Optimal Gossip with Direct Addressing

Transcription

1 Optimal Goip with Direct Addreing Bernhard Haeupler Microoft Reearch 1065 La Avenida, Mountain View Mountain View, CA Dahlia Malkhi Microoft Reearch 1065 La Avenida, Mountain View Mountain View, CA ABSTRACT Goip algorithm pread information in ditributed network by node repeatedly forwarding information to a few random contact. By their very nature, goip algorithm tend to be ditributed and fault tolerant. If done right, they can alo be fat and meage-efficient. A common model for goip communication i the random phone call model, in which in each ynchronou round each node can PUSH or PULL information to or from a random other node. For example, Karp et al. [FOCS 2000] gave algorithm in thi model that pread a meage to all node in Θ() round while ending only O(log ) meage per node on average. They alo howed that at leat Θ() round are neceary in thi model and that algorithm achieving thi round-complexity need to end ω(1) meage per node on average. Recently, Avin and Eläer [DISC 2013], tudied the random phone call model with the natural and commonly ued aumption of direct addreing. Direct addreing allow node to directly contact node whoe ID (e.g., IP addre) wa learned before. They how that in thi etting, one can break the barrier and achieve a goip algorithm running in O( ) round, albeit while uing O( ) meage per node. In thi paper we tudy the ame model and give a imple goip algorithm which pread a meage in only O(log ) round. We furthermore prove a matching Ω(log ) lower bound which how that thi running time i bet poible. In particular we how that any goip algorithm take with high probability at leat 0.99 log round to terminate. Latly, our algorithm can be tweaked to end only O(1) meage per node on average with only O() bit per meage. Our algorithm therefore imultaneouly achieve the optimal round-, meage-, and bit-complexity for thi etting. A all prior goip algorithm, our algorithm i alo robut againt failure. In particular, if in the beginning an obliviou adverary fail any F node our algorithm till, with high probability, inform all but o(f ) urviving node. Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. Copyright for component of thi work owned by other than the author() mut be honored. Abtracting with credit i permitted. To copy otherwie, or republih, to pot on erver or to reditribute to lit, require prior pecific permiion and/or a fee. Requet permiion from permiion@acm.org. PODC 14, July 15 18, 2014, Pari, France. Copyright i held by the owner/author(). Publication right licened to ACM. ACM /14/07...$ Categorie and Subject Decriptor F.2.2 [Theory of Computation]: ANALYSIS OF ALGO- RITHMS AND PROBLEM COMPLEXITY Nonnumerical Algorithm and Problem; F.1.2 [Theory of Computation]: COMPUTATION BY ABSTRACT DEVICES Mode of Computation Parallelim and concurrency General Term Algorithm, Theory, Performance, Reliability Keyword direct addreing, goip, information diemination, peerto-peer (P2P), pointer jumping, rumor preading 1. INTRODUCTION Goip algorithm, alo called rumor preading protocol, pread information in a network by having node repeatedly forward (limited amount of) information to a few, often randomly elected, contact. They have found countle application in olving coordination and information diemination tak in large ditributed ytem or network. Their advantage in thi etting over alternative approache i manifold: They are naturally ditributed and due to their randomized nature often very fault tolerant with repect to many kind of failure, uch a, permanent or tranient linkfailure or node crahe. If done right, they can alo be very efficient in their round-, meage-, and bit-complexity. Two model for goip communication that have been conidered and intenely tudied in the literature are the random phone call model and direct addreing. The random phone call model aume a complete network and ynchronou round in which each node i able to PUSH or PULL information to or from a random other node. Both variant pread a meage in O() round with O() meage ent per node [12] if every node communicate in every round. While the logarithmic running time i bet poible for thi etting, Karp et al. howed in an influential paper [10], that the meage-complexity could be reduced to O(log ) meage per node on average. They alo proved that any addre-obliviou algorithm with logarithmic running time need to end at leat Ω(log ) meage per node on average. Here addre-obliviou mean that the deciion of if and what a node end i independent of the ID of it communication partner() thi round (ee alo [1]). Latly, they demontrated the fault-tolerance of their goip algorithm. In particular, they remarked that any obliviou F

2 node failure till lead to all but O(F ) node being informed in the end, with high probability. Since then the random phone call model ha been generalized in many direction, mot notably to non-complete network topologie. For example, in [4 6] arbitrary topologie are aumed and the topping time of the uniform PUSH-PULL goip i tudied where in each round each node initialize a communication with a uniformly random neighbor. Direct addreing i another natural aumption that ha been intenely tudied. Direct addreing aume that node have unique ID (e.g., IP addree) and that in each ynchronou round a node can contact any node (or neighbor) whoe ID (or IP addre) i known to it. Since node do not know the topology and are only able to initiate one communication at a time, many algorithm for thi etting let node contact a random node known to them. For example, in the famou Name-Dropper algorithm of Harchol-Balter et al. [9], node repeatedly forward all ID they know to a random node known to them. They how that thi imple algorithm terminate in O(log 2 n) round (uing O(log 2 n) meage per node) if tarted from any (weakly) connected initial topology. Recently, [11] gave an improved determinitic algorithm which achieve O() round for the ame etting. A imilar etting but retricting a meage to only contain one contact wa conidered in [8]. Latly, direct addreing ha alo been extenively tudied in general topologie. For example in [2, 3, 7] it i aumed that all poible communication are given by an arbitrary unknown topology in which node only know their neighbor. Recently Avin and Eläer [1] howed that adding the natural direct addreing aumption in the random phone call model allow for goip algorithm that break the barrier. Their algorithm run in O( ) round however at the cot of having a larger meage-complexity