Distributed Broadcasting and Mapping Protocols in Directed Anonymous Networks

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1 Disribued Broadcasing and Mapping Proocols in Direced Anonymous Neworks Michael Langberg Moshe Schwarz Jehoshua Bruck Absrac We iniiae he sudy of disribued proocols over direced anonymous neworks ha are no necessarily srongly conneced. In such neworks, nodes are aware only of heir incoming and ougoing edges, have no unique ideniy, and have no knowledge of he nework opology or even bounds on is parameers, like he number of nodes or he nework diameer. Anonymous neworks are of ineres in various seings such as wireless ad-hoc neworks and peer o peer neworks. Our goal is o creae disribued proocols ha reduce he uncerainy by disribuing he knowledge of he nework opology o all he nodes. We consider wo basic proocols: broadcasing and unique label assignmen. These wo proocols enable a complee mapping of he nework and can serve as key building blocks in more advanced proocols. We develop disribued asynchronous proocols as well as derive lower bounds on heir communicaion complexiy, oal bandwidh complexiy, and node label complexiy. The resuling lower bounds are someimes surprisingly high, exhibiing he complexiy of opology exracion in direced anonymous neworks. Keywords: Anonymous neworks, direced neworks, disribued proocols. 1 Inroducion In his work we sudy he fundamenal problems of broadcasing and mapping (label assignmen and opology exracion) in direced anonymous neworks. In such a nework G, processors do no have unique idenifiers, hey execue idenical proocols, and hey have no knowledge of he opology of he nework (even he size or bounds on i are unknown). The only knowledge available o a verex is is own degree. Anonymous and unknown neworks have been exensively sudied during he las few decades. These sudies include graph exploraion [3, 1, 7], where a robo has o consruc a complee map of an unknown environmen; he sudy of nework communicaion or more specifically he ask of broadcasing and label assignmen [4, 2, 5]; and various characerizaions of aainable and unaainable asks wih respec o addiional symmery breaking assumpions [6, 8]. All sudies above consider he underlying nework o be undireced or direced bu srongly conneced. In such neworks, he ypical paradigm used in he conex of anonymous neworks is adapive message passing. Namely, verices in he nework learn of each oher via informaion ransmied up an down pahs This work was suppored in par by he Calech Lee Cener for Advanced Neworking and by NSF gran ANI Compuer Science Division, The Open Universiy of Israel, Raanana 43107, Israel. mikel@openu.ac.il Deparmen of Elecrical and Compuer Engineering, Ben-Gurion Universiy, Beer Sheva 84105, Israel. schwarz@ee.bgu.ac.il California Insiue of Technology, 1200 E California Blvd., Pasadena, CA 91125, U.S.A. bruck@paradise.calech.edu 1

2 ha connec hese verices, and consruc heir ougoing messages based on he informaion gained so far. In his work we address he design of disribued proocols when he underlying nework is direced bu no necessarily srongly conneced. This seing differs subsanially from he undireced case as a verex may only have a single chance o ransmi an ougoing message, and hus he proocols mus be designed accordingly. 1.1 Our conribuion This work iniiaes he sudy of disribued asynchronous proocols over direced anonymous neworks ha are no necessarily srongly conneced. To his end, we assume ha he nework G has wo special verices, a roo s and a erminal ; and we assume ha here is a pah connecing s and. Our objecive is o preform cerain asks on G. Roughly speaking, he proocols we presen proceed as follows. An iniial message is sen from he roo verex s o is children. This iniiaes he disribued proocol which ends when he erminal verex is in a final sae wih is sae as oupu. The algorihmic asks we presen in his work are he basic asks of broadcasing and label assignmen. For broadcasing, a message m is given a he roo s, and we wish o disribue m hroughou he enire nework G. This, in iself, seems a rivial ask obained by simple propagaion of m. However, we also wan he proocol o erminae iff all he verices of G have received m. This urns ou o be significanly more involved as our proocol canno use he sandard erminaion echniques used for graphs which are srongly conneced (namely, message passing). We sar by presening a simple broadcasing proocol ha will erminae correcly if G is essenially a ree. We hen presen he main resul of his work, a disribued asynchronous broadcasing proocol for general direced graphs. Finally we urn o design a proocol which assigns unique labels o all verices of G (hus enabling us o map he graph opology) and erminaes afer hese labels have been assigned. Our label-assigning proocol is srongly buil on our broadcasing proocol, and can be viewed as a sligh modificaion of i. All he proocols we presen are disribued and asynchronous. We consider several measures of complexiy (which will be discussed in greaer deail laer): oal communicaion complexiy he number of bis ransmied hroughou he proocol, required bandwidh maximal number of bis ransmied over a single edge, and he maximal number of bis in a label. Our resuls can be roughly summarized as follows: Broadcasing in acyclic neworks: A graph G is said o be a grounded ree if every verex of G has in-degree 1, excluding he source s wih no incoming edges and he erminal which may have several incoming edges. For grounded rees G, we describe an asynchronous disribued proocol ha broadcass a message m from he roo verex s o all of G, and hals iff all verices have received m. The proocol has required bandwidh O(log E ) + m and oal communicaion complexiy a mos O( E log E ) + E m (here m is he size of m). We show ha our resuls are igh by providing a maching lower bound on he oal communicaion complexiy of broadcasing in grounded rees. For general direced acyclic graphs we provide an asynchronous disribued proocol for broadcasing wih required bandwidh O( E ) + m and herefore oal communicaion complexiy O( E 2 ) + E m. Our resuls are he bes possible when considering cerain proocols we refer o as commodiy preserving. Broadcasing in general neworks: For general G, we describe an asynchronous disribued proocol ha broadcass a message m from he roo verex s o all of G, and hals iff all verices have received m. The proocol has oal communicaion complexiy a mos O( E 2 V log d ou )+ E m and O( E V log d ou )+ m required bandwidh (here and in wha follows, d ou is he maximal ou-degree in he given nework). 2

