K-set agreement bounds in round-based models through combinatorial topology Adam Shimi, Armando Castañeda

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Adam Shimi, Armando Castañeda. K-set agreement bounds in round-based models through combina- torial topology. 39th ACM Symposium on Principles of (PODC 2020), Aug 2020, Salerno, Italy. pp.395-404. ￿hal-02950742￿

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Official URL https://doi.org/10.1145/3382734.3405752

To cite this version: Shimi, Adam and Castañeda, Armando K-set agreement bounds in round-based models through combinatorial topology. (2020) In: 39th ACM Symposium on Principles of Distributed Computing (PODC 2020), 3 August 2020 - 7 August 2020 (Salerno, Italy).

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Adam Shimi Armando Castañeda IRIT, University of Toulouse UNAM Toulouse, France Mexico City, Mexico [email protected] [email protected]

A recent trend, with parallel results by Charron-Bost and Schiper [9] on one hand, and Afek and Gafni [1] and Raynal and Stainer [26] on the other hand, is using this concept of round for formalizing many different models within a common framework. But the techniques used for proving results in these models tend to be ad-hoc, very specific to some model or setting. What is required going forward is a general approach to proving impossibility results and bounds on round-based models. Actually, there is at least one example of a general mathematical ABSTRACT technique used in this context: the characterization of consensus Round-based models are very common message-passing models; solvability through point-set topology by Nowak et al. [24]. We combinatorial topology applied to distributed computing provides propose what might be seen as an extension to higher dimension sweeping results like general lower bounds. We combine both to of this intuition, by applying combinatorial topology (instead of study the computability of :-set agreement. point-set topology) to bear on :-set agreement (instead of just Among all the possible round-based models, we consider obliv- consensus). ious ones, where the constraints are given only round per round Combinatorial topology abstracts the reasoning around knowl- by a set of allowed graphs. And among oblivious models, we fo- edge and indistinguishability behind many impossibility results in cus on closed-above ones, that is models where the set of possible distributed computing. It thus provide generic mathematical tools graphs contains all graphs with more edges than some starting and methods for deriving such results [15]. Moreover, this approach graphs. These capture intuitively the underlying structure required is the only one that managed to prove impossibility results and by some communication model, like containing a ring. characterization of solvability of the :-set agreement [10], our focus We then derive lower bounds and upper bounds in one round for problem. :-set agreement, such that these bounds are proved using combina- Concretely, we look at closed-above round-models, that is mod- torial topology but stated only in terms of graph properties. These els where constraints happens round per round, and the set of bounds extend to multiple rounds when limiting our algorithms communication graphs allowed is the closure-above of a set of to be oblivious – recalling only pairs of processes and initial value, graphs. These models capture some safety properties, where we not who send what and when. require some underlying structure in communication, like having an underlying star, ring or tree. This is a strict generalization of the models with a fixed communication graph considered by Castañeda et al. [6]. For our models, we derive upper bound and lower bounds on the : for which :-set agreement is solvable. And although the proofs of the bounds use combinatorial topology, they are stated KEYWORDS in terms of variants of the domination number, a well-known and used combinatorial number on graphs. Distributed Computability, Set-agreeement, Round-based mod- els, Combinatorial Topology, Lower bounds, Upper Bounds 1.2 Overview • We start by defining closed-above models in Section 2. • Then we give various upper bounds for :-set agreement 1 INTRODUCTION in one round on those models in Section 3. These have the 1.1 Motivation advantage of not requiring any combinatorial topology. • Next, we introduce in Section 4 the combinatorial topology Rounds structure many models of distributed computing: they sim- necessary for our lower bounds, both the basic definitions plify algorithms, capture the distributed equivalent of time complex- and our main technical lemma. ity [13], and underly many fault-tolerant algorithms, like Paxos [23]. • We then go to lower bounds on round-based models for :- set agreement in one round in Section 5. Recall that these

https://doi.org/10.1145/3382734.3405752 bounds use combinatorial topology, but are stated in terms Here we abstract away all implementation details, and consider of graph properties. a model with rounds, parameterized with the allowed sequences • Finally, Section 6 generalize both upper and lower bounds of communication graphs – directed graphs where each node is to the case of multiple rounds. a process and each arrow correspond to a delivered message to Due to size constraints, most of the proof can be found in the the destination from the source. There are no crashes, just a spec- full version [28] ification of which message can be received at which round. This abstracts the Heard-Of model [9], synchronous message adver- 1.3 Related Works saries [1, 26] and dynamic networks [21, 22], as well as all other models relying only on the properties of rounds. Round-based models. The idea of using rounds for abstracting We fix Π = {?1, ..., ?= } as our set of = processes for the rest of many different models is classical in message-passing. This includes the paper. the synchronous adversary models of Afek and Gafni [1] and Ray- nal and Stainer [26]; the Heard-Of model of Charron-Bost and Definition 2.1 (Communication model). Let A0?