arXiv subjects

Thorsten Götte

Publications and source records attributed to Thorsten Götte.

13 recordsLinked to original sources

arXiv2026

Counting in Population Protocols on Graphs

We consider the problem of counting the number of agents in a population protocol where the agents are connected by an underlying graph $G=(V,E)$ with $|V|=n$ nodes. In each step, a random scheduler selects an edge uniformly at random, and the incident nodes make a state transition. As per standard assumptions, agents are identical and anonymous, that is, have no identifiers. To break symmetry, in each interaction one of the agents is declared as the initiator uniformly at random. Our size counting protocol uses $\tilde O(n)$ states and stabilizes in $O( B(G) \cdot \log^2(n) + L(G) \cdot \log(n))$ interactions with high probability, where $B(G)$ is the broadcast time and $L(G)$ is the load balancing time. Our protocol is based on novel protocols for sampling independent random bits (given that the scheduler determines an initiator and responder) and approximating $\log n$ up to an additive error of $O(\log \log n)$ with high probability. The latter uses $O(poly\log(n))$ states and $O(B(G)\cdot\log^2 n)$ interactions. Both results may be of independent interest. The main protocol for exact counting requires the presence of a unique leader, the other two do not. None of the protocols requires any knowledge about the graph $G$. We conclude with impossibility results for terminating uniform population protocols that compute graph-size properties (like counting nodes or determining parity) with and without a leader.

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arXiv2026

Discrete Incremental Voting: New Bounds for General Graphs and Expanders

We analyze the discrete incremental voting process (DIV) introduced by Cooper, Radzik, and Shiraga [OPODIS '23]. In this process, we consider a set $V$ of $n$ nodes connected in an undirected graph $G = (V, E)$ where each node has an integer opinion. In one step a randomly selected node interacts with its randomly selected neighbor and changes its opinion by $1$ in the direction of the neighbour's opinion. The process converges to a unique opinion that, in expectation, is the degree-weighted average of the initial opinions. We show that if the graph has conductance $Φ(G)$, the ratio of the average to smallest degree is $γ(G)$, and the maximal difference between initial opinions is $K$, then the expected convergence time is ${O}\left({n\left(K\log (Kn)+γ(G) n \right)}/{Φ(G)^2}\right)$. This bound is essentially optimal for a large class of graphs of bounded expansion. We also show that for regular graphs, if the second largest eigenvalue is $o(1/\log^2 n)$ and $K$ is $o\left({n}/{\log^2 n}\right)$, then w.h.p.\ DIV converges to the initial average opinion (rounded up or down).

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arXiv2025

A Space-Time Trade-off for Fast Self-Stabilizing Leader Election in Population Protocols

We consider the problem of self-stabilizing leader election in the population model by Angluin, Aspnes, Diamadi, Fischer, and Peralta (JDistComp '06). The population model is a well-established and powerful model for asynchronous, distributed computation with a large number of applications. For self-stabilizing leader election, the population of $n$ anonymous agents, interacting in uniformly random pairs, must stabilize with a single leader from any possible initial configuration. The focus of this paper is to develop time-efficient self-stabilizing protocols whilst minimizing the number of states. We present a parametrized protocol, which, for a suitable setting, achieves the asymptotically optimal time $O(\log n)$ using $2^{O(n^2\log n)}$ states (throughout the paper, ``time'' refers to ``parallel time'', i.e., the number of pairwise interactions divided by $n$). This is a significant improvement over the previously best protocol Sublinear-Time-SSR due to Burman, Chen, Chen, Doty, Nowak, Severson, and Xu (PODC '21), which requires $2^{O(n^{\log n}\log n)}$ states for the same time bound. In general, for $1\le r\le n/2$, our protocol requires $2^{O(r^2\log{n})}$ states and stabilizes in time $O((n\log{n})/r)$, w.h.p.; the above result is achieved for $r=Θ(n)$. For $r=\log^2n$ our protocol requires only sub-linear time using only $2^{O(\log^3 n)}$ states, resolving an open problem stated in that paper. Sublinear-Time-SSR requires $O(\log n\cdot n^{1/(H+1)})$ time using $2^{Θ(n^H) \cdot \log n}$ states for all $1\le H\leΘ(\log n)$. Similar to previous works, it solves leader election by assigning a unique rank from $1$ through $n$ to each agent. The principal bottleneck for self-stabilizing ranking usually is to detect if there exist agents with the same rank. One of our main conceptual contributions is a novel technique for collision detection.

