Modelling and Analysis Using GROOVE

In this paper we present case studies that describe how the graph transformation tool GROOVE has been used to model problems from a wide variety of domains. These case studies highlight the wide applicability of GROOVE in particular, and of graph transformation in general. They also give concrete templates for using GROOVE in practice. Furthermore, we use the case studies to analyse the main strong and weak points of GROOVE

Leader election in synchronous networks

Worst, best and average number of messages and running time of leader election algorithms of different distributed systems are analyzed. Among others the known characterizations of the expected number of messages for LCR algorithm and of the worst number of messages of Hirschberg-Sinclair algorithm are improve

Proving Distributed Algorithms by Combining Refinement and Local Computations

Distributed algorithms are considered to be very complex to design and to prove; our paper contributes to the design of correct-by-construction distributed algorithms. The main idea relies upon the development of distributed algorithms following a top/down approach, which is clearly well known in earlier works of Dijkstra, and to use refinement for proving the correctness of the resulting algorithms. However, the link between the problem and the first model remains to be expressed and the refinement is a real help to justify in a very progressive way the choices of design. We propose in this work a framework combining local computations models and refinement to prove the correctness of a large class of distributed algorithms. Local computations models define abstract computing processes for solving problems by distributed algorithms and can be integrated into a the Event-B modelling language to define proof-based patterns for the design of distributed algorithms. We illustrate our approach by examples like the leader election protocol or the distributed coloring algorithm. Our proposal is integrated into an environment called ViSiDiA

Distributed Computing on Core-Periphery Networks: Axiom-based Design

Inspired by social networks and complex systems, we propose a core-periphery network architecture that supports fast computation for many distributed algorithms and is robust and efficient in number of links. Rather than providing a concrete network model, we take an axiom-based design approach. We provide three intuitive (and independent) algorithmic axioms and prove that any network that satisfies all axioms enjoys an efficient algorithm for a range of tasks (e.g., MST, sparse matrix multiplication, etc.). We also show the minimality of our axiom set: for networks that satisfy any subset of the axioms, the same efficiency cannot be guaranteed for any deterministic algorithm

Distributed Computation and Reconfiguration in Actively Dynamic Networks

In this paper, we study systems of distributed entities that can actively modify their communication network. This gives rise to distributed algorithms that apart from communication can also exploit network reconfiguration in order to carry out a given task. At the same time, the distributed task itself may now require a global reconfiguration from a given initial network Gs to a target network Gf from a family of networks having some good properties, like small diameter. To formally capture costs associated with creating and maintaining connections, we define three reasonable edge-complexity measures: the total edge activations, the maximum activated edges per round, and the maximum activated degree of a node. We give (poly)log(n) time algorithms for the general task of transforming any Gs into a Gf of diameter (poly)log(n), while minimizing the edge-complexity. There is a natural trade-off between time and edge complexity. Our main lower bound shows that Ω(n) total edge activations and Ω(n/log n) activations per round must be paid by any algorithm (even centralized) that achieves an optimum of Θ(log n) rounds. On the positive side, we give three distributed algorithms for our general task. The first runs in O(log n) time, with at most 2n active edges per round, a total of O(n log n) edge activations, a maximum degree n − 1, and a target network of diameter 2. The second achieves bounded degree by paying an additional logarithmic factor in time and in total edge activations. It gives a target network of diameter O(log n) and uses O(n) active edges per round. Our third algorithm shows that if we slightly increase the maximum degree to polylog(n) then we can achieve a running time of o(log2n). This novel model of distributed computation and reconfiguration in actively dynamic networks and the proposed measures of the edge complexity of distributed algorithms, may open new avenues for research in the algorithmic theory of dynamic networks

Round- and Message-Optimal Distributed Graph Algorithms

Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication. In this paper we provide algorithms that are simultaneously round- and message-optimal for a number of well-studied distributed optimization problems. Our main result is such a distributed algorithm for the fundamental primitive of computing simple functions over each part of a graph partition. From this algorithm we derive round- and message-optimal algorithms for multiple problems, including MST, Approximate Min-Cut and Approximate Single Source Shortest Paths, among others. On general graphs all of our algorithms achieve worst-case optimal $\tilde{O}(D+\sqrt n)$ round complexity and $\tilde{O}(m)$ message complexity. Furthermore, our algorithms require an optimal $\tilde{O}(D)$ rounds and $\tilde{O}(n)$ messages on planar, genus-bounded, treewidth-bounded and pathwidth-bounded graphs.Comment: To appear in PODC 201