Distributed Computation in Dynamic Networks

Transcription

1 Dstrbuted Computaton n Dynamc Networks Faban Kuhn Facuty of Informatcs, Unversty of Lugano Lugano, Swtzerand 6904 faban.kuhn@us.ch Nancy Lynch Computer Scence and AI Laboratory, MIT Cambrdge, MA ynch@csa.mt.edu Rotem Oshman Computer Scence and AI Laboratory, MIT Cambrdge, MA rotem@csa.mt.edu ABSTRACT In ths paper we nvestgate dstrbuted computaton n dynamc networks n whch the network topoogy changes from round to round. We consder a worst-case mode n whch the communcaton nks for each round are chosen by an adversary, and nodes do not know who ther neghbors for the current round are before they broadcast ther messages. The mode captures mobe networks and wreess networks, n whch mobty and nterference render communcaton unpredctabe. In contrast to much of the exstng work on dynamc networks, we do not assume that the network eventuay stops changng; we requre correctness and termnaton even n networks that change contnuay. We ntroduce a stabty property caed T -nterva connectvty (for T 1), whch stpuates that for every T consecutve rounds there exsts a stabe connected spannng subgraph. For T = 1 ths means that the graph s connected n every round, but changes arbtrary between rounds. We show that n 1-nterva connected graphs t s possbe for nodes to determne the sze of the network and compute any computabe functon of ther nta nputs n O(n 2 ) rounds usng messages of sze O(og n + d), where d s the sze of the nput to a snge node. Further, f the graph s T -nterva connected for T > 1, the computaton can be sped up by a factor of T, and any functon can be computed n O(n + n 2 /T) rounds usng messages of sze O(og n+d). We aso gve two ower bounds on the token dssemnaton probem, whch requres the nodes to dssemnate k peces of nformaton to a the nodes n the network. The T-nterva connected dynamc graph mode s a nove mode, whch we beeve opens new avenues for research n the theory of dstrbuted computng n wreess, mobe and dynamc networks. Categores and Subject Descrptors: F.2.2 [Anayss of Agorthms and Probem Compexty]: Non-numerca Agorthms and Probems computatons on dscrete structures G.2.2 [Dscrete Mathematcs]: Graph Theory graph agorthms G.2.2 [Dscrete Mathematcs]: Graph Theory network probems Genera Terms: Agorthms, Theory Keywords: dstrbuted agorthms, dynamc networks Permsson to make dgta or hard copes of a or part of ths work for persona or cassroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commerca advantage and that copes bear ths notce and the fu ctaton on the frst page. To copy otherwse, to repubsh, to post on servers or to redstrbute to sts, requres pror specfc permsson and/or a fee. STOC 10, June 5 8, 2010, Cambrdge, Massachusetts, USA. Copyrght 2010 ACM /10/06...$ INTRODUCTION The study of dynamc networks has ganed mportance and popuarty over the ast few years. Drven by the growng ubquty of the Internet and a pethora of mobe devces wth communcaton capabtes, nove dstrbuted systems and appcatons are now wthn reach. The networks n whch these appcatons must operate are nherenty dynamc; typcay we thnk of them as beng arge and competey decentrazed, so that each node can have an accurate vew of ony ts oca vcnty. Such networks change over tme, as nodes jon, eave, and move around, and as communcaton nks appear and dsappear. In some networks, e.g., peer-to-peer, nodes partcpate ony for a short perod of tme, and the topoogy can change at a hgh rate. In wreess ad-hoc networks, nodes are mobe and move around unpredctaby. Much work has gone nto deveopng agorthms that are guaranteed to work n networks that eventuay stabze and stop changng; ths abstracton s unsutabe for reasonng about truy dynamc networks. The objectve of ths paper s to make a step towards understandng the fundamenta possbtes and mtatons for dstrbuted agorthms n dynamc networks n whch eventua stabzaton of the network s not assumed. We ntroduce a genera dynamc network mode, and study computabty and compexty of essenta, basc dstrbuted tasks. Under what condtons s t possbe to eect a eader or to compute an accurate estmate of the sze of the system? How effcenty can nformaton be dssemnated reaby