ℎBΠ be the set of l Schiper [9]; and the dynamic networks of Kuhn et al [21]. graphs. Then >< ⊆ (A0?ℎBΠ) is a communication model. Rounds are also used for building a distributed theory of time Any set of infinite sequences of graphs defines a model. In order complexity [13] and for structuring fault-tolerant algorithms like to make models more manageable, we focus on a restricted form, Paxos [23]. where the graph for each round is decided independently of the Previous work on the solvability of consensus and :-set agree- others. The model is thus entirely characterized by the set of allowed ment include the characterization of consensus solvability for oblivi- graphs. We call these communication models oblivious, following ous round-based models of Coulouma et al. [11], the failure-detector- Coulouma et al [11]. based approach of Jeanneau et al. [19], and the focus on graceful degradation in algorihtms for :-set agreement of Biely et al. [3]. Definition 2.2 (Oblivious communication models). Let >< be a communication model. Then >< is oblivious , ∃( ⊆ A0?ℎBΠ : Combinatorial Topology. Combinatorial Topology was first ap- >< = (l . plied to the problem of :-set agreement in wait-free shared memory by Herlihy and Shavit [18], Saks and Zaharoglou [27] and Borowsky Intuitively, oblivous models capture safety properties: bad things and Gafni [4]. that must not happen. Or equivalently, good things that must hap- Beyond these first forays, many other results got proved through pen at every round. Usually, these good properties are related to combinatorial topology. Among others, we can cite the lower bounds connectivity, like containing a cycle or a spanning tree. Since such for renaming by Castañeda and Rajsbaum [7] and the derivation of a property tends to be invariant when more messages are sent, we lower-bounds for message-passing by Herlihy and Rasjbaum [16]; can look at oblivious models defined by a set of subgraphs. There is even a result by Alistarh et al. [2] showing that traditional Definition 2.3 (Closed-above communication models). Let >< be proof techniques (dubed extension-based proofs) cannot prove the an oblivious communication model. Then >< is closed-above impossibility of :-set agreement in specific shared-memory models, , ∃( ⊆ A0?ℎBΠ : >< = ( ↑ )l , where ↑  , { | + () = whereas techniques from combinatorial topology can.  ∈( For a full treatment of combinatorial topology applied to dis- + () ∧ () ⊇ ()}. Ð tributed computing, see Herlihy et al. [15]. We call the graphs in ( the generators of ><. If ( is a singleton, then >< is simple closed-above. Combination of Topology and Round-based models. Two papers at least applied topology (combinatorial or not) to general round- Classical examples of closed-above models are the non-empty based models in order to study agreement problems: Godard and kernel predicate (only graphs where at least one process broad- Perdereau [14] used combinatorial topology to study :-set agree- casts) and the non-split predicate (only graphs where each pair of ment in models with omission failures; and Nowak et al. [24] char- processes hears from a common process), used notably by Charron- acterization of consensus for general round-based models (not nec- Bost et al. [8] for characterizing the solvability of approximate essarily oblivious) using point-set topology. consensus (the variant of consensus where the decided value should be less than Y apart, where Y > 0 is fixed beforehand). Another 2 DEFINITIONS closed-above model is the one satisfying the tournament property of Afek and Gafni [1], which they show is equivalent to wait-free 2.1 Communication models read-write shared memory. One common feature of many models of distributed computation is One example of an oblivous model which is not closed-above is the notion of rounds, or layers. Formally, rounds are communication- the one generated by all graphs containing a cycle, except the clique. closed as defined by Elrad and Francez [12]: at each round, a process More generally, the closure-above forces us to have all graphs with ? only takes into account the messages (or information) sent by more edges than our generators. other processes at this same round. Nonetheless, closed-above models capture a fundamental intu- Traditionally, rounds are thought of as synchronous: synchrony ition behind distributed computing models: specifying what should indeed provides a natural way to implement them. But asynchro- not happen. They also have a good tradeoff between expressivity nous rounds also exist, both in message-passing [9] and in shared- and simplicity, since the "combinatorial data" used to build them memory [5, 17]. is contained in a small number of graphs. Finally, the patterns expected by safety properties tend to be independent of which pro- 3.1 Simple closed-above models: almost too cesses play which roles – what matters is the existence of a ring or easy spanning tree, not who is where on it. Recall that the domination number of a graph is the size of its small- We call such closed-above models symmetric. est dominating set, that is the size of the smallest set of nodes whose Definition 2.4 (Symmetric models). Let >< be a closed-above set of outgoing neighbors is Π. Note that the outgoing neighbors model, and ( be the set of graphs generating it. Then >< is sym- of a set ( ⊆ Π contains ( – that is, we assume self-loop. metric , ( = (~<((), where (~<(() = {c () |  ∈ ( ∧ c : Π → Definition 3.1 (Domination number). Let  be a graph. Then its Π a permutation on Π}. domination number W () , <8={8 ∈ [1, =] | ∃% ⊆ Π : |% | = In the rest of the paper, we will limit ourselves to closed-above 8 ∧ $DC (?) = Π}. models, both symmetric and not. ? ∈% Ð Because the simple closed-above model generated by  only 2.2 Oblivious algorithms allows graphs containing , their domination number is at most Because most applications of combinatorial topology to distributed W (). This entails a very simple upper bound on :-set agreement. computing aim towards impossibility results, the traditional al- gorithms considered err on the side of power: full information T!"