arXiv subjects

James Aspnes

Publications and source records attributed to James Aspnes.

At least 19 recordsLinked to original sources

arXiv2025

Stochastic well-structured transition systems

Extending well-structured transition systems to incorporate a probabilistic scheduling rule, we define a new class of stochastic well-structured transition systems that includes population protocols, chemical reaction networks, and many common gossip models; as well as augmentations of these systems by an oracle that exposes a total order on agents as in population protocols in the comparison model or an equivalence relation as in population protocols with unordered data. We show that any implementation of a phase clock in these systems either stops or ticks too fast after polynomially many expected steps, and that any terminating computation in these systems finishes or fails in expected polynomial time. This latter property allows an exact characterization of the computational power of many stochastic well-structured transition systems augmented with a total order or equivalence relation on agents, showing that these compute exactly the languages in BPP, while the corresponding unaugmented systems compute just the symmetric languages in BPL.

cs.DC↗

arXiv2025

The consensus number of a shift register equals its width

The consensus number of a w-bit register supporting logical left shift and right shift operations is exactly w, giving an example of a class of types, widely implemented in practice, that populates all levels of the consensus hierarchy. This result generalizes to w-wide shift registers over larger alphabets. In contrast, a register providing arithmetic right shift, which replicates the most significant bit instead of replacing it with zero, is shown to solve consensus for any fixed number of processes as long as its width is at least two.

cs.DC↗

arXiv2025

Obfuscated Consensus

The classic Fischer, Lynch, and Paterson impossibility proof demonstrates that any deterministic protocol for consensus in either a message-passing or shared-memory system must violate at least one of termination, validity, or agreement in some execution. But it does not provide an efficient procedure to find such a bad execution. We show that for wait-free shared memory consensus, given a protocol in which each process performs at most $s$ steps computed with total time complexity at most $t$, there exists an adversary algorithm that takes the process's programs as input and computes within $O(st)$ time a schedule that violates agreement. We argue that this bound is tight assuming the random oracle hypothesis: there exists a deterministic obfuscated consensus protocol that forces the adversary to spend ${\Omega}(st)$ time to find a bad execution despite having full access to all information available to the protocol. This bound is based on a general reduction from constructing an obfuscated consensus protocol to constructing an obfuscated threshold function that provably costs $\Omega(t)$ time to evaluate on a single input, where $t$ is a tunable parameter, and for which an adversary with access to the threshold function implementation cannot extract the threshold any faster than by doing binary search. We give a particular implementation of such an obfuscated threshold function that is not very efficient but that is provably secure assuming the random oracle hypothesis. Since our obfuscated consensus protocol does not depend on the specific details of this construction, it may be possible to replace it with one that is more efficient or requires weaker cryptographic assumptions, a task we leave for future work.

cs.DC↗

arXiv2024

Stably computable relations and predicates

A population protocol stably computes a relation R(x,y) if its output always stabilizes and R(x,y) holds if and only if y is a possible output for input x. Alternatively, a population protocol computes a predicate R( ) on pairs if its output stabilizes on the truth value of the predicate when given as input. We consider how stably computing R(x,y) and R( ) relate to each other. We show that for population protocols running on a complete interaction graph with n>=2, if R( ) is a stably computable predicate such that R(x,y) holds for at least one y for each x, then R(x,y) is a stably computable relation. In contrast, the converse is not necessarily true unless R(x,y) holds for exactly one y for each x.

cs.DC↗

arXiv2023

Privacy in Population Protocols with Probabilistic Scheduling

The population protocol model introduced by Angluin et al. in 2006 offers a theoretical framework for designing and analyzing distributed algorithms among limited-resource mobile agents. While the original population protocol model considers the concept of anonymity, the issue of privacy is not investigated thoroughly. However, there is a need for time- and space-efficient privacy-preserving techniques in the population protocol model if these algorithms are to be implemented in settings handling sensitive data, such as sensor networks, IoT devices, and drones. In this work, we introduce several formal definitions of privacy, ranging from assuring only plausible deniability of the population input vector to having a full information-theoretic guarantee that knowledge beyond an agent's input and output bear no influence on the probability of a particular input vector. We then apply these definitions to both existing and novel protocols. We show that the Remainder-computing protocol given by Delporte-Gallet et al. in 2007 (which is proven to satisfy output independent privacy under adversarial scheduling) is not information-theoretically private under probabilistic scheduling. In contrast, we provide a new algorithm and demonstrate that it correctly and information-theoretically privately computes Remainder under probabilistic scheduling.