n the network? To what extent does stabty n the communcaton graph hep sove these probems? These and smar questons are the focus of our current work. 1.1 The Dynamc Graph Mode In the nterest of broad appcabty our dynamc network mode makes few assumptons about the behavor of the network, and we study t from the worst-case perspectve. In the current paper we consder a fxed set of nodes that operate n synchronzed rounds and communcate by broadcast. In each round the communcaton graph s chosen adversaray, under an assumpton of T -nterva connectvty: throughout every bock of T consecutve rounds there must exst a connected spannng subgraph that remans stabe. We consder the range from 1-nterva connectvty, n whch the communcaton graph can change competey from one round to the next, to -nterva connectvty, n whch there exsts some stabe connected spannng subgraph that s not known to the nodes n advance. We note that edges that do not beong to the stabe subgraph can st change arbtrary from one round to the next, and nodes do not know whch edges are stabe and whch are not. We do not assume that a neghbor-dscovery mechansm s avaabe to

2 the nodes; they have no means of knowng ahead of tme whch nodes w receve ther message. In ths paper we are mosty concerned wth determnstc agorthms, but we aso ncude a randomzed agorthm and a randomzed ower bound. The computaton mode s as foows. In every round, the adversary frst chooses the edges for the round; for ths choce t can see the nodes nterna states at the begnnng of the round. At the same tme and ndependent of the adversary s choce of edges, each node tosses prvate cons and uses them to generate ts message for the current round. Determnstc agorthms generate the message based on the ntera state aone. In both cases the nodes do not know whch edges were chosen by the advesary. Each message s then devered to the sender s neghbors, as chosen by the adversary; the nodes transton to new states, and the next round begns. Communcaton s assumed to be bdrectona, but ths s not essenta. We typcay assume that nodes ntay know nothng about the network, and communcaton s mted to O(og n) bts per message. To demonstrate the power of the adversary n the dynamc graph mode, consder the probem of oca token crcuaton: each node u has a oca Booean varabe token u, and f token u = 1, node u s sad to have the token. In every round exacty one node n the network has the token, and t can ether keep the token or pass t to one of ts neghbors. The goa s for a nodes to eventuay have the token n some round. Ths probem s mpossbe to sove n 1-nterva connected graphs: n every round, the adversary can see whch node u has the token, and provde that node wth ony one edge {u, v}. Node u then has no choce except to eventuay pass the token to v. After v receves t, the adversary can turn around and remove a of v s edges except {u, v}, so that v has no choce except to pass the token back to u. In ths way the adversary can prevent the token from ever vstng any node except u, v. Perhaps surprsngy gven our powerfu adversary, even n 1- nterva connected graphs t s possbe to reaby compute any computabe functon of the nta states of the nodes, and even have a nodes output the resut at the same tme (smutanety). The dynamc graph mode we suggest can be used to mode varous dynamc networks. Perhaps the most natura scenaro s mobe networks, n whch communcaton s unpredctabe due to the mobty of the agents. There s work on achevng contnua connectvty of the communcaton graph n ths settng (e.g., [14]), but currenty tte s known about how to take advantage of such a servce. The dynamc graph mode can aso serve as an abstracton for statc or dynamc wreess networks, n whch cosons and nterference make t dffcut to predct whch messages w be devered, and when. Fnay, dynamc graphs can be used to mode tradtona communcaton networks, repacng the tradtona assumpton of a bounded number of faures wth our connectvty assumpton. Athough we assume that the node set s statc, ths s not a fundamenta mtaton. We defer n-depth dscusson to future work; however, our technques are amenabe to standard methods such as ogca tme, whch coud be used to defne the permssbe outputs for a computaton wth a dynamc set of partcpants. 