3 Unique label assignmen in general neworks: For general G, we describe an asynchronous disribued proocol ha assigns unique labels o all verices of G, and hals iff all verices have been assigned labels. The proocol has oal communicaion complexiy a mos O( E 2 V log d ou ), and O( E V log d ou ) required bandwidh. The labels have lengh of O( V log d ou ) bis which is also shown o be igh by proving a lower bound. The resuling label complexiy in he las proocol is surprisingly high. One migh expec o be able o label he verices wih a mos O(log V ) bis, which is also achievable in boh he undireced case, and he direced bu srongly-conneced case. However, he lower bound we describe shows our resul o be igh, and he exponenial blowup necessary. 1.2 Proof Techniques We describe he main ideas behind our broadcasing proocols, label assignmen proocol and lower bounds. Broadcasing: Our objecive is wo-fold. Firs of all we wan o ransmi a message m from he roo s o all verices of G. This is done by propagaion, every verex ha receives m forwards i on is ougoing edges. Secondly, we wan he proocol o erminae iff all verices have received m. To his end, each inernal verex ha has received m will send ou an addiional message or messages ha evenually will reach he erminal verex. These addiional massages should be consruced in such a way ha will be able o decide wheher all verices of G have been visied or no. This would be a rivial ask if he verices of G were o have unique labels and he erminal would know he size of G (or even a bound on i), namely each verex could simply send is label which will evenually reach. However, in our model of sudy we assume boh ha verices in G do no have unique labels, and ha hey do no have an esimae on he size of G. We now urn o he ask of erminaion. Clearly, only using propagaion, he erminal has no idea when he enire graph has been visied. Thus, addiional informaion should be sen from he inernal verices in G o. Roughly speaking, his addiional informaion will represen a cerain commodiy. Furhermore, he acions aken by each verex in G are commodiy-preserving. Namely, each inernal verex pariions he commodiy i receives among is ougoing edges. Thus, if he source were o send a uni of commodiy ino he graph G and over ime he erminal were o receive a uni of commodiy, one could conclude ha all verices in G have been visied. For example, firs consider he case of grounded rees. Here he commodiy we use is a scalar value, and he inuiion behind our proocol is ha of commodiy-preserving nework flows. Namely, he source sars by ransmiing as erminaion informaion he value 1 (here we assume he source has a single ougoing edge). This corresponds o a flow of value 1 leaving he source. When a verex of ou-degree d receives he value x as erminaion informaion, i sends he value x/d on each of is ougoing edges (corresponding o he disribuion of he flow of value x i received). The erminal will declare erminaion once he values i receives on is incoming edges sum up o 1. Such a proocol will indeed work, however i will no achieve he igh communicaion complexiy saed earlier. In Secion 3, we improve on his proocol by using a differen rule ha governs he flow disribuion a inernal nodes. Bu wha happens when G is no a grounded ree? The main difficuly lies in he case in which G conains cycles. In his case an inernal verex v may receive muliple commodiies over ime, some of which need o be discarded as hey have already been viewed by v. To deal wih cyclic neworks, we use he commodiy-preserving paradigm as before, bu our commodiy is more involved. Insead of ransmiing as erminaion informaion a scalar value, we ransmi muliple values. Loosely speaking, he commodiy leaving he source is he uni inerval [0, 1) (represened by is end poins). Each verex, afer receiving an inerval will send on is d ougoing edges a d-pariion of he inerval received. The erminal will declare erminaion only afer i has received (over ime) a family of inervals whose union is exacly he inerval [0, 1). Each inernal verex can deec cycles by remembering which inervals i has seen so far. Once a cycle is deeced, addiional informaion is sen o he erminal which guaranees correc erminaion. A 3

4 naive implemenaion of hese ideas will lead o exponenial (in V ) communicaion complexiy. To obain he polynomial complexiy saed above, our proocols use several addiional ideas. Label assignmen: As menioned earlier, our unique label assignmen proocol is srongly based on he commodiy preserving paradigm used in he broadcas proocol discussed above. However, roughly speaking, now once a verex v in he graph receives a cerain commodiy, i pariions his commodiy no only among is ougoing edges bu raher a porion of he commodiy is also preserved for v iself. The commodiy preserved for v will ac as a disinc label for he verex v. To ensure correc erminaion of he proocol (once par of he commodiy is sored a inernal verices of he graph) some changes o he broadcas proocol are needed. Lower bounds: We presen boh lower bounds on he oal communicaion complexiy of any broadcasing proocol and lower bounds on he lengh of labels in any label assignmen proocol. For our lower bound on he communicaion complexiy, we firs bound he number of disinc messages ha mus be ransmied as erminaion informaion hroughou he proocol. Roughly speaking, we show ha if he erminaion informaion of a given proocol is limied o exis in a small message space (say Σ), hen one can design a graph G on which he proocol will eiher erminae before all verices have received he broadcas or no erminae a all. A deailed analysis of his line of proof appears in Secion 3.2. Once we have esablished a bound on Σ, a bound on he communicaion complexiy essenially follows by noicing ha (under any encoding) mos of he disinc messages σ Σ mus be represened by log Σ bis. For our lower bounds on he label size of any label assignmen proocol we use a sandard pruning echnique (e.g., [4]). Namely, we sar by considering an exponenially large graph for which any proocol mus assign labels of polynomial size. We hen prune he graph in such a way ha for cerain verices he proocol behaves exacly as if i were running on he original large graph. The pruned graph will on one hand have verices which are assigned large labels and on he oher hand have few verices. This will imply our resul. 1.3 Organizaion The paper is organized as follows. In Secion 2 we presen our model and working assumpions. We coninue in Secion 3 o presen our broadcasing proocol for grounded rees and DAGs, along wih he proof ha our proocol is opimal. In Secion 4, we presen our broadcasing proocol for general graphs. Finally, our label-assigning proocol is presened in Secion 5. We conclude wih a discussion and open problems. 2 The Model In his work we sudy anonymous proocols on direced graphs G = (V, E) wih wo special verices. The roo verex, denoed by s and he erminal verex denoed by. Verices in V \ {s, } will be referred o as inernal verices. We assume ha s has no incoming edges and only one ougoing edge, and has no ougoing edges. As our access o G is only hrough s and, if here are inernal verices which are no on a pah from s o bu are sill reachable from s or conneced o, our proocols will no erminae. Throughou, o simplify our presenaion, we assume ha all verices in G are reachable from s. All verices in G are assumed o know nohing of he opology of he nework (including is size) nor do hey have unique idenifiers. Each verex is assumed o know how many incoming and ougoing edges i has, and has he power o disinguish beween differen incoming/ougoing edges. The model we presen is asynchronous. Our resuls can be easily exended o he case in which here are muliple roo/erminal verices, he roo has muliple ougoing edges, he case in which here are verices in G ha are no reachable from s, and o he case ha he communicaion hroughou he nework is synchronous. 4

