Abstract
In this paper we consider a known variant of the standard population protocol model in which agents can be connected by edges, referred to as the network constructor model. During an interaction between two agents the relevant connecting edge can be formed, maintained or eliminated by the transition function. The state space of agents is fixed (constant size) and the size n of the population is not known, i.e., not hardcoded in the transition function.
Since pairs of agents are chosen uniformly at random the status of each edge is updated every Θ(n²) interactions in expectation which coincides with Θ(n) parallel time. This phenomenon provides a natural lower bound on the time complexity for any nontrivial network construction designed for this variant. This is in contrast with the standard population protocol model in which efficient protocols operate in O(polylog n) parallel time.
The main focus in this paper is on efficient manipulation of linear structures including formation, selfreplication and distribution (including pipelining) of complex information in the adopted model.
 We propose and analyse a novel edge based phase clock counting parallel time Θ(nlog n) in the network constructor model, showing also that its leader based counterpart provides the same time guaranties in the standard population protocol model. Note that all currently known phase clocks can count parallel time not exceeding O(polylog n).
 The new clock enables a nearly optimal O(nlog n) parallel time spanning line construction (a key component of universal network construction), which improves dramatically on the best currently known O(n²) parallel time protocol, solving the main open problem in the considered model [O. Michail and P. Spirakis, 2016].
 We propose a new probabilistic bubblesort algorithm in which random comparisons and transfers are allowed only between the adjacent positions in the sequence. Utilising a novel potential function reasoning we show that rather surprisingly this probabilistic sorting (via conditional pipelining) procedure requires O(n²) comparisons in expectation and whp, and is on par with its deterministic counterpart.
 We propose the first population protocol allowing selfreplication of a strand of an arbitrary length k (carrying a kbit message of size independent of the state space) in parallel time O(n(k+log n)). The pipelining mechanism and the time complexity analysis of the strand selfreplication protocol mimic those used in the probabilistic bubblesort. The new protocol permits also simultaneous selfreplication, where l copies of the strand can be created in time O(n(k+log n)log l). Finally, we discuss application of the strand selfreplication protocol to pattern matching. Our protocols are always correct and provide time guaranties with high probability defined as 1n^{η}, for a constant η > 0.
BibTeX  Entry
@InProceedings{gasieniec_et_al:LIPIcs.DISC.2022.44,
author = {G\k{a}sieniec, Leszek and Spirakis, Paul and Stachowiak, Grzegorz},
title = {{Brief Announcement: New Clocks, Fast Line Formation and SelfReplication Population Protocols}},
booktitle = {36th International Symposium on Distributed Computing (DISC 2022)},
pages = {44:144:3},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772556},
ISSN = {18688969},
year = {2022},
volume = {246},
editor = {Scheideler, Christian},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17235},
URN = {urn:nbn:de:0030drops172351},
doi = {10.4230/LIPIcs.DISC.2022.44},
annote = {Keywords: Population protocols, network constructors, probabilistic bubblesort, selfreplication}
}
Keywords: 

Population protocols, network constructors, probabilistic bubblesort, selfreplication 
Collection: 

36th International Symposium on Distributed Computing (DISC 2022) 
Issue Date: 

2022 
Date of publication: 

17.10.2022 