# Peter Robinson

Department of Computer ScienceNational University of Singapore

Singapore

My research focuses on distributed computing. This comprises designing new fault-tolerant and scalable algorithms and proving lower bounds for distributed computing problems. I'm also interested in developing new (more realistic) models for distributed computation.

I'm on the program committee of SIROCCO 2014, ICDCN 2015, and FOMC 2014.

## Keywords

«Asynchrony» «Big Data» «Byzantine Failures» «Churn» «Communication Complexity» «Distributed Agreement» «Distributed Storage» «Dynamic Network» «Fault-Tolerance» «Haskell» «Leader Election» «Local Algorithm» «Lower Bound» «Machine Learning» «Mobile Ad-Hoc Network» «Natural Language Processing» «P2P» «Randomization»## Publications

- 2015 (1)
- Distributed Computation of Large-scale Graph Problems

by Hartmut Klauck, Danupon Nanongkai, Gopal Pandurangan, Peter Robinson.

26th ACM-SIAM Symposium on Discrete Algorithms (SODA 2015). San Diego, California.

Abstract...Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have $k$ machines that jointly perform computation on an arbitrary $n$-node (typically, $n\gg k$) input graph. The graph is assumed to be

*randomly*partitioned among the $k\ge 2$ machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity, i.e., the number of communication rounds, of solving various fundamental graph problems.We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of $\Omega (n/k)$ rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an $\Omega (n/{k}^{2})$ lower bound for connectivity, ST verification and other related problems. Our lower bounds develop and use new bounds in

*random-partition*communication complexity.To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in $\stackrel{\u0303}{O}(n/k)$ time (the notation $\stackrel{\u0303}{O}$ hides $\mathrm{\text{polylog}}(n)$ factors and an additive $\mathrm{\text{polylog}}(n)$ term); this shows that one can achieve almost

*linear*(in $k$) speedup, whereas for shortest paths, we present algorithms that run in $\stackrel{\u0303}{O}(n/\sqrt{k})$ time (for $(1+\u03f5)$-factor approximation) and in $\stackrel{\u0303}{O}(n/k)$ time (for $O(\mathrm{log}n)$-factor approximation) respectively.Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.

- Distributed Computation of Large-scale Graph Problems
- 2014 (7)
- DEX: Self-Healing Expanders

by Gopal Pandurangan, Peter Robinson, Amitabh Trehan.

28th IEEE International Parallel Distributed Processing Symposium (IPDPS 2014). Phoenix, Arizona.

Abstract...We present a fully-distributed self-healing algorithm DEX, that maintains a constant degree expander network in a dynamic setting. To the best of our knowledge, our algorithm provides the first efficient distributed construction of expanders — whose expansion properties hold

*deterministically*— that works even under an all-powerful adaptive adversary that controls the dynamic changes to the network (the adversary has unlimited computational power and knowledge of the entire network state, can decide which nodes join and leave and at what time, and knows the past random choices made by the algorithm). Previous distributed expander constructions typically provide only*probabilistic*guarantees on the network expansion which*rapidly degrade*in a dynamic setting; in particular, the expansion properties can degrade even more rapidly under*adversarial*insertions and deletions.Our algorithm provides efficient maintenance and incurs a low overhead per insertion/deletion by an adaptive adversary: only $O(\mathrm{log}n)$ rounds and $O(\mathrm{log}n)$ messages are needed with high probability ($n$ is the number of nodes currently in the network). The algorithm requires only a constant number of topology changes. Moreover, our algorithm allows for an efficient implementation and maintenance of a distributed hash table (DHT) on top of DEX, with only a constant additional overhead.

Our results are a step towards implementing efficient self-healing networks that have

*guaranteed*properties (constant bounded degree and expansion) despite dynamic changes. - Distributed Symmetry Breaking in Hypergraphs

by Shay Kutten, Danupon Nanongkai, Gopal Pandurangan, Peter Robinson.

28th International Symposium on Distributed Computing (DISC 2014). Austin, Texas.

Abstract...Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and $\Delta +1$-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in hypergraphs. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs.

In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in $O({\mathrm{log}}^{2}n)$ rounds ($n$ is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper — an $O(\Delta^{\epsilon}\polylog n)$-round hypergraph MIS algorithm in the CONGEST model where $\Delta $ is the maximum node degree of the hypergraph and $\u03f5>0$ is any arbitrarily small constant.