5 Anonymous proocols An anonymous proocol on G is defined by he following primiives: a sae space Π, a message space Σ, an iniial sae π 0 Π, an iniial massage σ 0, a sae funcion f : Π Σ N Π, a message funcion g : Π Σ N N {Σ,φ}, and a sopping predicae S : Π {0, 1}. An anonymous proocol is execued as follows. Iniially we associae he sae π 0 wih every verex in G. The message σ 0 is sen on he ougoing edge of s. Each verex ha receives message σ on is i-h incoming edge while having curren sae π, moves o sae π = f(π,σ, i) and sends message g(π,σ, i, j) on is j-h ougoing edge. If in he scenario above g(π,σ, i, j) = φ, hen no message is sen on ougoing edge j. We say ha an anonymous proocol has erminaed if S(π) = 1 for he sae π of, in his case π is he oupu of he proocol. Qualiy There are several qualiy parameers ha may be considered when sudying anonymous proocols. The size of he sae space is relaed o he amoun of memory needed a each verex of he nework. The size of he message space is a bound on he maximum message lengh ransmied on edges on he nework (i.e., bandwidh). Muliplying he bandwidh by he oal number of messages sen over he nework before erminaion will imply an upper bound on he oal communicaion complexiy of he proocol. In a synchronous model one may also consider he ime i akes for he proocol o erminae. In his work we focus on he asynchronous model in which we seek o design anonymous proocols wih minimal oal communicaion complexiy. To his end, we sudy boh he oal number of messages ransmied hroughou he nework, and he maximum message size. There is an obvious rade-off beween he wo, and heir produc is he oal communicaion complexiy. 3 Broadcasing over Grounded Trees and DAGs 3.1 Broadcasing on grounded rees We sar wih he ask of broadcasing a message m over grounded rees. The proocol will erminae iff all verices in he nework have received he broadcas message m. Roughly speaking, he proocol sars when he roo verex sends he message m and addiional erminaion informaion on is ougoing edge. Each verex in he graph, will ransmi informaion on is ougoing edges only when i has received he message m from is incoming edge. A verex of ou-degree d will ransmi boh he message m and erminaion informaion which depends on he incoming erminaion informaion and he value d. As saed in he Inroducion, he inuiion behind he proocol is ha of commodiy-preserving nework flows. Namely, he source sars by ransmiing as erminaion informaion he value 1. This corresponds o a flow of value 1 leaving he source. When a verex of ou-degree d receives he value x as erminaion informaion, i sends he value x/d on each of is ougoing edges (corresponding o he disribuion of he flow of value x i received). A naive implemenaion of his proocol resuls in oal communicaion complexiy bounded by O( E 3/2 )+ E m. The E m facor corresponds o he message m and is ineviable. The remaining O( E 3/2 ) follows from he fac ha we mus be able o represen many differen x values sen as erminaion informaion. Using a slighly differen rule ha governs he flow disribuion a inernal nodes, we are able o reduce his complexiy o O( E log E )) (which we show o be opimal). In he rule we presen, he values x ransmied as erminaion informaion will always be a power of 2. This way, x can be represened efficienly. More specifically, he flow disribuion of our proocol proceeds as follows. When a verex of ou-degree x d receives he value x as erminaion informaion, i sends he value on is firs 2d 2 log d ougoing 2 x log d edges and he value on is remaining ougoing edges. I is no hard o verify ha such a disribuion 2 log d 1 rule is commodiy preserving and ha he x values ransmied will always be a power of 2. Indeed, for 5