1.2 Contrbuton In ths paper we focus on two probems n the context of dynamc graphs. The frst probem s countng, n whch nodes must determne the sze of the network. The second s k-token dssemnaton, n whch k peces of nformaton, caed tokens, are handed out to some nodes n the network, and a nodes must coect a k tokens. We are especay nterested n the varant of k-token dssemnaton where the number of tokens s equa to the number of nodes n the network, and each node starts wth exacty one token. Ths varant of token dssemnaton aows any functon of the nta states of the nodes to be computed. However, t requres countng, snce nodes do not know n advance how many tokens they need to coect. We show that both probems can be soved n O(n 2 ) rounds n 1-nterva connected graphs. Then we extend the agorthm for T -nterva connected graphs wth known T > 1, obtanng an O(n+n 2 /T)-round protoco for countng or a-to-a token dssemnaton. When T s not known, we show that both probems can be soved n O(mn n 2, n + n 2 og(n)/t ) rounds. Fnay, we gve a randomzed agorthm for approxmate countng that assumes an obvous adversary, and termnates wth hgh probabty n amost-near tme. We aso gve two ower bounds, both concernng token-forwardng agorthms for token dssemnaton. A token-forwardng agorthm s one that does not combne or ater tokens, ony stores and forwards them. Frst, we gve an Ω(nog k) ower bound on k-token dssemnaton n 1-nterva connected graphs. Ths ower bound hods even aganst centrazed agorthms, n whch each node s tod whch token to broadcast by some centra authorty that can see the entre state of the network. We aso gve an Ω(n + nk/t) ower bound on k-token dssemnaton n T -nterva connected graphs for a restrcted cass of randomzed agorthms, n whch the nodes behavor depends ony on the set of tokens they knew n each round up to the current one. Ths ncudes the agorthms n the paper, and other natura strateges such as round robn, choosng a token to broadcast unformy at random, or assgnng a probabty to each token that depends on the order n whch the tokens were earned. For smpcty, the resuts we present here assume that a nodes start the computaton n the same round. It s generay not possbe to sove any non-trva probem f some nodes are ntay aseep and do not partcpate. However, f 2-nterva connectvty s assumed, t becomes possbe to sove k-token dssemnaton and countng even when computaton s ntated by one node and the rest of the nodes are aseep. 1.3 Reated Work For statc networks, nformaton dssemnaton and basc network aggregaton tasks have been extensvey studed (see e.g. [5, 20, 34]). In partcuar, the token dssemnaton probem s anayzed n [40], where t s shown that k tokens can aways be broadcast n tme O(n + k) n a statc graph. The varous probems have aso been studed n the context of aternatve communcaton modes. A number of papers ook at the probem of broadcastng a snge message (e.g. [9, 27]) or mutpe messages [13, 30] n wreess networks. Gosspng protocos are another stye of agorthm n whch t s assumed that n each round each node communcates wth a sma number of randomy-chosen neghbors. Varous nformaton dssemnaton probems for the gosspng mode have been consdered [21, 23, 25]; gosspng aggregaton protocos that can be used to approxmate the sze of the system are descrbed n [24, 36]. The gosspng mode dffers from our dynamc graph mode n that the neghbors for each node are chosen at random and not adversaray, and n addton, parwse nteracton s usuay assumed where we assume broadcast. A dynamc network topoogy can arse from node and nk faures; faut toerance,.e., resence to a bounded number of fauts, has been at the core of dstrbuted computng research from ts very begnnng [5, 34]. There s aso a arge body of prevous work on genera dynamc networks. However, n much of the exstng work, topoogy changes are restrcted and assumed to be webehaved n some sense. One popuar assumpton s eventua sta-

