Structural information and communication complexity

Beating this bound is thefirst open problem we highlight: Open problem 1:Is there an algorithm that can solve the single-source shortest pathsor simply compute the distance between two gi

Jukka Suomela (Ed.)

123

23rd International Colloquium, SIROCCO 2016

Helsinki, Finland, July 19–21, 2016

Revised Selected Papers

Structural Information and Communication Complexity

Commenced Publication in 1973

Founding and Former Series Editors:

Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Jukka Suomela

Aalto University

Espoo

Finland

ISSN 0302-9743 ISSN 1611-3349 (electronic)

Lecture Notes in Computer Science

ISBN 978-3-319-48313-9 ISBN 978-3-319-48314-6 (eBook)

DOI 10.1007/978-3-319-48314-6

Library of Congress Control Number: 2016955506

LNCS Sublibrary: SL1 – Theoretical Computer Science and General Issues

© Springer International Publishing AG 2016

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro films or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a speci fic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

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This volume contains the papers presented at SIROCCO 2016, the 23rd InternationalColloquium on Structural Information and Communication Complexity, held duringJuly 19–21, 2016, in Helsinki, Finland.

This year we received 50 submissions in response to the call for papers Eachsubmission was reviewed by at least three reviewers; we had a total of 18 ProgramCommittee members and 57 external reviewers The Program Committee decided toaccept 25 papers: 24 normal papers and one survey-track paper Fabian Kuhn, YannicMaus, and Sebastian Daum received the SIROCCO 2016 Best Paper Award for theirwork“Rumor Spreading with Bounded In-Degree.” Selected papers will also be invited

to a special issue of the Theoretical Computer Science journal

In addition to the 25 contributed talks, the conference program included a keynotelecture by Yoram Moses, invited talks by Keren Censor-Hillel, Adrian Kosowski,Danupon Nanongkai, and Thomas Sauerwald, and the award lecture by Masafumi(Mark) Yamashita, the recipient of the 2016 SIROCCO Prize for Innovation in Dis-tributed Computing

I would like to thank all authors for their high-quality submissions and all speakersfor their excellent talks I am grateful to the Program Committee and all externalreviewers for their efforts in putting together a great conference program, to the SteeringCommittee chaired by Andrzej Pelc for their help and support, and to everyone who wasinvolved in the local organization for making it possible to have SIROCCO 2016 insunny Helsinki

Finally, I would like to thank our sponsors for their support: the Federation ofFinnish Learned Societies, Helsinki Institute for Information Technology HIIT, andHelsinki Doctoral Education Network in Information and Communications Technology(HICT) provided financial support, Springer not only helped with the publication

of these proceedings but also sponsored the best paper award, Aalto University vided administrative support and helped with the conference venue, and EasyChairkindly provided a free platform for managing paper submissions and the production ofthis volume

Program Committee

Leonid Barenboim Open University of Israel

Jérémie Chalopin LIF, CNRS and Aix Marseille Université, France

Paola Flocchini University of Ottawa, Canada

Pierre Fraigniaud CNRS and Université Paris Diderot, France

Janne H Korhonen Reykjavik University, Iceland

Evangelos Kranakis Carleton University, Canada

Christoph Lenzen MPI for Informatics, Germany

Friedhelm Meyer auf

Gopal Pandurangan University of Houston, USA

Peter Robinson Queen’s University Belfast, UK

Thomas Sauerwald University of Cambridge, UK

Stefan Schmid Aalborg University, Denmark

Jukka Suomela Aalto University, Finland

Przemysƚaw Uznański ETH Zürich, Switzerland

Jung, DanielKarousatou, ChristinaKling, Peter

Konrad, ChristianKonwar, KishoriKuszner, LukaszKuznetsov, Petr

Su, Hsin-HaoTonoyan, TigranTrehan, ChhayaTschager, ThomasYamauchi, Yukiko

Yu, Haifeng

It is a pleasure to award the 2016 SIROCCO Prize for Innovation in distributedcomputing to Masafumi (Mark) Yamashita Mark has presented many original ideasand important results that have enriched the theoretical computer science communityand the distributed computing community, such as his seminal work “Computing onAnonymous Networks” (with T Kameda), which introduced the notion of “view” andhas inspired all the subsequent investigations on computability in anonymousnetworks, as well as his work on coteries, on self-stabilization, and on polling games,among others.

The prize is awarded for his lifetime achievements, but especially for introducing thecomputational universe of autonomous mobile robots to the algorithmic community and

to the distributed community in particular This has opened a new and exciting researcharea that has now become an accepted mainstream topic in theoretical computer science(papers on“mobile robots” now appear in all major theory conferences and journals)and clearly in distributed computing The fascinating new area of research it opened isnow under investigation by many groups worldwide

The introduction of this area to the theory community was actually made in hisSIROCCO paper [1] The full version was then published in the SIAM Journal onComputing [2] (This paper currently has more than 500 citations.)

The paper deals with the problem of coordination among autonomous robotsmoving on a plane This and subsequent papers on this topic provided the firstindications about which tasks can be accomplished using multiple deterministic,autonomous, and identical robots in a collaborative manner The formal model formobile robots introduced in the paper (called the Suzuki–Yamashita or SYM model)provides a nice abstraction that makes it easy to analyze algorithms but still capturesmany of the difficulties of coordination between the robots Many of the recent results ondistributed robotics are based on either this model or extensions of it The paperprovided the characterization (in terms of geometric pattern formation) of all tasks thatcan be performed by such teams of deterministic robots and provided some fundamentalimpossibility results including the impossibility of gathering two oblivious robots

A more recent work [3] extends the characterization to the model where robots arememory-less, thus showing the exact difference between oblivious robots and robotshaving memory

The 2015 Award Committee1:

Thomas Moscibroda (Microsoft)

Guy Even (Tel Aviv University)

Magnús Halldórsson (Reykjavik University)

Shay Kutten (Technion)– Chair

Andrzej Pelc (Université du Québec en Outaouais)

1 We wish to thank the nominators for the nomination and for contributing greatly to this text.

Selected Publications Related to Masafumi (Mark) Yamashita’s Contribution:

1 Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots In: Proceedings

of the 3rd International Colloquium on Structural Information and CommunicationComplexity, Siena, Italy, 6–8 June, pp 313–330 (1996)

2 Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots SIAM J Comput.28(4), 1347–1363 (1999)

3 Yamashita, M., Suzuki, I.: Characterizing geometric patterns formable by obliviousanonymous mobile robots Theor Comput Sci 411(26–28), 2433–2453 (2010)

4 Dumitrescu, A., Suzuki, I., Yamashita, M.: Motion planning for metamorphicsystems: feasibility, decidability, and distributed reconfiguration IEEE Trans.Robot 20(3), 409–418 (2004)

5 Souissi, S., Defago, X., Yamashita, M.: Using eventually consistent compasses togather memory-less mobile robots with limited visibility ACM Trans Auton.Adapt Syst 4(1), #9 (2009)

6 Das, S., Flocchini, P., Santoro, N., Yamashita, M.: Forming sequences of geometricpatterns with oblivious mobile robots Distrib Comput 28(2), 131–145 (2015)

7 Fujinaga, N., Yamauchi, Y., Ono, H., Shuji, K., Yamashita, M.: Pattern formation

by oblivious asynchronous mobile robots SIAM J Comput 44(3), 740–785 (2015)

of solving distributed problems.

