Universal Communication, Universal Graphs, and Graph Labeling
aa r X i v : . [ c s . CC ] N ov Universal Communication, Universal Graphs, and Graph Labeling
Nathaniel Harms ∗ November 12, 2019
Abstract
We introduce a communication model called universal SMP , in which Alice and Bob receive a function f belonging to a family F , and inputs x and y . Alice and Bob use shared randomness to send a messageto a third party who cannot see f , x , y , or the shared randomness, and must decide f ( x, y ). Our mainapplication of universal SMP is to relate communication complexity to graph labeling, where the goalis to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices x and y can be determined from the labels ℓ ( x ) , ℓ ( y ). We give a universal SMP protocol using O ( k )bits of communication for deciding whether two vertices have distance at most k in distributive lattices(generalizing the k -Hamming Distance problem in communication complexity), and explain how thisimplies a O ( k log n ) labeling scheme for deciding dist ( x, y ) ≤ k on distributive lattices with size n ; incontrast, we show that a universal SMP protocol for determining dist ( x, y ) ≤ n / ) communication cost. On the other hand, wedemonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees,low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency.Trees also have an O ( k ) protocol for deciding dist ( x, y ) ≤ k and planar graphs have an O (1) protocol for dist ( x, y ) ≤
2, which implies a new O (log n ) labeling scheme for the same problem on planar graphs. In the simultaneous message passing (SMP) model of communication, introduced by Yao [Yao79], Alice andBob separately receive inputs x and y to a function f . They send messages a ( x ) , b ( y ) to a third party, calledthe referee, who knows f and must output f ( x, y ) (with high probability) using the messages a ( x ) , b ( y ). Butwhat if the referee doesn’t know f ? Can they still compute f ( x, y )? Yes: Alice can include in her messagea description of f , and then the referee knows it; however, if f is restricted, they can sometimes do muchbetter. Here is a simple example: the players receive vertices x, y ∈ { , . . . , n } in a graph G of maximumdegree 2, and want to decide if ( x, y ) is an edge in G . Sharing a source of randomness, Alice and Bobrandomly label each vertex of G with a number up to 200; Alice sends the label of both neighbors of x andBob sends the label of y . The referee says yes if one of Alice’s labels matches the label of y , no otherwise.They will be correct with probability at least 99 / G . This is also anexample where the referee can decide many problems using only one strategy. In this work we will see thatmore interesting families of graphs, such as trees, planar graphs, and distributive lattices, also exhibit thesephenomena, even when we wish to compute distances instead of just adjacency.To study this, we introduce the universal SMP model, which operates as follows. Fix some family F offunctions. Alice and Bob receive a function f ∈ F and inputs x, y , and they use shared randomness to eachsend one message to the referee. The referee knows the family F and the size of the inputs, but doesn’tknow f, x, y or the shared randomness, and must compute f ( x, y ) with high probability. By choosing thefamily F to be the singleton family, one sees that this model includes standard SMP. As in the earlierexample, we will be studying communication problems on graphs, but this is not a significant restriction: ∗ University of Waterloo. [email protected] f is equivalent to determining adjacency in some graph (use f as the adjacency matrix), so we will treat F as a family of graphs.A surprising but intuitive application of universal SMP is that it connects two apparently disjoint areas ofstudy: communication complexity and graph labeling. For a graph family F , the graph labeling problem(introduced by Kannan, Naor, and Rudich [KNR92]) asks how to assign the shortest possible labels ℓ ( v ) toeach vertex v of a graph G ∈ F , so that the adjacency (or some other function [Pel05]) of vertices x, y canbe computed from ℓ ( x ) , ℓ ( y ) by a decoder that knows F . We observe the following principle (Theorem 1.1): If there is a (randomized) universal SMP protocol for the graph family F with communication cost c , thenthere is a labeling scheme for graphs G ∈ F with labels of size O ( c log n ) , where n is the number of vertices. Common variants of graph labeling are distance labeling [GPPR04], where the goal is to compute dist ( x, y )from the labels, and small-distance labeling, where the goal is to compute dist ( x, y ) if it is at most k andoutput “ > k ” otherwise [KM01, ABR05]. This is similar to the well-studied k -Hamming Distance problemin communication complexity, where the players must decide if their vertices x, y have distance at most k in the Boolean hypercube graph. A natural generalization of the Boolean hypercube is the family ofdistributive lattices (which also include, for example, the hypergrids). We demonstrate that techniques fromcommunication complexity can be used to obtain new graph labelings, by adapting the k -Hamming Distanceprotocol of Huang et al. [HSZZ06] to the universal SMP model, achieving an O ( k ) protocol for computing dist ( x, y ) ≤ k and the corresponding k -distance labeling scheme with label size O ( k log n ). It is interestingto note that, in contrast to the standard application of communication complexity as a method for obtaininglower bounds, we are using it to obtain upper bounds.Generalizing in another direction, we ask: for which graphs other than the Boolean hypercube can we obtainefficient communication protocols for k -distance? For constant k , k -Hamming Distance can be computed withcommunication cost O (1); which other graphs admit a constant-cost protocol? To approach this question, weobserve that many (but not all) graph families known to have efficient O (log n ) adjacency labeling schemesalso admit an O (1) universal SMP protocol for adjacency. Commonly studied families in the adjacencyand distance labeling literature are trees [KNR92, KM01, ABR05, AGHP16, ADK17] and planar graphs[KNR92, GPPR04, GL07, GU16, AKTZ19]. We study the k -distance problem on these families and findthat trees admit an O ( k ) protocol, while planar graphs admit an O (1) protocol for 2-distance; this impliesa new labeling scheme for planar graphs.Further motivation for the universal SMP model comes from universal graphs . Introduced by Rado [Rad64],an induced-universal graph U for a set F is one that contains each G ∈ F as an induced subgraph. Anefficient adjacency labeling scheme for a set F implies a small induced-universal graph for that set [KNR92].Deterministic universal SMP protocols are equivalent to universal graphs (Theorem 1.7), and we introduce probabilistic universal graphs as the analogous objects for randomized univeral SMP protocols. We thinkprobabilistic universal graphs are worthy of study alongside universal graphs, especially since many non-trivial families admit one of constant-size .The universal SMP model is also related to a recent line of work studying communication between partieswith imperfect knowledge of each other’s “context”. The most relevant incarnation of this idea is the recentwork [GS17, GKKS18], who study the 2-way communication model where Alice and Bob receive functions f and g respectively, with inputs x and y , and must compute f ( x, y ) under the guarantee that f and g areclose in some metric. In other words, one party does not have full knowledge of the function to be computed.The universal SMP model provides a framework for studying a similar problem in the SMP setting, wherethe players know the function but the referee does not; the similarity is especially clear when we define thefamily F to be all graphs of distance δ to a reference graph G in some metric (we discuss this situation inmore detail at the end of the paper). This could model, for example, a situation where the clients of a serviceoperate in a shared environment but the server does not; or, a situation in which the clients want to keeptheir shared environment secret from the server, and their inputs secret from each other. This suggests apossible application to privacy and security. A relevant example is private proximity testing (e.g. [NTL + k from each other,without revealing to each other or the server their exact locations.2he Discussion at the end of the paper highlights some interesting questions and open problems. A universal SMP protocol decides k -distance for a family F if for all graphs G ∈ F and vertices x, y , theprotocol will correctly decide if dist ( x, y ) ≤ k , with high probability. A labeling scheme decides k -distanceif dist ( x, y ) ≤ k can be decided from the labels of x, y . Below, the variable n always refers to the number ofvertices in the input graph. Implicit graph representations.
The main principle connecting communication and graph labeling is:
Theorem 1.1.
Any graph family F with universal SMP cost m has an adjacency labeling scheme with labelsof size O ( m log n ) . In particular, if the universal SMP cost for F is O (1) then F has an O (log n ) adjacencylabeling scheme. Adjacency labeling schemes of size O (log n ) are of special interest because log n is the minimum number ofbits required to label each vertex uniquely, and they correspond to implicit graph representations , as definedby Kannan, Naor, and Rudich [KNR92] (we omit their requirement that the encoding and decoding becomputable in polynomial-time). Section 2.3 elaborates further. To obtain implicit representations, we canrelax our requirements: Corollary 1.2.
For any constant c , any graph family F where each G ∈ F has a public-coin 2-way commu-nication protocol computing adjacency with cost c has an implicit representation. Distributive & Modular Lattices.
Distributive and modular lattices are generalizations of the Booleanhypercube and hypergrids (see Section 3 for definitions). We define a weakly-universal
SMP protocol as onewhere the referee shares the randomness of Alice and Bob. For distributive lattices we get the following:
Theorem 1.3.
The k -distance problem on the family of distributive lattices has: a weakly-universal SMPprotocol with cost O ( k log k ) ; a universal SMP protocol with cost O ( k ) ; and a size O ( k log n ) labelingscheme. Modular lattices are a superset of distributive lattices, but they do not admit k -distance protocols witha cost independent of n ; we show that any universal SMP protocol (and any labeling scheme) deciding 2-distance must have cost Ω( n / ) (Theorem 3.14). To our knowledge, there are no known labeling schemes fordistributive or modular lattices. Our adjacency labeling scheme (i.e. for k = 1) requires O ( n log n ) space tostore the whole lattice; this can be compared to Munro and Sinnamon [MS18], who present a data structuresof size O ( n log n ) for distributive lattices that supports meet and join operations (and therefore distancequeries, due to our Lemma 3.5). However, these are not labelings, so the result is not directly comparable. Planar graphs and other efficiently-labelable families.
When they introduced graph labeling, Kan-nan, Naor, and Rudich [KNR92] studied trees, low-arboricity graphs (whose edges can be partitioned intoa small number of trees), and planar graphs, and interval graphs (whose vertices are intervals in R , withan edge if the intervals intersect), among others. These families have O (log n ) adjacency labeling schemes.Trees, low-arboricity graphs, and planar graphs have constant-cost universal SMP protocols for adjacency.Trees admit an efficient k -distance protocol: Theorem 1.4.
The family of trees has a universal SMP protocol deciding k -distance with cost O ( k ) and a O ( k log n ) labeling scheme deciding k -distance. Planar graphs admit an efficient 2-distance protocol, which implies a new 2-distance labeling scheme:3 heorem 1.5.
The 2-distance problem on the family of planar graphs has a universal SMP protocol withcost O (1) and a labeling scheme of size O (log n ) . On the other hand, a universal SMP protocol deciding 2-distance on the family of graphs with arboricity2 has cost at least Ω( √ n ) (Proposition 4.4), and a universal SMP protocol deciding adjacency in intervalgraphs has cost Θ(log n ) (Proposition 4.5).Gavoille et al. [GPPR04] showed that trees have an O (log n ) labeling allowing dist ( x, y ) to be computedexactly from labels of x, y , and gave a matching lower bound; Kaplan and Milo [KM01] and Alstrup et al [ABR05] studied k -distance for trees, with the latter achieveing a log n + O ( k (log log n + log k )) labelingscheme. For planar graphs, [GPPR04] gives a lower bound of Ω( n / ) for computing distances exactly, andan upper bound of O ( √ n log n ), which was later improved to O ( √ n ) in [GU16]. Communication Complexity.