To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the balanced minimal dominating set problem (left open in Harris et al. [ICALP 2013]) and the minimal connected dominating set problem. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.

Our work shows that while some local symmetry breaking problems such as coloring can be solved in polylogarithmic rounds in both the LOCAL and CONGEST models, for many other hypergraph problems such as MIS, hitting set, and maximal clique, it remains challenging to obtain polylogarithmic time algorithms in the CONGEST model. This work is a step towards understanding this dichotomy in the complexity of hypergraph problems as well as using hypergraphs to design fast distributed algorithms for problems in (standard) graphs.

- Scalable Byzantine Leader Election in Dynamic Networks

by John Augustine, Gopal Pandurangan, Peter Robinson.

(under review)

- Enabling Efficient and Robust Distributed Computation in Highly Dynamic Networks

by John Augustine, Gopal Pandurangan, Peter Robinson, Scott Roche, Eli Upfal.

(under review)

- Sublinear Bounds for Randomized Leader Election

by Shay Kutten, Gopal Pandurangan, David Peleg, Peter Robinson, Amitabh Trehan.

Special Issue of Theoretical Computer Science, Elsevier. (TCS).

Invited paper.

Abstract...This paper concerns randomized leader election in synchronous distributed networks. A distributed leader election algorithm is presented for complete $n$-node networks that runs in $O(1)$ rounds and (with high probability) uses only $O(\sqrt{n}{\mathrm{log}}^{3/2}n)$ messages to elect a unique leader (with high probability). When considering the “explicit” variant of leader election where eventually every node knows the identity of the leader, our algorithm yields the asymptotically optimal bounds of $O(1)$ rounds and $O(n)$ messages. This algorithm is then extended to one solving leader election on any connected non-bipartite $n$-node graph $G$ in $O(\tau (G))$ time and $O(\tau (G)\sqrt{n}{\mathrm{log}}^{3/2}n)$ messages, where $\tau (G)$ is the mixing time of a random walk on $G$. The above result implies highly efficient (sublinear running time and messages) leader election algorithms for networks with small mixing times, such as expanders and hypercubes. In contrast, previous leader election algorithms had at least linear message complexity even in complete graphs. Moreover, super-linear message lower bounds are known for time-efficient deterministic leader election algorithms. Finally, we present an almost matching lower bound for randomized leader election, showing that $\Omega (\sqrt{n})$ messages are needed for any leader election algorithm that succeeds with probability at least $1/e+\u03f5$, for any small constant $\u03f5>0$. We view our results as a step towards understanding the randomized complexity of leader election in distributed networks.

- Distributed Agreement in Dynamic Peer-to-Peer Networks

by John Augustine, Gopal Pandurangan, Peter Robinson, Eli Upfal.

Journal of Computer and System Sciences, Elsevier. (JCSS).

- The Generalized Loneliness Detector and Weak System Models for k-Set Agreement

by Martin Biely, Peter Robinson, Ulrich Schmid.

IEEE Transactions on Parallel and Distributed Systems 25(4): pp. 1078-1088 (IEEE TPDS).

Abstract...This paper presents two weak partially synchronous system models MAnti[n-k] and MSink[n-k], which are just strong enough for solving $k$-set agreement: We introduce the generalized $(n-k)$-loneliness failure detector $\mathcal{L}(k)$, which we first prove to be sufficient for solving $k$-set agreement, and show that $\mathcal{L}(k)$ but not $\mathcal{L}(k-1)$ can be implemented in both models. MAnti[n-k] and MSink[n-k] are hence the first message passing models that lie between models where $\Omega $ (and therefore consensus) can be implemented and the purely asynchronous model. We also address $k$-set agreement in anonymous systems, that is, in systems where (unique) process identifiers are not available. Since our novel $k$-set agreement algorithm using $\mathcal{L}(k)$ also works in anonymous systems, it turns out that the loneliness failure detector $\mathcal{L}=\mathcal{L}(n-1)$ introduced by Delporte et al. is also the weakest failure detector for set agreement in anonymous systems. Finally, we analyze the relationship between $\mathcal{L}(k)$ and other failure detectors suitable for solving $k$-set agreement.