of Θ( ) meage per node and alo a larger bit-complexity of O(n log 3/2 n+nb log ) (compared to the O(nb log ) bit-complexity of [10]). Their algorithm i fault-tolerant in the ame way a dicued for [10] above: Theorem 1 (Theorem 1 of [1]). In the random phone call model with direct addreing there i a randomized fault-tolerant goip algorithm that, with high probability, pread a meage of b = Ω() bit in O( ) round uing O( ) meage per node and a total bit-complexity of O(n log 3/2 n + nb log ) bit. 1.1 Our Reult In thi paper we tudy the ame etting and improve upon the tate of the art by providing a dratically fater algorithm with imultaneouly improved meage- and bitcomplexity: Theorem 2. In the random phone call model with direct addreing there i a randomized fault-tolerant goip algorithm that, with high probability, pread a meage of b = Ω() bit in O(log ) round uing O(1) meage per node on average and a total bit-complexity of O(nb) bit. In addition to the fater running time, it i eay to ee that the O(1) meage-complexity per node and the total O(nb) bit-complexity are optimal. In contrat to [1], which performed wore than [10] on thee complexity meaure, our algorithm actually improve on the bound of [10] and even break both meage-complexity lower bound given in [10] for the random phone call model without direct addreing. Latly, we alo prove a matching Ω(log ) lower bound on the running time of any goip algorithm in the random phone call model with direct addreing. Thi how that our goip algorithm achieve the optimal round-, meage-, and bit-complexity imultaneouly. Our lower bound i particularly trong in the following ene: Intead of howing that any algorithm that take c log round for a ufficiently mall contant c fail with non-negligible probability we prove that any algorithm that take 0.99 log round or le fail with high probability. Our lower bound furthermore hold irrepectively of many model aumption and even applie to non-goip algorithm which can contact more than one known node per round: Theorem 3. In the random phone call model with direct addreing any (randomized) goip algorithm fail to pread a meage to all node with high probability if le than 0.99 log round are ued. Thi remain true if node are allowed to ue arbitrarily large meage, communicate non addre-obliviouly and even contact an unbounded number of known node per round intead of jut one. A a lat contribution we conider bound on the number of communication done by a node per round. While in a goip algorithm each node initialize only one communication per round, node might have to repond to multiple requet. In fact, the number of imultaneou requet can be a large a n 1. While thi aymmetry i crucially ued in eentially all algorithm that ue direct addreing, it i unfortunately a quetionable aumption for at leat ome ytem and application (ee Section 7 for a more detailed dicuion). In thi regard, we dicu the eay to prove fact that a ub-logarithmic running time inherently require = ω(1) and how that the bet running time poible for any given i / log. We then give a general framework for working efficiently in a etting with bounded. In particular, we how how our goip algorithm can efficiently compute a -clutering which can then be ued to achieve any point on thi trade-off curve between running time and round complexity (ubject only to our Ω(log ) lower bound from Theorem 3). Thi how, for example, that one can till have goip algorithm with an optimal Θ(log ) running-time while keeping maller than any polynomial in n, that i, = n o(1). We believe thi to be a ueful etting of parameter which eem realitic in many etting. Theorem 4. For any = log ω(1) n there i a randomized goip algorithm that, with high probability, compute a -clutering in O(log ) round uing O(1)-meage per node on average but having no node repond to more than requet per round. Given uch a -clutering a meage broadcat can then be performed eaily with an optimal round-, meage-, and bit-complexity. Organization. The ret of the paper i organized a follow: In Section 3 we introduce clutering, which are the main abtraction

3 ued to deign our goip algorithm. In Section 4 we give the implet verion of our algorithm which demontrate the main idea needed to achieve the Θ(log ) round complexity. In Section 5 we then explain how to reduce the number of meage ent to O(1) per node on average and achieve an optimal bit-complexity which prove our main theorem. In Section 6 we prove Theorem 3, i.e., our Ω(log ) round lower bound. In Section 7 we provide our reult on goip algorithm with a bounded amount of communication per node. We defer the dicuion of the fault-tolerance of our algorithm to the full verion. 2. THE RANDOM PHONE CALL MODEL AND DIRECT ADDRESSING In thi ection we give the formal definition for the random phone call model and the direct addreing aumption ued throughout thi paper. The Network. The random phone call model aume a complete network coniting of n node. Each node ha a unique addre or ID coniting of O() bit, that i, we aume a polynomially large ID pace. Initially node know their own ID but not the ID of any other node in the ytem. For convenience we aume that node know the number of node n. Thi aumption i without lo of generality ince for all problem conidered in thi paper it i eay to tet with high probability whether the algorithm ucceeded. Thi allow for determining the parameter n uing the claical gue-tetand-double trategy without increaing the running time by more than a contant factor. Communication. Communication take place in ynchronou round. In each round a node can chooe to initialize one communication. In particular, it can either chooe to contact a random node or ue direct addreing to contact a node whoe ID it know. Each uch contact from a node u to a node v conit of either node u PUSHing a meage to v or of node u PULLing a meage from v. A meage can in principle contain any number of bit but all 1 algorithm throughout thi paper only ue meage of (minimal) ize containing O() bit. In particular, every meage carrie either the information to be broadcat, a node count, or O(1) node ID. Following [1] (and [10]) we ay a goip algorithm i addre-obliviou if the deciion whether to end or not and if o what to end (or repond with) i independent of the communication partner() in thi round. In particular thi mean that for every node u the meage PUSHed by u doe not depend on the ID of the communication partner and all repone u give to a PULL are the ame in one round. All algorithm preented in thi paper