raison de la faible expressivité des bips et des collisions, la conception des algorithmes est difficile. Tout au long de ce travail, nous surmontons ces deux difficultés afin de fournir des primitives de communication efficaces. La thèse s’intéresse particulièrement aux solutions déterministes, rapides (en temps) et indépendantes des paramètres du graphe de communication (c’est-à-dire uniformes). La première partie de la thèse considère un cadre dans lequel les nœuds se réveillent en même temps (c’est-à-dire que le réseau a été configuré a priori). Pour obtenir des solutions efficaces pour des problèmes fondamentaux de communication distribuée, nous nous concentrons d’abord sur la résolution efficace de problèmes de brisure locale de symétrie : ensemble indépendant maximal et coloration de sommets utilisant au plus Δ + 1 couleurs (où Δ est le degré maximal du graphe de communication). Nous élaborons des solutions à ces problèmes ainsi qu'à leurs variantes à distance deux. Cela nous permet de simuler une communication par messages. Enfin, nous obtenons le premier algorithme de coloration utilisant moins de Δ + 1 couleurs dans le modèle de bips. Ensuite, nous étudions des problèmes définis à l’échelle du réseau, tels que l’élection d'un leader et la diffusion multiple de messages. L'élection d'un leader est un élément essentiel dans la conception d'algorithmes distribués. Nous donnons les deux premiers algorithmes d’élection de leader optimaux en temps pour le modèle de bips. L'un est déterministe, mais nécessite des identifiants uniques. Le second n’a pas besoin d’identifiants (utile pour des raisons de sécurité et de confidentialité), mais est randomisé. S'appuyant sur une élection de leader optimale en temps, plusieurs algorithmes pour la diffusion multiple, efficaces en temps et en calcul, sont présentés. La deuxième partie de la thèse considère un cadre plus difficile mais plus général, dans lequel les nœuds se réveillent de façon asynchrone. Nous nous concentrons sur le problème de désynchronisation à distance deux, qui permet un contrôle de l'accès au support, primordial dans les réseaux sans fil. Nous montrons qu'il est possible pour les nœuds de communiquer de manière cohérente au-delà de leur voisinage immédiat. A cette fin, une primitive permettant aux nœuds de simuler une communication est présentée. Cette primitive est un élément central dans la conception de l'algorithme de désynchronisation à distance deux. Enfin, nous exploitons cette solution afin d'implémenter des primitives de haut niveau pour l’envoi et la réception de messages

Surmonter les interférences dans le modèle de communication par bips

Les petits appareils électroniques peu coûteux et à communication sans fil sont devenus largement disponibles. Bien que chaque entité ait des capacités limitées (par exemple, communication basique ou mémoire de taille constante), un déploiement à grande échelle de telles entités communiquantes constitue un réseau performant, en plus d’être peu coûteux. De tels systèmes distribués présentent toutefois des défis importants en ce qui concerne la conception d'algorithmes simples, efficaces et évolutifs. Dans cette thèse, nous nous intéressons à l’étude de ces systèmes, composés d’appareils dotés de capacités de communication très limitées, à base de simples impulsions d’énergie. Ces systèmes distribués peuvent être modélisés à l'aide du modèle de bips, dans lequel les nœuds communiquent en émettant un bip, un simple signal indifférencié, ou en écoutant leurs voisins (selon un graphe de communication non orienté). Les communications simultanées (c'est-à-dire les collisions) entraînent des interférences non destructives : un nœud, dont deux voisins ou plus émettent simultanément un bip, détecte seulement un bip. Ce mécanisme de communication simple, général et économe en énergie rend les résultats obtenus dans le modèle de bips applicables à de nombreuses situations différentes, avec cependant un challenge. En raison de la faible expressivité des bips et des collisions, la conception des algorithmes est difficile. Tout au long de ce travail, nous surmontons ces deux difficultés afin de fournir des primitives de communication efficaces. La thèse s’intéresse particulièrement aux solutions déterministes, rapides (en temps) et indépendantes des paramètres du graphe de communication (c’est-à-dire uniformes). La première partie de la thèse considère un cadre dans lequel les nœuds se réveillent en même temps (c’est-à-dire que le réseau a été configuré a priori). Pour obtenir des solutions efficaces pour des problèmes fondamentaux de communication distribuée, nous nous concentrons d’abord sur la résolution efficace de problèmes de brisure locale de symétrie : ensemble indépendant maximal et coloration de sommets utilisant au plus Δ + 1 couleurs (où Δ est le degré maximal du graphe de communication). Nous élaborons des solutions à ces problèmes ainsi qu'à leurs variantes à distance deux. Cela nous permet de simuler une communication par messages. Enfin, nous obtenons le premier algorithme de coloration utilisant moins de Δ + 1 couleurs dans le modèle de bips. Ensuite, nous étudions