6 d N and α = 2d 2 log d i holds ha α 2 log d + (d α) 2 log d +1 = 1. The erminal will declare erminaion iff he sum of values x i receives on is incoming edges equals 1. Theorem 3.1 The proocol above erminaes iff each verex of G is conneced o. On erminaion each verex in G will have received he message m. The oal communicaion complexiy of he proocol is bounded by O( E log E ) + E m (here m is he size of m). Proof. Le G be a grounded ree in which each verex in G is conneced o he erminal. During he execuion of our proocol, each edge (u, v) will carry a message only once: immediaely afer u receives he message (m, x) on is sole incoming edge. If u has ou-degree d, he messages leaving u are of he form x x (m, ) or (m, ). The erminaion values leaving u sum up o he incoming erminaion value of 2 log d 2 log d 1 x. This corresponds o he flow conservaion described earlier and implies ha, evenually, afer each verex has been visied, he sum of all erminaion values enering he erminal will be equal o 1. If a some poin in ime here is a verex ha has no ye been visied, hen for similar reasons he sum of all erminaion values ha have enered he erminal so far will be sricly less han 1 and hus he erminal will no be in an acceping sae. In he case ha here exis verices in G ha are no conneced o (bu reachable from s), we are again guaraneed ha he proocol will no erminae. Indeed, he erminaion value x > 0 enering such a verex will no reach, implying ha he sum of erminaion values enering he erminal will no reach he sum of 1. For he communicaion complexiy we bound he size of Σ (he message space), which consiss of he message m and he erminaion values x [0, 1]. However, in he design of our proocol, x is always a power of 2. Moreover, when applied o graphs wih E edges, he size of x is a leas 2 O( E ). This implies he assered complexiy. 3.2 Lower bounds for broadcasing on grounded rees We now urn o show ha our broadcasing proocol is opimal for grounded rees. The proofs of he claims ha follow appear in Appendix A. The ouline of our proof is given in Secion 1.2 of he Inroducion. Theorem 3.2 Any broadcasing proocol ha (1) erminaes on grounded rees G in which each verex of G is conneced o, and (2) does no erminae oherwise; mus have oal communicaion complexiy a leas Ω( E log E ) + E m. Namely, for any broadcasing proocol as above, here exiss an infinie family of graphs G for which he communicaion complexiy of he proocol when execued on G is a leas c E log E + E m for some universal consan c > 0. Proof. Consider any broadcasing proocol ha saisfies he requiremens specified in he saemen of he heorem wih alphabe Σ. For a given graph G, we denoe by Σ G Σ he se of symbols ransmied over he edges of G when he proocol is applied o G. Le Σ n be he union of Σ G over graphs G wih n edges. In wha follows we presen lower bounds on he size of Σ n. Namely, we presen for infiniely many values of n a family of graphs wih E = n for which he union of Σ G over hese graphs saisfies he lower bound. Once we have esablished a lower bound on Σ n, a bound on he oal communicaion complexiy easily follows. Lemma 3.3 In any grounded ree G we may assume w.l.o.g. ha during any broadcasing process, each verex of G ransmis a single message. Lemma 3.3 implies ha he decision o declare erminaion of he proocol is based solely on he mulise of symbols enering he erminal node. Thus, for a proocol A we define he erminaion se T A o be he se of all finie mulises over alphabe Σ which are erminaing, namely, if he erminal node were o see exacly hose symbols on is incoming edges i would declare erminaion. 6

7 Le σ A (e) Σ denoe he characer ransmied on edge e when he proocol A is execued. Similarly, le σ A (E ) denoe he mulise of symbols ransmied over edges in E E. We say ha verex v 1 V is an ancesor of verex v 2 V if here is a pah from v 1 o v 2. In ha case we also say v 2 is a descendan of v 1. The following definiion will help us consruc he main ool of he proof. Definiion 3.4 Le G = (V, E) be a direced acyclic graph wih source s and erminal node. A linear cu is a pariion of V ino wo disjoin non-empy ses V 1 and V 2 such ha here are no v 1 V 1 and v 2 V 2 where v 1 is a descendan of v 2. Obviously, in any linear cu s V 1 and V 2. Linear cus are useful since hey provide us wih possible snapshos of he symbols crossing he cu in some running of he proocol. This is shown in he following lemma. Lemma 3.5 Le E be he se of edges crossing some linear cu of G. Then σ A (E ) T A, i.e., any mulise of symbols crossing a linear cu when running proocol A, is erminaing. Having seen he connecion beween erminaing ses and linear cus, we can sae in he following heorem which is he main ool used in our proof. Theorem 3.6 Le us consider wo disinc linear cus (no necessarily even in he same grounded ree!) wih ses of edges crossing hem E and E respecively. Then for any proocol, σ A (E ) σ A (E ) (where he sric-subse noaion is aken in he mulise sense). Lemma 3.5 and he resuling Theorem 3.6 apply equally well (hough wih some exra work) o direced acyclic graphs, and are he main ools we use for proving lower bounds. A simple applicaion of Theorem 3.6 is given in he following lemma. Lemma 3.7 Le G = (V, E) be a grounded ree. If e E is an ancesor of e E, and a leas one of he verices on he pah beween hem has ou-degree a leas 2, hen σ A (e ) = σ A (e ). We are now a a posiion o complee he proof of Theorem 3.2. For any n N we consruc a grounded ree G n = (V n, E n ) wih V n = {s,, v 1,..., v n } and edges E n = {(s, v 1 )} {(v i, v i+1 ) 1 i n 1} {(v i, ) 1 i n}. Thus, G n has n + 2 verices and 2n edges (see Figure 5 of Appendix C). According o Lemma 3.7, any proocol running on G n requires an alphabe of size a leas n + 1. I follows ha Σ n = Ω(n). Now, no maer how we encode he elemens of Σ, i mus be he case ha a leas Ω(n) elemens will require a represenaion of a leas Ω(log n) bis. Thus, as E = Ω(n), he oal required bandwidh and communicaion complexiy of any proocol is Ω(log E ) + m and Ω( E log E ) + E m respecively (he addiional dependence on m follows from he need o broadcas he message m). 3.3 Broadcasing on direced acyclic graphs As we move on o direced acyclic graphs, he overall picure is no so well undersood. On he posiive side, a sraighforward modificaion of he commodiy-preserving proocol for grounded rees presened earlier (in which he commodiy is a scalar value) implies a broadcasing proocol for DAGs wih required bandwidh O( E ) + m and herefore oal communicaion complexiy O( E 2 ) + E m. For a lower bound: we can assume w.l.o.g. ha each node does no send ou any messages unil hearing a message on each of is incoming edges. (We omi he rigorous proof of boh claims above.) Under his assumpion, he linear-cu 7