Lamport and Lynch [1] claimed “although one usually speak of a distributedsystem, it is more accurate to speak of a distributed view of a system,” after defining theword distributed to mean spread across space This claim seems to imply that everysystem is a distributed system at least from the view of atoms or molecules, and may be

in the face of solving a distributed problem, when we concentrate on the distributedview, like seabirds and cars in the example above

An abstract distributed view, which we call a formal distributed system (FDS),describes how system elements interact logically Our final goal is to understand avariety of FDSs and compare them in terms of the solvability of distributed problems

Wefirst propose a candidate for the model of FDS in such a way that it can describe

a wide variety of FDSs, and explain that many of the models of distributed systems(including ones suitable to describe biological systems) can be described as FDSs.Compared with other distributed system models, FDSs have two features: First, thesystem elements are modeled by points in d-dimensional space, where d can be greaterthan 3 Second incomputable functions can be taken as transition functions (corre-sponding to distributed algorithms)

We next explain some of our ongoing works in three research areas, localization,symmetry breaking and self-organization In localization, we discuss the simplestproblem of locating a single element with limited visibility to the center of a linesegment In symmetry breaking, we observe how elements in 3D space can eliminatesome symmetries Finally in self-organization, we examine why natural systems appear

to have richer autonomous properties than artificial systems, despite that the latterwould have stronger interaction mechanisms, e.g., unique identifiers, memory, syn-chrony, and so on

1 Lamport, L., Lynch, N.: Distributed computing: models and methods, In: van Leeuwen, J (ed.)Handbook of Theoretical Computer Science Formal Models and Semantics, Chap 18, vol B,

pp 1157–1199 MIT Press/Elsevier (1990)

A Principled Way of Designing Ef ficient

by‘Ki’ KoP thus states that if u is a necessary condition for i performing α, then Kiu isalso a necessary condition for i performingα Thus, for example, a process the enters thecritical section (CS) in a mutual exclusion protocol must know that the CS is empty when

it enters Similarly, if an ATM must only provide cash to a customer that has a sufficientpositive balance, then the ATM must know that the customer has such a balance.The talk illustrates the design of an unbeatable protocol for Consensus based on theKoP, along the lines of [1] Based on the Validity property in the specification InConsensus, a process can decide 0 only if some initial value is 0 The KoP immediatelyimplies that following every correct protocol for Consensus, a process must know of aninitial value of 0 when it decides 0 We consider binary Consensus, in which values are

0 or 1 We seek the optimal rule for deciding 1 in a protocol in which deciding on 0 isfavored, by having every process that knows of a 0 decide 0 The Agreement property

of Consensus implies that a process cannot decide 1 at a point when other processesdecide 0 It follows by KoP that a process that decides 1 must know that nobody isdeciding 0 In particular, it must know that no active process knows of a 0 A com-binatorial analysis of when a process knows that nobody knows of a 0 is performed,yielding a natural condition that can be easily computed The outcome is an elegant and

efficient protocol that strictly dominates all known protocols for Consensus in thesynchronous crash-failure model, which cannot be strictly dominated

A video of a similar invited talk given in February 2016 appears in IHP talk

1 Castañeda, A., Gonczarowski, Y.A., Moses, Y.: Unbeatable consensus In: Kuhn, F (ed.)DISC 2014 LNCS, vol 8784, pp 91–106 Springer, Heidelberg (2014) Full versionavailable on arXiv

2 Moses,Y.: Relating knowledge and coordinated action: the knowledge of preconditionsprinciple In: Proceedings of the 15th Conference on Theoretical Aspects of Rationality andKnowledge, pp 207–216 (2015)

Invited Talks

for the Congest Model

D rounds suffice for solving these problems by gathering all information at a singlenode and solving the problem on its local processor, the Congest model imposesadditional bandwidth restrictions, making such problems harder Below we discusssome known lower bounds for global problems in Congest, glimpse into some newresults, and discuss open questions

Computing the Diameter One of the lead examples of a global graph problem is that ofcomputing the diameter In the Congest model, the diameter can be computed inO(n) rounds [7, 9], and a beautiful lower bound of Xðn= log nÞ, which we describenext, is known even for small values of D [5, 7]

In a nutshell, the lower bound is obtained through a reduction from the wellknown2-party communication complexity problem of set-disjointness, in which Alice andBob receive input vectors
x;
y of length k, respectively, and need to output whetherthere is an index 1 ≤ i ≤ k for which xi = yi = 1 The reduction is obtained byconstructing a graph of n nodes, with two sets of nodes that are connected by acomplete matching and some additional edges within each set Alice and Bob are eachresponsible for one of the two sets, in terms of simulating the distributed algorithm forthe nodes within that set Any message that needs to be sent within a set is simulatedlocally, and communication is only needed for messages that cross the cut between thetwo sets

The crux is that Alice and Bob add edges within their sets according to their inputvectors, where a 0 input for index i corresponds to adding the corresponding edge This

is done in a way that promises that the diameter of the resulting graph determines theanswer to the set-disjointness problem The parameters are taken such that k =Θ(n2),and since set-disjointness is known to requireΩ(k) bits of communication, and the size

of the cut between the two sets of nodes is of sizeΘ(n) and the message size is of log

n bits, the end result is a lower bound ofXðn= log nÞ rounds

Keren Censor-Hillel —Supported in part by the Israel Science Foundation (grant 1696/14).

Recently, Abboud et al [1] introduce a new construction that allows obtaining asimilar near-linear lower bound for computing the diameter The main technical con-tribution is a bit-gadget, which allows the cut between the sets of Alice and Bob to be

of size onlyΘ(log n) and allows taking k = Θ(n), giving a lower bound of Xðn= log2 nÞ.While this is worse than the previously mentioned bound by a logarithmic factor, thestrength of the bit-gadget is in reducing the size of the cut and having a sparse con-struction, which then allows improving the state-of-the-art for additional problems: Itgives thefirst near-linear lower bounds for a ð3=2  Þ-approximation for the diameter,for computing or approximating the radius, for approximating all eccentricities, and forverifying certain types of spanners These can also be made to work for constant degreegraphs

Constructing a Minimum Spanning Tree (MST) To exemplify another type of lowerbounds for Congest that uses set-disjointness albeit in a different manner, consider theproblem offinding an MST

We next describe the key idea of theXðpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin=log nþ DÞ-round lower bound of [11].This bound is given for the problem of subgraph connectivity, which can be easily beshown to reduce tofinding an MST A base graph is given and some of its edges aremarked to be in the subgraph H, according to the inputs of Alice and Bob It is shownthat H is connected iff the inputs are not disjoint To simulate the required distributedalgorithm, Alice and Bob need to exchange information on certain edges of the graph

in a dynamic way That is, there is no static partition of the nodes between the 2 playerswhich makes the complexity depend on the size of the cut, but rather the assignment

of the nodes to be simulated changes from round to round and is not a partition Thus,while the cut between Alice and Bob’s nodes in each round is large, the used cut isO(log n), and choosing k¼ Oðn1 =2Þ gives almost the claimed lower bound (for ease ofdescription, this is a slightly weakened simplification of the lower bound) In ourcontext, the interesting thing here is that although this is also a reduction from set-disjointness, the framework is entirely different from the distance computation lowerbounds

Constructing Additive Spanners Recently, another type of Congest lower bounds hasbeen introduced, for constructing additive spanners Previous work obtains variousspanners in the Congest model [2, 3, 10], and a lower bound ofΩ(D) is given in [10]

A +β-pairwise spanner of G is a subgraph S for which, given P  V, for every

u; v 2 P, it holds that dSðu; vÞ  dGðu; vÞ þ b In addition to algorithms for purelyadditive spanners, [4] give lower bounds, of which we describe theX p=n log nð Þ lowerbound for constructing (+2)-pairwise spanners with Pj j ¼ p Consider here p ¼ n3 =2.

Define the (p, m)-partial-complement problem as follows Alice receives a set x of

p elements in f1; ; mg and Bob needs to output a set y of m / 2 elements in

spanner, Bob must know that the corresponding pair on Alice’s side is not in P,otherwise its removal increases the distance between these nodes from 3 to 7, violatingthe +2 stretch requirement Since Bob must removeΘ(n3/2

) edges, this implies solvingthe (p, m)-partial-complement problem, hence requiresX p=n log nð Þ rounds This gives

a lower bound of a new flavor, where the graph is known to both players, and theuncertainty only comes from the unknown set of pairs

Discussion There are many additional lower bounds that are not described here.Many specific questions are still open in the above various settings and problems.One example is that, while our lower bounds for distance computations apply to sparsegraphs, they are far from being planar It is known that an MST can be computed inO(D log D) rounds in planar graphs [6], which raises the question of whether distancecomputations can be performed faster than the general lower bound as well Specifi-cally, can the diameter of planar graphs be computed in o(n/polylog n) rounds?

