arXiv subjects

Leszek Gąsieniec

Publications and source records attributed to Leszek Gąsieniec.

12 recordsLinked to original sources

arXiv2026

Efficient Stable Population Protocols for Parity and Beyond

For nearly two decades, population protocols have been extensively studied, yielding efficient solutions for central problems in distributed computing, including leader election, and majority computation, a predicate type in Presburger Arithmetic closely tied to population protocols. Surprisingly, no protocols have achieved both time- and space-efficiency for congruency predicates, such as parity computation, which are complementary in this arithmetic framework. This gap highlights a significant challenge in the field. To address this gap, we explore the parity problem, where agents are tasked with computing the parity of the given sub-population size. Then we extend the solution for parity to compute congruences modulo an arbitrary $m$. Previous research on efficient population protocols has focused on protocols that minimise both stabilisation time and state utilisation for specific problems. In contrast, this work slightly relaxes this expectation, permitting protocols to place less emphasis on full optimisation and more on universality, robustness, and probabilistic guarantees. This allows us to propose a novel computing paradigm that integrates population weights (or simply weights), a robust clocking mechanism, and efficient anomaly detection coupled with a switching mechanism (which ensures slow but always correct solutions). This paradigm facilitates universal design of efficient multistage stable population protocols. Specifically, the first efficient parity and congruence protocols introduced here use both $O(\log^3 n)$ states and achieve silent stabilisation in $O(\log^3 n)$ time. We conclude by discussing the impact of implicit conversion between unary and binary representations enabled by the weight system, with applications to other problems, including the computation and representation of (sub-)population sizes.

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arXiv2025

Selective Population Protocols

The model of population protocols provides a universal platform to study distributed processes driven by pairwise interactions of anonymous agents. While population protocols present an elegant and robust model for randomized distributed computation, their efficiency wanes when tackling issues that require more focused communication or the execution of multiple processes. To address this issue, we propose a new, selective variant of population protocols by introducing a partition of the state space and the corresponding conditional selection of responders. We demonstrate on several examples that the new model offers a natural environment, complete with tools and a high-level description, to facilitate more efficient solutions. In particular, we provide fixed-state stable and efficient solutions to two central problems: leader election and majority computation, both with confirmation. This constitutes a separation result, as achieving stable and efficient majority computation requires $Ω(\log n)$ states in standard population protocols, even when the leader is already determined. Additionally, we explore the computation of the median using the comparison model, where the operational state space of agents is fixed, and the transition function determines the order between (arbitrarily large) hidden keys associated with interacting agents. Our findings reveal that the computation of the median of $n$ numbers requires $Ω(n)$ time. Moreover, we demonstrate that the problem can be solved in $O(n\log n)$ time, both in expectation and with high probability, in standard population protocols. In contrast, we establish that a feasible solution in selective population protocols can be achieved in $O(\log^4 n)$ time.

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arXiv2025

Anonymous Self-Stabilising Localisation via Spatial Population Protocols

In the distributed localization problem (DLP), $n$ anonymous robots (agents) $a_0, a_1, ..., a_{n-1}$ begin at arbitrary positions $p_0, ..., p_{n-1}$ in $S$, where $S$ is an Euclidean space. The primary goal in DLP is for agents to reach a consensus on a unified coordinate system that accurately reflects the relative positions of all points, $p_0, ..., p_{n-1}$. Extensive research on DLP has primarily focused on the feasibility and complexity of achieving consensus when agents have limited access to inter-agent distances, often due to missing or imprecise data. In this paper, however, we examine a minimalist, computationally efficient model of distributed computing in which agents have access to all pairwise distances, if needed. Specifically, we introduce a novel variant of population protocols, referred to as the spatial population protocols model. In this variant each agent can memorise one or a fixed number of coordinates, and when agents $a_i$ and $a_j$ interact, they can not only exchange their current knowledge but also either determine the distance $d(i,j)$ between them in $S$ (distance query model) or obtain the vector $v(i,j)$ spanning points $p_i$ and $p_j$ (vector query model). We propose several localisation protocols, including: (1) Two leader-based protocols with distance queries, stabilizing silently in $o(n)$ time using an efficient multi-contact epidemic, a generalization of the one-way epidemic in population protocols; (2) A distance-based protocol self-stabilizing silently in $O(n(\log n/n)^{1/(k+1)}\log n)$ time in $k$-dimensions, leveraging a leader election mechanism; (3) An optimally fast protocol with vector queries, self-stabilizing silently in $O(\log n)$ time.

