Lower bounds in distributed computing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 167-170).Distributed computing is the study of achieving cooperative behavior between independent computing processes with possibly conflicting goals. Distributed computing is ubiquitous in the Internet, wireless networks, multi-core and multi-processor computers, teams of mobile robots, etc. In this thesis, we study two fundamental distributed computing problems, clock synchronization and mutual exclusion. Our contributions are as follows. 1. We introduce the gradient clock synchronization (GCS) problem. As in traditional clock synchronization, a group of nodes in a bounded delay communication network try to synchronize their logical clocks, by reading their hardware clocks and exchanging messages. We say the distance between two nodes is the uncertainty in message delay between the nodes, and we say the clock skew between the nodes is their difference in logical clock values. GCS studies clock skew as a function of distance. We show that surprisingly, every clock synchronization algorithm exhibits some execution in which two nodes at distance one apart have Q( lo~gD clock skew, where D is the maximum distance between any pair of nodes. 2. We present an energy efficient and fault tolerant clock synchronization algorithm suitable for wireless networks. The algorithm synchronizes nodes to each other, as well as to real time. It satisfies a relaxed gradient property. That is, it guarantees that, using certain reasonable operating parameters, nearby nodes are well synchronized most of the time. 3. We study the mutual exclusion (mutex) problem, in which a set of processes in a shared memory system compete for exclusive access to a shared resource. We prove a tight Q(n log n) lower bound on the time for n processes to each access the resource once. .(cont.) Our novel proof technique is based on separately lower bounding the amount of information needed for solving mutex, and upper bounding the amount of information any mutex algorithm can acquire in each step. We hope that our results offer fresh ways of looking at classical problems, and point to interesting new open problemsby Rui Fan.Ph.D

Tight Bounds for Set Disjointness in the Message Passing Model

In a multiparty message-passing model of communication, there are $k$ players. Each player has a private input, and they communicate by sending messages to one another over private channels. While this model has been used extensively in distributed computing and in multiparty computation, lower bounds on communication complexity in this model and related models have been somewhat scarce. In recent work \cite{phillips12,woodruff12,woodruff13}, strong lower bounds of the form $\Omega(n \cdot k)$ were obtained for several functions in the message-passing model; however, a lower bound on the classical Set Disjointness problem remained elusive. In this paper, we prove tight lower bounds of the form $\Omega(n \cdot k)$ for the Set Disjointness problem in the message passing model. Our bounds are obtained by developing information complexity tools in the message-passing model, and then proving an information complexity lower bound for Set Disjointness. As a corollary, we show a tight lower bound for the task allocation problem \cite{DruckerKuhnOshman} via a reduction from Set Disjointness

Being Fast Means Being Chatty: The Local Information Cost of Graph Spanners

We introduce a new measure for quantifying the amount of information that the nodes in a network need to learn to jointly solve a graph problem. We show that the local information cost ($\textsf{LIC}$) presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST-KT1 model, where each node has initial knowledge of its neighbors' IDs, we prove that $\Omega(\textsf{LIC}_\gamma(P)/ \log\tau \log n)$ bits are required for solving a graph problem $P$ with a $\tau$-round algorithm that errs with probability at most $\gamma$. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST-KT1 model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing a $(2t-1)$-spanner that consists of at most $O(n^{1+1/t + \epsilon})$ edges, where $\epsilon = \Theta(1/t^2)$. Our main result is that any $O(\textsf{poly}(n))$-time algorithm must send at least $\tilde\Omega((1/t^2) n^{1+1/2t})$ bits in the CONGEST model under the KT1 assumption. Previously, only a trivial lower bound of $\tilde \Omega(n)$ bits was known for this problem. A consequence of our lower bound is that achieving both time- and communication-optimality is impossible when designing a distributed spanner algorithm. In light of the work of King, Kutten, and Thorup (PODC 2015), this shows that computing a minimum spanning tree can be done significantly faster than finding a spanner when considering algorithms with $\tilde O(n)$ communication complexity. Our result also implies time complexity lower bounds for constructing a spanner in the node-congested clique of Augustine et al. (2019) and in the push-pull gossip model with limited bandwidth