References

1 Abboud, A., Censor-Hillel, K., Khoury, S.: Near-linear lower bounds for distributed distancecomputations, even in sparse networks In: Gavoille, C., Ilcinkas, D (eds.) DISC 2016.LNCS, vol 9888, pp 29–42 Springer, Heidelberg (2016)

2 Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: Additive spanners and (alpha, spanners ACM Trans Algorithms 7(1), 5 (2010)

beta)-3 Baswana, S., Sen, S.: A simple and linear time randomized algorithm for computing sparsespanners in weighted graphs Random Struct Algorithms 30(4), 532–563 (2007)

4 Censor-Hillel, K., Kavitha, T., Paz, A., Yehudayoff, A.: Distributed construction ofpurelyadditive spanners In: Gavoille, C., Ilcinkas, D (eds.) DISC 2016 LNCS, vol 9888,

Truly Local?

Adrian Kosowski

Inria and IRIF, CNRS— Université Paris Diderot, 75013 Paris, France

adrian.kosowski@inria.frAbstract In this talk we attempt to identify the characteristics of a task ofdistributed network computing, which make it easy (or hard) to solve by means

of fast local algorithms We look at specific combinatorial tasks within theLOCAL

model of distributed computation, and rephrase some recent algorithmic results

in a framework of constraint satisfaction Finally, we discuss the issue of efcient computability for relaxed variants of the LOCAL model, involving theso-called non-signaling property

fi-In distributed network computing, autonomous computational entities are represented

by the nodes of an undirected system graph, and exchange information by sendingmessages along its edges A major line of research in this area concerns the notion oflocality, and asks how much information about its neighborhood a node needs tocollect in order to solve a given computational task In particular, in the seminalLOCAL

model [19], the complexity of a distributed algorithm is measured in term of number ofrounds, where in each round all nodes synchronously exchange data along networklinks, and subsequently perform individual computations A t-round algorithm is thusone in which every node exchanges data with nodes at distance at most t (i.e., at most

t hops away) from it

Arguably, the most important class of local computational tasks concerns symmetrybreaking, and several forms of such tasks have been considered, including the con-struction of proper graph colorings [3–9, 11, 15, 17, 18, 22], of maximal independentsets (MIS) [1, 4, 5, 14, 16, 18], as well as edge-based variants of these problems (cf.e.g [21]) In this talk we address the following question: What makes somesymmetry-breaking problems in theLOCALmodel easier than others?

We note that theLOCALmodel has twoflavors, involving the design of deterministicand randomized algorithms, which are clearly distinct [8] When considering ran-domized algorithms, for n-node graphs of maximum degreeΔ, a hardness separationbetween the randomized complexities of the specific problems of MIS and (Δ + 1)-coloring has recently been observed [11, 14] No analogous separation is as yet knownwhen considering deterministic solutions to these problems We look at some partialevidence in this direction, making use of the recently introduced framework of conflictcoloring representations [9] for local combinatorial problems A conflict coloringrepresentation captures a distributed task through a set of local constraints on edges

This talk includes results of joint work with: P Fraigniaud, C Gavoille, M Heinrich, and

M Markiewicz.

of the system graph, thus constituting a special case of the much broader class ofconstraint satisfaction problems (CSP) with binary constraints Whereas all local tasksare amenable to a conflict coloring formulation, one may introduce a natural constraintdensity parameter, which turns out to be inherently smaller for some problems than forothers For example, for the natural representation of the (Δ + 1)-coloring task, theconstraint density is 1/(Δ + 1), while for any accurate representation of MIS, theconstraint density is at least 1/2 We discuss implications of how low constraint density(notably, much smaller than 1/Δ) may be helpful when finding solutions to a distributedtask, especially when applying the so-called shattering method [20] in a randomizedsetting, and more directly, when designing faster deterministic algorithms through adirect attack on the conflict coloring representation of the task [9].

We close this talk with a discussion of relaxed variants of theLOCALmodel, inspired

by the physical concept of non-signaling In a computational framework, thenon-signaling property can be stated as the following necessary (but not sufficient)property of theLOCALmodel: for any t > 0, given two subsets of nodes S1and S2of thesystem graph, such that the distance between the nearest nodes of S1and S2is greaterthan t, in any t-roundLOCALalgorithm, the outputs of nodes from S1must be (proba-bilistically) independent of the inputs of nodes from S2 We point out that for a number

of symmetry breaking tasks in theLOCALmodel, the currently best known asymptoticlower bounds can be deduced solely by exploiting the non-signaling property This isthe case for problems such as MIS [10, 14] or 2-coloring of the ring [10] On the otherhand, such an implication is not true for, e.g., theΩ(log*n) lower bound on the number

of rounds required to 3-color the ring [15]— this lower bound follows from different(stronger) properties of theLOCALmodel [12, 13] This leads us to look at the conversequestion: How to identify conditions under which non-signaling solutions to a dis-tributed task can be converted into an algorithm in theLOCALmodel? We note someprogress in this respect for quantum analogues of theLOCALmodel [2]

References

1 Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for themaximal independent set problem J Algorithms 7(4), 567–583 (1986)

2 Arrighi, P., Nesme, V., Werner, R.F.: Unitarity plus causality implies localizability

J Comput Syst Sci 77(2), 372–378 (2011)

3 Barenboim, L.: Deterministic (Δ + 1)-coloring in sublinear (in Δ) time in static, dynamic andfaulty networks In: Proceedings of the 34th ACM Symposium on Principles of DistributedComputing (PODC), pp 345–354 (2015)

4 Barenboim, L., Elkin, M.: Distributed (Δ + 1)-coloring in linear (in Δ) time In: Proceedings

of the 41th ACM Symposium on Theory of Computing (STOC), pp 111–120 (2009)

5 Barenboim, L., Elkin, M.: Distributed graph coloring: fundamentals and recent ments In: Synthesis Lectures on Distributed Computing Theory Morgan & ClaypoolPublishers (2013)

develop-6 Barenboim, L., Elkin, M., Kuhn, F.: Distributed (Δ + 1)-coloring in linear (in Δ) time SIAM

J Comput 43(1), 72–95 (2014)

7 Barenboim, L., Elkin, M., Pettie, S., Schneider, J.: The locality of distributed symmetrybreaking In: Proceedings of the 53rd IEEE Symposium on Foundations of Computer Sci-ence (FOCS), pp 321–330 (2012)

8 Chang, Y-J., Kopelowitz, T., Pettie, S.: An exponential separation between randomized anddeterministic complexity in the LOCAL model, In: Proceedings 57th IEEE Symposium onFoundations of Computer Science (FOCS) (2016, to appear).http://arxiv.org/abs/1602.08166

9 Fraigniaud, P., Heinrich, M., Kosowski, A.: Local conflict coloring, In: Proceedings of the57th IEEE Symposium on Foundations of Computer Science (FOCS) (2016, to appear)

15 Linial, N.: Locality in distributed graph algorithms SIAM J Comput 21(1), 193–201 (1992)

16 Luby M A simple parallel algorithm for the maximal independent set problem.SIAM J Comput 15, 1036–1053 (1986)

17 Naor M.: A lower bound on probabilistic algorithms for distributive ring coloring SIAM

J Discrete Math 4(3), 409–412 (1991)

18 Panconesi, A Srinivasan, A.: Improved distributed algorithms for coloring and networkdecomposition problems In: Proceedings of the 24th ACM Symposium on Theory ofComputing (STOC), pp 581–592 (1992)

19 Peleg, D.: Distributed computing: a locality-sensitive approach SIAM (2000) Philadelphia,PA

20 Schneider, J., Wattenhofer, R.: A new technique for distributed symmetry breaking In:Proceedings of the 29th ACM Symposium on Principles of Distributed Computing (PODC),

pp 257–266 (2010)