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arXiv2025

Silent Self-Stabilizing Ranking: Time Optimal and Space Efficient

We present a silent, self-stabilizing ranking protocol for the population protocol model of distributed computing, where agents interact in randomly chosen pairs to solve a common task. We are given $n$ anonymous agents, and the goal is to assign each agent a unique rank in $\{1, \dots, n\}$. Given unique ranks, it is straightforward to select a designated leader. Thus, our protocol is a self-stabilizing leader election protocol as well. Ranking requires at least $n$ states per agent; hence, the goal is to minimize the additional number of states, called overhead states. The core of our protocol is a space-efficient but non-self-stabilizing ranking protocol that requires only $n + O(\log n)$ states. Our protocol stabilizes in $O(n^2\log n)$ interactions w.h.p.\ and in expectation, using $n + O(\log^2 n)$ states in total. Our stabilization time is asymptotically optimal (see Burman et al., PODC'21). In comparison to the currently best known ranking protocol by Burman et al., which requires $n + Ω(n)$ states, our result exponentially improves the number of overhead states.

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arXiv2025

WalkSAT is linear on random 2-SAT

In an influential article Papadimitriou [FOCS 1991] proved that a local search algorithm called WalkSAT finds a satisfying assignment of a satisfiable 2-CNF with $n$ variables in $O(n^2)$ expected time. Variants of the WalkSAT algorithm have become a mainstay of practical SAT solving (e.g., [Hoos and Stützle 2000]). In the present article we analyse the expected running time of WalkSAT on random 2-SAT instances. Answering a question raised by Alekhnovich and Ben-Sasson [SICOMP 2007], we show that WalkSAT runs in linear expected time for all clause/variable densities up to the random 2-SAT satisfiability threshold.

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arXiv2024

Distributed And Parallel Low-Diameter Decompositions for Arbitrary and Restricted Graphs

We consider the distributed and parallel construction of low-diameter decompositions with strong diameter for (weighted) graphs and (weighted) graphs that can be separated through $k \in \tilde{O}(1)$ shortest paths. This class of graphs includes planar graphs, graphs of bounded treewidth, and graphs that exclude a fixed minor $K_r$. We present algorithms in the PRAM, CONGEST, and the novel HYBRID communication model that are competitive in all relevant parameters. Given $\mathcal{D} > 0$, our low-diameter decomposition algorithm divides the graph into connected clusters of strong diameter $\mathcal{D}$. For a arbitrary graph, an edge $e \in E$ of length $\ell_e$ is cut between two clusters with probability $O(\frac{\ell_e\cdot\log(n)}{\mathcal{D} })$. If the graph can be separated by $k \in \tilde{O}(1)$ paths, the probability improves to $O(\frac{\ell_e\cdot\log \log n}{\mathcal{D} })$. In either case, the decompositions can be computed in $\tilde{O}(1)$ depth and $\tilde{O}(kn)$ work in the PRAM and $\tilde{O}(1)$ time in the HYBRID model. In CONGEST, the runtimes are $\tilde{O}(HD + \sqrt{n})$ and $\tilde{O}(HD)$ respectively. All these results hold w.h.p. Broadly speaking, we present distributed and parallel implementations of sequential divide-and-conquer algorithms where we replace exact shortest paths with approximate shortest paths. In contrast to exact paths, these can be efficiently computed in the distributed and parallel setting [STOC '22]. Further, and perhaps more importantly, we show that instead of explicitly computing vertex-separators to enable efficient parallelization of these algorithms, it suffices to sample a few random paths of bounded length and the nodes close to them. Thereby, we do not require complex embeddings whose implementation is unknown in the distributed and parallel setting.