A General Characterization of Indulgence (Invited Paper)

An indulgent algorithm is a distributed algorithm that, besides tolerating process failures, also tolerates arbitrarily long periods of instability, with an unbounded number of timing and scheduling failures. In particular, no process can take any irrevocable action based on the operational status, correct or failed, of other processes. This paper presents an intuitive and general characterization of indulgence. The characterization can be viewed as a simple application of Murphy's law to partial runs of a distributed algorithm, in a computing model that encompasses various communication and resilience schemes. We use our characterization to establish several results about the inherent power and limitations of indulgent algorithms

Algorithms for self-healing networks

Many modern networks are reconfigurable, in the sense that the topology of the network can be changed by the nodes in the network. For example, peer-to-peer, wireless and ad-hoc networks are reconfigurable. More generally, many social networks, such as a company\u27s organizational chart; infrastructure networks, such as an airline\u27s transportation network; and biological networks, such as the human brain, are also reconfigurable. Modern reconfigurable networks have a complexity unprecedented in the history of engineering, resembling more a dynamic and evolving living animal rather than a structure of steel designed from a blueprint. Unfortunately, our mathematical and algorithmic tools have not yet developed enough to handle this complexity and fully exploit the flexibility of these networks. We believe that it is no longer possible to build networks that are scalable and never have node failures. Instead, these networks should be able to admit small, and, maybe, periodic failures and still recover like skin heals from a cut. This process, where the network can recover itself by maintaining key invariants in response to attack by a powerful adversary is what we call self-healing. Here, we present several fast and provably good distributed algorithms for self-healing in reconfigurable dynamic networks. Each of these algorithms have different properties, a different set of gaurantees and limitations. We also discuss future directions and theoretical questions we would like to answer

Distributed computation in wireless and dynamic networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 211-221) and index.Today's wireless networks tend to be centralized: they are organized around a fixed central backbone such as a network of cellular towers or wireless access points. However, as mobile computing devices continue to shrink in size and in cost, we are reaching the point where large-scale ad-hoc wireless networks, composed of swarms of cheap devices or sensors, are becoming feasible. In this thesis we study the theoretical computation power of such networks, and ask what tasks are they capable of carrying out. how long does solving particular tasks take. and what is the effect of the unpredictable network topology on the network's computation power. In the first part of the thesis we introduce an abstract model for dynamic networks. In contrast to much of the literature on mobile and ad-hoc networks, our model makes fairly minimalistic assumptions; it allows the network topology to change arbitrarily from round to round, as long as in each round the communication graph is connected. We show that even in this weak model, global computation is still possible, and any function of the nodes' initial inputs can be computed efficiently. Also, using tools from the field of epistemic logic, we analyze information flow in dynamic networks, and study the time required to achieve various notions of coordination. In the second part of the thesis we restrict attention to static networks, which retain an important feature of wireless networks: they are potentially (symmetric. We show that in this setting. classical data aggregation tasks become much harder. and we develop both upper and lower bounds on computing various classes of functions. Our main tool in this part of the thesis is communication complexity: we use existing lower bounds in two-player communication complexity, and also introduce a new problem, task allocation, and study its communication complexity in the two-player and multi-player settings.by Rotei Oshman.Ph.D

