Computing via synthetically engineered bacteria is a vibrant and active field with numerous applications in bio-production, bio-sensing, and medicine. Motivated by the lack of robustness and by resource limitation inside single cells, distributed approaches with communication among bacteria have recently gained in interest. In this paper, we focus on the problem of population growth happening concurrently, and possibly interfering, with the desired bio-computation. Specifically, we present a fast protocol in systems with continuous population growth for the majority consensus problem and prove that it correctly identifies the initial majority among two inputs with high probability if the initial difference is varOmega (sqrt{nlog n}) where n is the total initial population. We also present a fast protocol that correctly computes the Nand of two inputs with high probability. By combining Nand gates with the majority consensus protocol as an amplifier, it is possible to compute arbitrary Boolean functions. Finally, we extend the protocols to several biologically relevant settings. We simulate a plausible implementation of a noisy Nand gate with engineered bacteria. In the context of continuous cultures with a constant outflow and a constant inflow of fresh media, we demonstrate that majority consensus is achieved only if the flow is slower than the maximum growth rate. Simulations suggest that flow increases consensus time over a wide parameter range. The proposed protocols help set the stage for bio-engineered distributed computation that directly addresses continuous stochastic population growth.

Distributed programs are often designed with implicit assumptions about the underlying system. We focus on assumptions related to clock synchronization. When a program written with clock synchronization assumptions is monitored to determine if it satisfies its requirements, the monitor should also account for these assumptions precisely. Otherwise, the monitor will either miss potential bugs (false negatives) or find bugs that are inconsistent with these assumptions (false positives). However, if assumptions made by the program are implicit or change over time and are not immediately available to the monitor, such false positives and/or negatives are unavoidable. This paper characterizes precision (probability that the violation identified by the monitor is valid) and recall (probability that the monitor identifies an actual violation) of the monitor based on the gap between clock synchronization assumptions made by the program/application and the clock synchronization assumptions made by the monitor. Our analysis is based on the development of an analytical model for precision, recall and sensitivity of monitors detecting conjunctive predicates. We validate the model via simulations and experiments on the Amazon Web Services platform.

In this paper, we provide a family of bounds for the rate at which the functions of many inputs can be computed in-network on general topologies. Going beyond simple symmetric functions where the output is invariant to the permutation of the operands, e.g., average, parity, we describe an algorithm that is analyzed to provide throughput bounds (both lower and upper) for the general functions. In particular, we analyze our algorithm when the function to be computed is given as a binary tree schema. Our lower bounds depend on schema parameters like the number of operands and graph parameters like the second largest eigenvalue of the transition matrix of simple random walk on network graph, the maximum and minimum degree of any node in the network. The lower bounding technique that we have used is based on the network flows and can capture general multi-commodity flow settings. Our proposed algorithm uses the well-known simple random walk on a network as its basic primitive for routing. We show that the lower bound obtained on the rate of computation is tight for the complete network topology, the hypercube and the star topology. We also present an upper bound on the expected latency of any data operand in terms of the height of schema, well-studied random walk parameter like the hitting time, and the relative distance from the critical data rate. For the computation time of symmetric functions on the random geometric graph under the gossip model, our approach achieves an order-optimal \(\widetilde{O}(n)\) time despite enforcing a binary tree schema for function computation. In general, Big-O notation represents an upper bound and \(\widetilde{O}\) hides \(\text {poly}\log n\) factors.

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+varepsilon ). Let varDelta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log {varDelta } / log log varDelta ) rounds, for constants varepsilon in (0,1] and fin {mathbb {N}}^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and varepsilon , our algorithm improves over the (f+varepsilon )-approximation algorithm of Kuhn et al. (SODA, 2006)whose running time is O(log varDelta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(flog n) rounds, improving over the classical result of Khuller et al. (J Algorithms, 1994) that achieves a running time of O(flog ^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of Koufogiannakis and Young (Distrib Comput, 2011). We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (flceil log _2(M)+1 rceil +varepsilon )-approximate integral solution in O(1+f/logn)·logΔloglogΔ+(f·logM)1.01·logε-1·(logΔ)0.01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} O\\left( (1+f/\\log n)\\cdot \\left( {\\frac{\\log \\varDelta }{ \\log \\log \\varDelta } + ({f\\cdot \\log M})^{1.01}\\cdot \\log \\varepsilon ^{-1}\\cdot (\\log \\varDelta )^{0.01}}\\right) \\right) \\end{aligned}$$\\end{document}rounds, where f bounds the number of variables in a constraint, varDelta bounds the number of constraints a variable appears in, and M=max left{ 1, lceil 1/a_{min } rceil right} , where a_{min } is the smallest normalized constraint coefficient.