Our lower bounds are achieved by reduction from the family of all graphs,which has complexity Θ( n ), in contrast to the upper bound of ⌈ log n ⌉ for the standard SMP cost of computingadjacency in any graph (since Alice and Bob can send ⌈ log n ⌉ bits to identify their vertices). Theorem 1.6.
For the family G of all graphs, the universal SMP cost of computing adjacency in G is Θ( n ) . The basic relationships between universal SMP, standard SMP, and universal graphs are as follows. Below, weuse D k ( Adj ( G )) and R k ( Adj ( G )) for the deterministic and randomized (standard) SMP cost of computingadjacency on G , and D univ ( F ) , R univ ( F ) for the deterministic and randomized universal SMP cost for com-puting adjacency in the family F . We use the term “ ⊏ -universal graph” as opposed to “induced-universal”to denote a slightly different object that allows non-injective embeddings (see Section 2 for definitions). Theorem 1.7.
For a set F , the following relationships hold. Let U range over the set of all ⊏ -universalgraphs: max G ∈F D k ( Adj ( G )) ≤ D univ ( F i ) = min U D k ( Adj ( U )) = min U ⌈ log | U |⌉ , with equality on the left iff ∃ H ∈ F such that ∀ G ∈ F , G can be embedded in H . For e U ranging over theset of all probabilistic universal graphs : max G ∈F R k ( Adj ( G )) ≤ R univ ( F ) ≤ min e U D k ( Adj ( e U )) ≤ O (cid:0) R univ ( F ) (cid:1) . Randomized and deterministic universal SMP satisfy Ω (cid:18) D univ ( F )log n (cid:19) ≤ R univ ( F ) ≤ D univ ( F ) . The above results on graph labeling are proved through the relationship between randomized and deter-ministic universal SMP. We obtain this relationship by adapting Newman’s Theorem [New91], a standardderandomization result in communication complexity. Finally, we note the interesting fact that universalSMP characterizes the gap between standard SMP models where the referee does or does not share therandomness with Alice and Bob:
Proposition 1.8 (Informal) . Let F be a family of graphs and let Π be a weakly-universal SMP protocol for F , which defines a distribution over the referee’s decision functions F , which we interpret as the adjacencymatrices of graphs. Let U Π be the family on which this distribution is supported. Then, taking the minimumover all such protocols Π , R univ ǫ ( F ) = min Π D univ ( U Π ) . .2 Other Related Work Graph labeling.
Randomized labeling schemes for trees have been studied by Fraigniaud and Korman[FK09], who give a randomized adjacency labeling scheme of O (1) bits per label that has one-sided error (i.e. itcan erroneously report that x, y are adjacent when they are not), and they show that achieving one-sidederror in the opposite direction requires a randomized labeling with Ω(log n ) bits. They also give randomizedschemes for determining if x is an ancestor of y , but they do not address distance problems. Spinrad’sbook [Spi03] has a chapter on implicit graphs and Alstrup et al. [AKTZ19] for a recent survey on adjacencylabeling schemes and induced-universal graphs. We know of no labeling schemes for lattices, but Fraigniaudand Korman [FK16] recently studied adjacency labeling schemes for posets of low “tree-dimension”. Distance-preserving labeling studies an opposite problem to k -distance labeling, where distances must beaccurately reported when they are above some threshold D . Recent work includes Alstrup et al. [ADKP16].To our knowledge, k -distance or even 2-distance has not been studied for planar graphs, but there are manyresults on other types of planar graph labelings with restrictions at distance 2. An example is the frequencyassignment problem or L ( p, q ) -labeling problem, which asks how to construct a labeling ℓ assigning integers[ k ] to vertices of a planar graph so that dist ( x, y ) ≤ ⇒ | ℓ ( x ) − ℓ ( y ) | ≥ p and dist ( x, y ) ≤ ⇒| ℓ ( x ) − ℓ ( y ) | ≥ q , with various optimization goals. See [Cal11] for a survey. Uncertain communication.
There are several works studying communication problems where the partiesdo not agree on the function to be computed, starting with Goldreich, Juba, and Sudan [GJS12] who studiedcommunication where parties have different “goals”. Canonne et al. [CGMS17] study communication in theshared randomness setting where the randomness is shared imperfectly. Haramarty and Sudan [HS16] studycompression (´a la Shannon) in situations where the parties do not agree on a common distribution. Asmentioned earlier, Ghazi et al. [GKKS18] and Ghazi and Sudan [GS17] study 2-way communication wherethe parties do not agree on the function to be computed. [ k ] means { , . . . , k } . The letter n always denotes the number of vertices in a graph. We use the notation [ E ] = 1 iff the statement E holds, and [ E ] = 0 otherwise. For a graph G , V ( G ) is the set of vertices and E ( G ) is the set of edges. For vertices x, y , we write G ( x, y ) = [ x, y are adjacent in G ] for the entry in theadjacency matrix of G . For an undirected, unweighted graph G and vertices u, v, dist ( u, v ) is the length ofthe shortest path from u to v .For any graph G and integer k , we denote by G k the k -closure of G , where two vertices u, v are adjacent iff dist ( u, v ) ≤ k in G ; it is convenient to require that each vertex is adjacent to itself in G k . For a set of graphs F , F k = { G k : G ∈ F} . D k ( f ) is the deterministic SMP cost of the function f and R k ( f ) is the randomized SMP cost of the function f , in the model where Alice and Bob share randomness but the deterministic referee does not. In this paper we focus on deciding adjacency. Every Boolean communication problem f : X × Y → { , } on finite domains X , Y is equivalent to the adjacency problem on the graph G with vertex set X ∪ Y and G ( u, v ) = f ( u, v ). We may either allow self-loops in G if X = Y or take G to be bipartite. We will generallypermit graphs to have self-loops. 5 efinition 2.1. A family of graphs F = ( F i ) is a sequence of sets F i indexed by integers i , along with astrictly increasing size function n ( i ), so that F i is a set of graphs with vertex set [ n ( i )]. If F i has size n ( i ) = i then we write F n . Definition 2.2 (Universal SMP and Variations) . Let F be a family of graphs with size function n and letΦ be an operation taking size n ( i ) graphs to size n ( i ) graphs. Let c : N → N and let ǫ > ǫ -error, cost c sequence of universal SMP communication protocols for F is as follows. For any i ∈ N , aprotocol Π i for F i is a triple ( a i , b i , F i ) where: • Alice and Bob receive (
G, x ) , ( G, y ) respectively, where G ∈ F i and x, y ∈ V ( G ) = [ n ( i )]; • Alice and Bob share a random string r and compute messages a i ( r, G, x ) , b i ( r, G, y ) ∈ { , } c ( i ) , re-spectively; • For each i , the (deterministic) referee has a function F i : { , } c ( i ) × { , } c ( i ) → { , } , called the decision function . F i ( a i ( r, G, x ) , b i ( r, G, y )) must satisfy:1. If x, y are adjacent in Φ( G ) then P r [ F i ( a i ( r, G, x ) , b i ( r, G, y )) = 1] > − ǫ ; and2. If x, y are not adjacent in Φ( G ) then P r [ F i ( a i ( r, G, x ) , b i ( r, G, y ))] < ǫ .A universal SMP protocol is symmetric when the functions a i , b i computed by Alice and Bob are identicaland the function F i satisfies F i ( a, b ) = F i ( b, a ) for all messages a, b ∈ { , } c . We write R univ ǫ (Φ( F )) for thecommunication complexity in the universal SMP model of computing adjacency in graphs Φ( F ) = { Φ( G ) : G ∈ F} , where ǫ is the allowed probability of error. We write R univ (Φ( F )) for R univ / (Φ( F )). If no operationΦ is specified, it is assumed to be the identity.It is also convenient to define a weakly-universal SMP protocol as a universal SMP protocol where the refereecan see the shared randomness, so the choice function is of the form F i ( r, a ( r, G, x ) , b ( r, G, y )) for randomseed r , graph G ∈ F , and x, y ∈ V ( G ). We denote the ǫ -error complexity in this model with R weak ǫ (Φ( F )).Finally, we write D univ (Φ( F )) for the deterministic universal SMP complexity. Remark 2.3.
We include the operator Φ in the definition to emphasize that the players are given theoriginal graph G , not the graph Φ( G ); for example, the players are not given G k (from which it may bedifficult to compute G ), but are instead given G . We will show that a deterministic universal SMP protocol is equivalent to an embedding into a ⊏ -universalgraph, which we we define using the following notion of embedding (following the terminology of Rado[Rad64]): Definition 2.4.
For graphs
G, H , a mapping φ : V ( G ) → V ( H ) is an embedding iff ∀ u, v ∈ V ( G ), G ( u, v ) = H ( φ ( u ) , φ ( v )). If such a mapping exists we write G ⊏ H .For a set of graphs F i , a graph U is ⊏ -universal if ∀ G ∈ F i , G ⊏ U ; i.e. ∀ G ∈ F i there exists an embedding φ G : V ( G ) → V ( U ). For a family of graphs F = ( F i ), a sequence U = ( U i ) is a ⊏ -universal graph sequence if for each i , U i is ⊏ -universal for F i .Define an equivalence relation on V ( G ) by u ≡ v iff ∀ w ∈ V ( G ) , G ( u, w ) = G ( v, w ), i.e. u, v have identicalrows in the adjacency matrix. For a graph G , define the ≡ -reduction G ≡ as a graph on the equivalenceclasses C of V ( G ) with U, W ∈ C adjacent iff ∃ u ∈ U, w ∈ W such that u, w are adjacent.An embedding is not the same as a homomorphism since we must map non-edges to non-edges, and G ⊏ H isnot the same as G being an induced subgraph of H since the mapping is not necessarily injective. Thereforea universal graph by our definition is not the same as an induced-universal graph, where G must exist as aninduced subgraph. We could for example map the path a — b — c a ′ — b ′ — a ′ . This difference between6efinitions is captured by the ≡ relation between vertices. It is necessary to allow self-loops, otherwise the ⊏ relation is not transitive. The important properties of ⊏ , ≡ , and ≡ -reductions are stated in the nextproposition; the proofs are routine and for completeness are included in the appendix. The relation ≃ is theisomorphism relation on graphs. Proposition 2.5.