des problèmes définis à l’échelle du réseau, tels que l’élection d'un leader et la diffusion multiple de messages. L'élection d'un leader est un élément essentiel dans la conception d'algorithmes distribués. Nous donnons les deux premiers algorithmes d’élection de leader optimaux en temps pour le modèle de bips. L'un est déterministe, mais nécessite des identifiants uniques. Le second n’a pas besoin d’identifiants (utile pour des raisons de sécurité et de confidentialité), mais est randomisé. S'appuyant sur une élection de leader optimale en temps, plusieurs algorithmes pour la diffusion multiple, efficaces en temps et en calcul, sont présentés. La deuxième partie de la thèse considère un cadre plus difficile mais plus général, dans lequel les nœuds se réveillent de façon asynchrone. Nous nous concentrons sur le problème de désynchronisation à distance deux, qui permet un contrôle de l'accès au support, primordial dans les réseaux sans fil. Nous montrons qu'il est possible pour les nœuds de communiquer de manière cohérente au-delà de leur voisinage immédiat. A cette fin, une primitive permettant aux nœuds de simuler une communication est présentée. Cette primitive est un élément central dans la conception de l'algorithme de désynchronisation à distance deux. Enfin, nous exploitons cette solution afin d'implémenter des primitives de haut niveau pour l’envoi et la réception de messages.Small inexpensive inter-communicating electronic devices have become widely available. Although the individual device has severely limited capabilities (e.g., basic communication, constant-size memory or limited mobility), multitudes of such weak devices communicating together are able to form a low-cost, easily deployable, yet highly performant network. Such distributed systems present significant challenges however when it comes to the design of efficient, scalable and simple algorithms. In this thesis, we are interested in studying such systems composed of devices with severely limited communication capabilities - using only simple bursts of energy. These distributed systems may be modeled using the beeping model, in which nodes communicate by beeping or listening to their neighbors (according to some undirected communication graph). Simultaneous communications (i.e., collisions) result in non-destructive interference: a node with two or more neighbors beeping simultaneously detects a beep. Its simple, general and energy efficient communication mechanism makes the beeping model widely applicable. However, that simplicity comes at a cost. Due to the poor expressiveness of beeps and the interference caused by simultaneous communications, algorithm design is challenging. Throughout this work, we overcome both difficulties in order to provide efficient communication primitives. A particular focus of the thesis is on deterministic and time-efficient solutions independent of the communication graph's parameters (i.e., uniform). The first part of the thesis considers a setting in which nodes wake up at the same time (i.e., the network has been set up a priori). To obtain efficient solutions to fundamental distributed communication problems, we first focus on efficiently solving problems for local symmetry-breaking: (Δ+1)-vertex coloring and maximal independent set (where Δ is the maximum degree of the communication graph). The solutions we devise are particularly efficient when the communication graph is sparse. They are then used to solve the 2-hop variants of these problems and to simulate message-passing. Finally, combining this simulation with existing results, which assume message-passing, gives the first vertex coloring algorithm using less than Δ+1 colors in the beeping model. Then, we study problems defined on a global scale, such as leader election and multi-broadcast (i.e., information dissemination). Leader election is a crucial building block in the design of distributed algorithms. We give the first two time-optimal leader election algorithms for the beeping model. One is deterministic, but requires unique identifiers. The second one does not need identifiers (useful for security and privacy reasons), but is randomized. Building upon the time-optimal leader election solution, computationally efficient and time-optimal algorithms for multi-broadcast are presented. Although a previous time-optimal solution was available, it required computationally expensive methods. The second part of the thesis considers a more difficult but more general setting, in which nodes wake up at some arbitrary time rounds. We focus on the desynchronization problem, and more precisely on its 2-hop variant, which can be used as medium access control method. We show that it is possible for nodes to communicate in a coherent manner beyond their 1-hop neighborhood. More concretely, a primitive allowing nodes to simulate communication on the square of the communication graph is presented. This primitive is a centerpiece in the design of the 2-hop desynchronization algorithm. Finally, by leveraging this solution, we show that higher-level primitives for sending and receiving messages can be obtained in this difficult setting