- DEX: Self-Healing Expanders
- 2013 (6)
- Sublinear Bounds for Randomized Leader Election

by Shay Kutten, Gopal Pandurangan, David Peleg, Peter Robinson, Amitabh Trehan.

14th International Conference on Distributed Computing and Networking (ICDCN 2013). Mumbai.

Received best paper award.

Abstract...This paper concerns randomized leader election in synchronous distributed networks. A distributed leader election algorithm is presented for complete n-node networks that runs in $O(1)$ rounds and (with high probability) takes only $O(\sqrt{n}{\mathrm{log}}^{3/2}n)$ messages to elect a unique leader (with high probability). This algorithm is then extended to solve leader election on any connected non-bipartite n-node graph $G$ in $O(\tau (G))$ time and $O(\tau (G)\sqrt{n}{\mathrm{log}}^{3/2}n)$ messages, where $\tau (G)$ is the mixing time of a random walk on $G$. The above result implies highly efficient (sublinear running time and messages) leader election algorithms for networks with small mixing times, such as expanders and hypercubes. In contrast, previous leader election algorithms had at least linear message complexity even in complete graphs. Moreover, super-linear message lower bounds are known for time-efficient deterministic leader election algorithms. Finally, an almost-tight lower bound is presented for randomized leader election, showing that $\Omega (\sqrt{n})$ messages are needed for any $O(1)$ time leader election algorithm which succeeds with high probability. It is also shown that $\Omega ({n}^{1/3})$ messages are needed by any leader election algorithm that succeeds with high probability, regardless of the number of the rounds. We view our results as a step towards understanding the randomized complexity of leader election in distributed networks.

- Efficient Computation of Balanced Structures

by David G. Harris, Ehab Morsy, Gopal Pandurangan, Peter Robinson, Aravind Srinivasan.

40th International Colloquium on Automata, Languages and Programming (ICALP 2013). Riga.

Abstract...Basic graph structures such as maximal independent sets (MIS’s) have spurred much theoretical research in distributed algorithms, and have several applications in networking and distributed computing as well. However, the extant (distributed) algorithms for these problems do not necessarily guarantee fault-tolerance or load-balance properties: For example, in a star-graph, the central vertex, as well as the set of leaves, are both MIS’s, with the latter being much more fault-tolerant and balanced - existing distributed algorithms do not handle this distinction. We propose and study “low-average degree” or “balanced” versions of such structures. Interestingly, in sharp contrast to, say, MIS’s, it can be shown that checking whether a structure is balanced, will take substantial time. Nevertheless, we are able to develop good sequential and distributed algorithms for such “balanced” versions. We also complement our algorithms with lower bounds.

- On the Complexity of Universal Leader Election

by Shay Kutten, Gopal Pandurangan, David Peleg, Peter Robinson, Amitabh Trehan.

32nd ACM Symposium on Principles of Distributed Computing (PODC 2013). Montreal.

Abstract...Electing a leader is a fundamental task in distributed computing. In its

*implicit*version, only the leader must know who is the elected leader. This paper focuses on studying the message and time complexity of*randomized*implicit leader election in synchronous distributed networks. Surprisingly, the most “obvious” complexity bounds have not been proven for randomized algorithms. The “obvious” lower bounds of $\Omega (m)$ messages ($m$ is the number of edges in the network) and $\Omega (D)$ time ($D$ is the network diameter) are non-trivial to show for randomized (Monte Carlo) algorithms. (Recent results that show that even $\Omega (n)$ ($n$ is the number of nodes in the network) is*not*a lower bound on the messages in complete networks, make the above bounds somewhat less obvious). To the best of our knowledge, these basic lower bounds have not been established even for deterministic algorithms (except for the limited case of comparison algorithms, where it was also required that some nodes may not wake up spontaneously, and that $D$ and $n$ were not known).We establish these fundamental lower bounds in this paper for the general case, even for randomized Monte Carlo algorithms. Our lower bounds are universal in the sense that they hold for all universal algorithms (such algorithms should work for all graphs), apply to every $D$, $m$, and $n$, and hold even if $D$, $m$, and $n$ are known, all the nodes wake up simultaneously, and the algorithms can make any use of node’s identities. To show that these bounds are tight, we present an $O(m)$ messages algorithm. An $O(D)$ time algorithm is known.

