Abstract We consider the standard population protocol model, where ( a priori ) indistinguishable and anonymous agents interact in pairs according to uniformly random scheduling. The self-stabilizing leader election problem requires the protocol to converge on a single leader agent from any possible initial configuration. We initiate the study of time complexity of population protocols solving this problem in its original setting: with probability 1, in a complete communication graph. The only previously known protocol by Cai, Izumi, and Wada [Theor. Comput. Syst. 50] runs in expected parallel time $$\Theta (n^2)$$ Θ ( n 2 ) and has the optimal number of n states in a population of n agents. The existing protocol has the additional property that it becomes silent, i.e., the agents’ states eventually stop changing. Observing that any silent protocol solving self-stabilizing leader election requires $$\Omega (n)$$ Ω ( n ) expected parallel time, we introduce a silent protocol that uses optimal O ( n ) parallel time and states. Without any silence constraints, we show that it is possible to solve self-stabilizing leader election in asymptotically optimal expected parallel time of $$O(\log n)$$ O ( log n ) , but using at least exponential states (a quasipolynomial number of bits). All of our protocols (and also that of Cai et al.) work by solving the more difficult ranking problem: assigning agents the ranks $$1,\ldots ,n$$ 1 , … , n .

Understanding the algorithmic behaviors that are in principle realizable in a chemical system is necessary for a rigorous understanding of the design principles of biological regulatory networks. Further, advances in synthetic biology herald the time when we will be able to rationally engineer complex chemical systems and when idealized formal models will become blueprints for engineering. Coupled chemical interactions in a well-mixed solution are commonly formalized as chemical reaction networks (CRNs). However, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. Here, we study the following problem: What functions f : ℝ k → ℝ can be computed by a CRN, in which the CRN eventually produces the correct amount of the “output” molecule, no matter the rate at which reactions proceed? This captures a previously unexplored but very natural class of computations: For example, the reaction X 1 + X 2 → Y can be thought to compute the function y = min ( x 1 , x 2 ). Such a CRN is robust in the sense that it is correct whether its evolution is governed by the standard model of mass-action kinetics, alternatives such as Hill-function or Michaelis-Menten kinetics, or other arbitrary models of chemistry that respect the (fundamentally digital) stoichiometric constraints (what are the reactants and products?). We develop a reachability relation based on a broad notion of “what could happen” if reaction rates can vary arbitrarily over time. Using reachability, we define stable computation analogously to probability 1 computation in distributed computing and connect it with a seemingly stronger notion of rate-independent computation based on convergence in the limit t → ∞ under a wide class of generalized rate laws. Besides the direct mapping of a concentration to a nonnegative analog value, we also consider the “dual-rail representation” that can represent negative values as the difference of two concentrations and allows the composition of CRN modules. We prove that a function is rate-independently computable if and only if it is piecewise linear (with rational coefficients) and continuous (dual-rail representation), or non-negative with discontinuities occurring only when some inputs switch from zero to positive (direct representation). The many contexts where continuous piecewise linear functions are powerful targets for implementation, combined with the systematic construction we develop for computing these functions, demonstrate the potential of rate-independent chemical computation.

The online car-sharing problem finds many real-world applications. The problem, proposed by Luo, Erlebach and Xu in 2018, mainly focuses on an online model in which there are two locations: 0 and 1, and k total cars. Each request which specifies its pick-up time and pick-up location (among 0 and 1, and the other is the drop-off location) is released in each stage a fixed amount of time before its specified start (i.e. pick-up) time. The time between the booking (i.e. released) time and the start time is enough to move empty cars between 0 and 1 for relocation if they are not used in that stage. The model, called k S2L-F, assumes that requests in each stage arrive sequentially regardless of the same booking time and the decision (accept or reject) must be made immediately. The goal is to accept as many requests as possible. In spite of only two locations, the analysis does not seem easy and the (tight) competitive ratio (CR) is only known to be 2 for k = 2 and 1.5 for a restricted value of k , i.e., a multiple of three. In this paper, we remove all the holes of unknown CR's; namely we prove that the CR is 2 k k + ⌊ k / 3 ⌋ for all k ≥ 2 . Furthermore, if the algorithm can delay its decision until all requests have come in each stage, the CR is improved to roughly 4/3. We can take this advantage even further; precisely we can achieve a CR of 2 + R 3 if the number of requests in each stage is at most Rk , 1 ≤ R ≤ 2 , where we do not have to know the value of R in advance. Finally we demonstrate that randomization also helps to get (slightly) better CR's, and prove some lower bounds to show the tightness.