cs.CR↗

arXiv2023

Fast Convergence of $k$-Opinion Undecided State Dynamics in the Population Protocol Model

We analyze the convergence of the $k$-opinion Undecided State Dynamics (USD) in the population protocol model. For $k$=2 opinions it is well known that the USD reaches consensus with high probability within $O(n \log n)$ interactions. Proving that the process also quickly solves the consensus problem for $k>2$ opinions has remained open, despite analogous results for larger $k$ in the related parallel gossip model. In this paper we prove such convergence: under mild assumptions on $k$ and on the initial number of undecided agents we prove that the USD achieves plurality consensus within $O(k n \log n)$ interactions with high probability, regardless of the initial bias. Moreover, if there is an initial additive bias of at least $\Omega(\sqrt{n} \log n)$ we prove that the initial plurality opinion wins with high probability, and if there is a multiplicative bias the convergence time is further improved. Note that this is the first result for $k > 2$ for the USD in the population protocol model. Furthermore, it is the first result for the unsynchronized variant of the USD with $k>2$ which does not need any initial bias.

cs.DC↗

arXiv2020

Message complexity of population protocols

The standard population protocol model assumes that when two agents interact, each observes the entire state of the other agent. We initiate the study of $\textit{message complexity}$ for population protocols, where the state of an agent is divided into an externally-visible $\textit{message}$ and an internal component, where only the message can be observed by the other agent in an interaction. We consider the case of $O(1)$ message complexity. When time is unrestricted, we obtain an exact characterization of the stably computable predicates based on the number of internal states $s(n)$: If $s(n) = o(n)$ then the protocol computes semilinear predicates (unlike the original model, which can compute non-semilinear predicates with $s(n) = O(\log n)$), and otherwise it computes a predicate decidable by a nondeterministic $O(n \log s(n))$-space-bounded Turing machine. We then introduce novel $O(\mathrm{polylog}(n))$ expected time protocols for junta/leader election and general purpose broadcast correct with high probability, and approximate and exact population size counting correct with probability 1. Finally, we show that the main constraint on the power of bounded-message-size protocols is the size of the internal states: with unbounded internal states, any computable function can be computed with probability 1 in the limit by a protocol that uses only $\textit{1-bit}$ messages.

cs.DC↗

arXiv2020

Approximate Majority With Catalytic Inputs

Population protocols are a class of algorithms for modeling distributed computation in networks of finite-state agents communicating through pairwise interactions. Their suitability for analyzing numerous chemical processes has motivated the adaptation of the original population protocol framework to better model these chemical systems. In this paper, we further the study of two such adaptations in the context of solving approximate majority: persistent-state agents (or catalysts) and spontaneous state changes (or leaks). Based on models considered in recent protocols for populations with persistent-state agents, we assume a population with $n$ catalytic input agents and $m$ worker agents, and the goal of the worker agents is to compute some predicate over the states of the catalytic inputs. We call this model the Catalytic Input (CI) model. For $m = \Theta(n)$, we show that computing the exact majority of the input population with high probability requires at least $\Omega(n^2)$ total interactions, demonstrating a strong separation between the CI model and the standard population protocol model. On the other hand, we show that the simple third-state dynamics of Angluin et al. for approximate majority in the standard model can be naturally adapted to the CI model: we present such a constant-state protocol for the CI model that solves approximate majority in $O(n \log n)$ total steps w.h.p. when the input margin is $\Omega(\sqrt{n \log n})$. We then show the robustness of third-state dynamics protocols to the transient leaks events introduced by Alistarh et al. In both the original and CI models, these protocols successfully compute approximate majority with high probability in the presence of leaks occurring at each step with probability $\beta \leq O\left(\sqrt{n \log n}/n\right)$, exhibiting a resilience to leaks similar to that of Byzantine agents in previous works.

cs.DC↗

arXiv2020

Why Extension-Based Proofs Fail

We introduce extension-based proofs, a class of impossibility proofs that includes valency arguments. They are modelled as an interaction between a prover and a protocol. Using proofs based on combinatorial topology, it has been shown that it is impossible to deterministically solve k-set agreement among n > k > 1 processes in a wait-free manner in certain asynchronous models. However, it was unknown whether proofs based on simpler techniques were possible. We show that this impossibility result cannot be obtained for one of these models by an extension-based proof and, hence, extension-based proofs are limited in power.