An interesting fundamental problem is whether

*both*upper bounds (messages and time) can be reached*simultaneously*in the randomized setting for all graphs. (The answer is known to be negative in the deterministic setting). We answer this problem partially by presenting a randomized algorithm that matches both complexities in some cases. This already separates (for some cases) randomized algorithms from deterministic ones. As first steps towards the general case, we present several universal leader election algorithms with bounds that trade-off messages versus time. We view our results as a step towards understanding the complexity of universal leader election in distributed networks. - Fast Byzantine Agreement in Dynamic Networks

by John Augustine, Gopal Pandurangan, Peter Robinson

32nd ACM Symposium on Principles of Distributed Computing (PODC 2013). Montreal.

Abstract...We study Byzantine agreement in dynamic networks where topology can change from round to round and nodes can also experience heavy

*churn*(i.e., nodes can join and leave the network continuously over time). Our main contributions are randomized distributed algorithms that guarantee*almost-everywhere Byzantine agreement*with high probability even under a large number of Byzantine nodes and continuous adversarial churn in a number of rounds that is polylogarithmic in $n$ (where $n$ is the stable network size). We show that our algorithms are essentially optimal (up to polylogarithmic factors) with respect to the amount of Byzantine nodes and churn rate that they can tolerate by showing lower bound. In particular, we present the following results:An $O({\mathrm{log}}^{3}n)$ round randomized algorithm that achieves almost-everywhere Byzantine agreement with high probability under a presence of up to $O(\sqrt{n}/\mathrm{\text{polylog}}(n))$ Byzantine nodes and up to a churn of $O(\sqrt{n}/\mathrm{\text{polylog}}(n))$ nodes per round. We assume that the Byzantine nodes have knowledge about the entire state of network at every round (including random choices made by all the nodes) and can behave arbitrarily. We also assume that an adversary controls the churn — it has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power (but is oblivious to the topology changes from round to round). Our algorithm requires only polylogarithmic in $n$ bits to be processed and sent (per round) by each node.

We also present an $O({\mathrm{log}}^{3}n)$ round randomized algorithm that has same guarantees as the above algorithm, but works even when the churn and network topology is controlled by an adaptive adversary (that can choose the topology based on the current states of the nodes). However, this algorithm requires up to polynomial in $n$ bits to be processed and sent (per round) by each node.

We show that the above bounds are essentially the best possible, if one wants fast (i.e., polylogarithmic run time) algorithms, by showing that any (randomized) algorithm to achieve agreement in a dynamic network controlled by an adversary that can churn up to $\Theta (\sqrt{n\mathrm{log}n})$ nodes per round should take at least a

*polynomial*number of rounds.

Our algorithms are the first-known, fully-distributed, Byzantine agreement algorithms in highly dynamic networks. We view our results as a step towards understanding the possibilities and limitations of highly dynamic networks that are subject to malicious behavior by a large number of nodes.

- Search and Storage in Dynamic Peer-to-Peer Networks

by John Augustine, Anisur Molla, Ehab Morsy, Gopal Pandurangan, Peter Robinson, Eli Upfal.

25th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2013). Montreal.

Abstract...We study robust and efficient distributed algorithms for searching, storing, and maintaining data in dynamic Peer-to-Peer (P2P) networks. P2P networks are highly dynamic networks that experience heavy node churn (i.e., nodes join and leave the network continuously over time). Our goal is to guarantee, despite high node churn rate, that a large number of nodes in the network can store, retrieve, and maintain a large number of data items. Our main contributions are fast randomized distributed algorithms that guarantee the above with high probability even under high adversarial churn. In particular, we present the following main results:

A randomized distributed search algorithm that with high probability guarantees that searches from as many as $n-o(n)$ nodes ($n$ is the stable network size) succeed in $O(\mathrm{log}n)$-rounds despite $O(n/{\mathrm{log}}^{1+\delta}n)$ churn, for any small constant $\delta >0$, per round. We assume that the churn is controlled by an oblivious adversary (that has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power, but is oblivious to the random choices made by the algorithm).