We consider the standard population protocol model, where (a priori) indistinguishable and anonymous agents interact in pairs according to uniformly random scheduling. The self-stabilizing leader election problem requires the protocol to converge on a single leader agent from any possible initial configuration. We initiate the study of time complexity of population protocols solving this problem in its original setting: with probability 1, in a complete communication graph. The only previously known protocol by Cai, Izumi, and Wada [Theor. Comput. Syst. 50] runs in expected parallel time Θ(n2) and has the optimal number of n states in a population of n agents. The existing protocol has the additional property that it becomes silent, i.e., the agents' states eventually stop changing.

Population protocols [3] are used as a distributed model that captures the behavior of passively mobile agents. Leader election is one of the most well-studied problems in this model. In this paper, we focus on the self-stabilizing leader election (SSLE) problem proposed by Angluin et al. [5]. Previously, it is known that SSLE can be performed on arbitrary rings and tori with a constant number of states [11], but SSLE on complete graphs requires Ω(n) states [9].

In this paper, we study the self-stabilizing leader election (SSLE) problem in population protocols. We construct a non-deterministic population protocol that can solve SSLE on directed rings of all sizes. Our algorithm uses a constant number of states and can be converted to a deterministic population protocol on undirected rings using previous techniques [8]. Furthermore, we extend our algorithm to perform SSLE on directed and undirected tori of arbitrary sizes.

We study the role that parallelism plays in time complexity of variants of Winfree's abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the “hierarchical” aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the “seeded” aTAM, tiles attach one at a time to a growing assembly. Adleman et al. [Running time and program size for self-assembled squares, in Proceedings of the 33 rd Annual ACM Symposium on Theory of Computing (Hersonissos, Greece), ACM, New York, 2001, pp. 740--748] showed how to assemble an $n \times n$ square in $O(n)$ time in the seeded aTAM using $O(\frac{\log n}{\log \log n})$ unique tile types, where both of these parameters are optimal. They asked whether the hierarchical aTAM could allow a tile system to use the ability to form large assemblies in parallel before they attach to break the $\Omega(n)$ lower bound for assembly time. We show that there is a tile system with the optimal $O(\frac{\log n}{\log \log n})$ tile types that assembles an $n \times n$ square using $O(\log^2 n)$ parallel “stages,” which are close to the optimal $\Omega(\log n)$ stages, forming the final $n \times n$ square from four $n/2 \times n/2$ squares, which are themselves recursively formed from $n/4 \times n/4$ squares, etc. However, despite this nearly maximal parallelism, the system requires superlinear time to assemble the square. We extend the definition of partial order tile systems studied by Adleman et al. in a natural way to hierarchical assembly and show that no hierarchical partial order tile system can build any shape with diameter $D$ in less than time $\Omega(D)$, demonstrating that in this case the hierarchical model affords no speedup whatsoever over the seeded model. We also strengthen the $\Omega(D)$ time lower bound for deterministic seeded systems of Adleman et al. to nondeterministic seeded systems. Finally, we show that for infinitely many $n$, a tile system can assemble an $n \times n'$ rectangle, with $n > n'$, in time $O(n^{4/5} \log n)$, breaking the linear-time lower bound that applies to all seeded systems and partial order hierarchical systems.

We study the problem of routing traffic for independent selfish users in a congested network to minimize the total latency. The inefficiency of selfish routing motivates regulating the flow of the system to lower the total latency of the Nash Equilibrium by economic incentives or penalties. When applying tax to the routes, we follow the definition of [Christodoulou et al, Algorithmica, 2014] to define ePoA as the Nash total cost including tax in the taxed network over the optimal cost in the original network. We propose a simple tax scheme consisting of step functions imposed on the links. The tax scheme can be applied to routing games with parallel links, affine cost functions and single-commodity networks to lower the ePoA to at most 4/3 - epsilon, where epsilon only depends on the discrepancy between the links. We show that there exists a tax scheme in the two link case with an ePoA upperbound less than 1.192 which is almost tight. Moreover, we design another tax scheme that lowers ePoA down to 1.281 for routing games with groups of links such that links in the same group are similar to each other and groups are sufficiently different.

We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap. This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly and avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem.

Understanding the algorithmic behaviors that are in principle realizable in a chemical system is necessary for a rigorous understanding of the design principles of biological regulatory networks. Further, advances in synthetic biology herald the time when we'll be able to rationally engineer complex chemical systems, and when idealized formal models will become blueprints for engineering.