The following properties are satisfied by the ⊏ relation, the ≡ relation, and ≡ -reductions:1. ⊏ is transitive.2. For any graph G and u, v ∈ V ( G ) , u ≡ v iff there exists H and an embedding φ : G → H such that φ ( u ) = φ ( v ) .3. For any graph G, ( G ≡ ) ≡ ≃ G ≡ .4. For any graph G, G ⊏ G ≡ and G ≡ ⊏ G .5. For any graphs G, H , G ⊏ H iff G ≡ ⊏ H ≡ .6. For any graphs G, H , G ≡ ⊏ H ≡ iff G ≡ is an induced subgraph of H ≡ . These properties allows us to prove relationships between the standard SMP model, deterministic universalSMP, and ⊏ -universal graphs. First we show that deterministic universal SMP protocols can always be madesymmetric . Proposition 2.6. If Π is a deterministic universal SMP protocol for the set F , then there exists a deter-ministic universal SMP protocol Π ′ that is symmetric and has the same cost as Π .Proof. Let G ∈ F and let a, b : V ( G ) → { , } m be the encoding functions for G and F the decision functionfor graphs of size | G | . The restriction of b to the domain V ( G ≡ ) → { , } m is injective so it has an inverse b − : image( b ) → V ( G ≡ ) that satisfies b − b ( x ) ≡ x ; the same holds for a, a − . Define the encoding function b ′ : V ( G ) → { , } m as b ′ = ab − b and define the decision function F ′ ( p, q ) = F ( p, ba − ( q )). Then for any x, y ∈ V ( G ) , F ′ ( a ( x ) , b ′ ( y )) = F ( a ( x ) , ba − ab − b ( y )) = F ( a ( x ) , b ( y )) = G ( x, y ) so this is a valid protocol.Since image( b ′ ) ⊆ image( a ) we can write b ′ ( x ) = aa − b ′ ( x ) = aa − ab − b ( x ) = a ( x ) for every x so b ′ = a ,thus F ′ ( a ( x ) , a ( y )) = G ( x, y ) = G ( y, x ) = F ′ ( a ( y ) , a ( x )) so the protocol is symmetric.The standard deterministic SMP complexity measure can be expressed in terms of ≡ -reductions: Proposition 2.7.
For all graphs G , D k ( Adj ( G )) = ⌈ log | G ≡ |⌉ .Proof. It is well-known that for any function f : X × Y → { , } , D k ( f ) = ⌈ log min( r, c ) ⌉ where r is thenumber of distinct columns in the communication matrix of f , and c is the number of distinct rows [Yao79].The communication matrix of the function Adj ( G ) is the adjacency matrix of G , which is symmetric, andtwo rows (or columns) indexed by u, v are distinct iff u v ; so the number of distinct rows is the size of G ≡ .The analogous fact for universal SMP is that the deterministic universal SMP cost is determined by the sizeof the smallest universal graph. Proposition 2.8.
For any graph family F = ( F i ) , D univ ( F i ) = min U {⌈ log | U ≡ |⌉ : ∀ G ∈ F i , G ⊏ U ≡ } . Note that this does not imply that every deterministic SMP protocol is symmetric, since in this paper we are only concernedwith adjacency on an undirected graph, for which the communication matrix is symmetric. This proposition shows that forsymmetric communication matrices, the deterministic SMP protocol is symmetric. roof. Let U be any graph such that G ⊏ U ≡ for all G ∈ F i and for each G ∈ F i let g be the embedding G → U ≡ . Consider the protocol where on inputs ( G, x ) , ( G, y ), Alice and Bob send g ( x ) , g ( y ) using ⌈ log | U ≡ |⌉ bits and the referee outputs U ≡ ( g ( x ) , g ( y )). This is correct by definition so D univ ( F i ) ≤ ⌈ log | U ≡ |⌉ .Now suppose there is a protocol Π for F i with cost c and decision function F i , and let G ∈ F i . By Proposition2.6 we may assume that on inputs ( G, x ) , ( G, y ) Alice and Bob share the encoding function g : V ( G ) → { , } c .Let U be the graph with vertices { , } c and U ( u, v ) = F ( u, v ). Then U ( g ( x ) , g ( y )) = F ( g ( x ) , g ( y )) = G ( x, y )so G ⊏ U ⊏ U ≡ (by transitivity). Now | U ≡ | ≤ c so c ≥ log | U ≡ | .It is easy to see that D k can be used as a lower bound on D univ but such lower bounds are tight only whenthe family F is essentially a “trivial” family of equivalent graphs. Lemma 2.9.
For any family F = ( F i ) , let U = ( U i ) be the smallest ⊏ -universal graph sequence for F .Then max G ∈F i D k ( Adj ( G )) ≤ D univ ( F i ) = D k ( Adj ( U i )) , with equality holding on the left iff ∃ H ∈ F i such that ∀ G ∈ F i , G ≡ ⊏ H ≡ .Proof. The equality on the right holds by the two prior propositions. The lower bound follows from the factthat any protocol Π i for F i in the universal model can be used as a protocol in the SMP model. Now we mustshow the equality condition. Let U ∈ F i be a graph maximizing | U ≡ | over all graphs in F i , and suppose D univ ( F i ) = max G ∈F i D k ( Adj ( G )) = max G ∈F i ⌈ log | G ≡ |⌉ = ⌈ log | U ≡ |⌉ , so ⌈ log | U ≡ |⌉ = min {⌈ log | H ≡ |⌉ : ∀ G ∈ F i , G ⊏ H ≡ } . Then there exists H such that U ≡ ⊏ H ≡ and | U ≡ | = | H ≡ | . Since U ≡ is an inducedsubgraph of H ≡ and | U ≡ | = | H ≡ | we must have U ≡ ≃ H ≡ so ∀ G ∈ F i , G ≡ ⊏ U ≡ . Just as deterministic universal communication is equivalent to embedding a family into a universal graph, wewill define probabilistic universal graphs and show that they are tightly related to universal communicationwith shared randomness.
Definition 2.10.
For graphs
G, H , a random mapping φ : V ( G ) → V ( H ) (i.e. a distribution over suchmappings) is an ǫ -error embedding iff ∀ u, v ∈ V ( G ), P φ [ G ( u, v ) = H ( φ ( u ) , φ ( v ))] > − ǫ . We will write G ⊏ ǫ H if there exists an ǫ -error embedding G → H . A graph U is ǫ -error universal for a setof graphs S if ∀ G ∈ S, G ⊏ ǫ U . U = ( U i ) is an ǫ -error universal graph sequence for the family F = ( F i ) iffor each i , U i is ǫ -error universal for F i .In the randomized setting we obtain equivalence (up to a constant factor) between universal SMP protocolsand probabilistic universal graphs. Lemma 2.11.
For any graph family F = ( F i ) and any ǫ > , if there exists a ǫ -error universal SMPprotocols for F with cost c ( i ) , then there exists a ǫ -error symmetric universal SMP protocols for F withcost at most c ( i ) .Proof. On input G ∈ F i , x, y ∈ V ( G ), and random string r , Alice and Bob send the concatentations g r ( x ) := a i ( r, G, x ) b i ( r, G, x ) and g r ( y ) := a i ( r, G, y ) b i ( r, G, y ). Then the referee computes F ′ i ( g r ( x ) , g r ( y )) = max { F i ( a i ( r, G, x ) , b i ( r, G, y )) , F i ( a i ( r, G, y ) , b i ( r, G, x )) } . It is clear that F ′ i is symmetric. If x, y are adjacent then P r [ F ′ i ( g r ( x ) , g r ( y )) = 0] ≤ P r [ F i ( a i ( r, G, x ) , b i ( r, G, y )) = 0] < ǫ , x, y are not adjacent then, by the union bound, P r [ F ′ i ( g r ( x ) , g r ( y )) = 1] ≤ P r [ F i ( a i ( r, G, x ) , b i ( r, G, y )) = 1] + P r [ F i ( a i ( r, G, y ) , b i ( r, G, x )) = 1] < ǫ . Applying this symmetrization, we get a relationship between universal SMP protocols and probabilisticuniversal graphs.
Lemma 2.12.
Let F = ( F i ) be a graph family and ǫ > . Then1. There is an ǫ -error universal graph sequence of size at most R univ ǫ/ ( F ) ; and2. If there is an ǫ -error universal graph sequence of size c ( i ) then R univ ǫ ( F ) ≤ ⌈ log c ⌉ .Proof. If Π i is an ǫ -error symmetric universal protocol for F i then there exists a function F i such that forevery G ∈ F i there is a random g such that P g [ F i ( g ( x ) , g ( y )) = G ( x, y )] < ǫ . Using F i as an adjacencymatrix, we get a graph U i of size at most 2 c , where c is the cost of Π i , such that for all G ∈ F i , G ⊏ ǫ U i .Then U = ( U i ) is an ǫ -error probabilistic universal graph sequence. By Lemma 2.11 we obtain an ǫ -errorsymmetric protocol with cost 2 R univ ǫ/ ( F ), so we have proved the first conclusion. The second conclusionfollows by definition.The basic relationships to standard SMP models follow essentially by definition and from the above lemma. Lemma 2.13.
Let F be any graph family and let ǫ > . Let U = ( U i ) be an ⊏ -universal graph sequence for F , and e U = ( e U i ) an ǫ -error universal graph sequence. Then max G ∈F i R k ǫ ( Adj ( G )) ≤ R univ ǫ ( F i ) ≤ D k ( Adj ( e U i )) ≤ R univ ǫ/ ( F i ) and R univ ǫ ( F i ) ≤ R k ǫ ( Adj ( U i )) . Proof.
The inequalities on the left follow the definitions and from the above lemma. On the right, we canobtain a universal SMP protocol by choosing for each G ∈ F i a (deterministic) embedding g : G → U i andthen using the randomized SMP protocol for Adj ( U i ).Universal graphs describe an interesting relationship between weakly-universal and universal SMP protocols(and therefore between standard SMP protocols where the referee does and does not share the randomness);namely, the optimal universal protocol is obtained by finding the smallest universal graph for the family ofprotocol graphs (decision functions) defined by a weakly-universal protocol. Proposition (1.8) . Let F be a family of graphs, let ǫ > , and let W ǫ be the set of all ǫ -error weakly-universal SMP protocols for F . For each Π ∈ W ǫ let U Π = ( U Π ,i ) be the family of graphs U Π ,i = { F i ( r, · , · ) : r is a random seed for Π } where F i is the decision function of Π . Then R univ ǫ ( F ) = min Π ∈ W ǫ D univ ( U Π ) . Proof.