A storage and maintenance algorithm that guarantees, with high probability, data items can be efficiently stored (with only $\Theta (\mathrm{log}n)$ copies of each data item) and maintained in a dynamic P2P network with churn rate up to $O(n/{\mathrm{log}}^{1+\delta}n)$ per round. Our search algorithm together with our storage and maintenance algorithm guarantees that as many as $n-o(n)$ nodes can efficiently store, maintain, and search even under $O(n/{\mathrm{log}}^{1+\delta}n)$ churn per round. Our algorithms require only polylogarithmic in $n$ bits to be processed and sent (per round) by each node.

To the best of our knowledge, our algorithms are the first-known, fully-distributed storage and search algorithms that provably work under highly dynamic settings (i.e., high churn rates per step). Furthermore, they are localized (i.e., do not require any global topological knowledge) and scalable. A technical contribution of this paper, which may be of independent interest, is showing how random walks can be provably used to derive scalable distributed algorithms in dynamic networks with adversarial node churn.

- Robust Leader Election in a Fast-Changing World

by John Augustine, Tejas Kulkarni, Paresh Nakhe, Peter Robinson.

9th International Workshop on Foundations of Mobile Computing (FOMC 2013).

Abstract...We consider the problem of electing a leader among nodes in a highly dynamic network where the adversary has unbounded capacity to insert and remove nodes (including the leader) from the network and change connectivity at will. We present a randomized algorithm that (re)elects a leader in $O(D\mathrm{log}n)$ rounds with high probability, where $D$ is a bound on the dynamic diameter of the network and $n$ is the maximum number of nodes in the network at any point in time. We assume a model of broadcast-based communication where a node can send only $1$ message of $O(\mathrm{log}n)$ bits per round and is not aware of the receivers in advance. Thus our results also apply to mobile wireless ad-hoc networks, improving over the optimal (for deterministic algorithms) $O(Dn)$ solution presented at FOMC 2011. We show that our algorithm is optimal by proving that

*any*randomized algorithm takes at least $\Omega (D\mathrm{log}n)$ rounds to elect a leader with high probability, which shows that our algorithm yields the best possible (up to constants) termination time.

- Sublinear Bounds for Randomized Leader Election
- 2012 (2)
- Towards Robust and Efficient Computation in Dynamic Peer-to-Peer Networks

by John Augustine, Gopal Pandurangan, Peter Robinson, Eli Upfal.

23rd ACM-SIAM Symposium on Discrete Algorithms (SODA 2012). Kyoto.

Abstract...Motivated by the need for robust and fast distributed computation in highly dynamic Peer-to-Peer (P2P) networks, we study algorithms for the fundamental distributed agreement problem. P2P networks are highly dynamic networks that experience heavy node churn (i.e., nodes join and leave the network continuously over time). Our goal is to design fast algorithms (running in a small number of rounds) that guarantee, despite high node churn rate, that almost all nodes reach a stable agreement. Our main contributions are randomized distributed algorithms that guarantee stable almost-everywhere agreement with high probability even under high adversarial churn in a polylogarithmic number of rounds. In particular, we present the following results:

An $O({\mathrm{log}}^{2}n)$-round ($n$ is the stable network size) randomized algorithm that achieves almost-everywhere agreement with high probability under up to linear churn per round (i.e., $\u03f5n$, for some small constant $\u03f5>0$), assuming that the churn is controlled by an oblivious adversary (that has complete knowledge and control of what nodes join and leave and at what time and has unlimited computational power, but is oblivious to the random choices made by the algorithm).

An $O(\mathrm{log}m{\mathrm{log}}^{3}n)$-round randomized algorithm that achieves almost-everywhere agreement with high probability under up to $\u03f5\sqrt{n}$ churn per round (for some small $\u03f5>0$), where $m$ is the size of the input value domain, that works even under an adaptive adversary (that also knows the past random choices made by the algorithm).

We also show that no deterministic algorithm can guarantee almost-everywhere agreement (regardless of the number of rounds), even under constant churn rate.

Our algorithms are the first-known, fully-distributed, agreement algorithms that work under highly dynamic settings (i.e., high churn rates per step). Furthermore, they are localized (i.e., do not require any global topological knowledge), simple, and easy to implement. These algorithms can serve as building blocks for implementing other non-trivial distributed computing tasks in dynamic P2P networks.