cs.DC↗

arXiv2020

Notes on Randomized Algorithms

Lecture notes for the Yale Computer Science course CPSC 4690/5690 Randomized Algorithms. Suitable for use as a supplementary text for an introductory graduate or advanced undergraduate course on randomized algorithms. Discusses tools from probability theory, including random variables and expectations, union bound arguments, concentration bounds, applications of martingales and Markov chains, and the Lov\'asz Local Lemma. Algorithmic topics include analysis of classic randomized algorithms such as Quicksort and Hoare's FIND, randomized tree data structures, hashing, Markov chain Monte Carlo sampling, randomized approximate counting, derandomization, quantum computing, and some examples of randomized distributed algorithms.

cs.DS↗

arXiv2020

Notes on Theory of Distributed Systems

Notes for the Yale course CPSC 465/565 Theory of Distributed Systems. Table of Contents: 1 Introduction, 2 Model, 3 Broadcast and convergecast, 4 Distributed breadth-first search, 5 Leader election, 6 Causal ordering and logical clocks, 7 Synchronizers, 8 Coordinated attack, 9 Synchronous agreement, 10 Byzantine agreement, 11 Impossibility of asynchronous agreement, 12 Paxos, 13 Failure detectors, 14 Quorum systems, 15 Permissionless systems, 16 Model, 17 Distributed shared memory, 18 Mutual exclusion, 19 The wait-free hierarchy, 20 Atomic snapshots, 21 Lower bounds on perturbable objects, 22 Restricted-use objects, 23 Common2, 24 Randomized consensus and test-and-set, 25 Renaming, 26 Software transactional memory, 27 Obstruction-freedom, 28 BG simulation, 29 Topological methods, 30 Approximate agreement, 31 Overview, 32 Self-stabilization, 33 Distributed graph algorithms, 34 Mobile Robots, 35 Beeping, 36 Population protocols

cs.DC↗

arXiv2017

Space-Optimal Majority in Population Protocols

Population protocols are a model of distributed computing, in which $n$ agents with limited local state interact randomly, and cooperate to collectively compute global predicates. An extensive series of papers, across different communities, has examined the computability and complexity characteristics of this model. Majority, or consensus, is a central task, in which agents need to collectively reach a decision as to which one of two states $A$ or $B$ had a higher initial count. Two complexity metrics are important: the time that a protocol requires to stabilize to an output decision, and the state space size that each agent requires. It is known that majority requires $Ω(\log \log n)$ states per agent to allow for poly-logarithmic time stabilization, and that $O(\log^2 n)$ states are sufficient. Thus, there is an exponential gap between the upper and lower bounds. We address this question. We provide a new lower bound of $Ω(\log n)$ states for any protocol which stabilizes in $O( n^{1-c} )$ time, for any $c > 0$ constant. This result is conditional on basic monotonicity and output assumptions, satisfied by all known protocols. Technically, it represents a significant departure from previous lower bounds. Instead of relying on dense configurations, we introduce a new surgery technique to construct executions which contradict the correctness of algorithms that stabilize too fast. Subsequently, our lower bound applies to general initial configurations. We give an algorithm for majority which uses $O(\log n)$ states, and stabilizes in $O(\log^2 n)$ time. Central to the algorithm is a new leaderless phase clock, which allows nodes to synchronize in phases of $Θ(n \log{n})$ consecutive interactions using $O(\log n)$ states per node. We also employ our phase clock to build a leader election algorithm with $O(\log n )$ states, which stabilizes in $O(\log^2 n)$ time.

cs.DC↗

arXiv2017

Time-Space Trade-offs in Population Protocols

Population protocols are a popular model of distributed computing, in which randomly-interacting agents with little computational power cooperate to jointly perform computational tasks. Inspired by developments in molecular computation, and in particular DNA computing, recent algorithmic work has focused on the complexity of solving simple yet fundamental tasks in the population model, such as leader election (which requires stabilization to a single agent in a special "leader" state), and majority (in which agents must stabilize to a decision as to which of two possible initial states had higher initial count). Known results point towards an inherent trade-off between the time complexity of such algorithms, and the space complexity, i.e. size of the memory available to each agent. In this paper, we explore this trade-off and provide new upper and lower bounds for majority and leader election. First, we prove a unified lower bound, which relates the space available per node with the time complexity achievable by a protocol: for instance, our result implies that any protocol solving either of these tasks for $n$ agents using $O( \log \log n )$ states must take $Ω( n / \rm{polylog} n )$ expected time. This is the first result to characterize time complexity for protocols which employ super-constant number of states per node, and proves that fast, poly-logarithmic running times require protocols to have relatively large space costs. On the positive side, we give algorithms showing that fast, poly-logarithmic stabilization time can be achieved using $O( \log^2 n )$ space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between $O(\log \log n)$ and $Θ( \log^2 n )$ state space size for both majority and leader election in population protocols, and introduce new techniques, which should be applicable more broadly.