3 bzaton (e.g., [1, 7, 8, 41, 22]), whch asserts that changes eventuay stop occurng; agorthms for ths settng typcay guarantee safety throughout the executon, but progress s ony guaranteed to occur after the network stabzes. Sef-stabzaton s a usefu property n ths context: t requres that the system converge to a vad confguraton from any arbtrary startng state. We refer to [15] for a comprehensve treatment of ths topc. Another assumpton, studed for exampe n [26, 28, 35], requres topoogy changes to be nfrequent and spread out over tme, so that the system has enough tme to recover from a change before the next one occurs. Some of these agorthms use nk-reversa [18], an agorthm for mantanng routes n a dynamc topoogy, as a budng bock. Protocos that work n the presence of contnua dynamc changes have not been as wdey studed. Eary work (e.g., [6]) consdered the probem of end-to-end message devery n contnuay changng networks under an assumpton of eventua connectvty, whch asserts that the source and the destnaton are connected by a path whose nks appear nfntey often durng the executon. There s some work on handng nodes that jon and eave contnuay n peer-to-peer overay networks [19, 31, 33]. Most cosey reated to the probems studed here s [37], where a few basc resuts n a smar settng are proved; many t s shown that n 1-nterva connected dynamc graphs (the defnton n [37] s sghty dfferent), f nodes have unque dentfers, t s possbe to gobay broadcast a snge message and have a nodes eventuay stop sendng messages. The tme compexty s at east near n the vaue of the argest node dentfer. In [2], Afek and Hender gve ower bounds on the message compexty of goba computaton n asynchronous networks wth arbtrary nk faures. The tme requred for goba broadcast has been studed n a probabstc verson of the edge-dynamc graph mode, where edges are ndependenty formed and removed accordng to smpe Markov processes [10, 11, 12]. Smar edge-dynamc graphs have aso been consdered n contro theory terature, e.g. [38, 39]. In [12] the authors aso consder a worst-case dynamc graph mode whch s smar to ours, except that the graph s not aways connected and cosons are modeed expcty. Ths ower-eve mode does not admt a determnstc agorthm for goba broadcast; however, [12] gves a randomzed agorthm that succeeds wth hgh probabty. A varant of T -nterva connectvty was used n [29], where two of the authors studed cock synchronzaton n asynchronous dynamc networks. In [29] t s assumed that the network satsfes T -nterva connectvty for a sma vaue of T, whch ensures that a connected subgraph exsts ong enough for each node to send one message. Ths s anaogous to 1-nterva connectvty n synchronous dynamc networks. Fnay, a somewhat reated computatona mode resuts from popuaton protocos, ntroduced n [3], where the system s modeed as a coecton of fnte-state agents wth parwse nteractons. Popuaton protocos typcay (but not aways) rey on a strong farness assumpton whch requres every par of agents to nteract nfntey often n an nfnte executon. We refer to [4] for a survey. Unke our work, popuaton protocos compute some functon n the mt, and nodes do not know when they are done. Ths can make sequenta composton of protocos chaengng, snce t s not possbe to execute a protoco unt t termnates, then take the fna resut and use t as nput for some other computaton. (Instead, one may use sef-stabzng popuaton protocos, whch are resent to nputs that fuctuate and eventuay stabze to some vaue; but ths s not aways possbe [16]). In our mode nodes must eventuay output the resut of the computaton, and sequenta composton s straghtforward. 2. PRELIMINARIES We assume that nodes have unque dentfers (UIDs) drawn from a namespace U. We use x u(r) to denote the vaue of node u s oca varabe x at the begnnng of round r. A synchronous dynamc network s modeed as a dynamc graph G = (V, E), where V s a statc set of nodes, and E : N {{u, v} u, v V } s a functon mappng a round number r N to a set of undrected edges E(r). We make the foowng assumpton about connectvty n the network graph. DEFINITION 2.1 (T -INTERVAL CONNECTIVITY). We say a dynamc graph G = (V, E) s T -nterva connected for T 1 f for a r N, the statc graph G r,t := (V, T r+t 1 =r E(r)) s connected. The graph s sad to be -nterva connected f there s a connected statc graph G = (V, E ) such that for a r N, E E(r). For the current paper we are many nterested n the foowng