#$"% 3.2 (U&&"$ '#)*+ #* :,-"/ 01$""%"*/ '4 W ()). Let protocols, which exchange at each round the view of everything  be a graph. Then W ()-set agreement is solvable in one round on ever heard by the process. For example, after a couple of rounds, the simple closed-above model generated by . views will contain nested sets of views, themselves containing views, recursively until the initial values. P$##6. The algorithm is just slightly different from the one In contrast, we focus on oblivious algorithms. That is, we limit stated at the start of the section: after one round, each process each process to remember only the initial values it knows, not who decides the minimum value of the ones of a fixed minimum domi- sent them or when. This amounts to a function from Π to the set nating set of . Since  is known, this minimum dominating set of initial values (with a ⊥ when the value is not known). In turn, can be computed beforehand. And because it is a dominating set, these algorithms lose the ability to trace the path of the value. every process receives at least one value from it, so every process We can view oblivious algorithms as full-information protocol can decide. whose decision map (the function from final view to decision value) Finally, since the minimum dominating set has at most W () depends only on the set of known pairs (process,initial value). The distinct values, at most W () values are decided, and thus our algo-  full-information protocol might still be used for deciding when rithm solves W ()-set agreement. to apply the decision map, but this map loses everything except From Castañeda et al. [6, Thm 5.1], we know this bound is tight: the known pairs. That is, the decision map is constrained to decide the oblivious model with a single graph  cannot solve :-set agree- similarly in situations where it received the same values, even when ment in one round for : < W (). Hence the weaker simple closed- they were from different processes. above model generated by  cannot solve :-set agreement in one Definition 2.5 (Oblivious algorithm). Let A be a full-information round for : < W (). protocol, with decision map X. Then A is an oblivious algorithm Still, simple closed-above models are somewhat artificial, as can , ∀E a view : X (E) = X (5 ;0C (E)), be seen in the proof: we know exactly the subgraph that must be where 5 ;0C (E) = 5 ;0C ((?,E? )) contained in the actual communication graph. A more realistic take (?,E? ) ∈E requires to spread the uncertainty to the underlying subgraph; we Ð {(?,E? )} if E? is a singleton from +8= thus look next at general closed-above models. and 5 ;0C ((?,E? )) = 5 ;0C (E? ) otherwise  3.2 General closed-above models: tweaking of 3 ONE ROUND UPPER BOUNDS: A START upper bounds WITHOUT TOPOLOGY For general closed-above models, we must deal with a set of possible Although lower bounds are our targets, they require upper bounds underlying subgraphs. This makes our previous approach inappli- to gauge their strength. We thus start with upper bounds on :- cable: we cannot hardcode a dominating set because we don’t know set agreement [10] for closed-above models. Another advantage the underlying subgraph for sure. of starting with our upper bounds is that they rely on concrete This new issue motivates the definition of a weakening of the algorithms, and allow us to introduce generalizations of the classical domination number: the equal-domination number of a set of domination number that will be used for our lower bounds. graphs. Intuitively, any set of that much process is a dominating Lastly, we also start with bounds for the one round case in this set in all the graphs considered. section and the next one. Bounds for multiple rounds depend on these one round bounds. Definition 3.3 (Equal Domination number of a set of graphs). Let ( These bounds follow from a very simple algorithm for solving :- be a set of graphs. Then its equal domination number W4@ (() , set agreement. We assume the set of initial values is totally ordered. maxW4@ (), where W4@ () = <8={8 ∈ [1, =] | ∀% ⊆ Π : |% | = Then everybody sends its initial value for one round, and decide  ∈( 8 =⇒ $DC (?) = Π}. the minimum it received. ? ∈% Ð T!"#$"% 3.4 (U&&"$ '#)*+ #* :,-"/ 01$""%"*/ '4 W4@ (() ?2 ?2 6#$ 1"*"$08 98#-"+,0'#:" %#+"8-). Let ( be a set of graphs. Then W4@ (()-set agreement is solvable on the closed-above model generated by (. ?1 ?1 P$##6. Let % be a set of W4@ (() processes with the smallest initial values. They have thus at most W4@ (() distinct initial values. By ?4 ?3 ?4 ?3 definition of W4@ ((), % dominates every graph in (, and thus every graph in the closed-above model generated by (. Figure 1: Two examples communication graphs Thus taking the minimum after one round will result in deciding one of those initial values, and thus one of at most W4@ (() values. 4@ We conclude that our algorithm solves W (()-set agreement after C#$#880$4 3.8. Let ( be a set of graphs. Then ∀8 ∈ [1,W4@ (()[: one round on the closed-above model generated by (.  (8 + (= − 2>E8 (()))-set agreement is solvable on the oblivious closed- above model generated by (~<((). Since the equal-domination number is independent of which process does what, it is the same for any permutation of the graph. When is this new bound better than the one using the equal- This entails an upper bound on symmetric models as a corollary. domination number? When there is some 8 such that = − 2>E8 (() < 4@ W4@ (() − 8. Let us take the symmetric models generated by the two C#$#880$4 3.5. Let ( be a set of graphs. Then W (()-set agree- ment is solvable on the closed-above model generated by (~<((). graphs in Figure 1. 