are addre-obliviou. Round-, Meage-, and Bit-Complexity. The important figure of efficiency for uch a goip algorithm are the round-complexity, that i, the number of ynchronou round needed, the meage-complexity, that i, 1 The only exception are when we hare a b-bit meage (via a CluterShare) and for reizing cluter in the Algorithm Cluter1, which demontrate the idea for achieving the optimal Θ(log ) round complexity while ignoring meage count and meage ize for implicity. the number of meage ent per node on average throughout the execution, and the bit-complexity, that i, the total number of bit communicated in all meage throughput the execution. We only conider and preent randomized protocol which complete their tak with high probability in the round-, bit-, and meage-complexity tated, that i, with probability at leat 1 1 for ome large choen contant n C > 1. C The (Fault-Tolerant) Broadcat Tak. Goip algorithm can be ued for a multitude of different communication and coordination tak. The algorithm in thi paper compute a cluter containing all node or, in the cae of Theorem 4, a -clutering which can then be ued to perform any of thee tak eaily and efficiently. To be conitent with prior work we focu on one pecific tak, namely, broadcating. In thi tak there i a meage, often alo called rumor, coniting of b bit with b = Ω() which i initially known to one node (or multiple node). The goal i to inform all node in the network. In the full verion we alo addre the etting of node-failure. In thi etting an obliviou adverary fail any F node at the beginning of the execution. Follow [10] and [1], the deired guarantee in thi etting i that all but o(f ) urviving node are clutered/informed. We call an algorithm achieving thi fault-tolerant. 3. CLUSTERINGS In thi ection we introduce clutering, the main tool ued in thi paper to achieve efficient goip. Firt, in Section 3.1, we define the concept and implementation of a clutering. In Section 3.2 we then explain how thee clutering upport efficient implementation of everal primitive allowing for coordination and communication between cluter. Thee primitive are extenively ued throughout thi paper to implement our fat and meage efficient goip communication algorithm. 3.1 Clutering Definition and Implementation A clutering i a partition of all node v V in a network into dijoint cluter C 1, C 2,..., C k and the et U = V \ i Ci of unclutered node. Each cluter C ha one cluter leader l C C known to all node in C. We call all clutered node that are not leader cluter follower. We define the ID of a cluter to be the ID of it leader and the ize of a cluter C to be the number of node C it contain. Clutering are implemented by each node v V containing a variable follow v. For any clutered node v C thi variable contain the ID of the cluter leader ID(l c) while for all unclutered node v U it i et to. Thi alo allow node to determine whether they are a cluter leader of follower by imply comparing their own ID to their follow variable. 3.2 Cluter Coordination Primitive Having a central leader which i known to all node in a cluter allow a cluter to coordinate uing only a contant number of communication round. In one communication round, follower may PUSH information to the leader and thu have the leader collect input from the entire cluter. In a econd round, follower may PULL a repone from the leader. Below, we omit the detail concerning PUSHing and PULLing meage, and briefly refer to exchange uch a

4 above a follower ending information (if any) to the leader and pulling one directive from it. Following i a lit of cluter-macro that make ue of the power of cluter, and will be ued later in our contruction. All thee cluter primitive require only a contant number of round and therefore alo a contant number of meage per node. CluterActivate(p): Thi primitive allow cluter to be activated independently with probability p by having follower pull the outcome of a p-biaed coin flipped by their leader. Each cluter tay activated until the next call of CluterActivate or until it i explicitly deactivated. CluterSize: Each cluter can determine it ize in two round by each cluter follower ending it ID to the cluter leader, the cluter leader counting the received ID and all follower pulling thi count from the leader. CluterDiolve(): Thi primitive enure that all cluter are at leat of the given ize and run in two round: In the firt round each cluter follower puhe their ID to the leader. The cluter leader ue thee meage to determine the cluter ize. In the econd round follower pull a value indicating their new clutering tatu from their current leader and et their follow variable to thi value. If the cluter leader repond with it own ID and otherwie repond with to diolve the cluter. It alo et it own follow variable the ame way. CluterReize(): Thi primitive enure that all cluter are of ize at mot 2. It run in two round. In the firt round each cluter follower puhe their ID to the leader. Thi allow the leader of any cluter to determine the ize of it cluter. In the econd round follower pull a meage from their current leader and update their follow variable a follow: The cluter leader (in hi mind) re-cluter it follower into cluter of equal ize (up to one) declaring for each new cluter the node with the larget ID it leader. To announce 2 and etablih thi clutering the leader repond to each node pull requet with the ID of the new leader. Each follower then et it follow variable to the larget leader ID that i at leat a large a it own ID. It i eay to ee that after a cluter reizing tep all cluter have ize at mot 2 1. CluterPUSH(mg)/CluterPULL(mg): Thi i a imilar primitive to a random PUSH or PULL of a node except on a cluter level. Each cluter leader decide on whether a puh or pull i to be performed and what meage mg to end or with what meage mg to repond if pulled. Thi deciion i then hared within each cluter by follower pulling it from the leader. All node in cluter intructed to do now contact a random node and either relay the meage of the cluter 2 The meage ued in thi primitive are of ize O( ) bit. Thi call for enuring that = Θ(1) if a mall bit complexity i deired. We note that thi complication arie only for addre-obliviou algorithm a a non addreobliviou algorithm could imply repond to each follower eparately with the ID of it new leader. leader via a PUSH or requet a meage via a PULL. All meage received by a node through either a PULL or PUSH get then relayed to their cluter leader (if there i one). CluterMerge(new leader ID): Thi primitive allow to merge cluter in one round. For thi all follower imply requet the ID of the new leader from their current leader to update their follow variable. In addition to anwering thee pull the leader alo et it own follow variable to the new leader. CluterShare(.): Thi primitive allow cluter to hare information within a cluter in two round. In the firt round, follower who have relevant information end thi information to the leader, who aggregate the information. In a econd round, follower pull the aggregated information from the leader. 