A stochastic flow network is a directed graph with incoming edges (inputs) and outgoing edges (outputs), tokens enter through the input edges, travel stochastically in the network, and can exit the network through the output edges. Each node in the network is a splitter, namely, a token can enter a node through an incoming edge and exit on one of the output edges according to a predefined probability distribution. Stochastic flow networks can be easily implemented by beam splitters, or by DNA-based chemical reactions, with promising applications in optical computing, molecular computing and stochastic computing. In this paper, we address a fundamental synthesis question: Given a finite set of possible splitters and an arbitrary rational probability distribution, design a stochastic flow network, such that every token that enters the input edge will exit the outputs with the prescribed probability distribution. The problem of probability transformation dates back to von Neumann’s 1951 work and was followed, among others, by Knuth and Yao in 1976. Most existing works have been focusing on the “simulation” of target distributions. In this paper, we design optimal-sized stochastic flow networks for “synthesizing” target distributions. It shows that when each splitter has two outgoing edges and is unbiased, an arbitrary rational probability${{ {a}} \over { {b}}}$with${ {a}} \leq { {b}} \leq {{ 2}^{{n}}}$can be realized by a stochastic flow network of size${ {n}}$that is optimal. Compared to the other stochastic systems, feedback (cycles in networks) strongly improves the expressibility of stochastic flow networks.

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biology's fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in its length. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time polylogarithmic in the size of the shape plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the fundamental efficiency advantage of active self-assembly over passive self-assembly and motivates experimental effort to construct self-assembly systems with active molecular components.

While computer science has given us a framework for determining the complexity and difficulty of solving computational problems, we do not yet have a theoretical framework for knowing what actions, behaviors, and life-like qualities can emerge from a given set of simple modular units. There has been much interest in developing models for programming active self-assembly processes in both the reconfigurable robotics community and the nanotechnology community. With respect to materials science and nanotechnology, the models proposed to date are either not yet implementable with our current understanding of synthetic chemistry or those that are implementable are limited to a set of features that do not capture the power of active components. Prior implementable models of molecular assembly only considered the passive behaviors of attaching and detaching from a complex. Inspired by the algorithmic tile assembly model [Winfree, 1996] and the graph grammar assembly model [Klavins et al., 2004], we describe a formal model for studying the complexity of self-assembled structures with active molecular components. In particular, we add an insertion primitive and we show a direct mapping of our model to a molecular implementation using DNA. We show that the expressive power of this language is stronger than regular languages, but at most as strong as context free grammars. Here, we explore the trade-off between the complexity of the system (in terms of the number of unit types), and the behavior of the system and speed of its assembly. We find that we can grow a line of any given length n in expected time O(log3n) using O(log2n) monomers. If we grow a line with k insertion rules, either the expected final length is infinite or the expected length at time t is at most , which is polynomial in t.

We study the role that parallelism plays in time complexity of variants of Winfree's abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the “hierarchical” aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the “seeded” aTAM, tiles attach one at a time to a growing assembly. Adleman, Cheng, Goel, and Huang (Running Time and Program Size for Self-Assembled Squares, STOC 2001) showed how to assemble an n×n square in O(n) time in the seeded aTAM using unique tile types, where both of these parameters are optimal. They asked whether the hierarchical aTAM could allow a tile system to use the ability to form large assemblies in parallel before they attach to break the Ω(n) lower bound for assembly time. We show that there is a tile system with the optimal tile types that assembles an n×n square using O(log2 n) parallel “stages”, which is close to the optimal Ω(log n) stages, forming the final n×n square from four n/2 × n/2 squares, which are themselves recursively formed from n/4 × n/4 squares, etc. However, despite this nearly maximal parallelism, the system requires superlinear time to assemble the square. We extend the definition of partial order tile systems studied by Adleman et al. in a natural way to hierarchical assembly and show that no hierarchical partial order tile system can build any shape with diameter N in less than time Ω(N), demonstrating that in this case the hierarchical model affords no speedup whatsoever over the seeded model. We also strengthen the Ω(N) time lower bound for deterministic seeded systems of Adleman et al. to nondeterministic seeded systems. Finally, we show that for infinitely many n, a tile system can assemble an n × n′ rectangle, with n × n′, in time O(n4/5 log n), breaking the linear-time lower bound that applies to all seeded systems and partial order hierarchical systems.