Let Π ∈ W ǫ ; we will construct a universal SMP protocol as follows. On input ( G, x ) , ( G, y ),Alice and Bob use shared randomness r to simulate Π and obtain vertices a ( r, G, x ) , b ( r, G, y ) in somegraph U r ∈ U Π with P r [ U r ( a ( r, G, x ) , b ( r, G, y )) = G ( x, y )] < ǫ . They now simulate the determinis-tic universal SMP protocol, i.e. an embedding φ : V ( U r ) → U ′ for some graph U ′ that is ⊏ -universalfor { U r } , and send φ ( a ( r, G, x )) , φ ( b ( r, G, y )) to the referee who computes U ′ ( φ ( a ( r, G, x )) , φ ( b ( r, G, x ))) = U r ( a ( r, G, x ) , b ( r, G, y )).Now let Π be an ǫ -error universal SMP protocol. Then Π ∈ W ǫ and for each i , U Π ,i = { U i } , where U i isthe graph of the decision function. D univ ( U Π ) ≤ ⌈ log | U i |⌉ , which is the cost of Π, so min Π ∈ W ǫ D univ ( U Π ) ≤ R univ ǫ ( F ). 9ewman’s Theorem for public-coin randomized (2-way) protocols is a classic result that gives a bound onthe number of uniform random bits required to compute a function f : X × Y → { , } in terms of thesize of the input domain [New91]. In the universal model, the input size can be very large since the graph(function) itself is part of the input. However, the shared part of the input does not contribute to the numberof random bits required in the universal SMP model. Lemma 2.14 (Newman’s Theorem for universal SMP) . Let ǫ, δ > and suppose there is an ǫ -error universalSMP protocol Π for the family F = ( F i ) . Then there is an ( ǫ + δ ) -error universal SMP protocol for the family F that uses at most log log (cid:16) n ( i ) O ( ǫ/δ ) (cid:17) bits of randomness and has the same communication cost.Proof. Fix i , let F be the deterministic decision function for F i , and let a ( r, · , · ) , b ( r, · , · ) be Alice and Bob’sencoding functions for the random seed r . For G ∈ F i and x, y ∈ V ( G ) we will say a seed r is bad for G, x, y if F ( a ( r, G, x ) , b ( r, G, y )) = G ( x, y ), and we will call this event bad ( G, x, y, r ).Let r , . . . , r m be independent random seeds, and let i ∼ [ m ] be uniformly random, where m > ǫδ ln( n ).Then for every G , the expected number of vertex pairs x, y for which the strings r , . . . , r m fail is E r ,...,r m "X x,y (cid:20) P i ∼ [ m ] [ bad ( G, x, y, r i )] > ǫ + δ (cid:21) ≤ n max x,y E r ,...,r m h h P i [ bad ( G, x, y, r i )] > ǫ + δ ii = n max x,y P r ,...,r m h P i [ bad ( G, x, y, r i )] > ǫ + δ i = n max x,y P r ,...,r m " m X i =1 [ bad ( G, x, y, r i )] > m ( ǫ + δ ) . The sum has mean µ = P mi =1 E r i [ [ bad ( G, x, y, r i )]] < mǫ , so by the Chernoff bound, the probability is atmost n P r ,...,r m " m X i =1 [ bad ( G, x, y, r i )] > (1 + mδ/µ ) µ ≤ n exp (cid:18) − m δ µ (cid:19) ≤ n exp (cid:18) − mδ ǫ (cid:19) < . Since the expected number of pairs x, y where choosing i ∼ [ m ] fails with probability more than ǫ + δ is lessthan 1, there must be some values of r , . . . , r m with no bad pairs for G . So for every G ∈ F i we may choose r , . . . , r m so that choosing i uniformly at random is the only random step; since m = ǫδ ln n = log n O ( ǫ/δ ) this requires at most log m = log log (cid:16) n O ( ǫ/δ ) (cid:17) random bits.With this result, we can conclude the proof of Theorem 1.7 in the next lemma. Lemma 2.15.
For any family F = ( F i ) with size function n ( i ) , Ω (cid:18) D univ ( F i )log n ( i ) (cid:19) ≤ R univ ( F i ) ≤ D univ ( F i ) . Proof.
The upper bound is clear, so we prove lower bound. Let Π = (Π i ) be a sequence of randomizeduniversal SMP protocols for F . By Newman’s theorem, we may assume that Π i uses at most log log n ( i ) c random bits for some constant c and has error probability 3 /
8. Let F i be the decision function of Π i , let m ( i ) be the cost of Π i , and let k = ⌈ c log n ( i ) ⌉ . To obtain a deterministic protocol, we can define the decisionfunction F ′ i on messages of k · m ( i ) bits as F ′ i ( a , b , a , b , . . . , a k , b k ) = majority( F i ( a j , b j )) j . Alice and Bobiterate over all k = 2 log log n ( i ) c random strings r and send a ( r, G, x ) , b ( r, G, y ) for each. Since the probabilityof error is at most 3 / r is uniform, at least 5 k/ > k/ F i ( a j , b j ) will give the correctanswer. This proves that D univ ( F i ) = O ( R univ ( F i ) log n ( i )).10n this paper we show lower bounds for a family F by giving embeddings of an arbitrary graph G into F ,so we need to know the complexity of the family G = ( G n ) of all graphs with n vertices. For our purposes,it is convenient to require that each graph G ∈ G n has G ( u, u ) = 1 for all u (i.e. all self-loops are present).However, since equality can be checked with cost O (1), the presence or absence of self-loops does not affectthe complexity. Theorem (1.6) . R univ ( G ) = Θ( n ) .Proof. For the upper bound, consider the (deterministic) protocol where on input
G, x, y , Alice and Bobsend x and y and the respective rows of the adjacency matrix of G . This has cost n + ⌈ log n ⌉ = O ( n ) andthe referee can determine G ( x, y ) by finding y in the row sent by Alice.Let Π be any protocol for G n with cost c . By Lemma 2.11, we may assume that Π is symmetric. Let F bethe decision function for graphs on n vertices and let G ∈ G n with vertex set [ n ]. Π defines a distributionover functions g : [ n ] → { , } c so that for all x, y, P g [ F ( g ( x ) , g ( y )) = G ( x, y )] < ǫ . Therefore, for x, y drawn uniformly from [ n ], E f,x,y [ [ F ( f ( x ) , f ( y )) = G ( x, y )]] < ǫ . Therefore, for every graph G ∈ G n thereis a function f G such that for x, y ∼ [ n ] uniformly at random, P x,y [ F ( f G ( x ) , f G ( y )) = G ( x, y )] < ǫ . Write N = (cid:0) n (cid:1) . There are at most 2 cn functions [ n ] → { , } c and there are 2 N simple graphs on [ n ] so there issome function f : [ n ] → { , } c where the number of graphs G such that f G = f is at least N cn = 2 N − cn .Let G, G ′ be any two such graphs. Then P x,y ∼ [ n ] [ G ( x, y ) = G ′ ( x, y )] ≤ P x,y ∼ [ n ] [ G ( x, y ) = F ( f ( x ) , f ( y )) or G ′ ( x, y ) = F ( f ( x ) , f ( y ))] < ǫ . So G, G ′ differ on at most 2 ǫN pairs. However, the largest number of graphs that differ from any graph G on at most 2 ǫN pairs of vertices is at most ǫN X k =0 (cid:18) Nk (cid:19) ≤ ǫN (cid:18) N ǫN (cid:19) ≤ ǫN (cid:18) eN ǫN (cid:19) ǫN = 2 ǫN log( e/ ǫ )+log(2 ǫN ) . Therefore we must have N − cn ≤ ǫN log( e/ ǫ ) + log(2 ǫN )so c = Ω( n ).Recall the example in the first paragraph of the introduction, for which we observed that a single decisionfunction would work for many problems. We now make a note about this phenomenon. A communicationprotocol for a graph family F = ( F i ) is really a sequence of protocols, one for each set F i of graphs with n ( i )vertices. Our next proposition addresses the uniformity of the sequence of protocols, that is, the question ofhow the protocols are related to one another as the size of the input grows. In general, we ask the question:If the family F has some relationship between F i and F i +1 , what does this imply about the relationshipbetween the protocols for i and i + 1? The families of graphs we study in this paper have constant-costprotocols and they are also upwards families , which we define next. These families have enough structureso that there exists a single, one-size-fits-all probabilistic universal graph, into which all graphs can beembedded regardless of their size; in other words, the referee can be ignorant not only of the graph G andvertices x, y , but also of the size of the graph, without increasing the cost of the protocol. Definition 2.16.
We call a graph family F = ( F i ) an upwards family if for every i and every G ∈ F i thereexists G ′ ∈ F i +1 such that G is an induced subgraph of G ′ . Any family F with a constant-cost protocol can be turned into a protocol ignorant of the size by requiring that Alice andBob tell the referee which of the 2 c possible decision functions to use, where c = 2 R univ ( F ) . Proposition 2.17. If F is an upwards graph family with an ǫ -error randomized universal graph sequence U = ( U i ) satisfying | V ( U i ) | ≤ c for some constant c (which may depend on ǫ ), then there exists a graph U ∗ of size c such that ∀ G ∈ F , G ⊏ ǫ U ∗ . Furthermore, for any i < j and any G ∈ F i , there exists G ′ ∈ F j with ǫ -error embedding g ′ : V ( G ′ ) → V ( U ∗ ) such that G is an induced subgraph of G ′ and the restriction of g ′ tothe domain V ( G ) is an ǫ -error embedding V ( G ) → V ( U ∗ ) .Proof. Let G ∈ F i and let G ′ ∈ F i +1 be such that G is an induced subgraph of G ′ . Let g ′ : V ( G ′ ) → V ( U i +1 )the random function determined by the randomized universal graph sequence. Then g ′ restricted to thedomain V ( G ) ⊂ V ( G ′ ) satisfies P g ′ [ U i +1 ( g ′ ( x ) , g ′ ( y )) = G ( x, y )] = P g ′ [ U i +1 ( g ′ ( x ) , g ′ ( y )) = G ′ ( x, y )] > − ǫ . Therefore we may replace U i with U i +1 in the sequence, for any i .Since each U i has size at most c , there are at most 2 c graphs U i appearing in the sequence U . Thus thereis some graph U ∗ that occurs an infinite number of times in the sequence. For every i there exists j > i such that U j = U ∗ . By applying the above argument, we may replace U i with U j = U ∗ in the sequence. Wearrive at the sequence U ′ = ( U ′ i ) with U ′ i = U ∗ for every i . Kannan, Naor, and Rudich [KNR92] call a family of graphs an implicit graph family if each of the n verticescan be given a label of O (log n ) bits so that adjacency can be determined from the labels of two vertices.They observe that an implicit encoding gives an upper bound on the size of an induced-universal graph . Wedefine these terms below in slightly more generality (and omit the requirement that encoding and decodingbe done in polynomial time): Definition 2.18.