8 mehod presened for grounded rees applies, and we ge a lower bound on he required bandwidh from each edge, of Ω(log E ) bis. Unforunaely, closing his gap currenly seems ou of reach. However, as we will see, if we resric ourselves o commodiy-preserving proocols our resuls are igh (namely we obain a lower bound ha maches he upper bound saed above). We suspec ha an analogous lower bound can be proved for general proocols as well. The proof of he following heorem appears in Appendix B. Theorem 3.8 Any commodiy-preserving proocol requires bandwidh of a leas Ω( E ) bis. 4 Broadcasing over General Graphs We now urn o address he case of broadcasing over general graphs. Loosely speaking, he proocols we presen generalize he commodiy-preserving flow of he previous secion, only he commodiies are now disinc and uniquely idenifiable. This basic commodiy is given in he following definiion. Definiion 4.1 The inerval se over [0, 1) is defined as I[0, 1) = {[a, b) [0, 1) a, b R}. Similarly, he inerval-union over [0, 1), denoed as U[0, 1), is defined as he se of all finie unions of disjoin inervals from I[0, 1). Elemens of U[0, 1) will be called inerval-unions. A pariion of an inerval-unionα U[0, 1) is a finie se of disjoin inerval-unions whose union is α. For our purposes, an inerval [a, b) may be simply represened by he numbers a and b, and since we will have a choice in selecing a and b, we will choose hem o be binary-poin numbers of finie represenaion, i.e., a sum of powers of 2 wih a finie number of summands. By convenion, an inerval of he form [a, a) is he unique empy inerval which we consider a subse of any inerval. We are now in a posiion o sae our broadcasing proocol. Loosely speaking, boh he messages ransmied hroughou he proocol and he sae space consis of elemens α U[0, 1). The proocol sars when he roo verex s ransmis as is iniial message he inerval-union consising of he single inerval [0, 1). Each verex has a sae α which includes he union of all he inervals i has seen so far. When a verex receives as a message an inerval-union α, i adds α o is sae α and hen pariions is new sae among is ougoing edges. In such a way he inerval [0, 1) is pariioned among he verices of he graph (verices may have corresponding α s ha inersec). Our proocol will erminae once has seen he enire inerval [0, 1). However, if here exiss a cycle in he graph, par of he inerval may ge suck in a loop never reaching he erminal. Our proocol idenifies such cases and hrough simple propagaion noifies o disregard his missing par. A crucial aspec of our proocol is he so-called sae-monooniciy propery, which ensures ha for every verex v, he inerval-union which is he sae of v a ime T is included in he inerval-union which is he sae of v a any ime T > T. More specifically, our sae space Π consiss of he se {U[0, 1)}. Namely, each verex v of ou-degree d holds a sae π = (ᾱ,β), where ᾱ = (α 1,α 2,...,α d ) {U[0, 1)} d and β U[0, 1). Inuiively, each α j corresponds o he inerval-union previously sen on he j-h ougoing edge of v; and β corresponds o addiional informaion needed o cope wih cycles. The message space will consis of wo union-inervals, namely Σ = (α,β) U[0, 1) U[0, 1), and a message m (o be broadcas). In our proocol, each message sen includes he message m. In wha follows, we focus on he addiional informaion in U[0, 1) U[0, 1) sen in a message σ, and for simpliciy of noaion, we do no menion he message m. The iniial message sen by s is σ 0 = ([0, 1), [0, 0)). We assume ha he iniial sae of all verices is π 0 ({[0, 0)}, [0, 0)). Again, each verex v of ou-degree d holds iniial sae (α 0,β 0 ) = ([0, 0) d, [0, 0)). Now we come o define he funcions f and g. Consider a verex v of ou-degree d. Le π = (ᾱ,β) = ((α i ) i=1 d,β) be he sae of v, and le σ = (α,β ) be he message v receives on incoming edge i. We define he value π = (ᾱ,β ) = ((α j )d j=1,β ) of f(π,σ, i). We sar by defining a canonical pariion of he union-inervalα ino d pars. Leα = r k=1 I k where each I k is a non-empy inerval of he form [a, b). Le 8

9 I1 1,..., Id 1 1 be a pariion of he firs inerval I 1 ino d 1 pars. A canonical pariion of α ino d union inervals α1,...,α d is defined by α j = I j 1 for j {1,..., d 1} and α d = r k=2 I k. Now, if π = π 0 hen ᾱ is a canonical pariion of α ino d pars, and β = β. If π = π 0 hen for j {1,..., d 1} he inerval-union α j = α j (i.e., α j = α α d \ d 1 α d remains unchanged), and he inerval-union j=1 α j. The inerval-union β = β β d j=1 (α α j ). The funcion g is defined enirely by π, namely g(π,σ, i, j) = g((ᾱ,β), (α,β ), i, j) = (α j \α j,β \ β). We elaborae on he definiion of f and g. We sar by noicing ha he sae π = (ᾱ,β) of each verex is increasing (wih respec o se inclusion) over ime. Boh in every α j porion and in he β porion of (ᾱ,β). For j {1,..., d 1}, α j is changed only once in he enire running process, while α d may change many imes. β may also change many imes during he running of he proocol. I is imporan o noice ha a message is sen from v on ougoing edge j iff eiher α j changes or β changes. Finally, he predicae S(π) = 1 iff π = (α,β) and α β = [0, 1). Theorem 4.2 The proocol above erminaes iff each verex of G is conneced o. On erminaion each verex in G will have received he message m. The oal communicaion complexiy of he proocol is O( E 2 V log d ou ) + E m. Proof. Le (ᾱ,β) be he sae of a verex v in G a some ime T. Le a R. We say ha he j-h ougoing edge of v α-carries a a ime T if a α j a ha ime. We say ha he j-h ougoing edge of v β-carries a a ime T if a β a ha ime. Noice ha if an edge α-carries (β-carries) a value a a ime T, i coninues carrying i a ime T > T due o our sae-monooniciy. For a [0, 1), consider he edges α-carrying a over ime T. These edges form a (monoonically increasing over ime) subgraph G T (a) of G in which he roo has ou-degree 1, verices in G wih in-degree zero have ou-degree zero, and he remaining verices have ou-degree a mos 1, i.e., G T (a) is eiher a pah beween s and some verex v, or a pah saring a s which a some poin closes a loop. This follows from he fac ha for a sae π = ((α j ),β) of v, he inerval-unions α j have an empy inersecion, and heir union is equal o he union of all union-inervals α ha have passed hrough v. Thus for each a [0, 1) eiher here exiss a ime T for which he subgraph G T (a) is conneced o, in which case will evenually receive a, or here exiss a T for which G T (a) conains a cycle which prevens a form reaching. If for some a and T he corresponding subgraph G T (a) conains a cycle, hen i is no hard o verify ha by our definiions a will be β-carried hroughou he predecessors of he cycle. Specifically, a will evenually be β-carried on edges leading o. We conclude ha will evenually be in a sae π for which S(π) = 1 (namely, he proocol will always evenually hal). Assume ha he proocol erminaed a ime T, while here was sill a verex v in G ha has no been visied. Le P be a pah from s o v and le u be verex on his pah closes o s ha has no been visied. By our definiions here is a value a which is α-carried on he pah P up o u (including) bu no α-carried on any oher edge along he pah P beween u and v. Namely, a ime T he graph G T (a) consiss of a pah ha ends a he verex u. Thus, a has no been α-carried o ye a ime T. Theoreically, i may sill be he case ha a has been β-carried o, however in order for a o be β-carried on any edge e a ime T, he graph G T (a) mus conain a cycle, and hus no verices wih in-degree greaer han 1 and ou-degree 0, which conradics he properies of he verex u. I is lef o analyze he communicaion complexiy of our proocol. This is done by analyzing boh he oal number of messages sen along he edges of G, and by analyzing he bi complexiy of each message. We sar wih he laer. Theorem 4.3 A mos O( E V log d ou ) + m bis are required o represen a symbol from he edge alphabe. 9

