New Classes of Distributed Time Complexity

A number of recent papers -- e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) -- have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem $\Pi$ in which a solution can be verified by checking all radius-$O(1)$ neighbourhoods, and the question is what is the smallest $T$ such that a solution can be computed so that each node chooses its own output based on its radius-$T$ neighbourhood. Here $T$ is the distributed time complexity of $\Pi$. The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are $\Theta(1)$, $\Theta(\log^* n)$, $\Theta(\log n)$, $\Theta(n^{1/k})$, and $\Theta(n)$. It is also known that there are two gaps: one between $\omega(1)$ and $o(\log \log^* n)$, and another between $\omega(\log^* n)$ and $o(\log n)$. It has been conjectured that many more gaps exist, and that the overall time hierarchy is relatively simple -- indeed, this is known to be the case in restricted graph families such as cycles and grids. We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including $\Theta(\log^{\alpha}n)$ for any $\alpha\ge1$, $2^{\Theta(\log^{\alpha}n)}$ for any $\alpha\le 1$, and $\Theta(n^{\alpha})$ for any $\alpha <1/2$ in the high end of the complexity spectrum, and $\Theta(\log^{\alpha}\log^* n)$ for any $\alpha\ge 1$, $\smash{2^{\Theta(\log^{\alpha}\log^* n)}}$ for any $\alpha\le 1$, and $\Theta((\log^* n)^{\alpha})$ for any $\alpha \le 1$ in the low end; here $\alpha$ is a positive rational number

Locality of Graph Problems in Directed Toroidic Grids

Locally checkable labeling problems in the LOCAL model of distributed computation are known to have only three distinct complexity classes when the attention is restricted to problems on toroidic grids only: trivial with time complexity Θ(1), local with time complexity Θ(log∗ n) and global with time complexity Θ(n). Prior work shows that problems belonging to the trivial class are easy to recognize, but that local and global labeling problems are undecidable to separate. However, a method called algorithm synthesis exists for creating an asymptotically optimal normal-form algorithm for any locally checkable labeling problem with a time complexity of Θ(log∗ n). This process, when automated, can be used to process vast amounts of suitably encoded labeling problems in bulk, creating a more defined boundary for the undecidable class of local problems. As a proof-of-concept of this method, this work presents a new asymptotically optimal algorithm for a relaxed form of 3-coloring as well as methods for more general search of local problems

Sublinear-Time Distributed Algorithms for Detecting Small Cliques and Even Cycles

In this paper we give sublinear-time distributed algorithms in the CONGEST model for subgraph detection for two classes of graphs: cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be listed in sublinear time, O(n^{5/6+o(1)}) rounds and O(n^{21/22+o(1)}) rounds, respectively. Prior to our work, it was not known whether it was possible to even check if the network contains a 4-clique or a 5-clique in sublinear time. For even-length cycles, C_{2k}, we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from O~(n^{5/6}) to O~(n^{3/4}) rounds. We also show two obstacles on proving lower bounds for C_{2k}-freeness: First, we use the new connection to extremal combinatorics to show that the current lower bound of Omega~(sqrt{n}) rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant delta in (0,1/2) such that for any k, a Omega(n^{1/2+delta}) lower bound on C_{2k}-freeness implies new lower bounds in circuit complexity. For general subgraphs, it was shown in [Orr Fischer et al., 2018] that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Omega~(n^{2-Theta(1/k)}) rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in O(n^{2 - Theta(1/k)}) rounds, nearly matching the lower bound of [Orr Fischer et al., 2018]

Brief Announcement: Lower Bounds for Asymptotic Consensus in Dynamic Networks

In this work we study the performance of asymptotic and approximate consensus algorithms in dynamic networks. The asymptotic consensus problem requires a set of agents to repeatedly set their outputs such that the outputs converge to a common value within the convex hull of initial values. This problem, and the related approximate consensus problem, are fundamental building blocks in distributed systems where exact consensus among agents is not required, e.g., man-made distributed control systems, and have applications in the analysis of natural distributed systems, such as flocking and opinion dynamics. We prove new nontrivial lower bounds on the contraction rates of asymptotic consensus algorithms, from which we deduce lower bounds on the time complexity of approximate consensus algorithms. In particular, the obtained bounds show optimality of asymptotic and approximate consensus algorithms presented in [Charron-Bost et al., ICALP\u2716] for certain classes of networks that include classical failure assumptions, and confine the search for optimal bounds in the general case. Central to our lower bound proofs is an extended notion of valency, the set of reachable limits of an asymptotic consensus algorithm starting from a given configuration. We further relate topological properties of valencies to the solvability of exact consensus, shedding some light on the relation of these three fundamental problems in dynamic networks