4@ In the first model, = − 2>E8 (() < W (() − 8 never happens, Now, the natural question to ask is whether we can improve this because every covering number of a star equals 1 (the biggest set bound. Or equivalently, is it tight? of outgoing neighbors different from Π contains only one process), The answer depends on the graphs. To see it, let us look at and its equal-domination number equals = (because when taking another combinatorial number : covering numbers. Given fewer only = − 1 processes, the center of the star might not be in there). processes than the equal-domination number of the graph, they 4@ Thus = − 2>E8 (() = = − 1 ≥ W (() − 8 = = − 8. do not always form a dominating set. Nonetheless, they might still On the other hand, this is the case in the second model, because 4@ get heard by some minimum number of processes. We call such 2>E2 (() = 3 and W (() = 4. We thus we have = −2>E2 (() = 4 − 3 = minimums the covering numbers of the graph: the 8-th covering 1 < W4@ (() − 8 = 4 − 2 = 2. Hence the upper bound with covering number of  is, given any set of 8 processes, the minimum number numbers ensure 3-set agreement solvability while the upper bound of processes hearing this set in . with the equal-domination number only ensures 4-set agreement Definition 3.6 (Covering numbers of a set of graphs). Let ( be a set solvability. 4@ of graphs. Then ∀8 < W ((), its 8-th covering number 2>E8 (() , min2>E8 (), where 2>E8 () , min |( $DC (?))|. 3.3 Intuitions on upper and lower bounds  ∈( % ⊆Π ? ∈% |% |=8 Ð Why do our upper bounds hold? Because we can extract from the These numbers capture the ability of a set of processes to dissem- underlying graphs some minimal connectivity of sets of processes. inate their values in the graph. If we take the 8 processes with the Hence, we know from these combinatorial numbers how much the minimal values will spread in the worst case, and thus we bound smallest initial values, we can be sure that at least 2>E8 (() processes will hear, and thus choose one of these. This then gives a solution the maximum number of values decided. On the other hand, our lower bounds will follow from study- to (8 + (= − 2>E8 (()))-set agreement in one round. ing how much values can spread in the best case. Why? Because T!"#$"% 3.7 (U&&"$ '#)*+- #* :,-"/ 01$""%"*/ '4 9#:"$=*1 the more values can spread, the more processes can distinguish *)%'"$- 6#$ 1"*"$08 98#-"+,0'#:" %#+"8-). Let ( be a set of between initial configurations, and the more they have a chance to 4@ graphs. Then ∀8 ∈ [1,W (()[: (8 + (= − 2>E8 (()))-set agreement is decide correctly. Ensuring enough indistinguishability thus entails solvable on the oblivious closed-above model generated by (. an impossibility at solving :-set agreement. P$##6. For a set of 8 processes with the 8 smallest initial values, This indistinguishability is linked to higher-dimension connec- tivity in combinatorial topology [15, Thm. 10.3.1]; we thus turn to they will reach at least 2>E8 (() processes after the first round. Thus these processes will decide one of the 8 values when taking the the topological approach to distributed computing for our lower smallest value they received. bounds. As for the rest of the processes, we can’t say anything about what they will receive, and thus we consider the worst case, where 4 ELEMENTS OF COMBINATORIAL they all decide differently, and not one of the 8 smallest values. Then TOPOLOGY the number of decided values is at most 8 + (= − 2>E8 (()), and the 4.1 Preliminary definitions theorem follows.  First, we need to introduce the mathematical objects that this ap- The covering numbers are also independent of processes names; proach uses. These are simplexes and complexes. A simplex is sim- we thus get a similar upper bound on symmetric models as a corol- ply a set of values, and can be represented as a generalization of a lary. triangle in higher dimensions. Simplexes capture configurations in (?3, {?3 } (%2,E1) (%1,E2) ?3 ?6 ?2

(%3,E) ?2 ?5 ?1 (?1, {?1,?3 } (?2, {?1,?2 }

?3 (%1,E1) (%2,E2) ?1 ?4 (a) Graph (b) Uninterpreted simplex (a) Bipartite graph (b) Pseudosphere Figure 2: A graph and its uninterpreted simplex Figure 3: A bipartite graph and a pseudosphere general, be them initial configurations, intermediate configurations, or decision configurations. 4.2 Uninterpreted complexes of closed-above Definition 4.1 (Simplex). Let >;B and +84FB be sets. Then f ⊆ models >;B × +84FB is a simplex on >;B and +84FB (or colored simplex) It so happens that closed-above models give rise to uninterpreted , ∀? ∈ >;B : |{E ∈ +84FB|(?,E) ∈ f}| ≤ 1. complex that are easy to define and study. Indeed, they are unions We have 2>; (f) or =0<4B(f) , {? ∈ >;B | ∃E ∈ +84FB : of pseudospheres, where pseudospheres are colored complexes (?,E) ∈ f}. And we have E84FB(f) = {E ∈ +84FB | ∃? ∈ >;B : topologically equivalent to =-spheres. These pseudospheres have (?,E) ∈ f}. We also write E84Ff (?) for the E ∈ +84FB such that already been used in the literature to study multiple models of (?,E) ∈ f computation [15, Chap. 13]. The dimension of f is |B86<0| − 1. Definition 4.5 (Pseudospheres [15, Def 13.3.1]). Let +1,+2, ...,+= be Although we define Views to be any set for readability, the sets. Then the pseudosphere complex i (Π;+8 | 8 ∈ [1, =]) , traditional view is of sets of pairs, the first element being a process • ∀8, ∀E ∈ + : (% ,E) is a vertex of . name, and the second being either another view or an initial value. 8 8 • ∀ ⊆ [1, =] : {(% ,E ) | 9 ∈ , E ∈ + } is a simplex of  iff For more details, refer to [15]. 9 9 9 9 all % are distinct. Then a complex is a set of simplexes that is closed under inclusion. 9 It captures all considered configurations. We can think of these complexes as a generalization of complete Definition 4.2 (Complex). Let >;B and +84FB be sets. Then  ∈ bipartite graphs in = dimensions. Recall that a complete bipartite P(>;B × +84FB) is a simplicial complex on >;B and +84FB (or graph is a graph that can be split into two sets of nodes, the nodes colored simplicial complex) , of each set not linked to each other and each node of one set linked to all nodes of the other set. For example, Figure 3a is a bipartite • ∀(?,E) ∈ >;B × +84FB : {(?,E)} ∈ . graph. • ∀f, g simplexes on >;B × +84FB: f ∈  ∧ g ⊆ f =⇒ g ∈ . Now a pseudosphere is the same, except that nodes can be par- , The facets of  {f ∈  | ∀g ∈  : f ⊆ g =⇒ g = f}. titioned into = sets, no simplex contains more than one element The dimension of  is the maximum dimension of its facets.  