arXiv subjects

Armando Castañeda

Publications and source records attributed to Armando Castañeda.

25 records · Page 2Linked to original sources

arXiv2020

K set-agreement bounds in round-based models through combinatorial topology

Round-based models are very common message-passing models; combinatorial topology applied to distributed computing provides sweeping results like general lower bounds. We combine both to study the computability of k-set agreement. Among all the possible round-based models, we consider oblivious ones, where the constraints are given only round per round by a set of allowed graphs. And among oblivious models, we focus on closed-above ones, that is models where the set of possible graphs contains all graphs with more edges than some starting graphs. These capture intuitively the underlying structure required by some communication model, like containing a ring. We then derive lower bounds and upper bounds in one round for k-set agreement, such that these bounds are proved using combinatorial topology but stated only in terms of graph properties. These bounds extend to multiple rounds when limiting our algorithms to be oblivious -- recalling only pairs of processes and initial value, not who send what and when.

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arXiv2019

Tasks in Modular Proofs of Concurrent Algorithms

Proving correctness of distributed or concurrent algorithms is a mind-challenging and complex process. Slight errors in the reasoning are difficult to find, calling for computer-checked proof systems. In order to build computer-checked proofs with usual tools, such as Coq or TLA+, having sequential specifications of all base objects that are used as building blocks in a given algorithm is a requisite to provide a modular proof built by composition. Alas, many concurrent objects do not have a sequential specification. This article describes a systematic method to transform any task, a specification method that captures concurrent one-shot distributed problems, into a sequential specification involving two calls, Set and Get. This transformation allows system designers to compose proofs, thus providing a framework for modular computer-checked proofs of algorithms designed using tasks and sequential objects as building blocks. The Moir&Anderson implementation of renaming using splitters is an iconic example of such algorithms designed by composition.

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arXiv2018

Some Problems in Compact Message Passing

This paper seeks to address the question of designing distributed algorithms for the setting of compact memory i.e. sublinear bits working memory for arbitrary connected networks. The nodes in our networks may have much lower internal memory as compared to the number of their possible neighbours implying that a node may not be able to store all the IDs of its neighbours. These algorithms are useful for large networks of small devices such as the Internet of Things, for wireless or ad-hoc networks, and, in general, as memory efficient algorithms. We introduce the Compact Message Passing(CMP) model;an extension of the standard message passing model considered at a finer granularity where a node can interleave reads and writes with internal computations, using a port only once in a round. The interleaving is required for meaningful computations due to the low memory requirement and is akin to a distributed network with nodes executing streaming algorithms. Note that the internal memory size upper bounds the message sizes and hence e.g. for log-memory, the model is weaker than the Congest model; for such models our algorithms will work directly too. We present early results in the CMP model for nodes with log^2-memory. We introduce the concepts of local compact functions and compact protocols and give solutions for some classic distributed problems (leader election, tree constructions and traversals). We build on these to solve the open problem of compact preprocessing for the compact self-healing routing algorithm CompactFTZ posed in Compact Routing Messages in Self-Healing Trees(TCS2017) by designing local compact functions for finding particular subtrees of labeled binary trees. Hence, we introduce the first fully compact self-healing routing algorithm. We also give independent fully compact versions of the Forgiving Tree[PODC08] and Thorup-Zwick's tree based compact routing[SPAA01].

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arXiv2016

Unbeatable Set Consensus via Topological and Combinatorial Reasoning

The set consensus problem has played an important role in the study of distributed systems for over two decades. Indeed, the search for lower bounds and impossibility results for this problem spawned the topological approach to distributed computing, which has given rise to new techniques in the design and analysis of protocols. The design of efficient solutions to set consensus has also proven to be challenging. In the synchronous crash failure model, the literature contains a sequence of solutions to set consensus, each improving upon the previous ones. This paper presents an unbeatable protocol for nonuniform k-set consensus in the synchronous crash failure model. This is an efficient protocol whose decision times cannot be improved upon. Moreover, the description of our protocol is extremely succinct. Proving unbeatability of this protocol is a nontrivial challenge. We provide two proofs for its unbeatability: one is a subtle constructive combinatorial proof, and the other is a topological proof of a new style. These two proofs provide new insight into the connection between topological reasoning and combinatorial reasoning about protocols, which has long been a subject of interest. In particular, our topological proof reasons in a novel way about subcomplexes of the protocol complex, and sheds light on an open question posed by Guerraoui and Pochon (2009). Finally, using the machinery developed in the design of this unbeatable protocol, we propose a protocol for uniform k-set consensus that beats all known solutions by a large margin.

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arXiv2014

Unbeatable Consensus

The unbeatability of a consensus protocol, introduced by Halpern, Moses and Waarts in 2001, is a stronger notion of optimality than the accepted notion of early stopping protocols. Using a novel knowledge-based analysis, this paper derives the first practical unbeatable consensus protocols in the literature, for the standard synchronous message-passing model with crash failures. These protocols strictly dominate the best known protocols for uniform and for non-uniform consensus, in some case beating them by a large margin. The analysis provides a new understanding of the logical structure of consensus, and of the distinction between uniform and nonuniform consensus. Finally, the first (early stopping and) unbeatable protocol that treats decision values "fairly" is presented. All of these protocols have very concise descriptions, and are shown to be efficiently implementable.

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arXiv2013

Good, Better, Best! - Unbeatable Protocols for Consensus and Set Consensus

While the very first consensus protocols for the synchronous model were designed to match the worst-case lower bound, deciding in exactly t+1 rounds in all runs, it was soon realized that they could be strictly improved upon by early stopping protocols. These dominate the first ones, by always deciding in at most t+1 rounds, but often much faster. A protocol is unbeatable if it can't be strictly dominated. Unbeatability is often a much more suitable notion of optimality for distributed protocols than worst-case performance. Using a knowledge-based analysis, this paper studies unbeatability for both consensus and k-set consensus. We present unbeatable solutions to non-uniform consensus and k-set consensus, and uniform consensus in synchronous message-passing contexts with crash failures. The k-set consensus problem is much more technically challenging than consensus, and its analysis has triggered the development of the topological approach to distributed computing. Worst-case lower bounds for this problem have required either techniques based on algebraic topology, or reduction-based proofs. Our proof of unbeatability is purely combinatorial, and is a direct, albeit nontrivial, generalization of the one for consensus. We also present an alternative topological unbeatability proof that allows to understand the connection between the connectivity of protocol complexes and the decision time of processes. For the synchronous model, only solutions to the uniform variant of k-set consensus have been offered. Based on our unbeatable protocols for uniform consensus and for non-uniform k-set consensus, we present a uniform k-set consensus protocol that strictly dominates all known solutions to this problem in the synchronous model.

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arXiv2011

An Equivariance Theorem with Applications to Renaming (Preliminary Version)

In the renaming problem, each process in a distributed system is issued a unique name from a large name space, and the processes must coordinate with one another to choose unique names from a much smaller name space. We show that lower bounds on the solvability of renaming in an asynchronous distributed system can be formulated as a purely topological question about the existence of an equivariant chain map from a topological disk to a topological annulus. Proving the non-existence of such a map implies the non-existence of a distributed renaming algorithm in several related models of computation.

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