4. GOSSIP IN O(log log N) ROUNDS In thi ection we preent the implet verion of our algorithm to demontrate the principal idea needed achieve the optimal O(log ) round-complexity. The reulting algorithm Cluter1 i, for ake of implicity, not tuned to have a good meage- or bit-complexity. In particular, a contant fraction of the node i ending during mot of the Θ(log ) round and the meage ize occurring in the firt CluterReize call i hard to analyze. 4.1 Overview The idea of Cluter1 i imple, and a general outline can be inferred from the procedure call and comment at the top of Algorithm 1: To obtain an initial clutering, Procedure GrowInitialCluter ample a mall logarithmic fraction of the node to active ingleton-cluter. Then a tandard PUSH goip i performed for Θ(log ) round in which thee cluter recruit unclutered node. The uual exponential growth behavior of the PUSH goip protocol lead to cluter of ize at leat = C while alo enuring that all but a mall contant fraction of the node are recruited into ome cluter. In the main phae of the algorithm, namely Procedure SquareCluter, we repeatedly quare thi cluter ize by merging cluter in the following manner: We tart by uing CluterDiolve() and CluterReize() to enure that cluter have ize between and 2. Next, we determine the new cluter leader by activating each cluter with probability 1/. The inactive cluter are then prompted to join one of the new active cluter via two CluterPUSHe. Thi reult in each new cluter recruiting Θ() inactive cluter which lead to the deired new cluter ize of Θ( 2 ). In particular, the firt uch PUSH guarantee that each new cluter grow by a factor of Θ(), while the econd iteration enure that all old inactive cluter are merged into ome new, larger active cluter. The doubly exponential growth of quaring the cluter ize with every iteration guarantee that Θ(log ) iteration are ufficient to obtain cluter of ize n. Procedure MergeAllCluter ue two CluterPUSHe to merge all cluter into the cluter with the mallet ID. With one cluter compriing mot of the node in the network, Procedure UncluteredNodePull ue a tandard PULL goip to have the remaining unclutered node join the cluter which i known to require only Θ(log ) round. Latly, with

5 all node in one cluter preading any deired meage can be done via one CluterShare. A breakdown of the variou tep i given in the detailed procedure of Algorithm 1, uing the cluter communication primitive from Section 3.2. Algorithm 1 Cluter1 1: GrowInitialCluter cluter mot node 2: SquareCluter repeatedly merge cluter 3: MergeAllCluter merge cluter to one 4: UncluteredNodePull unclutered node join 5: CluterShare(meage) 6: procedure GrowInitialCluter 7: follow 8: With probability 1/C et follow ownid 9: for Θ(log ) time do 10: CluterPUSH(cluterID) 11: unclutered node: follow received ID (if any) 12: procedure SquareCluter 13: = C 14: CluterDiolve() 15: repeat 16: CluterReize() 17: CluterActivate(1/) 18: for two repetition do 19: active: CluterPUSH(follow) 20: inactive: CluterMerge(mallet received ID) 21: Θ( 2 ) 22: until > n 23: procedure MergeAllCluter 24: for two repetition do 25: CluterPUSH(follow) 26: CluterMerge(mallet received ID) 27: procedure UncluteredNodePull 28: for Θ(log ) time do 29: unclutered node PULL follow value 4.2 Correctne Proof In thi ection we prove the correctne and efficiency of Cluter1, which i decribed in Section 4.1 and Algorithm 1. We break the proof down to everal lemma, which then lead to the main reult. Lemma 5. At the end of Procedure GrowInitialCluter in Cluter1, at leat 0.9n node are contained in cluter of ize C with high probability. Proof. The clutering at the end of the procedure i created by Θ(log ) round of random PUSH goip tarting from cluter center that were obtained by ampling node independently with probability p = 1. By a Chernoff C ( ) n bound, we initially have the expected Θ cluter C center occur with high probability. By tandard goip analyi, the et containing all clutered node grow by a contant factor in each round, o long a ome contant fraction of the ytem remain unclutered. Furthermore, by chooing C C at mot an arbitrary mall contant fraction of node account for cluter that are maller than C n, namely about C C many. In total, one can guarantee that at leat a 90% fraction of node are good, that i, contained in a large cluter. Lemma 6. Every iteration inide Procedure SquareCluter in Cluter1 quare the cluter ize while enuring that all clutered node are contained in cluter of the new ize, with high probability. Proof. Initially, due to Theorem 5 which i followed by CluterDiolve(), we can aume that at leat 0.9n node are clutered and divided into cluter of ize between and two 2 after the CluterReize() in Line 16 i performed. The ame hold from there on by induction. Thee Θ(n/) cluter then get independently activated with probability 1/. Since n the expected number of activated cluter i Θ(n/ 2 ) = ω() and a Chernoff bound how that indeed Θ(n/ 2 ) cluter are activated with high probability. Now, in the firt CluterPUSH/CluterMerge iteration of Procedure SquareCluter, thee active cluter PUSH Θ() meage each for a total of Θ(n/) meage in order to recruit the Θ(n/) inactive cluter. By tandard ball and bin analyi, a linear fraction of every et of = Ω() PUSHe of each cluter land in a unique cluter and reult in a merge. Therefore, after the firt iteration all active cluter will recruit Θ() inactive cluter and thu grow to ize Θ( 2 ). However, there i alo a contant fraction of inactive cluter that did not get recruited. The econd CluterPUSH/CluterMerge iteration enure that thee cluter merge with ome active cluter to enure that all clutered node remain clutered. Thi i indeed the cae ince there are Θ(n) total number of node contained in the active cluter which PUSH out in the econd iteration. Every inactive cluter of ize = Ω() i therefore reached and merged into an active cluter with high probability. Lemma 7. After Procedure MergeAllCluter in Algorithm Cluter1, with high probability, there i a ingle cluter which contain at leat 0.9n node. Proof. Theorem 6 how that before Procedure Merge- AllCluter i invoked at leat 0.9n node are contained in cluter of ize Θ() with n. We now conider the cluter with the mallet ID. The firt CluterPUSH/Cluter- Merge iteration of Procedure MergeAllCluter will, with n high probability, grow thi cluter to ize Ω( ). The econd iteration then recruit all remaining cluter, with high log 2 n probability. Latly, it i well known (ee, e.g., [10]) that the tandard PULL goip take only O(log ) round to inform the remaining contant or even 1 1/ log Θ(1) n fraction of node. Thi i exactly what i ued in Procedure UncluteredNodePull to have all unclutered node join the ingle cluter in the end: Lemma 8. The PULL goip given in Algorithm Cluter1 a Procedure UncluteredNodePull reult in all