Simulating Real-Time Aspects of Wireless Sensor Networks

Wireless Sensor Networks (WSNs) technology has been mainly used in the applications with low-frequency sampling and little computational complexity. Recently, new classes of WSN-based applications with different characteristics are being considered, including process control, industrial automation and visual surveillance. Such new applications usually involve relatively heavy computations and also present real-time requirements as bounded end-to- end delay and guaranteed Quality of Service. It becomes then necessary to employ proper resource management policies, not only for communication resources but also jointly for computing resources, in the design and development of such WSN-based applications. In this context, simulation can play a critical role, together with analytical models, for validating a system design against the parameters of Quality of Service demanded for. In this paper, we present RTNS, a publicly available free simulation tool which includes Operating System aspects in wireless distributed applications. RTNS extends the well-known NS-2 simulator with models of the CPU, the Real-Time Operating System and the application tasks, to take into account delays due to the computation in addition to the communication. We demonstrate the benefits of RTNS by presenting our simulation study for a complex WSN-based multi-view vision system for real-time event detection

Large deviations of stochastic systems and applications

This dissertation focuses on large deviations of stochastic systems with applications to optimal control and system identification. It encompasses analysis of two-time-scale Markov processes and system identification with regular and quantized data. First, we develops large deviations principles for systems driven by continuous-time Markov chains with twotime scales and related optimal control problems. A distinct feature of our setup is that the Markov chain under consideration is time dependent or inhomogeneous. The use of two time-scale formulation stems from the effort of reducing computational complexity in a wide variety of applications in control, optimization, and systems theory. Starting with a rapidly fluctuating Markovian system, under irreducibility conditions, both large deviations upper and lower bounds are established first for a fixed terminal time and then for time-varying dynamic systems. Then the results are applied to certain dynamic systems and LQ control problems. Second, we study large deviations for identifications systems. Traditional system identification concentrates on convergence and convergence rates of estimates in mean squares, in distribution, or in a strong sense. For system diagnosis and complexity analysis, however, it is essential to understand the probabilities of identification errors over a finite data window. This paper investigates identification errors in a large deviations framework. By considering both space complexity in terms of quantization levels and time complexity with respect to data window sizes, this study provides a new perspective to understand the fundamental relationship between probabilistic errors and resources that represent data sizes in computer algorithms, sample sizes in statistical analysis, channel bandwidths in communications, etc. This relationship is derived by establishing the large deviations principle for quantized identification that links binary-valued data at one end and regular sensors at the other. Under some mild conditions, we obtain large deviations upper and lower bounds. Our results accommodate independent and identically distributed noise sequences, as well as more general classes of mixing-type noise sequences. Numerical examples are provided to illustrate the theoretical results

Fast Computation of Small Cuts via Cycle Space Sampling

We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph. In the model of distributed computing in a graph G=(V, E) with O(log V)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain simple O(Diam)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A simple extension of our work yields the first distributed algorithm with sub-linear time for 3-edge-connected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity. In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V) for finding cut pairs and 3-edge-connected components.Comment: Previous version appeared in Proc. 35th ICALP, pages 145--160, 200

Exact bounds for distributed graph colouring

We prove exact bounds on the time complexity of distributed graph colouring. If we are given a directed path that is properly coloured with $n$ colours, by prior work it is known that we can find a proper 3-colouring in $\frac{1}{2} \log^*(n) \pm O(1)$ communication rounds. We close the gap between upper and lower bounds: we show that for infinitely many $n$ the time complexity is precisely $\frac{1}{2} \log^* n$ communication rounds.Comment: 16 pages, 3 figure