21 Suomela, J.: Survey of local algorithms ACM Comput Surv 45(2), 24 (2013)

22 Szegedy, M., Vishwanathan, S.: Locality based graph coloring In: Proceedings of the 25thACM Symposium on Theory of Computing (STOC), pp 201–207 (1993)

Problems, A Survey

Danupon Nanongkai

KTH Royal Institute of Technology, Stockholm, Sweden

danupon@gmail.comhttps://sites.google.com/site/dannanongkai/

Abstract.In this article, we focus on the time complexity of computing tances and shortest paths on distributed networks (the CONGEST model) Wesurvey previous key results and techniques, and discuss where previous tech-niques fail and where major new ideas are needed This article is based on theinvited talk given at SIROCCO 2016 The slides used for the talk are available atthe webpage of SIROCCO 2016 (http://sirocco2016.hiit.fi/programme/#invited).Keywords:Shortest paths Graph algorithms  Distributed algorithms

dis-Our focus is on solving the single-source shortest paths problem on undirectedweighted distributed networks The network is modeled by the CONGEST model, andthe goal is for every node to know its distance to a given source node The algorithmshould run with the least number of rounds possible (known as time complexity) (See,e.g., [8] for detailed descriptions.) Through a series of studies (e.g [1, 3, 4, 8, 10, 11,

12, 14]), we now know that

1 any distributed algorithm with polynomial approximation ratio needs ~Xðpffiffiffin

þ DÞrounds [3]1, and

2 there is a deterministic ð1 þ Þ-approximation algorithm that takes ~OðOð1Þð ffiffiffi

n

pþDÞÞ rounds [1, 8]

Here, n and D are the number of nodes and the network diameter, respectively, and

~X and ~O hide logOð1Þn factors The above results imply that we already know the best

number of rounds an approximation algorithm can achieve, modulo some lower-orderterms The case of exact algorithm is, however, widely open The best exact algorithm

we know of takes OðnÞ rounds, due to the distributed version of the Bellman-Fordalgorithm Beating this bound is thefirst open problem we highlight:

Open problem 1:Is there an algorithm that can solve the single-source shortest paths(or simply compute the distance between two given nodes) exactly in time that issublinear in n, i.e in ~Oðn1 Þ rounds for some constant  [ 0?

Note that whether we can solve graph problems exactly in sublinear time (in n) isinteresting for many graph problems (e.g the minimum cut problem [6, 13])

1 This lower bound holds for randomized algorithms and, in fact, even for quantum algorithms [5].

An equally interesting question is whether we can solve the all-pairs shortest pathsproblem exactly in linear-time (in n) We already know that we can get a ð1 þ Þ-approximate solution in such running time.

One challenge in answering the above open problems is to avoid computing k-sourceh-hop distances The h-hop distance between nodes u and v, denoted by disthðu; vÞ, is the(weighted) length of the shortest path among paths between u and v containing at most hedges In the k-source h-hop distances problem, we are given k source nodes s1; s2; ; sk

and want to make every node u knows its distance to every source node si An ~Oðk þ nÞdistributed algorithm for solving this problem was presented in [12] and was animportant subroutine in subsequent algorithms (e.g [1, 8]) The drawback of this sub-routine is that it only providesð1 þ Þ-approximate distances Unfortunately, obtainingexact distances within the same running time is impossible, as Lenzen and Patt-Shamir[11] showed that such algorithm requires ~XðkhÞ rounds

Another open problem (raised before in [12]) is the directed case (referred to as theasymmetric case in [12]) This is when we think of each edgeðu; vÞ as two directededges, one from u to v and the other from v to u, and the weight of the two edges might

be different (Note that the directions and edge weight do not affect the communicationbetween u and v.) Obviously, the lower bound of ~Xðpffiffiffin

þ DÞ [3] for the undirectedcase also holds for this case Using the techniques in [12], we can get a ð1 þ Þ-approximation ~XðpffiffiffiffiffiffinD

þ DÞ-time algorithm If we do not care about the tion ratio, and simply want to know whether there is a directed path from the source toeach node (this problem is called single-source reachability), then the running time can

The main challenge in answering this open problem is to avoid the use of sparsespanner and related structures A spanner is a subgraph that approximately preservesthe distance between every pairs of nodes Spanner and other relevant structures, such

as emulator and hopset were used previously as the main tools to obtain tight upperbounds for the undirected case (see, e.g., [1, 8]) Unfortunately, similar structures donot exist on directed graphs A sparse spanner, for example, do not exist for a completebipartite graph with edges directed from left to right; removing any edgeðu; vÞ fromsuch graph will cause the distance from u to v to increase from one to infinity.The last open problem we highlight is on congested cliques, i.e when the network

is fully-connected For approximately solving the single-source shortest paths problem,

we already have a satisfying algorithm with polylogarithmic time and ð1 þ approximation ratio [1, 8] The best ð1 þ Þ-approximation algorithm for all-pairsshortest paths take ~Oðn0 :15715Þ time [2] For exact solutions, both single-source and all-pairs shortest paths have the best known running time of ~Oðn1 =3Þ [2]

Þ-Open problem 3:Can we improve the running time of [2] for solving single-sourceshortest paths exactly and all-pairs shortest pathsð1 þ Þ-approximately on congestedcliques?

The above problem is interesting because of its connection to algebraic techniques Itsanswer might lead us to understand these techniques better See [2, 9] for algebraictools developed so far on congested cliques.

3 Das Sarma, A., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D.,Wattenhofer R.: Distributed verification and hardness of distributed approximation SIAM

J Comput 41(5) (2012) Announced at STOC 2011

4 Elkin, M.: An unconditional lower bound on the time-approximation trade-off for the tributed minimum spanning tree problem SIAM J Comput 36(2) (2006) Announced atSTOC 2004

dis-5 Elkin, M., Klauck, H., Nanongkai, D., Pandurangan, G.: Can quantum communication speed

up distributed computation? In: Symposium on Principles of Distributed Computing, PODC(2014)

6 Ghaffari, M., Kuhn, F.: Distributed minimum cut approximation In: Afek, Y (ed.) DISC

2013 LNCS, vol 8205, pp 1–15 Springer, Heidelberg (2013)

7 Ghaffari, M., Udwani, R.: Brief announcement: distributed single-source reachability In:ACM Symposium on Principles of Distributed Computing, PODC, pp 163–165 (2015)

8 Henzinger, M., Krinninger, S., Nanongkai, D: A deterministic almost-tight distributedalgorithm for approximating single-source shortest paths In: Symposium on Theory ofComputing, STOC (2016)

9 Le Gall, F.: Further algebraic algorithms in the congested clique model and applications tograph-theoretic problems In: Gavoille, C., Ilcinkas, D (eds.) DISC 2016 LNCS, vol 9888,

Thomas Sauerwald

Computer Laboratory, University of Cambridge, USA

Abstract In this talk we will consider smoothing networks (a.k.a balancingnetworks) that accept an arbitrary stream of tokens on input and routes them tooutput wires Pairs of wires can be connected by balancers that direct arrivingtokens alternately to its two outputs Wefirst discuss some classical results andrelate smoothing networks to their siblings, including sorting and countingnetworks Then we will present some results on randomised smoothing net-works, where balancers are initialised randomly Finally, we will explorestronger notions of smoothing networks including a model where an adversarycan specify the input and the initialisation of all balancers

References

1 Aspnes, J., Herlihy, M., Shavit, N.: Counting networks J ACM 41(5), 1020–1048 (1994)

2 Batcher, K.E.: Sorting networks and their applications In: American Federation of mation Processing Societies: AFIPS Conference Proceedings (1968) Spring Joint ComputerConference, pp 307–314 (1968)

Infor-3 Busch, C., Mavronicolas, M.: A combinatorial treatment of balancing networks J ACM43(5), 794–839 (1996)