10 Proof. As described previously, an inerval [a, b) [0, 1) will be encoded by wriing down he numbers a and b. We will always make sure ha a and b are binary-poin numbers of finie represenaion. Given an inerval [a, b) [0, 1), say we wan o pariion i ino k 1 disjoin inervals. Le N be he smalles power of 2 such ha N k and denoe = (b a)/n. We pariion [a, b) in he following manner: [a, a + ), [a +, a + 2 ),..., [a + (k 2), a + (k 1) ), [a + (k 1), b) i.e., ino k 1 inervals of size and one inerval of size (b a) (k 1). Each of he new inervals requires addiional O(log k) bis o encode relaive o he encoding of [a, b). Consider any inerval-union ransmied on an edge in G. Le d ou be he maximal ou-degree of any verex in G. We show ha (a) he number of inervals in his union is bounded by E, and (b) he endpoins of any inerval in his inerval-union can be represened by O( V log d ou ) bis. This suffices o prove our asserion. Le a and b be he end poins of an inerval in an inerval-union ransmied in G. The bi represenaion of a and b is an arifac of how may imes he inervals ha included a and b were pariioned during he execuion of our proocol. As each verex in G only preforms inerval pariioning once during he execuion of he proocol (only when i received a message for he firs ime), and his pariion is ino a mos d ou pars, we conclude ha he oal bi represenaion of a and b is O( V log d ou ). We now bound he number of inervals which consiue an inerval-union. In general, his is done by bounding he oal number of differen inervals viewed in our proocol. A new inerval may be formed only by canonical pariioning of he α par, or by he inersecion of wo inervals in he β par. A mos O( E ) inervals may be obained by canonical pariioning (each verex v does he pariioning once, and produces d ou (v) new inervals). By observing ha he se of inerval end-poins in he β par mus be a subse of he inerval end-poins of he α par, we conclude ha he β par is also made up of a mos O( E ) inervals. Coninuing he proof of Theorem 4.2, an essenial observaion is he fac ha for any edge, any a R is α-carried (β-carried) by i a mos once. I follows from he previous claim ha he oal communicaion on any one edge is O( E V log d ou ) + m, which concludes he proof. 5 Label Assignmen on General Graphs The proocol from he previous secion, hough polynomial in he parameers of he graph, seems overcomplicaed for he ask achieved broadcasing over general graphs. However, his lays he foundaion for a proocol which assigns unique labels o verices of G. A sligh variaion o our previous proocol for broadcasing over general graphs allows us o assign unique id s o he verices of G. In he label-assignmen proocol, he sae of each verex will include an addiional union-inerval which represens is label. Namely, he sae of each verex of ou-degree d will now be π = (ᾱ,β) = ((α j ) d j=0,β) insead of π = ((α j) d j=1,β) as before. The addiional union-inerval α 0 will be he label of he verex v. We need o define he funcions f and g accordingly. f(π,σ, i) = π = (ᾱ,β ) = ((α j )d j=0,β ) is essenially defined as before, bu now if π = π 0 hen ᾱ is a canonical pariion ofα ino d + 1 pars (α j )d j=0, and β = β α 0. If π = π 0 hen f is exacly as defined previously wih he addiion ha he value of α 0 is idenical o ha of α 0. The funcion g is defined as before. Theorem 5.1 The unique labeling proocol above erminaes iff each verex of G is conneced o. On erminaion each verex in G will have a unique label. The oal communicaion complexiy of he proocol is O( E 2 V log d ou ). Each resuling verex label is of lengh O( V log d ou ) bis. Proof. The proof is essenially he same as ha of Theorem 4.2 so we only give he inuiion behind i. Wha his proocol does is send he inerval [0, 1) ino he graph. Verices receive inerval-unions on incoming 10