We study the composability of discrete chemical reaction networks (CRNs) that stably compute (i.e., with probability 0 of error) integer-valued functions $$f:\mathbb {N}^d\rightarrow \mathbb {N}$$. We consider output-oblivious CRNs in which the output species is never a reactant (input) to any reaction. The class of output-oblivious CRNs is fundamental, appearing in earlier studies of CRN computation, because it is precisely the class of CRNs that can be composed by simply renaming the output of the upstream CRN to match the input of the downstream CRN. Our main theorem precisely characterizes the functions f that are stably computable by output-oblivious CRNs with an initial leader. The key necessary condition is that for sufficiently large inputs, f is the minimum of a finite number of nondecreasing quilt-affine functions (an affine function is linear with a constant offset; a quilt-affine function is linear with a periodic offset).

In this paper we investigate the computational power of population protocols under some unreliable or weaker interaction models. More precisely, we focus on two features related to the power of interactions: omission failures and one-way communications. An omission failure, a notion that this paper introduces for the first time in the context of population protocols, is the loss by one or both parties of the information transmitted in an interaction. The failure may or may not be detected by either party. In one-way models, on the other hand, communication happens only in one direction: only one of the two agents can change its state depending on both agents’ states, and the other agent may or may not be aware of the interaction. These notions can be combined, obtaining one-way protocols with (possibly detectable) omission failures. We start our investigation by providing a complete classification of all the possible models arising from the aforementioned weaknesses, and establishing the computational hierarchy of these models. We then address for each model the fundamental question of what additional power is necessary and sufficient to completely overcome the model’s weakness and make it able to simulate faultless two-way protocols; by “simulator” we mean a wrapper protocol that, implementing an atomic communication of states between two agents, converts any standard two-way protocol into one that works in a weaker model. We answer this question by presenting simulators that work under certain assumptions (e.g., additional memory, unique IDs, etc.) and by proving that simulation is impossible without such assumptions.

This paper addresses the cornerstone family of local problems in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the n nodes can only send $$O(\log n)$$-bit messages in each round of communication. Our framework mostly follows by the derandomization approach of Luby (J Comput Syst Sci 47(2):250–286, 1993) combined with the power of all to all communication. Our key results are as follows: first, we show that in the congested clique model, which allows all-to-all communication, there is a deterministic maximal independent set algorithm that runs in $$O(\log ^2 {\varDelta })$$ rounds, where $${\varDelta }$$ is the maximum degree. When $${\varDelta }=O(n^{1/3})$$, the bound improves to $$O(\log {\varDelta })$$. In addition, we deterministically construct a $$(2k-1)$$-spanner with $$O(kn^{1+1/k}\log n)$$ edges in $$O(k \log n)$$ rounds in the congested clique model.

For many years, Herlihy’s elegant computability-based Consensus Hierarchy has been our best explanation of the relative power of various objects. Since real multiprocessors allow the different instructions they support to be applied to any memory location, it makes sense to consider combining the instructions supported by different objects, rather than considering collections of different objects. Surprisingly, this causes Herlihy’s computability-based hierarchy to collapse. In this paper, we suggest an alternative: a complexity-based classification of the relative power of sets of multiprocessor synchronization instructions, captured by the minimum number of memory locations of unbounded size that are needed to solve obstruction-free consensus when using different sets of instructions.

Shape formation is a basic distributed problem for systems of computational mobile entities. Intensively studied for systems of autonomous mobile robots, it has recently been investigated in the realm of programmable matter. Namely, it has been studied in the geometric Amoebot model, where the anonymous entities, called particles, operate on a hexagonal tessellation of the plane and have limited computational power (they have constant memory), strictly local interaction and communication capabilities (only with particles in neighboring nodes of the grid), and limited motorial capabilities (from a grid node to an empty neighboring node); their activation is controlled by an adversarial scheduler. Recent investigations have shown how, starting from a well-structured configuration in which the particles form a (not necessarily complete) triangle, the particles can form a large class of shapes. This result has been established under several assumptions: agreement on the clockwise direction (i.e., chirality), a sequential activation schedule, and randomization (i.e., particles can flip coins). In this paper we provide a characterization of which shapes can be formed deterministically starting from any simply connected initial configuration of $n$ particles. As a byproduct, if randomization is allowed, then any input shape can be formed from any initial (simply connected) shape by our algorithm, provided that $n$ is large enough. Our algorithm works without chirality, proving that chirality is computationally irrelevant for shape formation. Furthermore, it works under a strong adversarial scheduler, not necessarily sequential. We also consider the complexity of shape formation in terms of both the number of rounds and of moves performed by the particles. We prove that our solution has a complexity of $O(n^2)$ rounds and moves: this number of moves is also asymptotically optimal.