Self-stabilizing K-out-of-L exclusion on tree network

In this paper, we address the problem of K-out-of-L exclusion, a generalization of the mutual exclusion problem, in which there are $\ell$ units of a shared resource, and any process can request up to $\mathtt k$ units ($1\leq\mathtt k\leq\ell$). We propose the first deterministic self-stabilizing distributed K-out-of-L exclusion protocol in message-passing systems for asynchronous oriented tree networks which assumes bounded local memory for each process.Comment: 15 page

Distributed Queuing in Dynamic Networks

We consider the problem of forming a distributed queue in the adversarial dynamic network model of Kuhn, Lynch, and Oshman (STOC 2010) in which the network topology changes from round to round but the network stays connected. This is a synchronous model in which network nodes are assumed to be fixed, the communication links for each round are chosen by an adversary, and nodes do not know who their neighbors are for the current round before they broadcast their messages. Queue requests may arrive over rounds at arbitrary nodes and the goal is to eventually enqueue them in a distributed queue. We present two algorithms that give a total distributed ordering of queue requests in this model. We measure the performance of our algorithms through round complexity, which is the total number of rounds needed to solve the distributed queuing problem. We show that in 1-interval connected graphs, where the communication links change arbitrarily between every round, it is possible to solve the distributed queueing problem in O(nk) rounds using O(log n) size messages, where n is the number of nodes in the network and k <= n is the number of queue requests. Further, we show that for more stable graphs, e.g. T-interval connected graphs where the communication links change in every T rounds, the distributed queuing problem can be solved in O(n+ (nk/min(alpha,T))) rounds using the same O(log n) size messages, where alpha > 0 is the concurrency level parameter that captures the minimum number of active queue requests in the system in any round. These results hold in any arbitrary (sequential, one-shot concurrent, or dynamic) arrival of k queue requests in the system. Moreover, our algorithms ensure correctness in the sense that each queue request is eventually enqueued in the distributed queue after it is issued and each queue request is enqueued exactly once. We also provide an impossibility result for this distributed queuing problem in this model. To the best of our knowledge, these are the first solutions to the distributed queuing problem in adversarial dynamic networks.Comment: In Proceedings FOMC 2013, arXiv:1310.459

Local Mutual Exclusion for Dynamic, Anonymous, Bounded Memory Message Passing Systems

Mutual exclusion is a classical problem in distributed computing that provides isolation among concurrent action executions that may require access to the same shared resources. Inspired by algorithmic research on distributed systems of weakly capable entities whose connections change over time, we address the local mutual exclusion problem that tasks each node with acquiring exclusive locks for itself and the maximal subset of its "persistent" neighbors that remain connected to it over the time interval of the lock request. Using the established time-varying graphs model to capture adversarial topological changes, we propose and rigorously analyze a local mutual exclusion algorithm for nodes that are anonymous and communicate via asynchronous message passing. The algorithm satisfies mutual exclusion (non-intersecting lock sets) and lockout freedom (eventual success with probability 1) under both semi-synchronous and asynchronous concurrency. It requires ?(?) memory per node and messages of size ?(1), where ? is the maximum number of connections per node. We conclude by describing how our algorithm can implement the pairwise interactions assumed by population protocols and the concurrency control operations assumed by the canonical amoebot model, demonstrating its utility in both passively and actively dynamic distributed systems

Sandpile models

This survey is an extended version of lectures given at the Cornell Probability Summer School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and its connections to related models are presented. We discuss exactly computable results via Majumdar and Dhar's method. The main ideas of Priezzhev's computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice. We also discuss various questions arising on infinite graphs, such as convergence to a sandpile measure, and stabilizability of infinite configurations.Comment: 72 pages - v3 incorporates referee's comments. References closely related to the lectures were added/update

Time Versus Cost Tradeoffs for Deterministic Rendezvous in Networks

Two mobile agents, starting from different nodes of a network at possibly different times, have to meet at the same node. This problem is known as $\mathit{rendezvous}$. Agents move in synchronous rounds. Each agent has a distinct integer label from the set $\{1,\dots,L\}$. Two main efficiency measures of rendezvous are its $\mathit{time}$ (the number of rounds until the meeting) and its $\mathit{cost}$ (the total number of edge traversals). We investigate tradeoffs between these two measures. A natural benchmark for both time and cost of rendezvous in a network is the number of edge traversals needed for visiting all nodes of the network, called the exploration time. Hence we express the time and cost of rendezvous as functions of an upper bound $E$ on the time of exploration (where $E$ and a corresponding exploration procedure are known to both agents) and of the size $L$ of the label space. We present two natural rendezvous algorithms. Algorithm $\mathtt{Cheap}$ has cost $O(E)$ (and, in fact, a version of this algorithm for the model where the agents start simultaneously has cost exactly $E$) and time $O(EL)$. Algorithm $\mathtt{Fast}$ has both time and cost $O(E\log L)$. Our main contributions are lower bounds showing that, perhaps surprisingly, these two algorithms capture the tradeoffs between time and cost of rendezvous almost tightly. We show that any deterministic rendezvous algorithm of cost asymptotically $E$ (i.e., of cost $E+o(E)$) must have time $\Omega(EL)$. On the other hand, we show that any deterministic rendezvous algorithm with time complexity $O(E\log L)$ must have cost $\Omega (E\log L)$