Let F = ( F i ) be a graph family and m ( i ) a function of the graph size. The family F has an m -implicit encoding if ∀ i, ∃ F i : { , } m ( i ) × { , } m ( i ) → { , } such that F i is symmetric and ∀ G ∈ F i , ∃ g : V ( G ) → { , } m ( i ) satisfying ∀ x, y ∈ V ( G ) , F i ( g i ( x ) , g i ( y )) = G ( x, y ).For a graph family F = ( F i ), an induced-universal graph sequence is a sequence U = ( U i ) such that for each i and all G ∈ F i , G is an induced sugraph of U i .Our notion of ⊏ -universal graphs differs from induced-universal graphs, since the embedding relation G ⊏ U i allows non-injective mappings (two vertices of G may be mapped to the same vertex in U i ). This differenceaccounts for the extra factor n ( i ) in the next theorem. Theorem 2.19 ([Spi03]) . Let F = ( F i ) be a graph family with size n ( i ) . If there exists an m -implicit en-coding of F there is an induced-universal graph sequence U = ( U i ) such that | U i | ≤ n ( i )2 m ( i ) = 2 m ( i )+log n ( i ) . Due to the fact that a deterministic universal SMP protocol may always be assumed to be symmetric(Proposition 2.6), it follows by definition and from Lemma 2.15 that:
Theorem (1.1) . A graph family F = ( F i ) is m -implicit iff D univ ( F i ) ≤ m ( i ) for every i . Therefore, F is O ( R univ ( F ) · log n ) -implicit. If one’s goal is merely to obtain an O (1)-cost universal SMP protocol for a family F , the next observationshows that it suffices to find an O (1)-cost, public-coin, 2-way protocol for each member of F . Thereforethe family of all graphs with an O (1)-cost 2-way protocol is an implicit graph family with a polynomial-sizeinduced-universal graph. 12 orollary (1.2) . Let F = ( F i ) be a family of graphs with size n ( i ) and suppose that for every graph G ∈ F i there is an ǫ -error 2-way randomized communication protocol with cost at most c ( i ) . Then R univ ǫ ( F ) ≤ c ( i ) .Furthermore, for any fixed constant c , the family F of graphs with R ↔ ( Adj ( G )) ≤ c is O (log n ) -implicit.Proof. Every 2-way, deterministic cost c protocol can be represented as a binary tree with at most 2 c nodes,where each node is owned by either Alice or Bob and the message sent at each step is a 0 or 1 informingthe other player of which branch to take in the tree. A randomized 2-way protocol is a distribution oversuch trees. To obtain a universal SMP protocol for the family F , Alice and Bob do the following. On input G ∈ F and x, y ∈ V ( G ), Alice and Bob use shared randomness to draw the deterministic cost c protocolfor G from the distribution defined by the randomized 2-way protocol. Alice sends the size 2 c protocol treeand for each node she owns she identifies the branch to be taken. Bob does the same. The referee may thensimulate the protocol. The conclusion follows from Theorem 1.1. Distributive lattices and distances on these lattices will be defined in the next subsection, where we alsogive a necessary lemma characterizing the distances in terms of the meet and join . We will then presentan O ( k log k ) weakly-universal protocol and an O ( k ) universal communication protocol for the family D k ,where D are the distributive lattices. This implies a O ( k log n )-implicit encoding D k of the family D ofdistributive lattices. The O ( k log k ) weakly-universal protocol is optimal for sufficiently small values of k ,since it applies to the k -Hamming Distance problem as a special case, for which Saˇglam [Sa˘g18] recently gavea matching lower bound (even for 2-way communication). We obtain this result by adapting the optimal O ( k log k ) communication protocol for k -Hamming Distance originally presented by Huang et al. [HSZZ06].We also consider modular lattices, a generalization of distributive lattices, and show that deciding dist ( x, y ) ≤ n / ). A lattice is a type of partial order. We briefly review distributive lattices (see e.g. [CLM12] for a good intro-duction) and then give a characterization of distances in modular and distributive lattices. The undirectedgraphs we study are the cover graphs of partial orders. For x, y in a partial order P , we say that y covers x and write x ≺ y if ∀ z ∈ P : if x ≤ z < y then x = z . The cover graph (which is the undirected version of the transitive reduction ) is the graph cov( P ) on vertex set P with an edge { x, y } iff x ≺ y or y ≺ x .We will define a few types of lattices. Definition 3.1.
Let (
P, < ) be a partial order. For a pair x, y ∈ P : • If the set { z ∈ P : x, y ≥ z } has a unique maximum, we call that maximum the join of x, y and writeit as x ∧ y ; • If the set { z ∈ P : x, y ≤ z } has a unique minimum, we call that minimum the meet of x, y and writeit as x ∨ y .If ∀ x, y ∈ P the elements x ∧ y, x ∨ y exist, then P is a lattice . A lattice L is ranked if there exists a rankfunction such that x ≺ y = ⇒ rank( x ) + 1 = rank( y ) and the minimum element 0 L satisfies rank(0 L ) = 0.A finite lattice L is upper-semimodular if for every x, y ∈ L , x ∧ y ≺ x, y = ⇒ x, y ≺ x ∨ y . L is lower-semimodular if for every x, y ∈ L , x, y ≺ x ∨ y = ⇒ x ∧ y ≺ x, y . L is modular if it is both upper- andlower-semimodular. A lattice L is distributive if for all x, y, z ∈ L , x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ). Everydistributive lattice is modular and every modular lattice is ranked [CLM12].A point x in a lattice L is join-irreducible if there is no set S ⊆ L such that x = W S and meet-irreducible ifthere is no set S ⊆ L such that x = V S . Write J ( L ) for the set of join-irreducible elements.13 subset D of a partial order P is a downset or ideal if: for all x, y ∈ L , if x ∈ D and y ≤ x then y ∈ D . Wewill write D ( P ) for the set of ideals of P . Theorem 3.2 (Birkhoff (see e.g. [CLM12])) . Every distributive lattice L is isomorphic to the lattice ofdownsets of the partial order on its join-irreducible elements, ordered by inclusion; i.e. L ≃ D ( J ( L )) , withthe meet and join operations given by set union and intersection respectively. We need to prove some facts about distances in modular lattices.
Proposition 3.3.
Let L be a graded lattice and let x, y ∈ L . Then dist ( x, y ) ≥ | rank( x ) − rank( y ) | , withequality if x < y or y < x .Proof. This follows from the fact that for every edge u ≺ v in the path from x to y has rank( u ) + 1 =rank( v ).To prove our characterization of distance, we define inversions in the path. Definition 3.4.
Let L be a lattice and let c , . . . , c m be a path in cov( L ), so that c i ≺ c i +1 or c i +1 ≺ c i foreach i . If c i − , c i +1 ≺ c i or c i ≺ c i − , c i +1 we call c i an inversion on the path. Lemma 3.5.
The following holds for any x, y in a lattice M :1. If M is lower-semimodular then dist ( x, y ) = dist ( x, x ∧ y ) + dist ( y, x ∧ y ) ;2. If M is upper-semimodular then dist ( x, y ) = dist ( x, x ∨ y ) + dist ( y, x ∨ y ) ;3. If M is distributive then dist ( x, y ) = | X ∆ Y | where X, Y ∈ D ( J ( M )) are isomorphic images of x, y inBirkhoff ’s representation.Proof. It suffices to prove the first statement: the second follows by the analogous argument and the thirdfollows from the modulartiy of distributive lattices and Birkhoff’s representation.Let M be lower-semimodular, let x, y ∈ M , and let x = c , c , . . . , c m = y be a shortest path between x and y , so that dist ( x, y ) = dist ( x, c i ) + dist ( y, c i ) for any i . The statement holds trivially when x < y or y < x (since x ∧ y = x or x ∧ y = y ), so we assume x, y are incomparable. We prove the statement by induction onthe largest rank of an inversion of the form c i − , c i +1 ≺ c i in the path.First suppose that c i is any element of the path and assume for contradiciton that rank( c i ) < rank( x ∧ y ).Then dist ( x, x ∧ y ) = rank( x ) − rank( x ∧ y ) < rank( x ) − rank( c i ) ≤ dist ( x, c i ) , a contradiction. Thus rank( c i ) ≥ rank( x ∧ y ) for each element of the path.Suppose there are no inversions of the form c i − , c i +1 ≺ c i . Then c i < x, y and therefore c i ≤ x ∧ y sorank( c i ) ≤ rank( x ∧ y ), and by the above inequality we have rank( c i ) ≥ rank( x ∧ y ), so rank( c i ) = rank( x ∧ y ).Therefore, as desired, dist ( x, y ) = dist ( x, c i ) + dist ( y, c i ) = rank( x ) − rank( c i ) + rank( y ) − rank( c i )= rank( x ) − rank( x ∧ y ) + rank( y ) − rank( x ∧ y )= dist ( x, x ∧ y ) + dist ( y, x ∧ y ) . Now let c i be an inversion of the form c i − , c i +1 ≺ c i with rank( c i ) > rank( x ∧ y ). Then by lower-semimodulariity there is an element c ′ i = c i − ∧ c i +1 ≺ c i − , c i +1 . Then replacing c i with c ′ i maintainsthe length of the path. Performing the same operation on all such inversions of maximum rank reduces themaximum rank by 1 and the result holds by induction.14 .2 A Universal Protocol for Distributive Lattices Write D = ( D n ) for the family of cover graphs of distributive lattices on n vertices. We first give an optimalprotocol for distances in distributive lattices in the weak universal model (recall that in this model, thereferee sees the shared randomness). This protocol is adapted from a simplified presentation of Huang etal. ’s k -Hamming Distance protocol ([HSZZ06]) communicated to us by E. Blais. Theorem 3.6.
For any ǫ > and integer k , R weak ǫ ( D k ) = O ( k log( k/ǫ )) .Proof. For any distributive lattice L ≃ D ( J ( L )), identify each vertex x ∈ L with its ideal X ⊆ J ( L ) ofjoin-irreducibles. Write e , . . . , e m for the basis vectors of F m . Consider the following protocol. On thedistributive lattice L and vertices x, y , Alice and Bob perform the following:1. Define m = ⌈ ( k +2) ǫ ⌉ , q = ⌈ log ǫ + log P i =0 (cid:0) mi (cid:1) ⌉ .2. Let S = ( s , . . . , s m ) be a multiset of uniformly random elements of F q .3. For each join-irreducible element j ∈ J ( L ) assign a uniformly random index i j ∼ [ m ].4. For each vertex v ⊆ J ( L ) there is an indicator vector a ( v ) ∈ F m defined by a ( v ) = P j ∈ v e i j . Label v with ℓ ( v ) = P mi =1 a ( v ) i s i .5. Alice sends ℓ ( x ) and Bob sends ℓ ( y ) to the referee.6. The referee accepts iff ℓ ( x ) + ℓ ( y ) is a sum of at most k elements of S .By Lemma 3.5 and Birkhoff’s theorem, dist ( x, y ) = dist ( x, x ∧ y ) + dist ( x ∧ y, y ) = | X \ Y | + | Y \ X | = | X ∆ Y | , where ∆ denotes the symmetric difference. Suppose dist ( x, y ) = | X ∆ Y | ≤ k . Then ℓ ( x ) + ℓ ( y ) = P j ∈ X ∆ Y c ( j ) is a sum of at most k elements of S , so the protocol accepts with probability 1 (so this protocolhas 1-sided error).Now suppose dist ( x, y ) = | X ∆ Y | ≥ k + 1. The correctness of the protocol follows from the next two claimsalong with the observations that a ( x ) + a ( y ) = a ( x ∧ y ) and ℓ ( x ) + ℓ ( y ) = ℓ ( x ∧ y ) (with arithmetic in F )and that dist ( x, y ) ≥ k + 1 implies rank( x ∧ y ) ≥ k + 1. We will write | a ( v ) | for the number of 1’s in thevector a ( v ). Claim 3.7.
Any vertex v ⊆ J ( L ) with rank( v ) ≥ k + 1 has | a ( v ) | ≥ k + 1 with probability at least − ǫ/ .Proof of claim. If rank( v ) = k + 1, so v is a set of k + 1 join-irreducibles, then the probability that any twoindices i j , i j ′ collide, for j, j ′ ∈ v , is by the union bound at most (cid:18) k + 12 (cid:19) P [ i j = i j ′ ] = k ( k + 1)2 1 m ≤ ( k + 1) ǫ ( k + 2) = ǫ/ . For rank( v ) > k + 1 choose v ′ ≺ v so k + 1 ≤ rank( v ′ ) < rank( v ), so using induction and the assumption ǫ < / P [ | a ( v ) | ≤ k ] = k + 1 m P [ | a ( v ′ ) | = k + 1] + km P [ | a ( v ′ ) | ≤ k ] < ǫk + 2 + ǫk + 2 · ǫ ǫ (cid:18) k + 2 + ǫ k + 2) (cid:19) ≤ ǫ (cid:18)
13 + 112 (cid:19) < ǫ/ . Claim 3.8.