of each set as a vertex, and all the simplexes built from one ele- is called pure if all its facets have the same dimension. ment of each set are in the complex. Figure 3b is an example of How can we go from our round-based models, which are gener- a pseudosphere built from processes %1,%2,%3, and the three sets = = = ated by graphs, to simplexes and complexes? +1 {E1,E2}, +2 {E1,E2} and +3 {E}. Starting with a single graph, we define the uninterpreted simplex Among other things, pseudospheres are closed under intersec- induced by this graph. This simplex captures the configuration after tion, and are (= − 2) connected. a round using graph , simply in terms of who hears from whom. L"%%0 4.6 (I*/"$-"9/=#* #6 &-")+#-&!"$"- [15, F09/ 13.3.4]). It disregards input values, which makes it uninterpreted. i (Π;*8 | 8 ∈ [1, =]) ∩ i (Π;+8 | 8 ∈ [1, =]) = i (Π;*8 ∩ +8 | 8 ∈ Definition 4.3 (Uninterpreted simplex of a graph). Let  be a graph. [1, =]). Then the uninterpreted simplex of is f , the colored simplex  One advantage of pseudosphere is that they have high con- {(?, = (?) | ? ∈ Π }.  nectivity [15, Def. 3.5.6]. Intuitively, connectivity concerns the Given a set of graphs  representing the possible graphs, we (non-)existence of high-dimensional generalisation of holes in the generalize the previous definition to give the uninterpreted complex complexes. Since pseudospheres are topologically equivalent to of . spheres [15, Sect. 13.3], they only have these holes in the highest dimensions. Definition 4.4 (Uninterpreted complex of an oblivious model). Let  be an oblivous model defined by a set of graphs (. Then the L"%%0 4.7 (C#**"9/=:=/4 #6 &-")+#-&!"$"- [15, C#$. 13.3.7]). uninterpreted complex of  is  , the complex whose facets i (Π;+8 | 8 ∈ [1, =]) is (= − 2)-connected, where = , |{8 ∈ [1, =] | are exactly the {f |  ∈ (}. +8 ≠ ∅}|. The connectivity of the uninterpreted complex for a simple Now we can prove the connectivity of uninterpreted complexes closed-above model follows, because such a complex is a pseudo- for general closed-above models. sphere. Intuitively, for any process ?, its possible views are exactly T!"#$"% 4.12 (C#**"9/=:=/4 #6 /!" )*=*/"$&$"/"+ 9#%&8"B the upward closure of its view in the defining graph . Then the #6 0 98#-"+,0'#:" %#+"8). Let  be a closed-above model, and ( =-simplexes of the uninterpreted complex are exactly the simplex be the set of graphs from which it is built. you can build with one such view for each process. Then  is (|Π| − 2)-connected. L"%%0 4.8 (U*=*/"$&$"/"+ 9#%&8"B #6 0 -=%&8" 98#-"+,0'#:" %#+"8 =- 0 &-")+#-&!"$"). Let  be a simple closed-above model, P$##6. From the proof of Theorem 4.8, we know that  is a = = Π; union of pseudospheres:   . We want to apply the nerve and  be the graph from which it is built. Then  i ( {( |  ∈( Π = (%8 ) ⊆ ( ⊆ } | 8 ∈ [1, =]). lemma to this cover. First, by TheoremÐ 4.9,  is (= − 2)-connected. As for the intersection of any set  of  , we have two prop- P$##6. • (⊆). Let f be a =-simplex of  . By definition of   erties. First, it cannot be empty, since all  must contains the  , it is the uninterpreted simplex of a graph  ∈↑ . This   uninterpreted simplex of the complete graph on Π, by definition in turn means that ∀? ∈ Π : E84F (?) = = (?) ⊇ = (?). f   of ↑ . This gives us that the nerve complex is a simplex, and thus Thus f = {(% , = (% )) | 8 ∈ [1, =]} ⊆ i (Π; {( | = (% ) ⊆ 8  8  8 ∞-connected. ( ⊆ Π} | 8 ∈ [1, =]). And second, the intersection is also a pseudosphere, by applica- • (⊇). Let f be a n-simplex of i (Π; {( | = (% ) ⊆ ( ⊆ Π} |  8 tion of Lemma 4.6. Indeed, these are intersections of pseudospheres 8 ∈ [1, =]). Then ∀? ∈ Π : E84F (?) ⊇ = (?). Thus f is f  with the same processes which have an non-empty intersection for the uninterpreted simplex of a graph  such that ∀? ∈ Π : each color : the view of this process in the complete graph. = (?) ⊇ = (?).   We can thus conclude by application of the nerve lemma and We conclude that  ∈↑  and thus that f ∈    Theorem 4.9.

It follows instantly that the uninterpreted complexes of simple 4.3 Interpretation of uninterpreted complexes closed-above models are (|Π| − 2)-connected. We can only go so far with uninterpreted complexes; at some point, we need to consider initial values. C#$#880$4 4.9 (C#**"9/=:=/4 #6 /!" )*=*/"$&$"/"+ 9#%&8"B #6 0 -=%&8" 98#-"+,0'#:" %#+"8). Let  be a simple closed-above Definition 4.13 (Interpretation of uninterpreted simplex). Let f be Π ( − 1) model, and  be the graph from which it is built. Then  is (|Π| −2)- an uninterpreted simplex on and g be a = -simplex colored connected. by Π. Then the interpretation of f on g, f (g) , {(?,+ ) | ? ∈ Π ∧ (E ∈ + =⇒ (∃@ ∈ E84Ff (?) : E = E84Fg (@)))} From this corollary and the closure of pseudospheres by intersec- tion, we now deduce a similar characterization of the connectivity Then the same intuition can be applied to a full uninterpreted for general closed-above models. complex. But to do so, we need to first introduce the main tool in our Definition 4.14 (Interpretation of uninterpreted complex). Let A toolbox for studying connectivity of simplicial complexes: the nerve be an uninterpreted complex on Π and I be a pure (= − 1) com- lemma. This result uses a cover of a complex: a set of subcomplexes plex colored by Π. Then the interpretation of A on I, A(I) , such that their union gives the initial complex. f (g) Intuitively, the nerve lemma says that if you provide a cover of g a facet of I a complex that is "nice enough", then the connectivity of the initial f a facetÐ of A complex can be deduced from the way that the cover elements These interpretations give us protocol complexes, on which intersects. This is usually easier to determine than computing the known result on computability are applicable. connectivity directly. 