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arXiv2023

Distributed Construction of Near-Optimal Compact Routing Schemes for Planar Graphs

We consider the problem of computing compact routing tables for a (weighted) planar graph $G:= (V, E,w)$ in the PRAM, CONGEST, and the novel HYBRID communication model. We present algorithms with polylogarithmic work and communication that are almost optimal in all relevant parameters, i.e., computation time, table sizes, and stretch. All algorithms are heavily randomized, and all our bounds hold w.h.p. For a given parameter $ε>0$, our scheme computes labels of size $\widetilde{O}(ε^{-1})$ and is computed in $\widetilde{O}(ε^{-2})$ time and $\widetilde{O}(n)$ work in the PRAM and the HYBRID model and $\widetilde{O}(ε^{-2} \cdot HD)$ (Here, $HD$ denotes the network's hop-diameter) time in CONGEST. The stretch of the resulting routing scheme is $1+ε$. To achieve these results, we extend the divide-and-conquer framework of Li and Parter [STOC '19] and combine it with state-of-the-art distributed distance approximation algorithms [STOC '22]. Furthermore, we provide a distributed decomposition scheme, which may be of independent interest.

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arXiv2021

The Max-Line-Formation Problem

We consider n robots with limited visibility: each robot can observe other robots only up to a constant distance denoted as the viewing range. The robots operate in discrete rounds that are either fully synchronous (FSync) or semi-synchronized (SSync). Most previously studied formation problems in this setting seek to bring the robots closer together (e.g., Gathering or Chain-Formation). In this work, we introduce the Max-Line-Formation problem, which has a contrary goal: to arrange the robots on a straight line of maximal length. First, we prove that the problem is impossible to solve by robots with a constant sized circular viewing range. The impossibility holds under comparably strong assumptions: robots that agree on both axes of their local coordinate systems in FSync. On the positive side, we show that the problem is solvable by robots with a constant square viewing range, i.e., the robots can observe other robots that lie within a constant-sized square centered at their position. In this case, the robots need to agree on only one axis of their local coordinate systems. We derive two algorithms: the first algorithm considers oblivious robots and converges to the optimal configuration in time $\mathcal{O}(n^2 \cdot \log (n/\varepsilon))$ under the SSync scheduler. The other algorithm makes use of locally visible lights (LUMI). It is designed for the FSync scheduler and can solve the problem exactly in optimal time $Θ(n)$. Afterward, we show that both the algorithmic and the analysis techniques can also be applied to the Gathering and Chain-Formation problem: we introduce an algorithm with a reduced viewing range for Gathering and give new and improved runtime bounds for the Chain-Formation problem.

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arXiv2021

Beep-And-Sleep: Message and Energy Efficient Set Cover

We observe message-efficient distributed algorithms for the Set Cover problem. Given a ground set $U$ of $n$ elements and $m$ subsets of $U$, we aim to find the minimal number of these subsets that contain all elements. In the default distributed setup of this problem, each set has a bidirected communication link with each element it contains. Our first result is a $\tilde{O}(\log^2(Δ))$-time and $O(\sqrt{Δ)}(n+m))$-message algorithm with expected approximation ration of $O(\log(Δ))$ in the $KT_0$ model. The value $Δ$ denotes the maximal cardinality of each subset. Our algorithm is \emph{almost} optimal with regard to time and message complexity. Further, we present Set Cover algorithm in the Beeping model that only relies on carrier-sensing and can trade runtime for approximation ratio similar to the celebrated algorithm by Kuhn and Wattenhofer [PODC '03].