- Agreement in Directed Dynamic Networks

by Martin Biely, Peter Robinson, Ulrich Schmid.

19th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2012). Reykjavík.

Abstract...We study distributed computation in synchronous dynamic networks where an omniscient adversary controls the unidirectional communication links. Its behavior is modeled as a sequence of directed graphs representing the active (i.e. timely) communication links per round. We prove that consensus is impossible under some natural weak connectivity assumptions and introduce vertex-stable root components as a means for circumventing this impossibility. Essentially, we assume that there is a short period of time during which an arbitrary part of the network remains strongly connected, while its interconnect topology may keep changing continuously. We present a consensus algorithm that works under this assumption and prove its correctness. Our algorithm maintains a local estimate of the communication graphs and applies techniques for detecting stable network properties and univalent system configurations. Our possibility results are complemented by several impossibility results and lower bounds for consensus and other distributed computing problems like leader election, revealing that our algorithm is asymptotically optimal.

- Towards Robust and Efficient Computation in Dynamic Peer-to-Peer Networks
- 2011 (5)
- Weak System Models for Fault-Tolerant Distributed Agreement Problems

by Peter Robinson.

PhD Thesis. TU Vienna, Institute of Computer Engineering.

Abstract...This thesis investigates various aspects of weak system models for agreement problems in fault-tolerant distributed computing. In Part I we provide an introduction to the context of this work, discuss related literature and describe the basic system assumptions.

In Part II of this thesis, we introduce the Asynchronous Bounded-Cycle (ABC) model, which is entirely time-free. In contrast to existing system models, the ABC model does not require explicit time-based synchrony bounds, but rather stipulates a graph-theoretic synchrony condition on the relative lengths of certain causal chains of messages in the space-time graph of a run. We compare the ABC model to other models in literature, in particular to the classic models by Dwork, Lynch, and Stockmeyer. Despite Byzantine failures, we show how to simulate lock-step rounds, and therefore make consensus solvable, and prove the correctness of a clock synchronization algorithm in the ABC model. We then present the technically most involved result of this thesis: We prove that any algorithm working correctly in the partially synchronous $\Theta $-Model by Le Lann and Schmid, also works correctly in the time-free ABC model. In the proof, we use a variant of Farkas’ Theorem of Linear Inequalities and develop a non-standard cycle space on directed graphs in order to guarantee the existence of a certain message delay transformation for finite prefixes of runs. This shows that any time-free safety property satisfied by an algorithm in the $\Theta $-Model also holds in the ABC model. By employing methods from point-set topology, we can extend this result to liveness properties.

In Part III, we shift our attention to the borderland between models where consensus is solvable and the purely asynchronous model. To this end, we look at the $k$-set agreement problem where processes need to decide on at most $k$ distinct decision values. We introduce two very weak system models MAnti and MSink and prove that consensus is impossible in these models. Nevertheless, we show that $(n-1)$-set agreement is solvable in MAnti and MSink, by providing algorithms that implement the weakest failure detector $\mathcal{L}$. We also discuss how models MAnti and MSink relate to the $f$-source models by Aguilera et al. for solving consensus.

In the subsequent chapter, we present a novel failure detector $\mathcal{L}(k)$ that generalizes $\mathcal{L}$, and analyze an algorithm for solving $k$-set agreement with $\mathcal{L}(k)$, which works even in systems without unique process identifiers. Moreover, We explore the relationship between $\mathcal{L}(k)$ and existing failure detectors for $k$-set agreement. Some aspects of $\mathcal{L}(k)$ relating to anonymous systems are also discussed.

Next, we present a generic theorem that can be used to characterize the impossibility of achieving $k$-set agreement in various system models. This enables us to show that $({\Sigma}_{k},{\Omega}_{k})$ is not sufficient for solving $k$-set agreement. Furthermore, we instantiate our theorem with a partially synchronous system model.