For any vertex v ⊆ J ( L ) , if the indicator vector a ( v ) has weight ≥ k + 1 then, with probabilityat least − ǫ/ , ℓ ( v ) is not a sum of at most k vectors in S .Proof of claim. Write kS for the set of all sums of at most k vectors of S . Fix any a ( v ) with weight ≥ k + 1and let A = { i : a ( v ) i = 1 } so | A | ≥ k + 1. Let b ∈ kS be any sum of k vectors in S , and let B ⊂ [ m ] be aset of indices of size | B | ≤ k such that b = P i ∈ B s i . 15ince | B | ≤ k < | A | we must always have A \ B = ∅ and ℓ ( v ) + b = P i ∈ A \ B s i , so P [ ℓ ( v ) + b = 0] = 2 − q .Therefore, by the union bound over all such vectors b , P [ ℓ ( v ) ∈ kS ] ≤ k X i =0 (cid:18) mi (cid:19) − q < ǫ/ . We can put a bound on q by using k X i =0 (cid:18) mi (cid:19) ≤ k (cid:18) mk (cid:19) ≤ k (cid:16) emk (cid:17) k so q ≤ ǫ + log k + k log emk ≤ log 2 kǫ + k log ⌈ ekǫ ⌉ = O (cid:18) k log kǫ (cid:19) . Observe that the referee must see the set S for the above protocol to work. We can easily modify the aboveprotocol to get O ( k ). Theorem 3.9.
For any ǫ > and any integer k , R univ ǫ ( D k ) = O (cid:0) k log(1 /ǫ ) (cid:1) .Proof. The protocol is the same as above, with the following modification: Alice and Bob each send theindicator vectors a ( x ) , a ( y ) ∈ F m .The correctness of this protocol for error 1 / m = ⌈ k + 2) / ⌉ .This protocol is one-sided, so to achieve error ǫ we can run the protocol r = ⌈ log (1 /ǫ ) ⌉ times and take theAND of the results. The probability of failure is (1 / r = 3 − r < ǫ .Now we apply Theorem 1.1 to obtain Theorem 1.3.Since the family of distributive lattices is an upwards family (simply append a new least element to obtain alarger distributive lattice), we see from Proposition 2.17 that lattices in D k can be randomly embedded intoa constant-size graph, for any constant k . In fact, by inspection of the protocol, we see that the family D canbe randomly embedded into a small-dimensional hypercube, while D k can be embedded into the k -closureof the O ( k )-dimensional hypercube. Corollary 3.10.
For any ǫ > and any k , there exists a graph U of size O ( k log(1 /ǫ )) such that for all L ∈ D k , L ⊏ ǫ U . Since Lemma 3.5 works for any modular lattices, it is natural to ask whether we can achieve a similarconstant-cost protocol for computing distance thresholds in modular lattices. However, we show that this isimpossible.
Lemma 3.11.
There is a function m ( n ) = O ( n ) such that if G is any graph with n vertices (where G ( u, u ) = 1 for all u ), there exists a modular lattice M with size m ( n ) such that G is an induced subgraphof cov( M ) .Proof. Construct the lattice M as follows:1. Start with vertices V , which are all incomparable.2. For each edge e = { u, v } ∈ E , add vertices a e , b e such that a e < u, v < b e .16. ∀ e = { u, v } , e ′ = { u ′ , v ′ } ∈ E such that e ∩ e ′ = ∅ add a vertex c e,e ′ with a e , a e ′ < c e,e ′ < b e , b e ′ .4. Add vertices 0 M and 1 M such that 0 M < a e and b e < M for all e ∈ E .First we prove that M is a modular lattice and then we prove the bound on the size. Claim 3.12. M is a modular lattice.Proof of claim. Observe that all orderings < directly imposed by this process are covering orders ≺ . Let A = { a e } e ∈ E , B = { b e } e ∈ E , C = { c e } e ∈ E and V the original set of vertices. By construction, M is gradedwith rank(0 M ) = 0 , rank( A ) = 1 , rank( V ) = rank( C ) = 2 , rank( B ) = 3 , rank(1 M ) = 4. Note that for everypair of vertices x, y ∈ M, M ≤ x, y ≤ M so upper- and lower-bounds exist.Assume for contradiction that M is not a modular lattice, so there exist incomparable x, y ∈ M such thateither x ∧ y or x ∨ y does not exist, or such that x ∧ y ≺ x, y x ∨ y or x ∧ y x, y ≺ x ∨ y .Case 1: Suppose rank( x ) = rank( y ). Then x ∧ y = 0 M and x ∨ y = 1 M so x ∧ y x, y x ∨ y .Case 2: Suppose x, y ∈ A so x = a e , y = a e ′ . Then 0 M = a e ∧ a e ′ ≺ a e , a e ′ . If a e , a e ′ < u, v for u, v ∈ V then u, v ∈ e ∩ e ′ so u = v . If a e , a e ′ < v, c d,d ′ for v ∈ V and c d,d ∈ C then v ∈ e ∩ e ′ and c d,d ′ = c e,e ′ so e ∩ e ′ = ∅ , a contradiction. Finally, if a e , a e ′ < c d,d ′ , c d ′ ,d ′′ then c d,d ′ = c d ′ ,d ′′ = c e,e ′ . So a e ∨ a e ′ exists and a e ∧ a e ′ ≺ a e , a e ′ ≺ a e ∨ a e ′ . The same argument holds for x, y ∈ B .Case 3: Suppose x, y ∈ V and assume a e , a e ′ < x, y . Then x, y ∈ e ∩ e ′ so a e = a e ′ . A similar argumentholds for x, y < b e , b e ′ . So x ∧ y ≺ x, y ≺ x ∨ y .Case 4: Suppose x, y ∈ C so x = c e,e ′ , y = c d,d ′ . Suppose a s , a t < c e,e ′ , c d,d ′ . Then s, t ∈ { e, e ′ } ∩ { d, d ′ } soeither { e, e ′ } = { d, d ′ } or s = t . The same argument holds for c e,e ′ , c d,d ′ < b s , b t so x ∧ y ≺ x, y ≺ x ∨ y .Case 5: Suppose x ∈ V, y ∈ C so y = c e,e ′ which implies e ∩ e ′ = ∅ . If x / ∈ e ∪ e ′ then x ∧ c e,e ′ = 0 M and x ∨ c e,e ′ = 1 M so x ∧ c e,e ′ x, c e,e ′ x ∨ c e,e ′ ; so suppose x ∈ e ∪ e ′ . If a e , a e ′ < x, c e,e ′ then x ∈ e ∩ e ′ whichis a contradiction. Then x ∈ e or x ∈ e ′ ; say x ∈ e . Then a e = x ∧ c e,e ′ . The same argument holds for B so a e = x ∧ c e,e ′ ≺ x, c e,e ′ ≺ x ∨ c e,e ′ = b e . Claim 3.13. G is an induced subgraph of cov( M ) .Proof of claim. Suppose { u, v } ∈ E . Then there is a e ≺ u, v so dist ( u, v ) ≤ M ). Now let u, v ∈ V ( G )and suppose dist ( u, v ) ≤ M ) so that, by Lemma 3.5, u ∧ v ≺ u, v ≺ u ∨ w . By construction, either u = v so G ( u, v ) = G ( u, u ) = 1, or u ∧ v = a e for some e ∈ E ( G ) so u, v ∈ e and therefore G ( u, v ) = 1.The size of M is at most 2 + | E ( G ) | + | E ( G ) | = O ( n ). Let m ( n ) be the maximum size of a modular latticeobtained in this way from a graph of size n . We want all constructions to be of the same size, so repeatedlyappend new least elements until the size reaches m ( n ); this maintains the modular lattice property. Theorem 3.14.
Let M = ( M n ) be the family of cover graphs of modular lattices. R univ ( M ) ≥ Ω( n / ) .Proof. Suppose there is a protocol for M with cost o ( n / ). Given a graph G of size n , Alice and Bobconstruct the modular lattice of size m ( n ) = O ( n ) with G an induced subgraph of cov( M ) and run theprotocol for M with size m ( n ) (observe that all possible constructions must be of the same size, sincethe referee does not know which lattice Alice and Bob construct). This has cost o ( m ( n ) / ) = o ( n ), whichcontradicts Theorem 1.6. In this section we take inspiration from the field of implicit graphs and graph labeling and show that onemay often, but not always, obtain constant-cost adjacency and k -distance protocols for families that arecommonly studied in the graph labeling literature. 17 .1 Trees, Forests, and Interval Graphs In this section we pick the low-hanging fruit from trees and forests (and interval graphs). Applying Theorem1.1 with the next lemma, we get Theorem 1.4.
Lemma 4.1.
Let T = ( T n ) be the family of trees of size n . R univ ǫ ( T k ) = O (cid:0) k log ǫ (cid:1) , and this protocol willcorrectly compute the distance in the case dist ( x, y ) ≤ k .Proof. Consider the following protocol. On input (
T, x ) , ( T, y ) for a tree T , Alice and Bob perform thefollowing.1. Partition the vertices of T into sets T , . . . , T m such that T i = { v ∈ V ( T ) : ( i − k ≤ depth( v ) < ik } .For each v ∈ V ( T ) let t ( v ) be the index of the unique set satisfying v ∈ T t ( v ) .2. For each vertex v ∈ V ( T ) assign a uniformly random color ℓ ( v ) in [ m ] for m = ⌈ /ǫ ⌉ . Let x ′ be rootof the subtree of T t ( x ) that contains x , and let x ′′ be the root of the subtree of T t ( x ) − that contains x .Let x , x , . . . , x k , . . . , x k = x be the path from x ′′ to x (with x k = x ′ ) and let y , . . . , y k , . . . , y k bethe path from y ′′ to y . Alice and Bob send ℓ ( x ) , . . . , ℓ ( x k ) and ℓ ( y ) , . . . , ℓ ( y k ) respectively.3. If ℓ ( x ′ ) = ℓ ( y ′ ), let p be the maximum index such that ℓ ( x i ) = ℓ ( y i ) for each k < i ≤ p . Let d = ( k − p ) + ( k − p ). If ℓ ( x ′′ ) = ℓ ( y ′′ ), let p be the maximum index such that ℓ ( x i ) = ℓ ( y i ) foreach i ≤ p and let d = ( k − p ) + ( k − p ). If ℓ ( x ′′ ) = ℓ ( y ′ ) let p be the maximum index such that ℓ ( x i ) = ℓ ( y k + i ) for each i ≤ p and let d = ( k − p ) + ( k − k − p ). If ℓ ( x ′ ) = ℓ ( y ′′ ) do the same with x, y reversed. In each case, if d ≤ k , the referee outputs d , otherwise they output “ > k ”. If none of theabove cases hold, output “ > k ”.The cost of this protocol is 2 k ⌈ log m ⌉ = O ( k log(1 /ǫ )). With probability at least 1 − /m > − ǫ/
2, eachof the possible equalities x ′′ = y ′′ , x ′ = y ′ , x ′′ = y ′ , x ′ = y ′′ will be correctly observed by the referee. If { x ′ , x ′′ } ∩ { y ′ , y ′′ } = ∅ then x, y are not in the same subtree rooted at depth depth( x ′′ ), so the distancefrom x to any common ancestor of x, y is at least dist ( x, x ′′ ) > k . Therefore if dist ( x, y ) ≤ k , one ofthese equalities will hold. If x ′′ = y ′′ and q is the maximum integer such that x i = y i for all i ≤ q then dist ( x, y ) = ( k − q ) + ( k − q ), because the deepest common ancestor of x, y is at depth depth( x ) + q .Conditional on the 4 equalities being correctly observed, we will have d = ( k − p ) + ( k − p ) ≤ k since p ≥ q . If p > q then ℓ ( x q +1 ) = ℓ ( y q +1 ) even though x q +1 = y q +1 , which occurs with probability 1 /m < ǫ/ d = dist ( x, y ) is at most 2( ǫ/
2) = ǫ when dist ( x, y ) ≤ k . A similar argumentholds in the other 3 cases.If dist ( x, y ) > k then still with probability at least 1 − ǫ/ d = dist ( x, y ) with probability greater than 1 − ǫ/
2, for total error probability less that ǫ . If none of the 4equalities hold then the probability of error is at most ǫ/ arboricity of agraph is one such generalization, which measures the minimum number of forests required to partition allthe edges. Definition 4.2.