6 node joining the ingle cluter, with high probability, in O(log ) round. Proof. It i well known that the tandard uniform PULL goip terminate with high probability in Θ(log ) if one tart with at leat n/ log Θ(1) n node being informed. For completene we include the analyi here: We look at the fraction x of unclutered node and note that in one round any unclutered node (pulling a random node) remain unclutered with probability x. A long a the number of unclutered node i till at leat Θ() a Chernoff bound how that the quantity x hrink to at mot 2x 2 with high probability after each pull. Thi can happen at mot for 2 log round. Latly, with high probability, it take at mot a contant number of round until each remaining node pull from a clutered node. Thi how that the Θ(log ) round of PULL goip indeed, with high probability, lead to one cluter containing all node. The correctne and efficiency of Cluter1 now follow immediately: Theorem 9. Algorithm Cluter1 run for O(log ) round and, with high probability, pread any meage to all node. Proof. The CluterShare(meage) performed at the end of Cluter1 correctly pread any meage to all node ince according to Theorem 8 all node are contained in one ingle cluter. The running time of Cluter1 i furthermore eaily een to be O(log ) round. In particular, the loop inide Procedure GrowInitialCluter and UncluteredNodePull run explicitly for O(log ) iteration, while the loop inide Procedure SquareCluter quare the parameter in each iteration, and therefore run for at mot Θ(log ) iteration a well. To complete the proof, we refer to Section 3.2 for the fact that all cluter primitive performed in thee loop can be performed in a contant number of round. 5. LINEAR MESSAGE COMPLEXITY In thi ection we prove Theorem 2 by explaining how the Cluter1 algorithm can be tweaked to alo have optimal meage- and bit-complexity without compromiing it optimal O(log ) round-complexity. We call the reulting algorithm Cluter Overview Firt, we give the intuition behind achieving a linear meage complexity and decribe in which apect Cluter2 i different from Cluter1. The high-level recipe of Cluter2 i the ame a Cluter1. The key idea for achieving the optimal meage-complexity i in modifying Procedure GrowInitialCluter and SquareCluter to work with only Θ(n/ ) clutered node. Thi guarantee that even if all clutered node actively tranmit in thee round, only o(n) meage are ued for thee part. After merging all cluter, we are left with one large cluter of ize Θ(n/ ). While the tandard PULL goip would till finih in O(log ) round thi would again take to many meage (of unclutered node PULLing unclutered node). We therefore PUSH the cluter ID to Θ(n) node firt before doing the final PULL phae. With o many informed node each node expect to PULL at mot a contant number of time and it i eay to ee that in total at mot O(n) meage are ued with high probability. All in all a much more delicate control on the number of cluter, their ize and the number of clutered node i required. Thi i alo important to provably guarantee an optimal bit-complexity. The ubtle problem here i that a CluterReize() on a cluter of ize ω() lead to meage of ize ω() (ee footnote in Section 3.2). Thi control i achieved a follow: To control the number of node clutered in the Procedure GrowInitialCluter we have cluter meaure their growth. More preciely before and after each PUSH in the initial recruiting phae each cluter check it ize. If a cluter ha ize at leat Ω(log 3 n) but grew by le than a factor of 2 1 then it top recruiting. A Chernoff Bound how that no cluter will top recruiting if le than n/2 node are clutered. On the other hand, imilar argument a for Cluter1 how that indeed Θ(n/ ) node will get recruited of which mot lie in cluter of ize = Ω(log 3 n). A CluterReize(C ) called for every large cluter in every iteration furthermore enure that no individual cluter get too big. For the ProcedureSquareCluter we then again merge cluter. Due to the maller number of clutered node mot cluter requet now go to unclutered node and get ignored. Still, any iteration tarting with cluter of ize end up with larger cluter of ize Θ( 2 / ) = ω( 1.5 ). Growing the cluter ize from = Θ(log 3 n) to Ω(n/ log 2 n) therefore till take only O(log ) round. Chernoff bound over the larger Ω(log 3 n) ized cluter till guarantee that the new cluter ize can be predicted up to a contant with high probability which enure that no CluterReize() i applied to a too large (or too mall) cluter. The formal decription of Cluter2 i given a Algorithm Correctne Proof In thi ection we prove Theorem 2 by howing that Cluter2 pread a meage with the deired optimal round-, meage- and bit-complexity. The proof follow along the ame line a Theorem 9 while including the extra argument given in Section 5.1. We again break the proof into everal lemma, which then lead to the main reult. Lemma 10. If at mot n/f(n) node are clutered, then a CluterPUSH by ome cluter X, where X Ω(), grow X by factor at leat (2 Θ(1/f(n))) with high probability. Proof. Each PUSH requet of one of the X node i hit a unique unclutered node with probability of approximately (1 1 ). The expected number of node recruited f(n) for cluter X i therefore (1 1 ) X for a total number of 2f(n) X +(1 1 ) X and a growth factor of (2 Θ(1/2f(n))). 2f(n) Since the recruitment ucce of different node are eentially independent of each other applying a Chernoff type bound how that the growth factor i cloe to the expected and in particular i (2 Θ(1/f(n))), with high probability. Lemma 11. Procedure GrowInitialCluter of Cluter2 lead to Θ(n/ ) clutered node that are contained