Finally, we consider the $k$-set agreement problem in round-based systems. First, we introduce a novel abstraction that encapsulates the perpetual synchrony of a run, the so called stable skeleton graph, which allows us to express the solvability power of a system via graph-theoretic properties. We then present an approximation algorithm where processes output an estimate of their respective component of the stable skeleton graph. We define a class of communication predicates PSources(k) in this framework, and show that PSources(k) tightly captures the amount of synchrony necessary for $k$-set agreement, as $(k-1)$-set agreement is impossible with PSources(k). Based on the stable skeleton approximation, we present an algorithm that solves $k$-set agreement when PSources(k) holds.

- Optimal Regional Consecutive Leader Election in Mobile Ad-Hoc Networks

by Hyun Chul Chung, Peter Robinson, Jennifer L. Welch.

7th ACM SIGACT/SIGMOBILE International Workshop on Foundations of Mobile Computing (part of FCRC 2011). San Jose.

Abstract...The regional consecutive leader election (RCLE) problem requires mobile nodes to elect a leader within bounded time upon entering a specific region. We prove that any algorithm requires $\Omega (Dn)$ rounds for leader election, where D is the diameter of the network and $n$ is the total number of nodes. We then present a fault-tolerant distributed algorithm that solves the RCLE problem and works even in settings where nodes do not have access to synchronized clocks. Since nodes set their leader variable within $O(Dn)$ rounds, our algorithm is asymptotically optimal with respect to time complexity. Due to its low message bit complexity, we believe that our algorithm is of practical interest for mobile wireless ad-hoc networks. Finally, we present a novel and intuitive constraint on mobility that guarantees a bounded communication diameter among nodes within the region of interest.

- Solving k-Set Agreement with Stable Skeleton Graphs

by Martin Biely, Peter Robinson, Ulrich Schmid.

16th IEEE International Symposium on Parallel and Distributed Processing Workshops and PhD Forum (IPDPS 2011). Anchorage.

- Easy Impossibility Proofs for k-Set Agreement in Message Passing Systems

by Martin Biely, Peter Robinson, Ulrich Schmid.

15th International Conference On Principles Of Distributed Systems (OPODIS 2011). Toulouse.

Abstract...Despite of being quite similar agreement problems, distributed consensus ($1$-set agreement) and general $k$-set agreement require surprisingly different techniques for proving their impossibility in asynchronous systems with crash failures: Rather, than the relatively simple bivalence arguments as in the impossibility proof for consensus in the presence of a single crash failure, known proofs for the impossibility of $k$-set agreement in shared memory systems with $f\ge k>1$ crash failures use algebraic topology or a variant of Sperner’s Lemma. In this paper, we present a generic theorem for proving the impossibility of $k$-set agreement in various message passing settings, which is based on a reduction to the consensus impossibility in a certain subsystem resulting from a partitioning argument.

We demonstrate the broad applicability of our result by exploring the possibility/impossibility border of $k$-set agreement in several message passing system models: (i) asynchronous systems with crash failures, (ii) partially synchronous processes with (initial) crash failures, and, most importantly, (iii) asynchronous systems augmented with failure detectors. Furthermore, by extending the algorithm for initial crashes of Fisher, Lynch and Patterson (1985) to general $k$-set agreement, we show that the impossibility border of (i) is tightly matched.

The impossibility proofs in cases (i), (ii), and (iii) are instantiations of our main theorem. Regarding (iii), applying our technique reveals the exact border for the parameter $k$ where $k$-set agreement is solvable with the failure detector class $({\Sigma}_{k},{\Omega}_{k}{)}_{1\le k\le n-1}$ of Bonnet and Raynal. As ${\Sigma}_{k}$ was shown to be necessary for solving $k$-set agreement, this result yields new insights on the quest for the weakest failure detector

- The Asynchronous Bounded-Cycle Model

by Peter Robinson and Ulrich Schmid.

Theoretical Computer Science 412 (2011) 5580–5601. (TCS).

Invited paper.