Rendezvous in Networks in Spite of Delay Faults

Two mobile agents, starting from different nodes of an unknown network, have to meet at the same node. Agents move in synchronous rounds using a deterministic algorithm. Each agent has a different label, which it can use in the execution of the algorithm, but it does not know the label of the other agent. Agents do not know any bound on the size of the network. In each round an agent decides if it remains idle or if it wants to move to one of the adjacent nodes. Agents are subject to delay faults: if an agent incurs a fault in a given round, it remains in the current node, regardless of its decision. If it planned to move and the fault happened, the agent is aware of it. We consider three scenarios of fault distribution: random (independently in each round and for each agent with constant probability 0 < p < 1), unbounded adver- sarial (the adversary can delay an agent for an arbitrary finite number of consecutive rounds) and bounded adversarial (the adversary can delay an agent for at most c consecutive rounds, where c is unknown to the agents). The quality measure of a rendezvous algorithm is its cost, which is the total number of edge traversals. For random faults, we show an algorithm with cost polynomial in the size n of the network and polylogarithmic in the larger label L, which achieves rendezvous with very high probability in arbitrary networks. By contrast, for unbounded adversarial faults we show that rendezvous is not feasible, even in the class of rings. Under this scenario we give a rendezvous algorithm with cost O(nl), where l is the smaller label, working in arbitrary trees, and we show that \Omega(l) is the lower bound on rendezvous cost, even for the two-node tree. For bounded adversarial faults, we give a rendezvous algorithm working for arbitrary networks, with cost polynomial in n, and logarithmic in the bound c and in the larger label L

Hazard-free clock synchronization

The growing complexity of microprocessors makes it infeasible to distribute a single clock source over the whole processor with a small clock skew. Hence, chips are split into multiple clock regions, each covered by a single clock source. This poses a problem for communication between these clock regions. Clock synchronization algorithms promise an advantage over state-of-the-art solutions, such as GALS systems. When clock regions are synchronous the communication latency improves significantly over handshake-based solutions. We focus on the implementation of clock synchronization algorithms. A major obstacle when implementing circuits on clock domain crossings are hazardous signals. We can formally define hazards by extending the Boolean logic by a third value u. In this thesis, we describe a theory for designing and analyzing hazard-free circuits. We develop strategies for hazard-free encoding and construction of hazard-free circuits from finite state machines. Furthermore, we discuss clock synchronization algorithms and a possible combination of them. In the end, we present two implementations of the GCS algorithm by Lenzen, Locher, and Wattenhofer (JACM 2010). We prove by rigorous analysis that the systems implement the algorithm. The theory described above is used to prove that our clock synchronization circuits are hazard-free (in the sense that they compute the most precise output possible). Simulation of our GCS system shows that it achieves a skew between neighboring clock regions that is smaller than a few inverter delays.Aufgrund der zunehmenden Komplexität von Mikroprozessoren ist es unmöglich, mit einer einzigen Taktquelle den gesamten Prozessor ohne großen Versatz zu takten. Daher werden Chips in mehrere Regionen aufgeteilt, die jeweils von einer einzelnen Taktquelle abgedeckt werden. Dies stellt ein Problem für die Kommunikation zwischen diesen Taktregionen dar. Algorithmen zur Taktsynchronisation bieten einen Vorteil gegenüber aktuellen Lösungen, wie z.B. GALS-Systemen. Synchronisiert man die Taktregionen, so verbessert sich die Latenz der Kommunikation erheblich. In Schaltkreisen zwischen zwei Taktregionen können undefinierte Signale, sogenannte Hazards auftreten. Indem wir die boolesche Algebra um einen dritten Wert u erweitern, können wir diese Hazards formal definieren. In dieser Arbeit zeigen wir eine Methode zum Entwurf und zur Analyse von hazard-freien Schaltungen. Wir entwickeln Strategien für Kodierungen die Hazards vermeiden und zur Konstruktion von hazard-freien Schaltungen. Darüber hinaus stellen wir Algorithmen Taktsynchronisation vor und wie diese kombiniert werden können. Zum Schluss stellen wir zwei Implementierungen des GCS-Algorithmus von Lenzen, Locher und Wattenhofer (JACM 2010) vor. Oben genannte Mechanismen werden verwendet, um formal zu beweisen, dass diese Implementierungen korrekt sind. Die Implementierung hat keine Hazards, das heißt sie berechnet die bestmo ̈gliche Ausgabe. Anschließende Simulation der GCS Implementierung erzielt einen Versatz zwischen benachbarten Taktregionen, der kleiner als ein paar Gatter-Laufzeiten ist