Lower bounds in distributed computing

In this thesis I study the complexity theory of distributed computing in synchronous message passing models. The focus is on highly local problems, that is, problems in which very little communication is required. In this setting the underlying communication network is also the input graph. The distributed system must collectively compute a solution to a problem related to the structure of this network, with each computer producing its own part of the output. We study the LOCAL model, one of the standard models in distributed computing. It abstracts away faults, congestion, computational requirements, memory requirements, and many other challenges in distributed computing. We study this model to understand the locality aspect of distributed computing: How far does information have to propagate in distributed problem solving? How many communication rounds is required? Many interesting problems, such as finding a spanning tree in the communication network, are inherently global. We are interested in the other extreme: problems that can be solved in time that is weakly dependent or completely independent of the size of the communication network. Typical problems include classical symmetry breaking tasks such as colouring or maximal independent set. Typically the time complexity of such problems depends on two parameters: the size and the maximum degree of the input graph. The connection with the first parameter is well understood with tight upper and lower bounds. The connection with the maximum degree is much less well understood, with an exponential gap between the upper and lower bounds. We develop a new lower bound technique to give the first lower bound for a natural graph problem that is linear in the maximum degree. In addition we study the power of unique in several contexts. We show that while usually unique identifiers are unhelpful for constant-time algorithms, there are certain special cases in which they do help. In the context of local decision, where the task is essentially to verify a given solution instead of constructing a new one, we characterize the minimal information that is sufficient to replace unique identifiers. Finally, we also study the power of nondeterminism in local decision. We develop a local hierarchy in a manner analogous to the polynomial hierarchy in centralized computing, and study the structure of this hierarchy. The focus of this thesis is on structural results, especially lower bounds. The lower bounds as a function of the maximum degree of the graph employ a completely novel proof technique based on graph symmetries.Tutkin tässä väitöskirjatyössä hajautetun laskennan vaativuusteoriaa synkronisissa viestinvälitysmalleissa. Työ keskittyy paikallisiin ongelmiin, eli ongelmiin, joiden ratkaisu voidaan löytää käyttäen vain vähäinen määrä kommunikaatiota. Tutkitussa asetelmassa kommunikaatioverkko on myös ongelman syöte. Hajautetun järjestelmän täytyy löytää ratkaisu, joka liittyy tämän verkon rakenteeseen ja jokaisen verkon solmun täytyy tuottaa oma osansa tulosteesta. Työssä tutkitaan hajautetun laskennan standardimalleihin kuuluvaa LOCAL-mallia. Tämä malli ei huomioi erilaisia hajautetun laskennan haasteita, kuten vikoja, ruuhkautumista, tai laskenta- ja muistivaatimuksia. Tätä mallia tutkitaan paikallisuuden ymmärtämiseksi: kuinka kauas informaatiota täytyy propagoida, jotta ongelma voidaan ratkaista hajautetusti? Kuinka monta kommunikaatiokierrosta ongelman ratkaiseminen vaatii? Monet kiinnostavat ongelmat, kuten kommunikaatioverkon virittävän puun löytäminen, ovat globaaleja ongelmia. Tässä työssä tarkastelemme toista ääripäätä: ongelmia, jotka voidaan ratkaista ajassa, joka on kokonaan riippumaton tai korkeintaan heikosti riippuvainen kommunikaatioverkon koosta. Tällaisia ongelmia ovat esimerkiksi klassiset symmetrianrikkomisongelmat kuten väritys ja riippumattomat joukot. Tyypillisesti näiden ongelmien aikavaativuus riippuu kahdesta tekijästä, verkon koosta ja sen maksimiasteluvusta. Yhteys verkon kokoon on melko hyvin ymmärretty, mutta riippuvuus maksimiasteluvusta ei. Tunnettujen ylä- ja alarajojen erotus on eksponentiaalinen. Kehittämämme alarajatekniikka antaa ensimmäisen maksimiasteluvun suhteen lineaarisen alarajan luonnolliselle verkko-ongelmalle. Lisäksi olemme tutkineet solmuille annettavien uniikkien nimien vaikutusta. Vaikka tyypillisesti uniikit tunnisteet eivät auta vakioaikaisia algoritmeja, näytämme, että tietyissä tilanteissa niitä voidaan hyödyntää. Paikallisten päätösongelmien tapauksessa, jossa solmujen täytyy tarkastaa annettu ratkaisu uuden rakentamisen sijaan, karakterisoimme sen informaation määrän, joka riittää ja vaaditaan uniikkien tunnisteiden korvaamiseen. Viimeiseksi tutkimme epädeterministisyyden vaikutusta paikallisiin päätösongelmiin. Kehitämme paikallisen päätöshierarkian, joka on analoginen keskitetyn laskennan polynomisen hierarkian kanssa, ja tutkimme tämän hierarkian rakennetta. Väitöskirjatyö keskittyy rakenteellisiin tuloksiin, erityisesti mahdottomuustuloksiin. Verkon asteluvusta riippuvat alarajatulokset edustavat täysin uudenlaista alarajatekniikkaa, joka hyödyntää verkossa olevaa symmetriaa