4 Dowd, M., Perl, Y., Rudolph, L., Saks, M.E.: The periodic balanced sorting network

11 Mavronicolas, M., Sauerwald, T.: The impact of randomization in smoothing networks.Distrib Comput 22(5–6), 381–411 (2010)

12 Sauerwald, T., Sun, H.: Tight bounds for randomized load balancing on arbitrary networktopologies In: Proceedings of the 53rd Annual IEEE Symposium on Foundations ofComputer Science (FOCS 2012), pp 341–350 (2012)

Setting Ports in an Anonymous Network: How to Reduce

the Level of Symmetry? 35Ralf Klasing, Adrian Kosowski, and Dominik Pajak

Fooling Pairs in Randomized Communication Complexity 49Shay Moran, Makrand Sinha, and Amir Yehudayoff

Public vs Private Randomness in Simultaneous Multi-party

Communication Complexity 60Orr Fischer, Rotem Oshman, and Uri Zwick

Message Lower Bounds via Efficient Network Synchronization 75Gopal Pandurangan, David Peleg, and Michele Scquizzato

Recent Results on Fault-Tolerant Consensus in Message-Passing Networks 92Lewis Tseng

Shared Memory

Asynchronous Coordination Under Preferences and Constraints 111Armando Castañeda, Pierre Fraigniaud, Eli Gafni, Sergio Rajsbaum,

and Matthieu Roy

Concurrent Use of Write-Once Memory 127James Aspnes, Keren Censor-Hillel, and Eitan Yaakobi

In the Search for Optimal Concurrency 143Vincent Gramoli, Petr Kuznetsov, and Srivatsan Ravi

The F-Snapshot Problem 159Gal Amram

t-Resilient Immediate Snapshot Is Impossible 177Carole Delporte, Hugues Fauconnier, Sergio Rajsbaum,

and Michel Raynal

Mobile Agents

Linear Search by a Pair of Distinct-Speed Robots 195Evangelos Bampas, Jurek Czyzowicz, Leszek Gąsieniec, David Ilcinkas,

Ralf Klasing, Tomasz Kociumaka, and Dominik Pająk

Deterministic Meeting of Sniffing Agents in the Plane 212Samir Elouasbi and Andrzej Pelc

Distributed Evacuation in Graphs with Multiple Exits 228Piotr Borowiecki, Shantanu Das, Dariusz Dereniowski,

andŁukasz Kuszner

Universal Systems of Oblivious Mobile Robots 242Paola Flocchini, Nicola Santoro, Giovanni Viglietta,

and Masafumi Yamashita

Collaborative Delivery with Energy-Constrained Mobile Robots 258Andreas Bärtschi, Jérémie Chalopin, Shantanu Das, Yann Disser,

Barbara Geissmann, Daniel Graf, Arnaud Labourel, and Matúš Mihalák

Communication Problems for Mobile Agents Exchanging Energy 275Jurek Czyzowicz, Krzysztof Diks, Jean Moussi, and Wojciech Rytter

Data Dissemination and Routing

Asynchronous Broadcasting with Bivalent Beeps 291Kokouvi Hounkanli and Andrzej Pelc

Sparsifying Congested Cliques and Core-Periphery Networks 307Alkida Balliu, Pierre Fraigniaud, Zvi Lotker, and Dennis Olivetti

Rumor Spreading with Bounded In-Degree 323Sebastian Daum, Fabian Kuhn, and Yannic Maus

Whom to Befriend to Influence People 340Manuel Lafond, Lata Narayanan, and Kangkang Wu

Approximating the Size of a Radio Network in Beeping Model 358Philipp Brandes, Marcin Kardas, Marek Klonowski, Dominik Pająk,

and Roger Wattenhofer

An Approximation Algorithm for Path Computation and Function

Placement in SDNs 374Guy Even, Matthias Rost, and Stefan Schmid

Transiently Consistent SDN Updates: Being Greedy is Hard 391Saeed Akhoondian Amiri, Arne Ludwig, Jan Marcinkowski,

and Stefan Schmid

Author Index 407

Message Passing

Othon Michail1,2(B)and Paul G Spirakis1,2

1 Department of Computer Science, University of Liverpool, Liverpool, UK

{Othon.Michail,P.Spirakis}@liverpool.ac.uk

2 Computer Technology Institute and Press “Diophantus” (CTI), Patras, Greece

Abstract In this work, we study the following basic question: “How

much parallelism does a distributed task permit?” Our definition of allelism (or symmetry) here is not in terms of speed, but in terms of

par-identical roles that processes have at the same time in the execution.

We initiate this study in population protocols, a very simple model thatnot only allows for a straightforward definition of what a role is, butalso encloses the challenge of isolating the properties that are due to

the protocol from those that are due to the adversary scheduler, who controls the interactions between the processes We (i) give a partial

characterization of the set of predicates on input assignments that can

be stably computed with maximum symmetry, i.e., Θ(N min), whereN min

is the minimum multiplicity of a state in the initial configuration, and

(ii) we turn our attention to the remaining predicates and prove a strong

impossibility result for the parity predicate: the inherent symmetry of

any protocol that stably computes it is upper bounded by a constant that

depends on the size of the protocol.

is assigned to more than one process Moreover, we are interested in quantifying

the degree of parallelism (also called symmetry in this paper) that a task is

susceptible of

Leader election is a task of outstanding importance for distributed rithms One of the oldest [Ang80] and probably still one of the most commonlyused approaches [Lyn96,AW04,AAD+06,KLO10] for solving a distributed task

algo-in a given settalgo-ing, is to execute a distributed algorithm that manages to elect

Supported in part by the School of EEE/CS of the University of Liverpool, NeSTinitiative, and the EU IP FET-Proactive project MULTIPLEX under contract no

317532 The full version can be found at:https://arxiv.org/abs/1604.07187

c

 Springer International Publishing AG 2016

J Suomela (Ed.): SIROCCO 2016, LNCS 9988, pp 3–18, 2016.

a unique leader (or coordinator ) in that setting and then compose this (either

sequentially or in parallel) with a second algorithm that can solve the task byassuming the existence of a unique leader Actually, it is quite typical, that thetasks of electing a leader and successfully setting up the composition enclose thedifficulty of solving many other higher-level tasks in the given setting

Due to its usefulness in solving other distributed tasks, the leader electionproblem has been extensively studied, in a great variety of distributed settings[Lyn96,AW04,FSW14,AG15] Still, there is an important point that is much less

understood, concerning whether an election step is necessary for a given task and to what extent it can be avoided Even if a task T can be solved in a given

setting by first passing through a configuration with a unique leader, it is still

valuable to know whether there is a correct algorithm for T that avoids this In

particular, such an algorithm succeeds without the need to ever have less than

k processes in a given “role”, and we are also interested in how large k can be

without sacrificing solvability

Depending on the application, there are several ways of defining what the

“role” of a process at a given time in the execution is In the typical approach

of electing a unique leader, a process has the leader role if a leader variable in

its local memory is set to true and it does not have it otherwise In other cases,the role of a process could be defined as its complete local history In such cases,

we would consider that two processes have the same role after t steps iff both have the same local history after each one of them has completed t local steps It

could also be defined in terms of the external interface of a process, for example,

by the messages that the process transmits, or it could even correspond to thebranch of the program that the process executes In this paper, as we shall see,

we will define the role of a process at a given time in the execution, as the entire

content of its local memory So, in this paper, two processes u and v will be regarded to have the same role at a given time t iff, at that time, the local state

of u is equal to the local state of v.