probems. Countng. An agorthm s sad to sove the countng probem f whenever t s executed n a dynamc graph comprsng n nodes, a nodes eventuay termnate and output n. k-verfcaton. Cosey reated to countng, the k-verfcaton probem requres nodes to determne whether or not n k. A nodes begn wth k as ther nput, and must eventuay termnate and output yes or no. Nodes must output yes f and ony f there are at most k nodes n the network. k-token dssemnaton. An nstance of k-token dssemnaton s a par (V, I), where I : V P (T ) assgns a set of tokens from some doman T to each node, and S u V I(v) = k. An agorthm soves k-token dssemnaton f for a nstances (V, I), when the agorthm s executed n any dynamc graph G = (V, E), a nodes eventuay termnate and output S u V I(u). We assume that each token n the nodes nput s represented usng O(og n) bts. Nodes may or may not know k, dependng on the context. A-to-a token dssemnaton. A restrcted cass of k-token dssemnaton n whch k = n and for a u V we have I(u) = 1. The nodes know that each node starts wth a unque token, but they do not know n. k-commttee eecton. As a usefu step towards sovng countng and token dssemnaton we ntroduce a new probem caed k-commttee eecton. In ths probem, nodes must partton themseves nto sets, caed commttees, such that (a) The sze of each commttee s at most k, and (b) If k n, then there s just one commttee contanng a nodes. Each commttee has a unque commttee ID, and the goa s for a nodes to eventuay output a commttee ID such that the two condtons are satsfed. 3. BASIC FACTS In ths secton we state severa basc propertes of the dynamc graph mode, whch we ater use n our agorthms. The frst key fact pertans to the way nformaton spreads n connected dynamc networks. PROPOSITION 3.1. It s possbe to sove 1-token dssemnaton n 1-nterva connected graphs n n 1 rounds, f nodes are not requred to hat after they output the token. 1 1 Prop. 3.1 s ntended ony as an ustraton; n the rest of our agorthms nodes can hat after they perform the output acton.

4 PROOF SKETCH. We smpy have a nodes that know the token broadcast t n every round; when a node receves the token, t outputs t mmedatey, but contnues broadcastng t. In any gven round, consder a cut between the nodes that aready receved the token and those that have not. From 1-nterva connectvty, there s an edge n the cut; the token s broadcast on that edge and some new node receves t. Snce one node ntay knows the message and there are n nodes, after n 1 rounds a nodes have the token. To make ths ntuton more forma we use Lamport s causa order [32], whch formazes the noton of one node nfuencng another through a chan of messages orgnatng at the frst node and endng at the atter (possby gong through other nodes n between). We use (u, r) (v, r ) to denote the fact that node u s state n round r nfuences node v s state n round r, and the forma defnton s as foows. DEFINITION 3.1 (LAMPORT CAUSALITY). Gven a dynamc graph G = (V, E) we defne an order (V N) 2, where (u, r) (v, r ) ff r = r + 1 and {u, v} E(r). The causa order (V N) 2 s defned to be the refexve and transtve cosure of. The foowng emma shows that 1-nterva connectvty s suffcent to guarantee that the number of nodes that have nfuenced a node u grows by at east one n every round, and so does the number of nodes that u tsef has nfuenced. LEMMA 3.2. For any node u V and round r 0 we have (a) {v V : (u,0) (v, r)} mn {r + 1, n}, and (b) {v V : (v,0) (u, r)} mn {r + 1, n}. The proof of the emma s smar to that of Proposton 3.1, and t s omtted here. We can now re-state the prncpe behnd Proposton 3.1 as a coroary of Lemma 3.2. COROLLARY 3.3. (u, n 1). For a u, v V t hods that (v,0) PROOF. Lemma 3.2 shows that n round r = n 1 we have {v V : (v,0) (u, n 1)} n, and the cam foows. If we have an upper bound on the sze of the network, we can use Coroary 3.3 to compute smpe functons whch serve as budng bocks for agorthms. PROPOSITION 3.4. Gven an upper bound N on the sze of the network, functons such as the mnmum or maxmum of nputs to the nodes can be computed n N 1 rounds. Coroary 3.3 guarantees that f nodes aways broadcast the smaest (resp. argest) vaue they have heard, a nodes w have the