A graph G = ( V, E ) has arboricity α iff there exists an edge partition of G into forests T , . . . , T α . Equivalently, for S ranging over the set of subgraphs of G , G hasmax S (cid:24) E ( S ) V ( S ) − (cid:25) ≤ α . Low-arboricity graphs easily admit an efficient universal SMP protocol for adjacency.
Proposition 4.3.
Let F be any family of graphs with arboricity at most α . For all ǫ > , R univ ǫ ( F ) = O (cid:0) α log αǫ (cid:1) . roof. On the graph G and vertices x, y , Alice and Bob perform the following:1. Compute a partition of G into α forests T , . . . , T α .2. Assign to each vertex v a uniformly random number ℓ ( v ) ∼ [ m ] for m = ⌈ α/ǫ ⌉ .3. Let x i be the parent of x in tree i and let y i be the parent of y . Alice sends ℓ ( x ) and ℓ ( x i ) for each i ,and Bob does this same with y .4. The referee accepts iff ℓ ( x ) = ℓ ( y i ) or ℓ ( y ) = ℓ ( x i ) for any i .This protocol has one-sided error since if x, y are adjacent then either x i = y or y i = x for some i , so thereferee will accept with probability 1. If x, y are not adjacent then the referee will accept with probabilityat most 2 α · m < ǫ .However, even graphs of arboricity 2 do not admit efficient protocols or labeling schemes for distance 2,which we can show by embedding an arbitrary graph of size Ω( √ n ) into the 2-closure of an arboricity 2graph of size n : Proposition 4.4.
Let F be the family of arboricity-2 graphs. Then R univ ( F ) ≥ Ω( √ n ) .Proof. The lower bound is obtained via Theorem 1.6 in the same way as in Theorem 3.14, using the followingconstruction. For all simple graphs G = ( V, E ) with n vertices, there exists a graph A of size n + (cid:0) n (cid:1) andarboricity 2 such that G is an induced subgraph of A . Let A be the graph defined as follows:1. Add each vertex v ∈ V to A ;2. For each pair of vertices { u, v } add a vertex e { u,v } and add edges { u, e { u,v } } , { v, e { u,v } } iff { u, v } ∈ E .This graph has arboricity 2 since for each e { u,v } we may assign each of its 2 incident edges a color in { , } (if the edges exist). Then the edges with color i ∈ { , } form a forest with roots in V .Now we give an example of a family, the interval graphs, with size O (log n ) adjacency labels but with noconstant-cost universal SMP protocol; in fact, randomization does not give more than a constant-factorimprovement for this family. An interval graph of size n is a graph G where for each vertex x there is aninterval X ⊂ [2 n ] such that any two vertices x, y are adjacent in G iff X ∩ Y = ∅ . These have an O (log n )adjacency labeling scheme [KNR92] (one can simply label a vertex with its two endpoints in [2 n ]).There is a simple reduction from the Greater-Than communication problem, in which Alice and Bobreceive integers x, y ∈ [ n ] and must decide if x < y . It is known that the one-way public-coin communicationcost of Greater-Than is Ω(log n ) [MNSW98], so R k ( Greater-Than ) = Ω(log n ). Proposition 4.5.
For the family F of interval graphs, R univ ( F ) = Ω(log n ) .Proof. We can use a universal SMP protocol for F to get a protocol for Greater-Than as follows. Aliceand Bob construct the interval graph with intervals [1 , i ] , [ i, n ] for each i ∈ [ n ], so there are 2 n vertices in G . On input x, y ∈ [ n ], Alice and Bob compute adjacency on the intervals [1 , x ] , [1 , y ] and then again on[1 , x ] , [ y, n ]. Assume both runs of the protcol succeed. Then when the output is 1 for both runs we musthave y ∈ [1 , x ] so y ≤ x and otherwise we have y / ∈ [1 , x ] so x < y . Write P n for the set of planar graphs of size n and write P = ( P n ) for the family of planar graphs. Gavoille et al. [GPPR04] gave an O ( √ n log n ) labeling scheme where dist ( x, y ) can be computed from the labels of x, y , and Gawrychowski and Uzna´nski [GU16] improved this to O ( √ n ). These labeling schemes recursivelyidentify size- O ( √ n ) sets S and record the distance of each vertex v to each u ∈ S , so the √ n factor isunavoidable using this technique. We want to solve k -distance with a cost independent of n , so we need anew method. Our main tool is Schnyder’s elegant decomposition of planar graphs into trees:19 heorem 4.6 (Schnyder [Sch89], see [Fel12]) . Define the dimension dim( G ) of a graph G as is the minimum d such that there exist total orders < , . . . , < d on V ( G ) satisfying:(*) For every edge { u, v } ∈ E and w / ∈ { u, v } there exists < i such that u, v < i w . G is planar iff dim( G ) ≤ . If G is planar then there exists a partition T , T , T of the edges into directedtrees satisfying the following. Let T − i be edge-induced directed graph on V ( G ) obtained by reversing thedirection of each edge in T i . The graphs with edges T i ∪ T − i − ∪ T − i +1 have linear extensions < i such that < , < , < satisfy (*). Schnyder’s Theorem implies that the arboricity of planar graphs is at most 3, so we may use the protocolfor low-arboricity graphs (Proposition 4.3) to determine adjacency in P , so we move on to P , which mayhave large arboricity (arboricity is within a constant factor of degeneracy ): Theorem 4.7 ([AH03]) . There are planar graphs P for which the degeneracy of P is Θ(deg P ) , where deg P is the maximum degree of any vertex in P . We avoid this blowup in arboricity by treating edges of the form a ← b → c separately (with directions takenfrom the Schnyder wood). The proof uses the following split operation: Definition 4.8.
Let G ∈ P and fix a planar map and a Schnyder wood T , T , T . Define the graph split ( G )by the following procedure (see Figure 1):1. For each vertex s ∈ V ( G ) add vertices s, s , s , s to split ( G ) (excluding s i if s has no incoming edgein T i ). Add edges ( s i , s ) to T ′ i ;2. For each (directed) edge ( u, v ) ∈ T i add the edges ( u i − , v i ) , ( u i +1 , v i ) (arithmetic mod 3) to T ′ i ;3. For the unique (directed) edge ( v, u ) ∈ T i add the edges ( v i − , u ) , ( v i +1 , u ) to T ′ i . s ss s s Figure 1: Splitting vertex s , with T , T , T in blue, red, and green respectively (1,2, and 3 arrowheads). Proposition 4.9. split ( G ) is planar.Proof. We prove that splitting any vertex s results in a planar graph. By induction we may then split eachvertex in sequence and obtain a planar graph. Let < i be any total order on V ( G ) extending T i ∪ T − i − ∪ T − i +1 ,which satisfies condition (*) by Schnyder’s theorem. Let < ′ , < ′ , < ′ be the same total orders, extending T ′ , T ′ , T ′ , and augmented to include s , s , s as follows:1. For each u ∈ V ( G ), s i < ′ j u iff s < j u and u < ′ j iff u < j s ;2. For each i , set s i < ′ i s < ′ i s i +1 < ′ i s i − . This is possible since { s i } do not have a defined ordering in < i and remain incomparable after the previous step.Note that for any edge ( u, v ) ∈ T ′ i we have u < ′ i v and v < ′ j u for j = i . It suffices to prove that condition (*)is satisfied by the new orders. Let { u, v } ∈ E ( split ( G )) and let w / ∈ { u, v } . We will show that there exists i such that u, v < ′ i w . 20f u, v, w ∈ V ( G ) then we are done since the orders < ′ i are the same as < i on these vertices.If u = s i then either v ∈ V ( G ) \ s , in which case v < i s so v < ′ i u and therefore u < ′ j v for j = i , or v = s so u < ′ i v and therefore v < ′ j u for j = i . Let v = s . For any w ∈ V ( G ) \ { v } we have, by (*), either v, s < i w so v < ′ i u < ′ i s < ′ i w , or v, s < j w so u < ′ j v < ′ j w . If v = s then by construction there exists ( u ′ , u ) ∈ T i . By(*), either u ′ , v < i w so u < ′ i v < ′ i w , or u ′ , v < j w so v < ′ j u < ′ j u ′ < ′ j w .The only case remaining is if w = s i and u, v ∈ V ( G ). By construction there exists ( w ′ , w ) ∈ T i . Either u, v < i w ′ < ′ i w < ′ i s or by (*) there exists j such that u, v < j s and since ( w, s ) is an edge in T ′ i , s < ′ j w for j = i . Definition 4.10.
Let G = ( V, E ) be a planar graph. Fix a planar map and a Schnyder wood T , T , T .For each i , define the graph G i = ( V, E \ T i ) as the graph obtained by removing each edge in T i . Definethe head-to-head closure of G i , written G ←→ i , as the graph with an edge { u, v } iff there exists w ∈ V suchthat u ← w → v in G i . (Observe that the two outgoing edges of w must be in T i − , T i +1 .) Let G ←→ be thesubgraph of G containing all edges occuring in G ←→ i for each i . Lemma 4.11.
Let G be a planar graph. For any graph M , if M is a minor of G ←→ i then M is a minor of split ( G ) .Proof. We will prove the following claim.
Claim 4.12.