cs.DC↗

arXiv2016

Time and Space Optimal Counting in Population Protocols

This work concerns the general issue of combined optimality in terms of time and space complexity. In this context, we study the problem of (exact) counting resource-limited and passively mobile nodes in the model of population protocols, in which the space complexity is crucial. The counted nodes are memory-limited anonymous devices (called agents) communicating asynchronously in pairs (according to a fairness condition). Moreover, we assume that these agents are prone to failures so that they cannot be correctly initialized. This study considers two classical fairness conditions, and for each we investigate the issue of time optimality of counting given the optimal space per agent. In the case of randomly interacting agents (probabilistic fairness), as usual, the convergence time is measured in terms of parallel time (or parallel interactions), which is defined as the number of pairwise interactions until convergence, divided by n (the number of agents). In case of weak fairness, where it is only required that every pair of agents interacts infinitely often, the convergence time is defined in terms of non-null transitions, i.e, the transitions that affect the states of the interacting agents.First, assuming probabilistic fairness, we present a "non-guessing" time optimal protocol of O(n log n) expected time given an optimal space of only one bit, and we prove the time optimality of this protocol. Then, for weak fairness, we show that a space optimal (semi-uniform) solution cannot converge faster than in $Ω$(2^n) time (non-null transitions). This result, together with the time complexity analysis of an already known space optimal protocol, shows that it is also optimal in time (given the optimal space constrains).

cs.DC↗

arXiv2012

A one-bit swap object using test-and-sets and a max register

We describe a linearizable, wait-free implementation of a one-bit swap object from a single max register and an unbounded array of test-and-set bits. Each swap operation takes at most three steps. Using standard randomized constructions, the max register and test-and-set bits can be replaced by read-write registers, at the price of raising the cost of a swap operation to an expected O(max(log n, min(log t, n))) steps, where t is the number of times the swap object has previously changed its value and n is the number of processes.

cs.DC↗

arXiv2010

Slightly smaller splitter networks

The classic renaming protocol of Moir and Anderson (1995) uses a network of Theta(n^2) splitters to assign unique names to n processes with unbounded initial names. We show how to reduce this bound to Theta(n^{3/2}) splitters.

cs.DC↗

arXiv2006

The computational power of population protocols

We consider the model of population protocols introduced by Angluin et al., in which anonymous finite-state agents stably compute a predicate of the multiset of their inputs via two-way interactions in the all-pairs family of communication networks. We prove that all predicates stably computable in this model (and certain generalizations of it) are semilinear, answering a central open question about the power of the model. Removing the assumption of two-way interaction, we also consider several variants of the model in which agents communicate by anonymous message-passing where the recipient of each message is chosen by an adversary and the sender is not identified to the recipient. These one-way models are distinguished by whether messages are delivered immediately or after a delay, whether a sender can record that it has sent a message, and whether a recipient can queue incoming messages, refusing to accept new messages until it has had a chance to send out messages of its own. We characterize the classes of predicates stably computable in each of these one-way models using natural subclasses of the semilinear predicates.

cs.CC↗

arXiv2006

Path-independent load balancing with unreliable machines

We consider algorithms for load balancing on unreliable machines. The objective is to optimize the two criteria of minimizing the makespan and minimizing job reassignments in response to machine failures. We assume that the set of jobs is known in advance but that the pattern of machine failures is unpredictable. Motivated by the requirements of BGP routing, we consider path-independent algorithms, with the property that the job assignment is completely determined by the subset of available machines and not the previous history of the assignments. We examine first the question of performance measurement of path-independent load-balancing algorithms, giving the measure of makespan and the normalized measure of reassignments cost. We then describe two classes of algorithms for optimizing these measures against an oblivious adversary for identical machines. The first, based on independent random assignments, gives expected reassignment costs within a factor of 2 of optimal and gives a makespan within a factor of O(log m/log log m) of optimal with high probability, for unknown job sizes. The second, in which jobs are first grouped into bins and at most one bin is assigned to each machine, gives constant-factor ratios on both reassignment cost and makespan, for known job sizes. Several open problems are discussed.

cs.DS↗