11 edges and send subses of hose on ougoing edges. Bu unlike he proocol described in Secion 4, hey do no pariion he incoming inerval-unions among he ougoing edges, bu raher pariion hem among he ougoing edges and hemselves, keeping a sub-inerval o be used as a unique idenificaion. Thus, like he proof of Theorem 4.2, we follow he pah which α-carries some value a [0, 1). Like before, his pah may eiher end in an oupu verex, or close up on iself because of a cycle, and hen a is β-carried o he oupu hrough simple propagaion. Only now we have a hird opion, where a sops being α-carried no because of a cycle or because of reaching an oupu verex, bu raher because a verex has aken an inerval conaining a as is label. From hen on, a is being β-carried by propagaion o he oupu. The monooniciy and he pariioning done by each verex ensure ha he resuling idenificaion inervals are disjoin. For he bi complexiy of a label, noice ha each label is a single inerval, and use he analysis of Theorem 4.2. One migh hink ha a labeling of V verices by labels of O( V log d ou ) is inefficien, bu he following heorem shows ha his number is acually required by any labeling proocol. The ouline of our proof is given in Secion 1.2 of he Inroducion. Theorem 5.2 Any labeling proocol giving unique labels o he verices produces labels each of lengh Ω( V log d ou ) bis. Proof. Le us consider a full ree of heigh h and degree d where he edges are direced away from he roo s. Le be a verex conneced o all leaves of he ree. Any labeling proocol ha gives unique labels o he verices will use a leas d h disinc labels. Thus, here exiss some leaf verex v, which receives a label of Ω(h log d) bis. The main observaion is ha in such a graph wihou cycles, each verex receives a label which depends only on he messages sen o i along a pah from he roo. Therefore, we can remove all verices of he original ree no on he pah from he roo o v, and connecing all oher d 1 edges from each of he verices on he pah o he erminal (see Figure 6 of Appendix C). Thus we ge a new graph wih a oal of h + 3 verices and maximal ou-degree d. By our previous observaion, v sill receives a label of Ω(h log d) = Ω( V log d ou ) bis. We sress ha he execuion of he labeling proocol on he pah o v in he new pruned graph is idenical o he execuion of he labeling proocol on he pah o v in he original graph. Indeed, le u be a verex along he pah o v. The label of u and he messages leaving u depend solely on he incoming messages o u and he number of ougoing edges of u: boh remain unchanged. 6 Conclusion and Open Problems In his work we explored asynchronous disribued proocols running on anonymous direced neworks. By showing how o broadcas and assign labels on such neworks, we can ransform anonymous neworks o labeled neworks and even map he whole opology by flooding local informaion available o nodes. This is a crucial sep since mos exising proocols require eiher some knowledge of he nework or unique labeling of he nodes. By comparing he resuls of his work wih hose over undireced anonymous neworks, we can see a wide gap in performance which may be aribued mainly o he problem of erminaion, and he possible lack of feedback due o he direcionaliy of edges. This is srongly pronounced in he case of assigning labels, where insead of O(log V ) bis for labels in undireced anonymous neworks (or even direced bu srongly-conneced neworks) we require a leas Ω( V log d ou ), an exponenial gap. This is direcly aribued, as seen from he lower-bound proof, o he lack of feedback from erminal o source. 11

12 Anoher possible gap can be seen in he broadcasing proocol when moving from he resriced case of grounded rees, o he slighly more general case of direced acyclic graphs. Proocols which are commodiy preserving suffer from an exponenial increase in required bandwidh over he edges, from O(log E ) o Ω( E ) bis per edge. However, we do no know how o improve he lower bound for DAGs, and so, we pose as an open quesion: wha is he required bandwidh per edge for broadcasing over DAGs? and over general graphs? References [1] S. Albers and M. R. Henzinger. Exploring unknown environmens. In STOC 97: Proceedings of he weny-ninh annual ACM symposium on Theory of compuing, pages , New York, NY, USA, ACM Press. [2] B. Awerbuch, O. Goldreich, D. Peleg, and R. Vainish. A rade-off beween informaion and communicaion in broadcas proocols. Journal of he ACM, 37(2): , [3] X. Deng and C. H. Papadimiriou. Exploring an unknown graph. J. of Graph Th., 32: , [4] P. Fraigniaud, A. Pelc, D. Peleg, and S. Pérennes. Assigning labels in unknown anonymous neworks. Disribued Compuing, 14: , [5] L. Gargano, A. Pelc, S. Pérennes, and U. Vaccaro. Efficien communicaion in unknown neworks. Neworks, 38(1):39 49, [6] T. Kameda and M. Yamashia. Compuing on anonymous neworks: Par I - characerizing he solvable cases. IEEE Transacions on parallel and disribued sysems, 7(1):69 89, [7] P. Panaie and A. Pelc. Exploring unknown undireced graphs. J. of Algorihms, 33: , [8] N. Sakamoo. Comparison of iniial condiions for disribued algorihms on anonymous neworks. In PODC: 18h ACM SIGACT-SIGOPS Symposium on Principles of Disribued Compuing, pages ,

13 A Proof of Theorem 3.2 A.1 Proof of Lemma 3.3 Proof. The proof is by a simple inducion on he disance from he source s. The inducion basis is obvious: he source sends only one message on each of is ougoing edges. For he inducion sep, i is easy o see ha inner verices (i.e., no he erminal verex ), have only one incoming edges which, by he inducion hypohesis, carries jus one message. Thus, inner verices send exacly one message on each of heir ougoing edges. A.2 Proof of Lemma 3.5 Proof. Le G be a grounded ree, and E a se of edges crossing some linear cu which pariions V ino V 1 V 2. By he definiion of a linear cu, and due o he fac ha we use he asynchronous model, we can easily hink of V 1 as he se of verices which already processed he symbols on heir incoming edges (and sending symbols on heir ougoing edges), and V 2 as he se of verices which did no ye process incoming symbols. Thus, he linear cu resembles a snapsho in ime of a possible running of he proocol. A his poin we can consruc a differen grounded ree G = (V, E ) wih V = V 1 {}, all he edges beween verices of V 1 remain he same, and he edges crossing he linear cu in G, all connec o he erminal node (see Figure 1). s V 1 s V 1 V 2 (a) (b) Figure 1: (a) The grounded ree G wih he linear cu pariioning i ino V 1 and V 2. (b) The grounded ree G Obviously, G is also a grounded ree, and furhermore, all inner verices are on a pah from s o. I follows ha he proocol should run on i correcly as well, and erminae. The sequence of running he verices on G unil reaching he linear cu could be repeaed when running on G and so he mulise of symbols on edges crossing he cu in G is exacly he same as ha in G. Thus, he proocol should erminae upon geing σ A (E ) when running on G, which by definiion means ha σ A (E ) T A, as we waned o prove. A.3 Proof of Theorem 3.6 Proof. Assume o he conrary haσ A (E ) σ A (E ). Le G = (V, E) be he grounded ree from which E was obained, and le V 1 V 2 be he appropriae linear cu. In a similar fashion o he proof of Lemma 3.5, we consruc a new grounded ree G = (V, E ), wih V = V 1 {, } where is he erminal verex and is an auxiliary verex. All he edges beween verices of V 1 remain he same. Now, here exiss a pariion of E ino wo disjoin non-empy ses E 1 E 2 such haσ A(E 1 ) = σ A(E ). We connec he edges E 1 o he verex, and he verices of E 2 o (see Figure 2). 13