For any set P = { P j } of simple paths P j ⊆ V ( G ←→ i ) , with endpoints { ( s j , t j ) } such that notwo paths P j , P k have the same endpoints and P j ∩ P k ⊆ { s j , s k , t j , t k } , there exists a set of paths Q = { Q j } of paths in split ( G ) with the same endpoints such that Q j ∩ Q k ⊆{ s j , s k , t j , t k } ∪ { ( s j ) i − , ( s k ) i − , ( t j ) i − , ( t k ) i − } ∪ { ( s j ) i +1 , ( s k ) i +1 , ( t j ) i +1 , ( t k ) i +1 } , where the vertices s i , s i +1 , s i − are defined as in the split operation.Proof of claim. For each path P j , perform the following. For each edge { u, w } in the path P j , there is some(not necessarily unique) vertex v such that either ( v, u ) ∈ T i − and ( v, w ) ∈ T i +1 , or the same holds with u, w reversed. Add the edges { u, u i − } , { u i − , v i } , { v i , w i +1 } , { w i +1 , w } to Q j . If P j is a singleton P j = { u } so s j = t j then add u to Q j .Consider two paths Q j , Q k constructed this way. G ←→ i has vertex set V and split ( G ) has vertex set V ′ ⊃ V .By construction, P j ⊆ Q j and P k ⊆ Q k and ( Q j ∩ V ) = P j . Suppose there exists z ∈ Q j ∩ Q k that is not anendpoint, so z / ∈ { s j , s k , t j , t k } . If z ∈ V then z ∈ P j ∩ P k ⊆ { s j , s k , t j , t k } , so we only need to worry about z ∈ V ′ \ V .If z = v i for some vertex v then there are unique distinct vertices u i − , w i +1 ∈ V ′ adjacent to v i such that u i − , w i +1 ∈ Q j ∩ Q k . Then u, w ∈ Q j ∩ Q k also, so u, w ∈ P j ∩ P k ; but then u = w are the start and endpoints of P j , P k , so P j = P k , a contradiction.If z = v i − for some vertex v ∈ V then v ∈ Q j ∩ Q k , so by the case above, v ∈ { s j , s k , t j , t k } and z ∈ { ( s j ) i − , ( s k ) i − , ( t j ) i − , ( t k ) i − } . Likewise for z = v i +1 .Let M be a minor of G ←→ i , so a subdivision of M occurs as a subgraph of G ←→ i . Therefore there is a set ofpaths P in G ←→ i satisfying the conditions of the claim, so that by contracting each path into a single edge,and deleting the rest of the graph, we obtain M . Let Q = { Q j } be the set of paths given by the claim.For endpoints s j , t j ∈ Q j , contract the edges { s j , ( s j ) i ± } and { t j , ( t j ) i ± } . The result is a contraction of split ( G ) and a set of paths Q ′ that is a subdivision of M , so M is a minor of split ( G ), which proves thelemma. Corollary 4.13. G ←→ i is planar and G ←→ has arboricity at most 9. roof. A graph is planar iff it does not contain K or K , as a minor (Kuratowski’s Theorem). If G ←→ i isnot planar then it contains K or K , as a minor, so by the above lemma, split ( G ) contains K or K , asa minor, so split ( G ) is not planar, a contradiction. Since planar graphs have arboricity at most 3, the edgeunion G ←→ of 3 planar graphs has arboricity at most 9.By separating the ←→ edges from the remaining edges of P , we obtain a constant-cost universal SMPprotocol for P , and then by applying Theorem 1.1 we obtain Theorem 1.5. Lemma 4.14.
For all ǫ > , R univ ǫ ( P ) = O (cid:0) log ǫ (cid:1) .Proof. For a planar graph G = ( V, E ) with a fixed planar map and a Schnyder wood T , T , T , define thegraph G i = ( V, E \ T i ) as the graph obtained by removing the edges in tree T i .On planar graph G ∈ P n and vertices x, y , Alice and Bob perform the following:1. For each i define x i , y i to be the parents of x, y in T i . Run the protocol for adjacency with error ǫ/ x, y i ) and ( x i , y ) for each i .2. Run the protocol for low-arboricity graphs on G ←→ with error ǫ/ G ←→ has arboricity at most 9, we may apply the protocol for low-arboricity graphs instep 2. If dist ( x, y ) > − ǫ since there are 7applications of ǫ/ dist ( x, y ) = 2 then the algorithm will accept.Suppose x, y are of distance 2. Then the paths between them are of the following forms (with edge directionstaken from the Schnyder wood).1. x → v → y or x → v ← y . This is covered by step 1.2. x ← v → y . This is covered by step 2.Since planar graphs are an upwards family (just insert a new vertex), we obtain a constant-size probabilisticuniversal graph for P . Corollary 4.15.
For any ǫ > , there is a graph U of size O (log(1 /ǫ )) such that for every G ∈ P , G ⊏ ǫ U . Error-tolerance.
In the introduction we mentioned that the universal SMP model allows us to studyerror-tolerance in the SMP model. This could be done as follows: suppose the referee knows a referencegraph G and the players are guaranteed to see a graph that is “close” to G by some metric. How muchdoes this change the complexity of the problem, compared to computing G ? One common distance metricin, say, the property testing literature, is to count the number of edges that one must add or delete. Thatis, for two graphs G, H on vertex set [ n ], write dist ( G, H ) = n P i,j ∈ [ n ] [ G ( i, j ) = H ( i, j )]. The distance isusually thought of as a constant. We can easily give a strong negative result for this situation: Proposition 5.1.
Let F be any family of graphs and F δ the family of graphs G such that min F ∈F dist ( G, F ) ≤ δ . Then R univ ( F δ ) = Ω( √ δn ) . Proof.
Let G be any graph on √ δn vertices and let F ∈ F . Choose any set S ⊆ V ( F ) with | S | = | G | .Construct F ′ by replacing the subgraph induced by S with the graph G . Then dist ( F, F ′ ) ≤ | G | n = δ so F ′ ∈ F δ . Then the conclusion follows from Theorem 1.6.22his suggests that this is not the correct way to model contextual uncertainty in the SMP model, butuniversal SMP gives a framework for studying many other error tolerance settings. For example, we couldsuppose that the referee knows a reference planar graph G , and the players are guaranteed to see a graph G ′ that is close to G and also planar; this would not increase the cost of the protocol due to our results onplanar graphs. Implicit graph conjecture.
A major open problem in graph labeling is the implicit graph conjecture ofKannan, Naor, and Rudich [KNR92], which asks if every hereditary graph family F (where for each G ∈ F ,every induced subgraph of G is also in F ) containing at most 2 O ( n log n ) graphs of size n has an O (log n )adjacency labeling scheme. Not much progress has been made on this conjecture (see e.g. [Spi03, Cha16]).We ask a weakened version of this conjecture: Question 5.2.
For every hereditary family F = ( F n ) such that |F n | ≤ O ( n log n ) , is R univ ( F ) = O (log n ) ? Good candidates for disproving the implicit graph conjecture are geometric intersection graphs, like diskgraphs (intersections of disks in R ) or k -dot product graphs (graphs whose vertices are vectors in R k ,with an edge if the inner product is at least 1) [Spi03]. These are good candidates because encoding thecoordinates of the vertices as integers will fail [KM12]. Randomized communication techniques may be ableto make progress. Modular lattices.
We have shown that there is no constant-cost universal protocol for distance 2 inmodular lattices but, like low-arboricity graphs, adjacency (and therefore O (log n )-implicit encodings) maystill be possible. Planar graphs.
Our protocol for computing distance 2 on planar graphs did not generalize in a straight-forward fashion to distance 3. Nevertheless, we expect that there is a method for computing k -distance onplanar graphs with complexity dependent only on k ; given that a Schnyder wood partitions each edge into 3groups, we expect that e O (3 k ) should be possible, and maybe only poly( k ), considering that there is a O ( √ n )distance-labeling scheme. Sharing randomness with the referee.
Finally, it seems to be unknown what the relationship is betweenSMP protocols where the referee shares the randomness, and protocols where the referee is deterministic,even though both models are used extensively in the literature. Our Proposition 1.8 relates these twomodels via universal SMP but does not yet give a general upper bound on the universal cost in terms of theweakly-universal cost.
Question 5.3.
What general upper bounds can we get on universal SMP in terms of weakly-universal SMP?
Acknowledgments
Thanks to Eric Blais for comments on the structure of this paper; Amit Levi for helpful discussions andcomments on the presentation; Anna Lubiw for an introduction to planar graphs and graph labeling; CorwinSinnamon for comments on distributive lattices; and Sajin Sasy for observing the possible applications toprivacy. Thanks to the anonymous reviewers for their comments. This work was supported in part by theDavid R. Cheriton and GO-Bell Graduate Scholarships.23 eferences [ABR05] Stephen Alstrup, Philip Bille, and Theis Rauhe. Labeling schemes for small distances in trees.
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Appendix
Proof of Proposition 2.5.
1. If A ⊏ B and B ⊏ C with φ, ψ being the respective embeddings then for all u, v ∈ V ( A ) we have C ( ψφ ( u ) , ψφ ( v )) = B ( φ ( u ) , φ ( v )) = A ( u, v ).2. In the “only if” direction, it suffices to choose G ≡ . In the other direction, if φ : V ( G ) → V ( H ) isan embedding and φ ( u ) = φ ( v ) then for all w ∈ V ( G ) , G ( u, w ) = H ( φ ( u ) , φ ( w )) = H ( φ ( v ) , φ ( w )) = G ( v, w ) so u ≡ v .3. Let g map a vertex of G to its equivalence class and let u, v ∈ V ( G ). If G ( u, v ) = 1 then G ≡ ( g ( u ) , g ( v )) =1 by definition. If G ≡ ( g ( u ) , g ( v )) = 1 then there exists u ′ ∈ g ( u ) , v ′ ∈ g ( v ) such that G ( u ′ , v ′ ) = 1, so G ( u, v ) = G ( u ′ , v ) = G ( u ′ , v ′ ) = 1.4. Let g map vertices in V ( G ) to their equivalence class and let g ( u ) , g ( v ) ∈ V ( G ≡ ). If g ( u ) ≡ g ( v ) thenfor any w, G ( u, w ) = G ≡ ( g ( u ) , g ( w )) = G ≡ ( g ( v ) , g ( w )) = G ( v, w ) so u ≡ v and therefore g ( u ) = g ( v ).Therefore the map g ( u )
7→ { g ( u ) } is an isomorphism G ≡ → ( G ≡ ) ≡ .5. If G ⊏ H then by transitivity, G ≡ ⊏ G ⊏ H ⊏ H ≡ . Likewise, if G ≡ ⊏ H ≡ then G ⊏ G ≡ ⊏ H ≡ ⊏ H .6. If G ≡ is an induced subgraph of H ≡ then clearly there is an embedding. On the other hand, let g ( u ) , g ( v ) ∈ V ( G ≡ ) be the equivalence classes of u, v ∈ V ( G ) and suppose there is an embedding φ : G ≡ → H ≡ . If φ ( g ( u )) = φ ( g ( v )) then g ( u ) ≡ g ( v ) so g ( u ) = g ( v ) since ( G ≡ ) ≡ ≃ G ≡ . Therefore G ≡ is an induced subgraph of H ≡≡