7 Algorithm 2 Cluter2 1: GrowInitialCluter 2: SquareCluter 3: MergeAllCluter a in Algorithm 1 4: BoundedCluterPuh 5: UncluteredNodePull a in Algorithm 1 6: CluterShare(meage) 7: procedure GrowInitialCluter 8: follow 9: With probability 1/C log 4 n et follow ownid 10: CluterActivate(1) 11: for Θ(log ) time do 12: active cluter: CluterPUSH(follow) 13: unclutered node: follow any received ID 14: if CluterSize C log 3 n then 15: if cluter growth factor < (2 1/ ) then 16: deactivate cluter 17: ele 18: CluterReize(C log 3 n) 19: procedure quarecluter 20: = C log 3 n 21: CluterDiolve() 22: repeat 23: CluterReize() 24: CluterActivate(1/) 25: for two repetition do 26: active: CluterPUSH(follow) 27: inactive: CluterMerge(random received ID) 28: Θ( 2 / ) 29: until > n log 2 n 30: procedure BoundedCluterPuh 31: CluterActivate(1) 32: for Θ(log ) round do 33: CluterPUSH(follow) 34: unclutered node: follow any received ID 35: CluterSize 36: if cluter growth factor < 1.1 then 37: deactivate cluter in cluter of ize (1 + Θ(1))(C ), with high probability, while at mot O(n) meage of Θ() bit each are ent. Proof. For the initial ampling of ingleton cluter, for which the ampling probability i 1/C log 4 n, a Chernoff n bound how that Θ( ) uch ingleton cluter are created, with high probability. We would like to argue a in C log 4 n Theorem 5 that the growth of the et of clutered node i exponential until at leat Ω(n/ ) node are clutered. We need to how that the newly introduced ize-control doe not prevent thi exponential growth. By Theorem 10, a long a at leat n(1 O(1/ )) node are unclutered each cluter will grow by a factor of at leat 2 1/, with high probability. Hence, the ize control mechanim doe not inhibit the exponential growth until at leat Θ(n/ ) node are n clutered. Since at mot about C log 4 n (C log 3 n) node account for cluter that are maller than C log 3 n, indeed at leat Θ(n/ ) node are contained in large cluter. To guarantee that the meage complexity incurred by the initialization i mall we note that only clutered node are tranmitting. Thi lead to at mot O(n log / ) = O(n) meage ent. Since every cluter can grow by at mot a factor of two in each iteration it i furthermore clear that all meage ize ued in a CluterReize contain at mot two ID. Thi reult in meage of ize Θ(), a claimed. Lemma 12. Before and after every iteration inide Procedure SquareCluter in Cluter2, Θ(n/ ) node are contained in cluter of ize at leat, with high probability. Furthermore, the total number of meage ent during Procedure SquareCluter i O(n), each containing Θ() bit. Proof. We prove the lemma by induction. The bae cae i already given by Theorem 11. Here, we how that each iteration of the loop increae the cluter ize from to = Θ( 2 ) while keeping all clutered node clutered. In the firt iteration each active cluter PUSHe to Θ() random node and hit with high probability Θ(/ ) inactive cluter (alone). Once thee cluter merge, any active cluter therefore grow to = Θ( 2 ) a deired. The econd PUSH, performed by the larger cluter which cover Θ(n/ ) node, guarantee that no inactive cluter of ize i left behind without merging to a new larger cluter of ize at leat. Furthermore, ince inactive cluter randomly join the large active cluter during thi econd iteration no active cluter grow by more than a contant factor. The new cluter are therefore all of the deired ize = Θ( 2 ) while all clutered node remain clutered. Latly, ince > log 3 n we have that = ω( 1.5 ) which how that the number of iteration until the final cluter ize of Ω( n ) i reached i at mot Θ(log ). log 2 n All in all, only Θ(n/ ) node are communicating in each of the O(log ) round, which lead to the deired O(n) meage-complexity. Regarding the bit-ize of thee meage, the only thing to worry about i the call to Cluter- Reize(). However, ince the cluter ize are, up to contant, exactly what they are reized to, CluterReize() produce only Θ() bit meage containing Θ(1) ID. Theorem 7 applie without change to Cluter2 and we do not repeat it. Lemma 13. Procedure BoundedPuhCluter expand the ingle large cluter to Θ(n) node in O(log ) round, with high probability, while at mot O(n) meage are ent.

8 Proof. At the beginning of the procedure, we have from Theorem 7 that Θ(n/ ) belong to a ingle giant cluter. From here, we ue a tandard analyi of PUSH-PULL goip: So long a at leat.1n node are unclutered, during each iteration the number of clutered node grow exponentially (and therefore for at mot O(log ) round) until a large contant fraction of the node i clutered. By Theorem 10, the cluter detect thi event uccefully. Since the number of meage ent out in every round i exponentially increaing until it top the total number of meage form a geometric um which um to O(n) meage ent in total. Putting thee claim together, it follow that Cluter2 ha a bit-complexity of O(n + nb) = O(nb) and meage complexity of O(1) meage per node on average while maintaining the optimal O(log ) round complexity of Cluter1. Thi prove Theorem THE Ω(log log N) ROUND LOWER BOUND Thi ection i dedicated to prove the Ω(log ) roundcomplexity lower bound of Theorem 3. We firt remark that the imple lower bound provided in Section 7 a Theorem 16 doe not give any uper contant number of round for the mot intereting etting of = n Θ(1). Here we how that the round-complexity of broadcating a meage ha to be at leat 0.99 log with high probability. Thi hold for any and remain true even if one ignore many retriction of the model regarding meage ize, addre-oblivioune, or the number of known node that can be contacted per round. The main implication of thi lower bound i that both Cluter1 and Cluter2 achieve the optimal aymptotic running time. We need ome notation: Intead of each node ampling (up to) one random contact in each round we aume that thee node are ampled in advance. In particular, let u v,t be the uniformly random node given to node v if it ample an node at time t. Baed on thee ample we define G t to be the union of all edge (node-pair) that are potentially ampled at time t, that i, G t = (V, E t) with E t = {{u, v} v : u = u v,t}. Lemma 14. For every t = 0, 1,... let K t be the graph indicating which node know whom at the beginning of round t. It hold that: K 0 = ; K t+1 (K t G t+1) 2, and therefore alo that: ( t ) 2 t K t G i. i=1 Here G j i the graph that connect a node u to any node for which there i a path of length at mot j in G. Proof. The firt equation i imply tating that node do not know each other initially. To ee the econd equation we note that there are two way node can learn about new