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arXiv2021

Time-Optimal Construction of Overlay Networks

We show how to construct an overlay network of constant degree and diameter $O(\log n)$ in time $O(\log n)$ starting from an arbitrary weakly connected graph. We assume a synchronous communication network in which nodes can send messages to nodes they know the identifier of, and new connections can be established by sending node identifiers. If the initial network's graph is weakly connected and has constant degree, then our algorithm constructs the desired topology with each node sending and receiving only $O(\log n)$ messages in each round in time $O(\log n)$, w.h.p., which beats the currently best $O(\log^{3/2} n)$ time algorithm of [Götte et al., SIROCCO'19]. Since the problem cannot be solved faster than by using pointer jumping for $O(\log n)$ rounds (which would even require each node to communicate $Ω(n)$ bits), our algorithm is asymptotically optimal. We achieve this speedup by using short random walks to repeatedly establish random connections between the nodes that quickly reduce the conductance of the graph using an observation of [Kwok and Lau, APPROX'14]. Additionally, we show how our algorithm can be used to efficiently solve graph problems in \emph{hybrid networks} [Augustine et al., SODA'20]. Motivated by the idea that nodes possess two different modes of communication, we assume that communication of the \emph{initial} edges is unrestricted, whereas only polylogarithmically many messages can be communicated over edges that have been established throughout an algorithm's execution. For an (undirected) graph $G$ with arbitrary degree, we show how to compute connected components, a spanning tree, and biconnected components in time $O(\log n)$, w.h.p. Furthermore, we show how to compute an MIS in time $O(\log d + \log \log n)$, w.h.p., where $d$ is the initial degree of $G$.

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arXiv2021

Always be Two Steps Ahead of Your Enemy

We investigate the maintenance of overlay networks under massive churn, i.e. nodes joining and leaving the network. We assume an adversary that may churn a constant fraction $αn$ of nodes over the course of $\mathcal{O}(\log n)$ rounds. In particular, the adversary has an almost up-to-date information of the network topology as it can observe an only slightly outdated topology that is at least $2$ rounds old. Other than that, we only have the provably minimal restriction that new nodes can only join the network via nodes that have taken part in the network for at least one round. Our contributions are as follows: First, we show that it is impossible to maintain a connected topology if adversary has up-to-date information about the nodes' connections. Further, we show that our restriction concerning the join is also necessary. As our main result present an algorithm that constructs a new overlay -- completely independent of all previous overlays -- every $2$ rounds. Furthermore, each node sends and receives only $\mathcal{O}(\log^3 n)$ messages each round. As part of our solution we propose the Linearized DeBruijn Swarm (LDS), a highly churn resistant overlay, which will be maintained by the algorithm. However, our approaches can be transferred to a variety of classical P2P Topologies where nodes are mapped into the $[0,1)$-interval.

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arXiv2019

A Loosely Self-stabilizing Protocol for Randomized Congestion Control with Logarithmic Memory

We consider congestion control in peer-to-peer distributed systems. The problem can be reduced to the following scenario: Consider a set $V$ of $n$ peers (called clients in this paper) that want to send messages to a fixed common peer (called server in this paper). We assume that each client $v \in V$ sends a message with probability $p(v) \in [0,1)$ and the server has a capacity of $σ\in \mathbb{N}$, i.e., it can recieve at most $σ$ messages per round and excess messages are dropped. The server can modify these probabilities when clients send messages. Ideally, we wish to converge to a state with $\sum p(v) = σ$ and $p(v) = p(w)$ for all $v,w \in V$. We propose a loosely self-stabilizing protocol with a slightly relaxed legimate state. Our protocol lets the system converge from any initial state to a state where $\sum p(v) \in \left[σ\pm ε\right]$ and $|p(v)-p(w)| \in O(\frac{1}{n})$. This property is then maintained for $Ω(n^{\mathfrak{c}})$ rounds in expectation. In particular, the initial client probabilities and server variables are not necessarily well-defined, i.e., they may have arbitrary values. Our protocol uses only $O(W + \log n)$ bits of memory where $W$ is length of node identifers, making it very lightweight. Finally we state a lower bound on the convergence time an see that our protocol performs asymptotically optimal (up to some polylogarithmic factor).

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arXiv2018

On Underlay-Aware Self-Stabilizing Overlay Networks

We present a self-stabilizing protocol for an overlay network that constructs the Minimum Spanning Tree (MST) for an underlay that is modeled by a weighted tree. The weight of an overlay edge between two nodes is the weighted length of their shortest path in the tree. We rigorously prove that our protocol works correctly under asynchronous and non-FIFO message delivery. Further, the protocol stabilizes after $\mathcal{O}(N^2)$ asynchronous rounds where $N$ is the number of nodes in the overlay.

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