A stochastic flow network is a directed graph with incoming edges (inputs) and outgoing edges (outputs), tokens enter through the input edges, travel stochastically in the network and can exit the network through the output edges. Each node in the network is a splitter, namely, a token can enter a node through an incoming edge and exit on one of the output edges according to a predefined probability distribution. We address the following synthesis question: Given a finite set of possible splitters and an arbitrary rational probability distribution, design a stochastic flow network, such that every token that enters the input edge will exit the outputs with the prescribed probability distribution. The problem of probability synthesis dates back to von Neummann's 1951 work and was followed, among others, by Knuth and Yao in 1976, who demonstrated that arbitrary rational probabilities can be generated with tree networks; where minimizing the expected path length, the expected number of coin tosses in their paradigm, is the key consideration. Motivated by the synthesis of stochastic DNA based molecular systems, we focus on designing optimal-sized stochastic flow networks (the size of a network is the number of splitters). We assume that each splitter has two outgoing edges and is unbiased (probability 1/2 per output edge). We show that an arbitrary rational probability a/b with a ≤ b ≤ 2ncan be realized by a stochastic flow network of size n, we also show that this is optimal. We note that our stochastic flow networks have feedback (cycles in the network), in fact, we demonstrate that feedback improves the expressibility of stochastic flow networks, since without feedback only probabilities of the form a/(2n) (a an integer) can be realized.

Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network cost-sharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge cost-sharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal cost-sharing protocols for undirected and directed graphs, single-sink and multicommodity networks, and different measures of the inefficiency of equilibria. Our primary technical tool is a precise characterization of the cost-sharing protocols that induce only network games with pure-strategy Nash equilibria. We use this characterization to prove, among other results, that the Shapley protocol is optimal in directed graphs and that simple priority protocols are essentially optimal in undirected graphs.

Load balancing is a common approach for task assignment in distributed architectures. In this paper, we show that the degree of inefficiency in load balancing designs is highly dependent on the scheduling discipline used at each of the back-end servers. Traditionally, the back-end scheduler can be modeled as processor sharing (PS), in which case the degree of inefficiency grows linearly with the number of servers. However, if the back- end scheduler is changed to shortest remaining processing time (SRPT), the degree of inefficiency can be independent of the number of servers, instead depending only on the heterogeneity of the speeds of the servers. Further, switching the back-end scheduler to SRPT can provide significant improvements in the overall mean response time of the system as long as the heterogeneity of the server speeds is small.

This paper studies task assignment in a network of resource constrained computing platforms (called microservers). A task is an abstraction of a computational agent or data that is hosted by the microservers. For example, in an object tracking scenario, a task represents a mobile tracking agent, such as a vehicle location update computation, that runs on microservers, which can receive sensor data pertaining to the object of interest. Due to object motion, the microservers that can observe a particular object change over time and there is overhead involved in migrating tasks among microservers. Furthermore, communication, processing, or memory constraints, allow a microserver to only serve a limited number of objects at the same time. Our overall goal is to assign tasks to microservers so as to minimize the number of migrations, and thus be kinetically stable, while guaranteeing that as many tasks as possible are monitored at all times. When the task trajectories are known in advance, we show that this problem is NP-complete (even over just two time steps), has an integrality gap of at least 2, and can be solved optimally in polynomial time if we allow tasks to be assigned fractionally. When only probabilistic information about future movement of the tasks is known, we propose two algorithms: a multi-commodity flow based algorithm and a maximum matching algorithm. We use simulations to compare the performance of these algorithms against the optimum task allocation strategy.

We consider a model of game-theoretic network design initially studied by Anshelevich et al. [2], where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its path is the fixed cost of the edge divided by the number of players using it. In this special case, Anshelevich et al. [2] proved that pure-strategy Nash equilibria always exist and that the price of stability--the ratio in costs of a minimumcost Nash equilibrium and an optimal solution--is Θ(log k), where k is the number of players. Little was known about the existence of equilibria or the price of stability in the general weighted version of the game. Here, each player i has a weight wi ≥ 1, and its cost share of an edge in its path equals wi times the edge cost, divided by the total weight of the players using the edge.This paper presents the first general results on weighted Shapley network design games. First, we give a simple example with no pure-strategy Nash equilibrium. This motivates considering the price of stability with respect to α-approximate Nash equilibria--outcomes from which no player can decrease its cost by more than an α multiplicative factor. Our first positive result is that O(log wmax)-approximate Nash equilibria exist in all weighted Shapley network design games, where wmax is the maximum player weight. More generally, we establish the following trade-off between the two objectives of good stability and low cost: for every α = Ω(log wmax), the price of stability with respect to O(α)- approximate Nash equilibria is O((log W)/α), where W is the sum of the players' weights. In particular, there is always an O(logW)-approximate Nash equilibrium with cost within a constant factor of optimal.Finally, we show that this trade-off curve is nearly optimal: we construct a family of networks without o(log wmax/ log log wmax)-approximate Nash equilibria, and show that for all α = Ω(logwmax/ log log wmax), achieving a price of stability of O(log W/α) requires relaxing equilibrium constraints by an Ω(α) factor.