Abstract...This paper shows how synchrony conditions can be added to the purely asynchronous model in a way that avoids any reference to message delays and computing step times, as well as any global constraints on communication patterns and network topology. Our Asynchronous Bounded-Cycle (ABC) model just bounds the ratio of the number of forward- and backward-oriented messages in certain “relevant” cycles in the space-time diagram of an asynchronous execution. We show that clock synchronization and lock-step rounds can easily be implemented and proved correct in the ABC model, even in the presence of Byzantine failures. Furthermore, we prove that any algorithm working correctly in the partially synchronous $\Theta $-Model also works correctly in the ABC model. In our proof, we first apply a novel method for assigning certain message delays to asynchronous executions, which is based on a variant of Farkas’ theorem of linear inequalities and a non-standard cycle-space of graphs. Using methods from point-set topology, we then prove that the existence of this delay assignment implies model indistinguishability for time-free safety and liveness properties. Finally, we introduce several weaker variants of the ABC model and relate our model to the existing partially synchronous system models, in particular, the classic models of Dwork, Lynch and Stockmayer. Furthermore, we discuss aspects of the ABC model’s applicability in real systems, in particular, in the context of VLSI Systems-on-Chip.

- Weak System Models for Fault-Tolerant Distributed Agreement Problems
- 2010 (1)
- Regional Consecutive Leader Election in Mobile Ad-Hoc Networks

by Hyun Chul Chung, Peter Robinson, Jennifer L. Welch.

6th ACM SIGACT/SIGMOBILE Workshop on Foundations of Mobile Computing (DIALM-POMC 2010). Boston.

- Regional Consecutive Leader Election in Mobile Ad-Hoc Networks
- 2009 (1)
- Weak Synchrony Models and Failure Detectors for Message Passing k-Set Agreement

by Martin Biely, Peter Robinson, Ulrich Schmid.

13th International Conference On Principles Of Distributed Systems (OPODIS 2009). Nimes.

- Weak Synchrony Models and Failure Detectors for Message Passing k-Set Agreement
- 2008 (1)
- The Asynchronous Bounded-Cycle Model

by Peter Robinson and Ulrich Schmid.

10th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS 2008). Detroit.

Received best paper award.

Abstract...This paper shows how synchrony conditions can be added to the purely asynchronous model in a way that avoids any reference to message delays and computing step times, as well as any global constraints on communication patterns and network topology. Our Asynchronous Bounded-Cycle (ABC) model just bounds the ratio of the number of forward- and backward-oriented messages in certain “relevant” cycles in the space-time diagram of an asynchronous execution. We show that clock synchronization and lock-step rounds can easily be implemented and proved correct in the ABC model, even in the presence of Byzantine failures. Furthermore, we prove that any algorithm working correctly in the partially synchronous $\Theta $-Model also works correctly in the ABC model. In our proof, we first apply a novel method for assigning certain message delays to asynchronous executions, which is based on a variant of Farkas’ theorem of linear inequalities and a non-standard cycle-space of graphs. Using methods from point-set topology, we then prove that the existence of this delay assignment implies model indistinguishability for time-free safety and liveness properties. Finally, we introduce several weaker variants of the ABC model and relate our model to the existing partially synchronous system models, in particular, the classic models of Dwork, Lynch and Stockmayer. Furthermore, we discuss aspects of the ABC model’s applicability in real systems, in particular, in the context of VLSI Systems-on-Chip.

- The Asynchronous Bounded-Cycle Model
- 2006 (1)
- Log File Processing by Machine Learning and Information Extraction

by Peter Robinson.

Master Thesis. TU Vienna, Institute of Computer Languages, 2006. Nominated for Distinguished Young Alumnus Award.

Abstract...In today’s computer network systems lots of events are constantly written to log files. Unfortunately there is no common standard defining the structure of these event messages which are partly in human readable natural language form. Obviously, this lack of structure makes automatic processing a lot more difficult.

This master thesis describes the architecture and implementation of the LoP-System, a system that attempts to create machine readable event structures from ordinary log file events by natural language processing. The thesis explains implementational details as well as the theoretical concepts used.

The core of the system consists of a series of cascaded but independent components, partly enhanced with machine learning techniques. The raw input is first processed by a simple recursive descent parser which recognizes syntactical features (e.g. IP addresses) and is then passed on to a part-of-speech tagger based on a hidden Markov model. Applying regular expression patterns to the tagged words is used to combine them to basic word groups (e.g. noun groups), which are subsequently semantically analyzed. The final step is the construction of the output events by a rule based event constructor.

All components are implemented in Haskell, a purely functional programming language. Some of the components developed during this thesis, especially the part-of-speech tagger, are general natural language processing tools and can be applied to other domains.

- Log File Processing by Machine Learning and Information Extraction