4.4 A Powerful Tool Definition 4.10 (Nerve complex). Let  be a simplicial complex, On the combinatorial topology front, our results leverage two main (8 )8 ∈ a cover of . Then the nerve complex of this cover, N(8 | ) , the complex generated by tools: the impossibility result on :-set agreement based on connec- tivity [15, Thm. 10.3.1], and a way to compute the connectivity of a • the vertices are the  ; 8 complex from the way it is built. This section develops the second • and the simplexes are the sets { | 8 ∈ } for ⊆  such 8 idea. that 8 ≠ ∅ 8 ∈ Let A be a pure complex of dimension 3. We say that A is Ñ shellable if there is an ordering i1, . . . , iA of its facets such that for L"%%0 4.11 (N"$:" 8"%%0 [20, T!% 15.24]). Let  be a simplicial every 1 ≤ C ≤ A − 1, complex, (8 )8 ∈ a cover of  and : ≥ 0. Then C 38<( 8 ) ≥ (: − | | + 1) i8 ∩ iC+1 8 ∈ ∀ ⊆  : 8=1 ! 8Ñ= ∅ Ø is a pure subcomplex of dimension 3 − 1 of the boundary complex © © 8 ∈ ªª ­= ­ ®® skel3−1 ­⇒ ( is­:-connectedÑ ⇐⇒ N (8 | ) is®®:-connected). of iC+1, i.e., of iC+1. « « ¬¬ ′ C ?2 ?2 ?5 (1) For every facet i of A and every pure3-subcomplex 8=1 i8 ⊆ A satisfying that C ′ = B ′ Ð 8=1 i8 ∩i 8=1 (i ∩ f8 ) for some of A’s facets f1, . . . , fB , ′ ?1 ?4 ?3 with each f8 and i sharing a (3 − 1)-face, it holds that #ÐC  Ð ′ = B ′ 8=1 U (i8 ) ∩ U (i ) 8=1 (U (i ) ∩ U (f8 )). (2) For every C ≥ 0 and every collection i0, i1, . . . , iC of C + 1 #Ð  Ð ?3 ?1 ?4 facets of A with each i8 and i0 sharing a (3 − 1)-face, it holds C that 8=0 U (i8 ) is least (ℓ − C)-connected. (a) Shellable complex (b) Not-shellable com- Then, B is ℓ-connected. plex Ñ 5 ONE ROUND LOWER BOUNDS FOR ONE Figure 4: Examples of a complex that is shellable and one ROUND: A TOUCH OF TOPOLOGY that is not As before, we start with the simple closed-above case, where the model is the closure of a single graph. In this case the tight lower The intuition here is that the complex is the union of simplexes of bound follows from Castañeda et al. [6, Thm 5.1], as mentionned dimension 3, and there is an order in which to add simplexes, so that above. the new simplex is connected to the rest by 3 − 1 simplexes, some of its own facets. In the concrete case of 2-simplexes (triangles), T!"#$"% 5.1 (L#G"$ '#)*+ #* :,-"/ 01$""%"*/ 6#$ -=%&8" they must be connected to the rest by 1-simplexes (edges). 98#-"+,0'#:" %#+"8-). Let  a simple closed-above model gen- Here, unions and intersections apply to the complexes induced erated by the graph . Let : ≤ W (). Then :-set agreement is not by the facet and all its faces. Such a sequence of facets is a shelling solvable on  in a single round. order of A. For example, the complex in Figure 4a is shellable, but the one We thus focus on general closed-above models. Here we have to in Figure 4b is not. leverage the underlying structure of the protocol complex. We do C so through two tools : the main theorem from Section 4.4, as well Given a shelling order i1, . . . , iA of a complex A, ( =1 i8 )∩iC+1 8 as two graph parameters: the equal-domination number over a set is the union of the complexes induced by some (3−1)-facesg1, . . . , gB Ð of graphs, and the max-covering numbers of a set of graphs. of iC+1, by definition of shellability. Each g9 is a face of a facet f9 C 1 of ∪8=1i8 , hence iC+1 and f9 share a (3 − )-face. Then, Definition 5.2 (Distributed domination number of a set of graphs). C B Let ( be a set of graphs. Then the distributed domination num- = i8 ∩ iC+1 (iC+1 ∩ f9 ). ∀% ⊆ Π, ∀(8 ⊆ ( : =1 ! =1 8 9 38BC , > (|% | = 8 ∧ |(8 | = min(8, |(|)) Ø Ø ber (, W (() <8= 8 0 . We also use the following technical result.  =⇒ $DC (%) = Π       ∈(8  L"%%0 4.15 (S!"880'=8=/4 #6 -=%&8"B '#)*+0$4 [15, T!% Ð   13.2.2]). Let A be a pure (3 − 1)-dimensional sub-complex of the The difference between W4@ (() andW38BC (() is that a set ofW4@ (()   boundary complex of a simplex of dimension 3. Then A is shellable, processes dominates each graph of ( separately, whereas a set and any sequence of its facets is a shelling order for A. of W38BC (() processes might not dominate any graph of (, but it dominates every subset of 8 graphs of ( together. Thus W38BC (() ≤ Finally, we rely on the straightforward corollary of the nerve W4@ ((). Fitting, considering the former is used in lower bounds and lemma for a cover with two elements. the latter in upper bounds. C#$#880$4 4.16 (TG# "8"%"*/- *"$:" 8"%%0). Let  and be Next, the max-covering numbers are quite subtle. For 8 < W38BC ((), :-connected complexes. If  ∩ is (: − 1)-connected, then  ∪ is the 8-th max-covering number of ( is the maximum number of pro- :-connected. cesses hearing a set of 8 procs, summed over 8 graphs in (. We now state the main technical result of the section. It extends That is, the max-covering numbers capture how much values the result from Castañeda et al. [6] and adapts it to the interpretation can be disseminated in the best case. They serve in lower bounds of complexes we need here. While Castañeda et al. studied the by giving a best case scenario on which we can focus to prove complex given by the interpretation of a single graph (to capture impossibility. models like LOCAL and CONGEST [25]), we care about the Definition 5.3 (Max covering numbers of a set of graphs). Let ( complex resulting of the interpretation of a set of graphs. be a set of graphs and 8 < W38BC ((). Then the 8-th max-covering We thus send each input simplex into a complex, and show number of (, max-cov8 (() , max |( $DC (%))|. Π = that if both the output complexes and the mapping are "nice", the % ⊆ ,|% | 8  ∈(8 = interpreted complex is highly connected. (8 ⊆(,|(8 | min(8,|( |) Ð $DC (%)≠Π L"%%0 4.17. Let A be a pure shellable complex of dimension 3, ∈(8 We also define the 8-th max-coveringÐ coefficients on (, "8 (() , B a complex, (B8 )8 ∈ a cover of B, and ℓ ≥ 0 an integer. Suppose =−8−1 if <0G-2>E8 (() > 8 that there is a bijection U between the facets of A and the elements <0G-2>E8 (()−8 of (B8 )8 ∈ such that: ( =j − 8 k if <0G-2>E8 (() = 8 T!"