node. The firt one i by contacting a random node. Thi only add edge in G t+1 to the knowledge graph K t. By initiating a contact along an edge in G t+1 or along an edge in K t correponding to a node known from previou round, node can alo indirectly learn about node known to node they know. Even if all uch neighbor are contacted and every node repond with everything they know (which would reult in many meage of very large ize), a node doe not learn about more node than it 2-hop neighborhood. Thi i exactly what i tated in the econd equation. The lat equation now follow from the other two by induction and taking into account that for any two graph G, G and any j, j > 1 we have G i G (G G ) i and of coure (G j ) j = G jj. We can now prove Theorem 3: Theorem 15 (Theorem 3 retated). Suppoe we allow unlimited meage ize, non addre-obliviou communication, and have no retriction on how many communication a node can be involved in or even to how many known node a node contact in one round. Still, any algorithm running in le than T = log log log ω(1) round fail to pread a meage that tart at one node u to all node with high probability. Proof. Firt we note that preading a meage tarting at one node u can be ued to elect u a a cluter leader by imply attaching it ID to the meage pread. Thi could then be ued to make K T the complete graph by informing all node about each other in at mot two additional round, at leat if the meage ize i unbounded. We how that thi i, with high probability, impoible in le than T round by uing Theorem 14. In particular, it certainly hold that K T K 2T where K = ( T i=1 Gi). Note that K i imply a random graph that i created by each node ampling T neighbor independently and adding an undirected edge to thoe neighbor. It i clear that K T i a complete graph if and only if the diameter of K i at mot 2 T. It i eay to ee that for T = log log log ω(1) the probability that uch a random graph ha diameter o(/ log ) i extremely mall which complete the proof. An eay way to get uch a high probability reult i to ay that the probability that there i a node in K with degree = ω() i at mot n ω(1) and any graph with maximum degree Θ() necearily ha a diameter of Ω(/ log ). 7. BOUNDED COMMUNICATIONS In thi ection we dicu and prove tradeoff between the running time of any goip algorithm and the maximum number of communication a node can be involved in per round. In the tandard goip model which i alo employed in thi paper, each node can only initiate one PUSH or PULL communication per round. Still, a node might be involved in multiple communication initiated by other node. The poibility of thi happening i preent in the model of all prior goip work (dicued in Section 1). In few cae, in particular the random phone call model with an underlying complete graph, having a large number of communication per round in one node i not crucial. For example, the running time of the uniformly random goip on the complete graph remain O() even if each node only accept one communication per round (but the reult of [12] which concentrate on the leading contant and lower order term change). For mot goip work however uch a concentration of communication i unavoidable. For example the work [4 6] which generalize the random phone call model to arbitrary topologie how that goip i fat on any topology with good vertex or edge expanion. In particular, they imply that goip complete in log O(1) n round on a tar

9 graph becaue the tar graph ha a contant edge expanion Φ = O(1) and contant vertex expanion α = O(1). Thi i obviouly not poible without the center node talking to at leat = n/ log O(1) n node per round on average. Similarly, all reult that ue direct addreing crucially rely on each node being able to repond to many node and many of them can have one node communicate with (almot) all other node in the network in one round. Thi i alo the cae for Cluter1 and Cluter2. In fact our entire et of cluter-operation are baed on having a cluter-center coordinating ω(1) cluter-follower in a ingle communication tep. While in ome etting having = n may be acceptable, in other etting one would like to have to be not much more than a mall polynomial or even (ignificantly) le. In what follow we explicitly addre thi parameter and deign optimal algorithm for any. We firt remark that having a uper-contant i unavoidable if one want to achieve a ub-logarithmic round complexity. In particular, it i eay to ee that any global communication with the retriction that no node repond to more than requet per round take at leat / log round: Lemma 16. Any algorithm in which no node participate in more than communication per round require at leat / log round to inform all node of a meage. Proof. Since each informed node can participate in at mot communication the number of informed node cannot grow by more than a factor per round. Inductively the the number of informed node in round r i therefore at mot r 1 and / round are needed to bring thi quantity to n. In the remainder of thi ection we will how that the algorithmic idea developed in thi paper can be ued to achieve any point on the thi tradeoff curve between and the required number of round while alo maintaining the optimal meage- and bit-complexity. We do thi in two tep. We firt define a -clutering and how that any uch clutering eaily allow -bounded communication in the deired / log round complexity by running imple and robut goip algorithm on top of thi clutering. We then how that a -clutering can be computed in Θ(log ) round: Definition 1. A -clutering i a clutering in which every node i clutered and all cluter have a ize between and 2. We will how in Theorem 17 that uch a -clutering indeed allow to efficiently pread information in the deired O(/ log ) round uing the CluterPUSH-PULL protocol given in Algorithm 3. For ake of implicity we focu on the cae = log ω(1) n ince for maller the improvement of log i at mot O(log ) and therefore eentially negligible. Similarly, in thi ection we aume for convenience that < n 0.9 a otherwie one can might a well et = n and run Cluter2. Lemma 17. For any bound = log ω(1) n the algorithm CluterPUSH-PULL( ) given in Algorithm 3 pread a meage to all node, with high probability. It furthermore ue only O( ) round and O(n) meage once the - log clutering i computed. Algorithm 3 CluterPUSH-PULL( ) 1: Compute -Clutering 2: CluterShare(meage) 3: for Θ( ) iteration do log 4: if newly informed cluter then 5: CluterPUSH(meage), CluterShare(meage) 6: uninformed node: PULL meage from a random node 