Understanding the parallelism that a distributed task allows, is of tal importance for the following reasons First of all, usually, the more parallelism

fundamen-a tfundamen-ask fundamen-allows, the more efficiently it cfundamen-an be solved Moreover, the less symmetry

a solution for a given problem has to achieve in order to succeed, the more nerable it is to faults For an extreme example, if a distributed algorithm elects

vul-in every execution a unique leader vul-in order to solve a problem, then a svul-inglecrash failure (of the leader) can be fatal

1.1 Our Approach

We have chosen to initiate the study of the above problem in a very minimal

distributed setting, namely in Population Protocols of Angluin et al [AAD+06](see Sect.1.2for more details and references) One reason that makes populationprotocols convenient for the problem under consideration, is that the role of aprocess at a given step in the execution can be defined in a straightforwardway as the state of the process at the beginning of that step So, for example,

if we are interested in an execution of a protocol that stabilizes to the correct

answer without ever electing a unique leader, what we actually require is anexecution that, up to stability, never goes through a configuration in which a

state q is the state of a single node, which implies that, in every configuration

of the execution, every state q is either absent or the state of at least two nodes Then, it is straightforward to generalize this to any symmetry requirement k, by requiring that, in every configuration, every state q is either absent or the state

of at least k nodes.

What is not straightforward in this model (and in any model with

adver-sarially determined events), is how to isolate the symmetry that is only due to

the protocol For if we require the above condition on executions to be satisfied

for every execution of a protocol, then most protocols will fail trivially, because

of the power of the adversary scheduler In particular, there is almost always away for the scheduler to force the protocol to break symmetry maximally, forexample, to make it reach a configuration in which some state is the state of

a single node, even when the protocol does not have an inherent mechanism of

electing a unique state Moreover, though for computability questions it is cient to assume that the scheduler selects in every step a single pair of nodes tointeract with each other, this type of a scheduler is problematic for estimatingthe symmetry of protocols The reason is that even fundamentally parallel oper-ations, necessarily pass through a highly-symmetry-breaking step For example,

suffi-consider the rule (a, a) → (b, b) and assume that an even number of nodes are initially in state a The goal is here for the protocol to convert all as to bs If the scheduler could pick a perfect matching between the as, then in one step all as would be converted to bs, and additionally the protocol would never pass trough a configuration in which a state is the state of fewer than n nodes Now,

observe that the sequential scheduler can only pick a single pair of nodes in

each step, so in the very first step it yields a configuration in which state b is

the state of only 2 nodes Of course, there are turnarounds to this, for example

by taking into account only equal-interaction configurations, consisting of thestates of the processes after all processes have participated in an equal number

of interactions, still we shall follow an alternative approach that simplifies thearguments and the analysis

In particular, we will consider schedulers that can be maximally parallel.Such a scheduler, selects in every step a matching (of any possible size) of thecomplete interaction graph, so, in one extreme, it is still allowed to select onlyone interaction but, in the other extreme, it may also select a perfect matching

in a single step Observe that this scheduler is different both from the sequentialscheduler traditionally used in the area of population protocols and from the

fully parallel scheduler which assumes that Θ(n) interactions occur in parallel

in every step Actually, several recent papers assume a fully parallel schedulerimplicitly, by defining the model in terms of the sequential scheduler and thenperforming their analysis in terms of parallel time, defined as the sequential time

divided by n.

Finally, in order to isolate the inherent symmetry, i.e., the symmetry that

is only due to the protocol, we shall focus on those schedules1 that achieve ashigh symmetry as possible for the given protocol Such schedules may look intothe protocol and exploit its structure so that the chosen interactions maximizeparallelism It is crucial to notice that this restriction does by no means affectcorrectness Our protocols are still, as usual, required to stabilize to the cor-

rect answer in any fair execution (and, actually, in this paper against a more

generic scheduler than the one traditionally assumed) The above restriction is

only a convention for estimating the inherent symmetry of a protocol designed

to operate in an adversarial setting On the other hand, one does not expect

this measure of inherent symmetry to be achieved by the majority of tions If, instead, one is interested in some measure of the observed symmetry, then it would make more sense to study an expected observed symmetry under

execu-some probabilistic assumption for the scheduler We leave this as an interestingdirection for future research (see Sect.5for more details on this)

For a given initial configuration, we shall estimate the symmetry breakingperformed by the protocol not in any possible execution but an execution inwhich the scheduler tries to maximize the symmetry In particular, we shall

define the symmetry of a protocol on a given initial configuration c0 as the

maximum symmetry achieved over all possible executions on c0 So, in order to

lower bound by k the symmetry of a protocol on a given c0, it will be sufficient

to present a schedule in which the protocol stabilizes without ever “electing” fewer than k nodes On the other hand, to establish an upper bound of h on symmetry, we will have to show that in every schedule (on the given c0) the

protocol “elects” at most h nodes Then we may define the symmetry of the

protocol on a set of initial configurations as the minimum of its symmetriesover those initial configurations The symmetry of a protocol (as a whole) shall

be defined as a function of some parameter of the initial configuration and isdeferred to Sect.2

Observation 1 The above definition leads to very strong impossibility results,

as these upper bounds are also upper bounds on the observed symmetry In ticular, if we establish that the symmetry of a protocol A is at most h then, it is clear that under any scheduler the symmetry of A is at most h.

par-Section2brings together all definitions and basic facts that are used out the paper In Sect.3, we give a set of positive results The main result here

through-is a partial characterization, showing that a wide subclass of semilinear

pred-icates is computed with symmetry Θ(N min), which is asymptotically optimal.Then, in Sect.4, we study some basic predicates that seem to require much

symmetry breaking In particular, we study the majority and the parity

predi-cates For majority we establish a constant symmetry, while for parity we prove

a strong impossibility result, stating that the symmetry of any protocol thatstably computes it, is upper bounded by an integer depending only on the size

of the protocol (i.e., a constant, compared to the size of the system) The latter

1 By “schedule” we mean an “execution” throughout.

implies that there exist predicates which can only be computed by protocols

that perform some sort of leader-election (not necessarily a unique leader but atmost a constant number of nodes in a distinguished leader role) In Sect.5, wegive further research directions that are opened by our work All omitted detailsand proofs can be found in the full version

1.2 Further Related Work

In contrast to static systems with unique identifiers (IDs) and dynamic systems,

the role of symmetry in static anonymous systems has been deeply investigated

[Ang80,YK96,Kra97,FMS98] Similarity as a way to compare and contrast

dif-ferent models of concurrent programming has been defined and studied in [JS85].One (restricted) type of symmetry that has been recently studied in systems with

IDs is the existence of homonyms, i.e., processes that are initially assigned the

same ID [DGFG+11] Moreover, there are several standard models of distributedcomputing that do not suffer from a necessity to break symmetry globally (e.g.,

to elect a leader) like Shared Memory with Atomic Snapshots [AAD+93,AW04],Quorums [Ske82,MRWW01], and the LOCAL model [Pel00,Suo13]

Population Protocols were originally motivated by highly dynamic networks

of simple sensor nodes that cannot control their mobility The first papers focused

on the computational capabilities of the model which have now been almost pletely characterized In particular, if the interaction network is complete (as isalso the case in the present paper), i.e., one in which every pair of processesmay interact, then the computational power of the model is equal to the class of

com-the semilinear predicates (and com-the same holds for several variations) [AAER07].Interestingly, the generic protocol of [AAD+06] that computes all semilinearpredicates, elects a unique leader in every execution and the same is true forthe construction in [CDS14] Moreover, according to [AG15], all known genericconstructions of semilinear predicates “fundamentally rely on the election of

a single initial leader node, which coordinates phases of computation” linearity of population protocols persists up to o(log log n) local space but not

Semi-more than this [CMN+11] If additionally the connections between processes canhold a state from a finite domain, then the computational power dramatically

increases to the commutative subclass of NSPACE(n2) [MCS11a] The

for-mal equivalence of population protocols to chemical reaction networks (CRNs), which model chemistry in a well-mixed solution, has been recently demonstrated

[Dot14] Moreover, the recently proposed Network Constructors extension of

population protocols [MS16] is capable of constructing arbitrarily complex

sta-ble networks Czyzowicz et al [CGK+15] have recently studied the relation ofpopulation protocols to antagonism of species, with dynamics modeled by dis-crete Lotka-Volterra equations Finally, in [CCDS14], the authors highlightedthe importance of executions that necessarily pass through a “bottleneck” tran-sition (meaning a transition between two states that have only constant counts

in the population, which requires Ω(n2) expected number of steps to occur), byproving that protocols that avoid such transitions can only compute existencepredicates To the best of our knowledge, our type of approach, of computing

predicates stably without ever electing a unique leader, has not been followed

before in this area (according to [AG15], “[DH15] proposes a leader-less work for population computation”, but this should not be confused with what

frame-we do in this paper, as it only concerns the achievement of dropping the

require-ment for a pre-elected unique leader that was assumed in all previous results

for that problem) For introductory texts to population protocols, the interestedreader is encouraged to consult [AR09,MCS11b]

A population protocol (PP) is a 6-tuple (X, Y, Q, I, O, δ), where X, Y , and Q are all finite sets and X is the input alphabet, Y is the output alphabet, Q is the set

of states, I : X → Q is the input function, O : Q → Y is the output function, and

δ : Q × Q → Q × Q is the transition function.