#$"% 5.4 (L#G"$ '#)*+ #* :,-"/ 01$""%"*/ 6#$ 1"*"$08 6.1 Closure-above is not invariant by product, 98#-"+,0'#:" %#+"8-). Let  a closed-above model generated by but its still works the set of graphs (. What is the pitfall mentionned above? Quite simply, that the product = 38BC 38BC Let ; min(W (() − 2, min{C + "C (() − 2 | C ∈ [1,W (() − 1]}) of two closed-above models does not necessarily gives a closed- Then (; + 1)-set agreement is not solvable on  in a single round. above model. This follows from the fact that the closure-above of a product of graphs doesn’t always equal the product of the The term depending on W38BC (() in the lower bound serves when closure-above of the graphs. the max-covering numbers are not sufficient to distinguish adver- Let’s take an example: the product of a cycle with itself. saries with different properties. Consider for example the symmetric models of all unions of B stars, with B ≤ =. Then for those graphs, ?1 ?2 ?1 ?2 ?1 ?2 for C < W38BC ((), we have <0G-2>E (() = t, and thus " (() = = − C. C C ?6 ?3 ?6 ?3 = ?6 ?3 Hence the minimum over the C + "C (() − 2 is = − 2. But this would mean that (= − 1)-set agreement is impossible for Ë ?5 ?4 ?5 ?4 ?5 ?4 B < =, whereas we can clearly solve 2-set agreement for B = = − 1, 38BC for example. What depends on B is W (() itself. More precisely, Then we cannot build the following graph by extending the 38BC W (() = = − B + 1, because given %, we can consider only the cycles and taking the product: graph where the B centers of stars are in Π \ %, up until the point ? ? where |% | > = − B. 1 2 Hence our lower bound shows that for the symmetric union of B stars, (= − B)-set agreement is impossible in one round. Given ?6 ?3 that our upper bounds above tell that (= − B + 1)-set agreement is possible in one round for this model, the bound is tight. Finally, the bound can be specialized for symmetric models. ?5 ?4

C#$#880$4 5.5 (L#G"$ '#)*+- 6#$ -4%%"/$=9 98#-"+,0'#:" Why? Simply put, adding the new edge to either of the two %#+"8). Let  be a graph. Let ; = cycles necessarily creates other edges in the product. Adding an 38BC min(W ((~<()) − 2, min edge from ?2 to any other node than ?3 and ?4 also creates new C ∈[1,W38BC ((~<())−1] edges; so does adding an edge to ?4 and then an edge from ?4 to =−C−1 C + − 2 if <0G-2>EC ({}) > C ?6, or an edge from ?3 to ?6 in the second graph. C (<0G-2>EC ({ })−C) = −j2 k if <0G-2>EC ({}) = C ! Hence the product of the closure above of this cycle with itself Then (; + 1)-set agreement is not solvable on (~<(↑ ) in a single is not the closure-above of the squared cycle. To put it differently, round. closure-above is not invariant by the product operation. Nonetheless, the bell does not toll for our hopes of extending Notice that all these lower bounds are valid for general algo- our properties. What is used in the lower bound proofs above rithms, not only oblivious ones. The reason is that a one round full is not closure-above itself, but its consequences: being a union information protocol is an oblivious algorithm. of pseudospheres containing the full simplex, such that for each pseudosphere, all graphs contain the smallest graph. All three properties are present in a specific subset of the product 6 MULTIPLE ROUNDS of two simple closed-above models: all products where edges might Given that we focus on oblivious algorithms, a natural approach to be added to the last graph in the product but not to the other. Each extending our lower bounds to the multiple rounds case is to look added edge only alters the view of its destination, since it is in the at the product of our graphs. By product, we mean the graph of second graph, and multiple added edges don’t interfere because the paths with one edge per graph. Thus the products of A graphs they are all added to the same graph. Hence we can change the capture who will hear who after A corresponding communication views of processes one at a time, and thus we get a pseudosphere. rounds. Since adding no edge gives the original product and adding all missing edges gives the clique, we get the other two properties. Definition 6.1 (Graph path product). Let  and  be graphs with Then taking this subset of the product of two general closed-above auto-loops (∀E ∈ Π : (E,E) ∈ () ∧ (E,E) ∈ ()). Then their models result in a union of pseudosphres, one for each product of graph path product   , the graph (Π, ) such that ∀D,E ∈ the underlying graphs. Π : (D,E) ∈ Π =⇒ ∃F ∈ Π : (D,F) ∈ () ∧ (F,E) ∈ (). Therefore we can extract relevant subcomplexes from the prod- Ë uct of closed-above models, and then the lower bounds only depends Since we have a graph as the result, we can apply our lower on the properties of the underlying product of graphs. bounds for one round. At least, if the resulting graph still satisfy the hypotheses of our lower bounds. It does, although product 6.2 Upper bounds for multiple rounds doesn’t maintain closure-above. This subtlety is explained in the Even if we just explained how to deal with lower bounds for multiple next subsection. bounds, we still start by giving upper bounds for multiple rounds. A This is for the same reason as in the one round case: the upper- then implies by Lemma 6.2 that it is solvable on ↑ 8 , bounds require no combinatorial topology, and they allow us to 1,...,A ∈( 8=1  introduce concepts needed for the lower bounds. that is on . Ð Ë First, we need to prove a little result that is enough for our upper One issue with these bounds is that they require the computa- bounds: that the product of closed-above models is included in the tion of possibly many products, as well as the computation of the closure-above of the product. combinatorial numbers for a lot of graphs. One alternative is to L"%%0 6.2 (P$#+)9/ 0*+ =*98)-=#* 6#$ 98#-"+,0'#:"). Let forsake the best bound we can get for one that can be computed  and  be two graphs. Then ↑  ↑  ⊆↑ ( ). using only the numbers for the initial graphs. This hinges on covering number sequences. Recall that the 8-th Ë ′ Ë′ : = P$##6. Let ∈↑  ↑ . Thus ∃ ∈↑ , ∃ ∈↑  