arXiv subjects

John Michael Robson

Publications and source records attributed to John Michael Robson.

7 recordsLinked to original sources

arXiv2021

Revisiting the Role of Coverings in Anonymous Networks: Spanning Tree Construction and Topology Recognition

This paper revisits two classical distributed problems in anonymous networks, namely spanning tree construction and topology recognition, from the point of view of graph covering theory. For both problems, we characterize necessary and sufficient conditions on the communication graph in terms of directed symmetric coverings. These characterizations answer along-standing open question posed by Yamashita and Kameda [YK96], and shed new light on the connection between coverings and the concepts of views and quotient graphs developed by the same authors. Characterizing conditions in terms of coverings is significant because it connects the field with a vast body of classical literature in graph theory and algebraic topology. In particular, it gives access to powerful tools such as Reidemeister's theorem and Mazurkiewicz's algorithm. Combined together, these tools allow us to present elegant proofs of otherwise intricate results, and their constructive nature makes them effectively usable in the algorithms. This paper also gives us the opportunity to present the field of covering theory in a pedagogical way, with a focus on the two aforementioned tools, whose potential impact goes beyond the specific problems considered in this work.

cs.DC↗

arXiv2018

Deterministic Leader Election Takes $Θ(D + \log n)$ Bit Rounds

Leader election is, together with consensus, one of the most central problems in distributed computing. This paper presents a distributed algorithm, called \STT, for electing deterministically a leader in an arbitrary network, assuming processors have unique identifiers of size $O(\log n)$, where $n$ is the number of processors. It elects a leader in $O(D +\log n)$ rounds, where $D$ is the diameter of the network, with messages of size $O(1)$. Thus it has a bit round complexity of $O(D +\log n)$. This substantially improves upon the best known algorithm whose bit round complexity is $O(D\log n)$. In fact, using the lower bound by Kutten et al. (2015) and a result of Dinitz and Solomon (2007), we show that the bit round complexity of \STT is optimal (up to a constant factor), which is a significant step forward in understanding the interplay between time and message optimality for the election problem. Our algorithm requires no knowledge on the graph such as $n$ or $D$, and the pipelining technique we introduce to break the $O(D\log n)$ barrier is general.

cs.DC↗

arXiv2018

Design Patterns in Beeping Algorithms: Examples, Emulation, and Analysis

We consider networks of processes which interact with beeps. In the basic model defined by Cornejo and Kuhn (2010), processes can choose in each round either to beep or to listen. Those who beep are unable to detect simultaneous beeps. Those who listen can only distinguish between silence and the presence of at least one beep. We refer to this model as $BL$ (beep or listen). Stronger models exist where the nodes can detect collision while they are beeping ($B_{cd}L$), listening ($BL_{cd}$), or both ($B_{cd}L_{cd}$). Beeping models are weak in essence and even simple tasks are difficult or unfeasible within. We present a set of generic building blocks (design patterns) which seem to occur frequently in the design of beeping algorithms. They include multi-slot phases: the fact of dividing the main loop into a number of specialised slots; exclusive beeps: having a single node beep at a time in a neighbourhood (within one or two hops); adaptive probability: increasing or decreasing the probability of beeping to produce more exclusive beeps; internal (resp. peripheral) collision detection: for detecting collision while beeping (resp. listening). Based on these patterns, we provide algorithms for a number of basic problems, including colouring, 2-hop colouring, degree computation, 2-hop MIS, and collision detection (in $BL$). The patterns make it possible to formulate these algorithms in a rather concise and elegant way. Their analyses are more technical; one of them improves significantly upon that of the best known MIS algorithm by Jeavons et al. (2016). Finally, inspired by a technique from Afek et al. (2013), our last contribution is to show that any Las Vegas algorithm relying on collision detection can be transposed into a Monte Carlo algorithm without collision detection at the cost of a logarithmic slowdown, which we prove is optimal.

cs.DC↗

arXiv2017

Robustness in Highly Dynamic Networks

We investigate a special case of hereditary property that we refer to as {\em robustness}. A property is {\em robust} in a given graph if it is inherited by all connected spanning subgraphs of this graph. We motivate this definition in different contexts, showing that it plays a central role in highly dynamic networks, although the problem is defined in terms of classical (static) graph theory. In this paper, we focus on the robustness of {\em maximal independent sets} (MIS). Following the above definition, a MIS is said to be {\em robust} (RMIS) if it remains a valid MIS in all connected spanning subgraphs of the original graph. We characterize the class of graphs in which {\em all} possible MISs are robust. We show that, in these particular graphs, the problem of finding a robust MIS is {\em local}; that is, we present an RMIS algorithm using only a sublogarithmic number of rounds (in the number of nodes $n$) in the ${\cal LOCAL}$ model. On the negative side, we show that, in general graphs, the problem is not local. Precisely, we prove a $Ω(n)$ lower bound on the number of rounds required for the nodes to decide consistently in some graphs. This result implies a separation between the RMIS problem and the MIS problem in general graphs. It also implies that any strategy in this case is asymptotically (in order) as bad as collecting all the network information at one node and solving the problem in a centralized manner. Motivated by this observation, we present a centralized algorithm that computes a robust MIS in a given graph, if one exists, and rejects otherwise. Significantly, this algorithm requires only a polynomial amount of local computation time, despite the fact that exponentially many MISs and exponentially many connected spanning subgraphs may exist.

cs.DC↗

arXiv2009

Spanning Trees of Bounded Degree Graphs

We consider lower bounds on the number of spanning trees of connected graphs with degree bounded by $d$. The question is of interest because such bounds may improve the analysis of the improvement produced by memorisation in the runtime of exponential algorithms. The value of interest is the constant $β_d$ such that all connected graphs with degree bounded by $d$ have at least $β_d^μ$ spanning trees where $μ$ is the cyclomatic number or excess of the graph, namely $m-n+1$. We conjecture that $β_d$ is achieved by the complete graph $K_{d+1}$ but we have not proved this for any $d$ greater than 3. We give weaker lower bounds on $β_d$ for $d\le 11$.

cs.DM↗

arXiv2003

Cantorian Tableaux and Permanents

This article could be called "theme and variations" on Cantor's celebrated diagonal argument. Given a square nxn tableau T=(a_i^j) on a finite alphabet A, let L be the set of its row-words. The permanent Perm(T) is the set of words a_{π(1)}^1 a_{π(2)}^2 ...a_{π(n)}^n, where πruns through the set of permutations of n elements. Cantorian tableaux are those for which Perm(T)\cap L=\emptyset. Let s=s(n) be the cardinality of A. We show in particular that for large n, if s(n) <(1-ε) n/log n then most of the tableaux are non-Cantorian, whereas if s(n) >(1+ε) n/log n then most of the tableaux are Cantorian. We conclude our article by the study of infinite tableaux. Consider for example the infinite tableaux whose rows are the binary expansions of the real algebraic numbers in the unit interval. We show that the permanent of this tableau contains exactly the set of binary expansions of all the transcendental numbers in the unit interval.

math.CO↗

arXiv2000

Hard Tiling Problems with Simple Tiles

It is well-known that the question of whether a given finite region can be tiled with a given set of tiles is NP-complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right tromino alone. In the process, we show that Monotone 1-in-3 Satisfiability is NP-complete for planar cubic graphs. In higher dimensions, we show NP-completeness for the domino and straight tromino for general regions on the cubic lattice, and for simply-connected regions on the four-dimensional hypercubic lattice.

math.CO↗