14 s V 1 s V 1 V 2 (a) (b) Figure 2: (a) The grounded ree G wih he linear cu pariioning i ino V 1 and V 2. (b) The grounded ree G and auxiliary node As before, running proocol A on G produces symbols idenical o hose produced when A is run on G. By Lemma 3.5, σ A (E ) T A, bu we consruced G so ha σ A (E 1 ) = σ A(E ), and so i follows ha proocol A will erminae when running on G. However, verex is reachable from s bu is no conneced o he erminal verex, and so a proper proocol should no erminae a conradicion. Thus, σ A (E ) σ A (E ). A.4 Proof of Lemma 3.7 Proof. Le us choose arbirarily a linear cu which he edge e crosses, and le he appropriae pariion of V be V 1 V 2. Denoe by E he edges crossing he cu. Since e is an ancesor of e, here exiss a pah v 0, v 1,..., v k 1, v k such ha (v 0, v 1 ) = e and (v k 1, v k ) = e. We noe ha by he definiion of a linear cu, necessarily v 0 V 1 and {v 1,..., v k } V 2. We can now consider anoher pariion of V ino V 3 V 4 where, V 3 = V 1 {v 1, v 2,..., v k 1 } V 4 = V 2 \ {v 1, v 2,..., v k 1 }. This pariion is clearly anoher linear cu (see Figure 3). Denoe by E he verices crossing his cu. Now e E bu e E. Also, since we required a leas one verex from {v 1,..., v k 1 } o have an ou-degree a leas 2, i follows ha E > E. 1 s cu 2 nd cu s v 0 v k Figure 3: The wo linear cus of Lemma 3.7 Now, assume o he conrary ha σ A (e ) = σ A (e ). In ha case, necessarily, σ A (E ) σ A (E ), which is impossible by Theorem

15 B Proof of Theorem 3.8 As menioned, we consider proocols in which each node does no send ou any messages unil hearing a message on each of is incoming edges. For such proocols we change he definiion of he funcion g ha governs he ougoing messages. Namely, for a verex of in degree d in, g is now defined as a funcion from Π Σ d in N o Σ. For incoming messages σ = {σ 1,...,σ din } and sae π, g(π, σ, i) is he message ransmied on ougoing edge i given incoming messages σ and sae π. Definiion B.1 A proocol is commodiy-preserving if here exiss a real funcion q : Σ R + such ha for any in-degree d in, any ou-degree d ou, any sae π Π, and any se of incoming messages σ = {σ 1,...,σ din } d in d ou q(σ i ) = q (g(π, σ, i)). i=1 i=1 Proof. Unlike previous proofs which considered a graph and manipulaed i, we consider a se of graphs of some general form, and prove ha a leas one of hose graphs causes he proocol o require a bandwidh of Ω( E ) bis. Le us fix a commodiy-preserving proocol A. We build hese graphs sep by sep, saring wih he wo verices s and. We connec s o v 0. Since we can easily scale he commodiy, le us assume w.l.o.g., ha he source s sends σ 0 wih q(σ 0 ) = 1 o v 0 under proocol A. We now draw wo ougoing edges from v 0. Proocol A mus divide he commodiy σ 0 beween he wo edges, and for convenience of illusraion, le us assume he smaller quaniy is ransmied over he lef edge while he larger is sen over he righ edge. Namely, if v 0 ransmis σ 1 o v 1 and σ 2 o u 0 hen q(σ 1 ) q(σ 2 ). We connec he lef edge o a verex we call v 1. We now coninue in he same fashion: verex v i has wo ougoing edges and he proocol divides he quaniy enering v i so ha he smaller quaniy flows over he lef ougoing edge o v i+1. We connec he righ ougoing edge from v i o a verex we call u i. We coninue his spliing unil we reach v 2n 1 which we do no spli anymore, bu raher connec o. We also connec all u i wih i 1 (mod 2) o. The resuling graph is seen in Figure 4 (a). v s 0 v 1 v 0 s 1 v 2 u 0 u 1 v 2 u 1 v 3 u 2 u 0 3 v 4 u 2 u 3 v 4 u 3 u 4 v 5 u 4 v 5 w (a) (b) Figure 4: (a) The skeleon ree for n = 3 and (b) afer choosing he subse {u 0, u 2 } We inroduce an auxiliary verex w which we connec o. We can now choose arbirarily any subse S {u 0, u 2,..., u 2n 2 }, and connec he verices of S o w, while he res we connec o (see Figure 4 (b)). For each choice of S we ge a differen graph. We noe ha if no for verex w, he resuling graphs would all be grounded rees. 15

16 Le us denoe by q(v) = q(σ v ) he quaniy flowing ino verex v under Proocol A. We can wrie he following general inequaliy chain: q(u 2i+2 ) < q(v 2i+2 ) 1 2 q(v 2i+1) 1 2 q(u 2i). (1) Since proocol A is a commodiy-preserving proocol, he quaniy ransmied from w o is simply v S q(v). Bu by inequaliy (1), for wo subses S, S {u 0, u 2,..., u 2n 2 }, S = S, i follows ha q(v) = q(v) v S v S and so we ge as many differen quaniies flowing from w o as here are subses, i.e., 2 n differen quaniies implying 2 n differen symbols σ Σ. I follows ha o encode Σ we need Ω(n) bis. However, he graph iself conains O(n) verices and O(n) edges. Hence he proof is complee. 16

17 C Figures v 1 s v n Figure 5: The grounded ree G n s s v v (a) (b) Figure 6: (a) The full ree and (b) he pruned ree, which are used in Theorem