covering number of a graph is the minimum number of processes  ′  ′. Let D,E ∈ Π such that (D,E) ∈  . We show that Ë hearing a set of 8 processes that do not broadcast. In a sense, it gives (D,E) ∈ ; this will entail that ∈↑ ( ). Ë Ë the guaranty of propragation of information by a set of 8 processes. Because (D,E) ∈  , ∃F ∈ Π : (D,F) ∈ () ∧ (F,E) ∈ ′ ′ Ë ′ That’s the whole story for one round. But what happens when (). But ∈↑  and  ∈↑ , therefore (D,F) ∈ ( )∧(F,E) ∈ you do multiple rounds? Then, if the 8-th covering number of the ′ Ë ′ ′ =  ( ). We conclude that (D,E) ∈   . graph is greater than 8, this means that in the next rounds, the What this means is that taking theË closure-above of the products minimum number of people who will hear the value of the 8 initial of our graphs over-approximate the actual model after A rounds. processes is the 2>E8 -th covering number. And if this number is And thus, algorithms working on these approximations work on greater than 2>E8 , this repeats. the actual model. Covering number sequences capture this process. One can also Now, let us start with simple closed-above models. Just like see them as the sequences of covering numbers for powers of the for the one round case, they are completely characterized by the graph. domination number of their underlying graph. Definition 6.6 (Covering number sequences). Let  be a graph. , 8 T!"#$"% 6.3 (U&&"$ '#)*+ (%)8/=&8" $#)*+-) 6#$ -=%&8" Then the 8-th covering numbers sequence of  (B9 )9 ∈N∗ such 98#-"+,0'#:" %#+"8-). |Π| 8 ≥ 4@ ( ) Let  be a simple closed-above model de- 8 8 if B: W  A thatB = 2>E8 () and ∀: ≥ 1 : B = 8 4@ fined by the graph . Let A > 0. Then W ( )-set agreement is solvable 1 :+1 2>E 8 () if B < W () B : ! in A rounds in . : Armed with these sequences, we get an upper bound directly A A P$##6. We have that W ( )-set agreement is solvable on ↑  from . by Theorem 3.2. This then implies by Lemma 6.2 that it is solvable on (↑ )A , that is on .  T!"#$"% 6.7 (U&&"$ '#)*+- #* :,-"/ 01$""%"*/ '4 9#:"$=*1 *)%'"$- -"H"*9"-). Let  be a simple closed-above model defined But for general closed-above models, one cannot use the domi- by the graph  on Π. Then if the 8-th covering sequence of  reaches nation number itself, because one cannot know which of the un- = at some point, 8-set agreement is solvable on the model . derlying graphs will be there. As in the one round case, we use the equal-domination number and covering numbers. We can adapt this bound for general closed-above models by generalizing the covering numbers sequences to a set of graphs. T!"#$"% 6.4 (U&&"$ '#)*+ (%)8/=&8" $#)*+-) #* :,-"/ 01$"", %"*/ '4 W4@ (() 6#$ 1"*"$08 98#-"+,0'#:" %#+"8-). Let  be Definition 6.8 (Covering numbers sequences for sets of graphs). Let a general closed-above model generated by the set of graphs (. Let B be set of graphs. Then the 8-th covering numbers sequence of > 4@ A ( , (B9 )9 ∈N∗ such that B1 = min2>E8 () and A 0. Then W (( )-set agreement is solvable in A rounds on .  ∈( 4@ A = if B: ≥ max:4@−3>< () P$##6. We have that W (( )-set agreement is solvable on 1 : =  ∈( A ∀: ≥ B:+1 < min2>EB: () if B: max :4@−3>< () ↑ 8 by Theorem 3.4. This then implies by Lemma 6.2  ∈(  ∈( ∈ =1 © ª 1,...,A ( 8 ­ ® Ë A T!"#$"% 6.9 (U&&"$ '#)*+- #* :,-"/ 01$""%"*/ '4 9#:"$=*1 Ð  that it is solvable on ↑ 8 , that is on . *)%'"$- -"H"*9"-« 6#$ 1"*"$08 98#-"+,0'#:" %#+"8-).¬Let (  ,..., ∈( =1 1 A 8 be a set of graph on Π. Then if the 8-th covering sequence of ( reaches = Ð Ë T!"#$"% 6.5 (U&&"$ '#)*+- (%)8/=&8" $#)*+-) #* :,-"/ 01$"", at some point, 8-set agreement is solvable on the oblivious closed-above %"*/ '4 9#:"$=*1 *)%'"$- 6#$ 1"*"$08 98#-"+,0'#:" %#+"8-). model generated by (. Let  be a general closed-above model generated by the set of graphs (. 4@ A A P$##6. If the 8-th covering number sequence of ( reaches = after Let A > 0. Then ∀8 ∈ [1,W (( )[: (8 + (= −2>E8 (( )))-set agreement is solvable on the oblivious closed-above model generated by ( in A step A, this means that every set of 8 processes is heard by everyone rounds. after A rounds. In particular, the 8 processes with the smallest initial values will be heard by everyone. 4@ A A P$##6. We have that ∀8 ∈ [1,W (( )[: (8 + (= − 2>E8 (( )))-set Hence sending all the values heard for now for A rounds, and then A deciding the smallest value received, ensures that one of the 8-th agreement is solvable on ↑ 8 by Theorem 3.7. This  1,...,A ∈( 8=1 smallest values will be chosen, and thus solves 8-set agreement. Ð Ë 6.3 Lower bounds for multiple rounds algorithms. In Structural Information and Communication Complexity, pages 3–18, 2019. doi:10.1007/978-3-030-24922-9_1. T!"#$"% 6.10 (L#G"$ '#)*+ (%)8/=&8" $#)*+-) #* :,-"/ [7] Armando Castañeda and Sergio Rajsbaum. New combinatorial topology bounds 01$""%"*/ 6#$ -=%&8" 98#-"+,0'#:" %#+"8-). Let A > 0 and for renaming: the lower bound. Distributed Computing, 22(5):287–301, Aug 2010. let  a simple closed-above model generated by the graph . doi:10.1007/s00446-010-0108-2. [8] Bernadette Charron-Bost, Matthias Függer, and Thomas Nowak. Approximate Then (W () − 1)-set agreement is not solvable on  in A rounds by an consensus in highly dynamic networks: The role of averaging algorithms. In oblivious algorithm. Automata, Languages, and Programming, pages 528–539, 2015. doi:10.1007/ 978-3-662-47666-6_42. T!"#$"% 6.11 (L#G"$ '#)*+ (%)8/=&8" $#)*+-) #* :,-"/ [9] Bernadette Charron-Bost and André Schiper. The heard-of model: computing in distributed systems with benign faults. Distributed Computing, 22(1):49–71, 01$""%"*/ 6#$ 1"*"$08 98#-"+,0'#:" %#+"8-). Let A > 0 and April 2009. doi:10.1007/s00446-009-0084-6. let  be a closed-above model generated by the set of graphs (. [10] Soma Chaudhuri. More choices allow more faults: Set consensus problems in = 38BC A A 38BC A totally asynchronous systems. Information and Computation, 105(1):132–158, Let ; min(W (( )−2, min{C+"C (( )−2 | C ∈ [1,W (( )−1]}) 1993. doi:https://doi.org/10.1006/inco.1993.1043. 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