7: CluterShare(meage) Proof. We firt addre the round and meage complexity and then prove correctne: The claimed round complexity of O( ) i eaily verified ince the only loop of the algorithm in Line 3 take 3 log round per iteration and run for Θ( ) iteration. All log other intruction take only O(1) round. It i alo eay to ee that each node procee at mot O(1) meage over the coure of the algorithm. Next we prove correctne, that i, that CluterPUSH- PULL pread any meage to all node with high probability. The firt CluterShare (Line 5) pread the meage to = ω() node. From there on, o long a at mot n/ω( ) node are informed, each CluterPUSH and CluterShare in the main loop (of Line 3) increae the number of informed node with high probability by a /O(1) factor. Hence, the et of informed node grow exponentially until Θ( it reache n/ > ( /O(1)) log ) node. To ee that thi uffice for the CluterPULL/CluterShare in Line 6 7 to inform all remaining node with high probability we focu on one uninformed node. Thi node get informed during the CluterPULL if at leat one of the node in it cluter PULL the meage from a node in Line 6. Since there are at leat = log ω(1) n uch node of which each ha an independent chance of 1/ to PULL from an informed node thi happen with high probability. Thi how that CluterPUSH-PULL indeed inform all node about the meage with high probability uing only ) round and O(n) meage. O( log What remain to how i that a Θ( )-clutering can be computed efficiently. The proce for generating the clutering i eentially a generalization of the Cluter2 algorithm, where we top growing cluter at around ize. Then for additional O(log ) round, we join all unclutered node to exiting cluter. All the while, we never connect any leader to more than follower, nor tranmit more than O(n) meage. The detailed peudo-code decription of the reulting algorithm Cluter3( ) i given in Algorithm 4. Theorem 18. For any given = log ω(1) n the algorithm Cluter3( ), with high probability, compute a Θ( )-clutering uing O(log )-round and O(n) meage while no node communicate with more than node in any round. Proof. A before, the Procedure GrowInitialCluter and SquareCluter reult in Θ(n/ ) clutered node grouped into cluter of ize Θ(). Here the cluter

10 Algorithm 4 Cluter3( ) with = log ω(1) n 1: GrowInitialCluter a in Algorithm 2 2: SquareCluter(to ize /C ) a in Algorithm 2 3: MergeCluter 4: BoundedCluterPuh 5: UncluteredNodePull a in Algorithm 1 6: CluterReize( /C ) 7: procedure MergeCluter 8: CluterActivate(10 ) /C 9: active: CluterPUSH(cluterID) 10: inactive: CluterMerge(random received ID) 11: procedure BoundedCluterPuh 12: CluterActivate(1) 13: for Θ(log ) round do 14: CluterReize( /C ) 15: CluterPUSH(follow) 16: unclutered node: follow any received ID 17: CluterSize 18: if cluter growth factor < 1.1 then 19: deactivate cluter ize i = Ω( /C ). The proof for thi follow the ame way a for Cluter2, and we do not repeat the argument here. Procedure MergeCluter of Cluter3 i a bit different from Cluter2, though the technique are imilar. It activate every cluter with probability 10 followed /C by a CluterPUSH of the active cluter. Since i mall enough a Chernoff bound how that roughly a 10C fraction of cluter i activated. Since = log ω(1) n each inactive cluter get furthermore PUSHed by multiple active cluter to merge. Importantly, and differently to other merging phae, inactive cluter chooe one to merge with among thee cluter uniformly at random. A Chernoff type-bound how that thi way inactive cluter get aigned to active cluter evenly, i.e., each active cluter get choen by Θ( /C ) inactive cluter and will therefore grow to ize about /10C. Subequently, thee cluter then recruit unclutered node in the BoundedCluterPuh call. After each recruiting tep cluter meaure whether they grew by at leat a factor of 1.1 and reize. A before thi guarantee that all but a mall contant fraction of the node join a cluter while the number of meage i O(n). Reizing active cluter guarantee that no cluter get too big and no leader ha to communicate too much. A a lat tep, we run UncluteredNodePull on the unclutered node and make them join the firt cluter they PULL from. Up to contant every unclutered node join a uniformly random cluter. Even without any ize control every cluter grow by at mot a mall contant in thi round. The final CluterReize( /C ) enure that all cluter have the ame ize (up to a factor of two) a deired by a Θ( )- clutering. Acknowledgment We thank Chen Avin and Robert Eläer for haring an early verion of [1] and for intereting dicuion. 8. REFERENCES [1] C. Avin and R. Eläer. Fater Rumor Spreading: Breaking the Barrier. In Proceeding of the International Sympoium on Ditributed Computing, DISC 13, page , [2] K. Cenor-Hillel, B. Haeupler, J. Kelner, and P. Maymounkov. Global Computation in a Poorly Connected World: Fat Rumor Spreading with no Dependence on Conductance. In Proceeding of the ACM Sympoium on Theory of Computing, STOC 12, page , [3] K. Cenor-Hillel and H. Shachnai. Fat information preading in graph with large weak conductance. In Proceeding of the 22nd ACM-SIAM Sympoium on Dicrete Algorithm (SODA), page , [4] G. Giakkoupi. Tight bound for rumor preading in graph of a given conductance. In Proceeding of the 28th International Sympoium on Theoretical Apect of Computer Science, STACS, page 57 68, [5] G. Giakkoupi and T. Sauerwald. Rumor preading and vertex expanion. In Proceeding of the ACM-SIAM Sympoium on Dicrete Algorithm, SODA 12, page , [6] B. Haeupler. Analyzing network coding goip made eay. In Proceeding of the ACM Sympoium on Theory of Computing, STOC, page , [7] B. Haeupler. Simple, fat and determinitic goip and rumor preading. In Proceeding of the ACM-SIAM Sympoium on Dicrete Algorithm, SODA 13, page , [8] B. Haeupler, G. Pandurangan, D. Peleg, R. Rajaraman, and Z. Sun. Dicovery through goip. In Proceeding of the 24th ACM Sympoium on Parallelim in Algorithm and Architecture, SPAA, page , [9] M. Harchol-Balter, T. Leighton, and D. Lewin. Reource dicovery in ditributed network. In Proceeding of the ACM Sympoium on Principle of Ditributed Computing, PODC 99, page , [10] R. Karp, C. Schindelhauer, S. Shenker, and B. Vocking. Randomized rumor preading. In Proceeding of the 41t Annual Sympoium on Foundation of Computer Science, FOCS, page , [11] S. Kutten, D. Peleg, and U. Vihkin. Determinitic reource dicovery in ditributed network. Theory of Computing Sytem, 36(5): , [12] B. Pittel. On preading a rumor. SIAM Journal on Applied Mathematic, page , 1987.