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arXiv2025

Improving Efficiency in Near-State and State-Optimal Self-Stabilising Leader Election Population Protocols

We investigate leader election problem via ranking within self-stabilising population protocols. In this scenario, the agent's state space comprises $n$ rank states and $x$ extra states. The initial configuration of $n$ agents consists of arbitrary arrangements of rank and extra states, with the objective of self-ranking. Specifically, each agent is tasked with stabilising in a unique rank state silently, implying that after stabilisation, each agent remains in its designated state indefinitely. In this paper, we present several new self-stabilising ranking protocols, greatly enriching our comprehension of these intricate problems. All protocols ensure self-stabilisation time with high probability (whp), defined as $1-n^{-η},$ for a constant $η>0.$ We delve into three scenarios, from which we derive stable (always correct), either state-optimal or almost state-optimal, silent ranking protocols that self-stabilise within a time frame of $o(n^2)$ whp, including: - Utilising a novel concept of an agent trap, we derive a state-optimal ranking protocol that achieves self-stabilisation in time $O(min(kn^{3/2},n^2\log^2 n)),$ for any $k$-distant starting configuration. - Furthermore, we show that the incorporation of a single extra state ($x=1$) ensures a ranking protocol that self-stabilises in time $O(n^{7/4}\log^2 n)=o(n^2)$, regardless of the initial configuration. - Lastly, we show that extra $x=O(\log n)$ states admit self-stabilising ranking with the best currently known stabilisation time $O(n\log n)$, when whp and $x=O(\log n)$ guarantees are imposed.

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arXiv2024

Simple approximation algorithms for Polyamorous Scheduling

In Polyamorous Scheduling, we are given an edge-weighted graph and must find a periodic schedule of matchings in this graph which minimizes the maximal weighted waiting time between consecutive occurrences of the same edge. This NP-hard problem generalises Bamboo Garden Trimming and is motivated by the need to find schedules of pairwise meetings in a complex social group. We present two different analyses of an approximation algorithm based on the Reduce-Fastest heuristic, from which we obtain first a 6-approximation and then a 5.24-approximation for Polyamorous Scheduling. We also strengthen the extant proof that there is no polynomial-time $(1+δ)$-approximation algorithm for the Optimisation Polyamorous Scheduling problem for any $δ< \frac1{12}$ unless P = NP to the bipartite case. The decision version of Polyamorous Scheduling has a notion of density, similar to that of Pinwheel Scheduling, where problems with density below the threshold are guaranteed to admit a schedule (cf. the recently proven 5/6 conjecture, Kawamura, STOC 2024). We establish the existence of a similar threshold for Polyamorous Scheduling and give the first non-trivial bounds on the poly density threshold.

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arXiv2024

Perpetual maintenance of machines with different urgency requirements

A garden $G$ is populated by $n\ge 1$ bamboos $b_1, b_2, ..., b_n$ with the respective daily growth rates $h_1 \ge h_2 \ge \dots \ge h_n$. It is assumed that the initial heights of bamboos are zero. The robotic gardener maintaining the garden regularly attends bamboos and trims them to height zero according to some schedule. The Bamboo Garden Trimming Problem (BGT) is to design a perpetual schedule of cuts to maintain the elevation of the bamboo garden as low as possible. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. The objective is to design a perpetual schedule of servicing which minimizes the maximum (weighted) waiting time for servicing. We consider two variants of BGT. In discrete BGT the robot trims only one bamboo at the end of each day. In continuous BGT the bamboos can be cut at any time, however, the robot needs time to move from one bamboo to the next. For discrete BGT, we show tighter approximation algorithms for the case when the growth rates are balanced and for the general case. The former algorithm settles one of the conjectures about the Pinwheel problem. The general approximation algorithm improves on the previous best approximation ratio. For continuous BGT, we propose approximation algorithms which achieve approximation ratios $O(\log \lceil h_1/h_n\rceil)$ and $O(\log n)$.

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arXiv2024

Polyamorous Scheduling

Finding schedules for pairwise meetings between the members of a complex social group without creating interpersonal conflict is challenging, especially when different relationships have different needs. We formally define and study the underlying optimisation problem: Polyamorous Scheduling. In Polyamorous Scheduling, we are given an edge-weighted graph and try to find a periodic schedule of matchings in this graph such that the maximal weighted waiting time between consecutive occurrences of the same edge is minimised. We show that the problem is NP-hard and that there is no efficient approximation algorithm with a better ratio than 4/3 unless P = NP. On the positive side, we obtain an $O(\log n)$-approximation algorithm; indeed, a $O(\log Δ)$-approximation for $Δ$ the maximum degree, i.e., the largest number of relationships of any individual. We also define a generalisation of density from the Pinwheel Scheduling Problem, "poly density", and ask whether there exists a poly-density threshold similar to the 5/6-density threshold for Pinwheel Scheduling [Kawamura, STOC 2024]. Polyamorous Scheduling is a natural generalisation of Pinwheel Scheduling with respect to its optimisation variant, Bamboo Garden Trimming. Our work contributes the first nontrivial hardness-of-approximation reduction for any periodic scheduling problem, and opens up numerous avenues for further study of Polyamorous Scheduling.