If δ(a, b) = (a  , b  ), we call (a, b) → (a  , b  ) a transition A transition

(a, b) → (a  , b  ) is called effective if x = x  for at least one x ∈ {a, b} and ineffective otherwise When we present the transition function of a protocol we only present the effective transitions The system consists of a population V

of n distributed processes (also called nodes) In the generic case, there is an underlying interaction graph G = (V, E) specifying the permissible interactions

between the nodes Interactions in this model are always pairwise In this work,

G is a complete directed interaction graph.

Let Q be the set of states of a population protocol A A configuration c of

A on n nodes is an element of IN |Q| ≥0 , such that, for all q ∈ Q, c[q] is equal to the number of nodes that are in state q in configuration c and it holds that



q∈Q c[q] = n For example, if Q = {q0, q1, q2, q3} and c = (7, 12, 52, 0), then, in

c, 7 nodes of the 7 + 12 + 52 + 0 = 71 in total, are in state q0, 12 nodes in state

q1, and 52 nodes in state q2

Execution of the protocol proceeds in discrete steps and it is determined by

an adversary scheduler who is allowed to be parallel, meaning that, in every step,

it may select one or more pairwise interactions (up to a maximum matching) tooccur at the same time This is an important difference from classical populationprotocols where the scheduler could only select a single interaction per step More

formally, in every step, a non-empty matching (u1, v1), (u2, v2), , (u k , v k) from

E is selected by the scheduler and, for all 1 ≤ i ≤ k, the nodes u i , v i interact

with each other and update their states according to the transition function δ.

A fairness condition is imposed on the adversary to ensure the protocol makes progress An infinite execution is fair if for every pair of configurations c and c  such that c → c  (i.e., c can go in one step to c  ), if c occurs infinitely often in

the execution then so does c 

In population protocols, we are typically interested in computing predicates

on the inputs, e.g., N a ≥ 5, being true whenever there are at least 5 as in the

input.2 Moreover, computations are stabilizing and not terminating, meaning

2 We shall use throughout the paper N i to denote the number of nodes with

input/statei.

that it suffices for the nodes to eventually converge to the correct output We

say that a protocol stably computes a predicate if, on any population size, any

input assignment, and any fair execution on these, all nodes eventually stabilizetheir outputs to the value of the predicate on that input assignment

We define the symmetry s(c) of a configuration c as the minimum plicity of a state that is present in c (unless otherwise stated, in what fol-

multi-lows by “symmetry” we shall always mean “inherent symmetry”) That is,

s(c) = min q∈Q : c[q]≥1 {c[q]} For example, if c = (0, 4, 12, 0, 52) then s(c) = 4, if

c = (1, ) then s(c) = 1, which is the minimum possible value for symmetry, and if c = (n, 0, 0, , 0) then s(c) = n which is the maximum possible value for

symmetry So, the range of the symmetry of a configuration is{1, 2, , n}.

LetC0(A) be the set of all initial configurations for a given protocol A Given

an initial configuration c0∈ C0(A), denote by Γ (c0) the set of all fair executions

of A that begin from c0, each execution being truncated to its prefix up to stability.3

Given any initial configuration c0 and any execution α ∈ Γ (c0), define

the symmetry breaking of A on α as the difference between the symmetry

of the initial configuration of α and the minimum symmetry of a tion of α, that is, the maximum drop in symmetry during the execution For- mally, b( A, α) = s(c0)− min c∈α {s(c)} Also define the symmetry of A on α as s(A, α) = min c∈α {s(c)} Of course, it holds that s(A, α) = s(c0)− b(A, α) Moreover, observe that, for all α ∈ Γ (c0), 0 ≤ b(A, α) ≤ s(c0)− 1 and

configura-1≤ s(A, α) ≤ s(c0) In several cases we shall denote s(c0) by N min

The symmetry breaking of a protocol A on an initial configuration c0 can

now be defined as b( A, c0) = minα∈Γ (c0 {b(A, α)} and:

Definition 1 We define the symmetry of A on c0 as s(A, c0) =maxα∈Γ (c0 {s(A, α)}.

Remark 1 To estimate the inherent symmetry with which a protocol computes

a predicate on a c0, we execute the protocol against an imaginary scheduler who

is a symmetry maximizer.

Now, given the set C(N min ) of all initial configurations c0 such that

s(c0) = N min , we define the symmetry breaking of a protocol A on C(N min)

as b( A, N min) = maxc0∈C(N min){b(A, c0)} and:

Definition 2 We define the symmetry of A on C(N min ) as s( A, N min) =minc0∈C(N min){s(A, c0)}.

Observe again that s( A, N min ) = N min −b(A, N min) and that 0≤ b(A, N min)≤

N min − 1 and 1 ≤ s(A, N min)≤ N min

3 In this work, we only require protocols to preserve their symmetry up to stability.

This means that a protocol is allowed to break symmetry arbitrarily after stability,e.g., even elect a unique leader, without having to pay for it We leave as an inter-esting open problem the comparison of this convention to the apparently harderrequirement of maintaining symmetry forever

This means that, in order to establish that a protocolA is at least g(N min)

symmetric asymptotically (e.g., for g(N min ) = Θ(log N min)), we have to show

that for every sufficiently large N min, the symmetry breaking of A on C(N min)

is at most N min − g(N min), that is, to show that for all initial configurations

c0 ∈ C(N min ) there exists an execution on c0 that drops the initial symmetry

by at most N min − g(N min ), e.g., by at most N min − log N min for g(N min) =

log N min , or that does not break symmetry at all in case g(N min ) = N min On

the other hand, to establish that the symmetry is at most g(N min), e.g., at most

1 which is the minimum possible value, one has to show a symmetry breaking

of at least N min − g(N min ) on infinitely many N mins

In this section, we try to identify predicates that can be stably computed withmuch symmetry We first give an indicative example, then we generalize to arrive

at a partial characterization of the predicates that can be computed with imum symmetry, and, finally, we highlight the role of output-stable states insymmetric computations

max-3.1 An Example: Count-to-x

Protocol Count-to-x: X = {0, 1}, Q = {q0, q1, q2, , q x }, I(σ) = q σ, for all

σ ∈ X, O(q x ) = 1 and O(q) = 0, for all q ∈ Q\{q x }, and δ: (q i , q j)→ (q i+j , q0),

if i + j < x, (q i , q j)→ (q x , q x), otherwise

Proposition 1 The symmetry of Protocol Count-to-x, for any x = O(1), is at

least (2/3)N min /x − (x − 1)/3, when x ≥ 2, and N min , when x = 1; i.e., it is Θ(N min ) for any x = O(1).

Proof The scheduler4partitions the q1s, let them be N1(0) initially and denoted

just N1 in the sequel, into N1/x groups of x q1s each, possibly leaving an

incomplete group of r ≤ x − 1 q1s residue Then, in each complete group, it

performs a sequential gathering of x − 3 other q1s to one of the nodes, which

will go through the states q1, q2, , q x−1 The same gathering is performed inparallel to all groups, so every state that exists in one group will also exist inevery other group, thus, its cardinality never drops belowN1/x In the end, at step t, there are many q0s, N x−1 (t) = N1/x , and N1(t) = N1/x  + r, where

0 ≤ r ≤ x − 1 is the residue of q1s That is, in all configurations so far, thesymmetry has not dropped belowN1/x .