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arXiv2022

A time and space optimal stable population protocol solving exact majority

We study population protocols, a model of distributed computing appropriate for modeling well-mixed chemical reaction networks and other physical systems where agents exchange information in pairwise interactions, but have no control over their schedule of interaction partners. The well-studied *majority* problem is that of determining in an initial population of $n$ agents, each with one of two opinions $A$ or $B$, whether there are more $A$, more $B$, or a tie. A *stable* protocol solves this problem with probability 1 by eventually entering a configuration in which all agents agree on a correct consensus decision of $\mathsf{A}$, $\mathsf{B}$, or $\mathsf{T}$, from which the consensus cannot change. We describe a protocol that solves this problem using $O(\log n)$ states ($\log \log n + O(1)$ bits of memory) and optimal expected time $O(\log n)$. The number of states $O(\log n)$ is known to be optimal for the class of polylogarithmic time stable protocols that are "output dominant" and "monotone". These are two natural constraints satisfied by our protocol, making it simultaneously time- and state-optimal for that class. We introduce a key technique called a "fixed resolution clock" to achieve partial synchronization. Our protocol is *nonuniform*: the transition function has the value $\left \lceil {\log n} \right \rceil$ encoded in it. We show that the protocol can be modified to be uniform, while increasing the state complexity to $Θ(\log n \log \log n)$.

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arXiv2021

Towards the 5/6-Density Conjecture of Pinwheel Scheduling

Pinwheel Scheduling aims to find a perpetual schedule for unit-length tasks on a single machine subject to given maximal time spans (a.k.a. frequencies) between any two consecutive executions of the same task. The density of a Pinwheel Scheduling instance is the sum of the inverses of these task frequencies; the 5/6-Conjecture (Chan and Chin, 1993) states that any Pinwheel Scheduling instance with density at most 5/6 is schedulable. We formalize the notion of Pareto surfaces for Pinwheel Scheduling and exploit novel structural insights to engineer an efficient algorithm for computing them. This allows us to (1) confirm the 5/6-Conjecture for all Pinwheel Scheduling instances with at most 12 tasks and (2) to prove that a given list of only 23 schedules solves all schedulable Pinwheel Scheduling instances with at most 5 tasks.

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arXiv2021

Time and Space Optimal Exact Majority Population Protocols

In this paper we study population protocols governed by the {\em random scheduler}, which uniformly at random selects pairwise interactions between $n$ agents. The main result of this paper is the first time and space optimal {\em exact majority population protocol} which also works with high probability. The new protocol operates in the optimal {\em parallel time} $O(\log n),$ which is equivalent to $O(n\log n)$ sequential {\em pairwise interactions}, where each agent utilises the optimal number of $O(\log n)$ states. The time optimality of the new majority protocol is possible thanks to the novel concept of fixed-resolution phase clocks introduced and analysed in this paper. The new phase clock allows to count approximately constant parallel time in population protocols.

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arXiv2018

Doing-it-All with Bounded Work and Communication

We consider the Do-All problem, where $p$ cooperating processors need to complete $t$ similar and independent tasks in an adversarial setting. Here we deal with a synchronous message passing system with processors that are subject to crash failures. Efficiency of algorithms in this setting is measured in terms of work complexity (also known as total available processor steps) and communication complexity (total number of point-to-point messages). When work and communication are considered to be comparable resources, then the overall efficiency is meaningfully expressed in terms of effort defined as work + communication. We develop and analyze a constructive algorithm that has work $O( t + p \log p\, (\sqrt{p\log p}+\sqrt{t\log t}\, ) )$ and a nonconstructive algorithm that has work $O(t +p \log^2 p)$. The latter result is close to the lower bound $Ω(t + p \log p/ \log \log p)$ on work. The effort of each of these algorithms is proportional to its work when the number of crashes is bounded above by $c\,p$, for some positive constant $c < 1$. We also present a nonconstructive algorithm that has effort $O(t + p ^{1.77})$.

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arXiv2018

Almost logarithmic-time space optimal leader election in population protocols

The model of population protocols refers to a large collection of simple indistinguishable entities, frequently called {\em agents}. The agents communicate and perform computation through pairwise interactions. We study fast and space efficient leader election in population of cardinality $n$ governed by a random scheduler, where during each time step the scheduler uniformly at random selects for interaction exactly one pair of agents. We propose the first $o(\log^2 n)$-time leader election protocol. Our solution operates in expected parallel time $O(\log n\log\log n)$ which is equivalent to $O(n \log n\log\log n)$ pairwise interactions. This is the fastest currently known leader election algorithm in which each agent utilises asymptotically optimal number of $O(\log\log n)$ states. The new protocol incorporates and amalgamates successfully the power of assorted {\em synthetic coins} with variable rate {\em phase clocks}.

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