Now, we cannot pick, as a symmetry maximizing choice of the scheduler, a

perfect bipartite matching between the q1s and the q x−1s converting them all

to the alarm state q x, because this could possibly leave the symmetry-breaking

residue of q1s What we can do instead, is to match in one step as many as

4 Always meaning the imaginary symmetry-maximizing scheduler when

lower-bounding the symmetry

we can so that, after the corresponding transitions, N x (t )≥ N1(t ) is satisfied.

In particular, if we match y of the (q1, q x−1 ) pairs we will obtain N x (t  ) = 2y,

N x−1 (t ) =N1/x − y, and N1(t ) =N1/x − y + r and what we want is

2y ≥ N1/x  − y + r ⇒ 3y ≥ N1/x  + r ⇒ y ≥ N1/x  + r

which means that if we match approximately 1/3 of the (q1, q x−1) pairs then we

will have as many q x as we need in order to eliminate all q1s in one step and all

remaining q x−1s in another step

The minimum symmetry in the whole course of this schedule is

N min − ((2/3)N1/x − (x − 1)/3) = N min − ((2/3)N min /x − (x − 1)/3) Next,

we consider the case in which there are some q0s in the initial configuration

Observe that in this protocol the q0s can only increase, so their minimum

car-dinality is precisely their initial carcar-dinality N0 Consequently, in case N0 ≥ 1 and N1≥ 1, and if N min = min{N0, N1}, the symmetry breaking of the sched- ule defined above is N min − min{N0, N x−1 (t )} If, for some initial configura- tion, N0≥ N x−1 (t  ) then the symmetry breaking is N min − N x−1 (t )≤ N min − ((2/3) N1/x−(x−1)/3) This gives again N min −((2/3)N min /x−(x−1)/3), when N1 ≤ N0, and less than N min − ((2/3)N min /x − (x − 1)/3), when

N1 > N0 = N min If instead, N0 < N x−1 (t  ) < N1, then, in this case, the

symmetry breaking is N min − min{N0, N x−1 (t )} = N0 − N0 = 0 Finally, if

N0 = n, then the symmetry breaking is 0 We conclude that for every

ini-tial configuration, the symmetry breaking of the above schedule is at most

N min − N x−1 (t ) ≤ N min − ((2/3)N min /x − (x − 1)/3), for all x ≥ 2, and

0, for x = 1 Therefore, the symmetry of the Count-to-x protocol is at least (2/3) N min /x + (x − 1)/3 = Θ(N min ), for x ≥ 2, and N min , for x = 1

3.2 A General Positive Result

Theorem 1 Any predicate of the form 

i∈[k] a i N i ≥ c, for integer constants

k ≥ 1, a i ≥ 1, and c ≥ 0, can be computed with symmetry more than

N min /(c/

j∈L a j+ 2) − 2 = Θ(N min ).

Proof We begin by giving a parameterized protocol (Protocol 1) that stablycomputes any such predicate, and then we shall prove that the symmetry of thisprotocol is the desired one

Take now any initial configuration C0 on n nodes and let L ⊆ [k] be the set of indices of the initial states that are present in C0 Let also q min be the

Protocol 1 Positive-Linear-Combination

Q = {q0, q1, q2, , q c }

I(σ i) =q a i, for allσ i ∈ X

O(q c) = 1 andO(q) = 0, for all q ∈ Q\{q c }

j∈L a j  copies of each initial state Observe that

each group has total sum

j∈L a j x = x

j∈L a j= 

j∈L a j (j∈L a j)≥ c Moreover, state q min has a residue r min of at most x and every other state q i has a residue r i ≥ r min Finally, keep y = min + r min )/(x + 1)  − 1 from

those groups and drop the otherN min /x − y groups making their nodes part

of the residue, which results in new residue values r j  = x( N min /x − y) + r j,

for all j ∈ L It is not hard to show that y ≤ r 

j , for all j ∈ L.

We now present a schedule that achieves the desired symmetry The schedule

consists of two phases, the gathering phase and the dissemination phase In the

dissemination phase, the schedule picks a node of the same state from every groupand starts aggregating to that node the sum of its group sequentially, performing

the same in parallel in all groups It does this until the alarm state q c firstappears When this occurs, the dissemination phase begins In the disseminationphase, the schedule picks one after the other all states that have not yet been

converted to q c For each such state q i , it picks a q c which infects one after the

other (sequentially) the q i s, until N c (t) ≥ N i (t) is satisfied for the first time Then, in a single step that matches each q i to a q c , it converts all remaining q is

to q c

We now analyze the symmetry breaking of the protocol in this schedule

Clearly, the initial symmetry is N min As long as a state appears in the groups,

its cardinality is at least y, because it must appear in each one of them When a state q i first becomes eliminated from the groups, its cardinality is equal to its

residue r  i Thus, so far, the minimum cardinality of a state is

min{y, min

Finally, we must also take into account the dissemination phase In this phase,

the q c s are 2y initially and can only increase, by infecting other states, until they become n and the cardinalities of all other states decrease until they all become

0 Take any state q i = q c with cardinality N i (t) when the dissemination phase begins What the schedule does is to decrement N i (t), until N c (t ) ≥ N i (t ) is

first satisfied, and then to eliminate all occurrences of q i in one step Due to

the fact that N iis decremented by one in each step resulting in a corresponding

increase by one of N c , when N c (t )≥ N i (t  ) is first satisfied, it holds that N i (t )≥

N c (t )− 1 ≥ N c (t) − 1 ≥ 2y − 1 ≥ y for all y ≥ 1, which implies that the lower bound of y on the minimum cardinality, established for the gathering phase, is

not violated during the dissemination phase

We conclude that the symmetry of the protocol in the above schedule is morethanN min /(c/

3.3 Output-Stable States

Informally, a state q ∈ Q is called output-stable if its appearance in an execution guarantees that the output value O(q) must be the output value of the execution More formally, if q is output-stable and C is a configuration containing q, then the set of outputs of C  must contain O(q), for all C  such that C  C , where

‘’ means reaches in one or more steps Moreover, if all executions under

con-sideration stabilize to an agreement, meaning that eventually all nodes stabilize

to the same output, then the above implies that if an execution ever reaches a

configuration containing q then the output of that execution is necessarily O(q).

A state q is called reachable if there is an initial configuration C0 and an

execution on C0 that can produce q We can also define reachability just in terms of the protocol, under the assumption that if Q0⊆ Q is the set of initial states, then any possible combination of cardinalities of states from Q0 can be

part of an initial configuration A production tree for a state q ∈ Q, is a directed binary in-tree with its nodes labeled from Q such that its root has label q, if

a is the label of an internal node (the root inclusive) and b, c are the labels of

its children, then the protocol has a rule of the form {b, c} → {a, ·} (that is,

a rule producing a by an interaction between a b and a c in any direction)5,

and any leaf is labeled from Q0 Observe now that if a path from a leaf to the

root repeats a state a, then we can always replace the subtree of the highest appearance of a by the subtree of the lowest appearance of a on the path and still have a production tree for q This implies that if q has a production tree, then q also has a production tree of depth at most |Q|, that is, a production

tree having at most 2|Q|−1leaves, which is a constant number, when compared

to the population size n, that only depends on the protocol Now, we can call a state q reachable (by a protocol A) if there is a production tree for it These are

summarized in the following proposition

Proposition 2 Let A be a protocol, C0 be any (sufficiently large) initial figuration of A, and q ∈ Q any state that is reachable from C0 Then there is

con-an initial configuration C0 which is a sub-configuration of C0of size n  ≤ 2 |Q|−1

such that q is reachable from C0

Proposition2is crucial for proving negative results, and will be invoked in Sect.4

5 Whenever we use an unordered pair in a rule, like{b, c}, we mean that the property

under consideration concerns both (b, c) and (c, b).