true mn or max vaue after n 1 rounds; the upper bound N s needed for nodes to know when they have the true mn or max. One appcaton s eader eecton, whch can be mpemented by choosng the node wth the smaest UID as the unque eader. We aso note that havng an upper bound on the sze aows the use of randomzed agorthms for data aggregaton whch rey on computng the max or the mn of random varabes chosen by the nodes [17, 36]; see Secton 7. The remander of the paper focuses on countng and sovng the token dssemnaton probem. The two probems are ntertwned, and both are usefu as a startng pont for dstrbuted computng n dynamc networks. We remark that when message szes are not mted, both probems can be soved n near tme by havng nodes constanty broadcast a the nformaton they have coected so far. PROPOSITION 3.5. Countng and a-to-a token dssemnaton can be soved n O(n) rounds n 1-nterva connected graphs, usng messages of sze O(n og n). PROOF. Consder a smpe protoco for countng. Each node mantans a set A contanng a the UIDs t has heard about so far, where ntay A u(0) = {u} for a u V. In each round r, node u broadcasts A u and adds to A u any UIDs t receves from other nodes. If r > A u, node u hats and outputs A u ; otherwse node u contnues on to the next round. It s not hard to see that for a u, v V and rounds r, f (v,0) (u, r) then v A u(r). Thus, {v V : (v,0) (u, r)} A u(r). Correctness of the protoco foows from Lemma 3.2: f node u hats n round r, then r > A u(r) {v V : (v,0) (u, r)}, and Lemma 3.2 shows that r > n. Next, usng Coroary 3.3 we have that n ths case V A u(r). And fnay, snce obvousy A u(r) V, t foows that A u(r) = V and node u s output s correct. Termnaton aso foows from Lemma 3.2 and the fact that A u(r) V n every round r. To sove a-to-a token dssemnaton, we have nodes attach every token they have heard so far to every message they send. In the seque we descrbe soutons whch use ony O(og n) bts per message. 4. COUNTING THROUGH k-committee In ths secton we show how k-commttee eecton can be used to sove countng and token dssemnaton. Our countng agorthm works by successve doubng: at each pont the nodes have a guess k for the sze of the network, and attempt to verfy whether or not k n. If t s dscovered that k < n, the nodes doube k and repeat; f k n, the nodes hat and output the count. We defer the probem of determnng the exact count unt the end of the secton, and focus for now on the k-verfcaton probem, that s, checkng whether or not k n. Suppose that nodes start out n a state that represents a souton to k-commttee eecton: each node has a commttee ID, such that no more than k nodes have the same ID, and f k n then a nodes have the same commttee ID. The probem of checkng whether k n s then equvaent to checkng whether there s more than one commttee: f k n there must be one commttee ony, and f k < n there must be more than one. Nodes can therefore check f k n by executng a smpe k-round protoco that checks f there s more than one commttee n the graph. The k-verfcaton protoco. Each node has a oca varabe x, whch s ntay set to 1. Whe x u = 1, node u broadcasts ts commttee ID. If t hears from some neghbor a dfferent commttee ID from ts own, or the speca vaue, t sets x u 0 and broadcasts n a subsequent rounds. After k rounds, a nodes output the vaue of ther x varabe. LEMMA 4.1. If the nta state of the executon represents a souton to k-commttee eecton, at the end of the k-verfcaton protoco each node outputs 1 ff k n. PROOF SKETCH. Frst suppose that k n. In ths case there s ony one commttee n the graph; no node ever hears a dfferent commttee ID from ts own. After k rounds a nodes st have x = 1, and a output 1. Next, suppose k < n. We can show that after the th round of the protoco, at east nodes n each commttee have x = 0. In any round of the protoco, consder a cut between the nodes that beong to a partcuar commttee and st have x = 1, and the rest

5 of the nodes, whch ether beong to a dfferent commttee or have x = 0. From 1-nterva connectvty, there s an edge n the cut, and some node u n the commttee that st has x u = 1 hears ether a dfferent commttee ID or. Node u then sets x u 0, and the number of nodes n the commttee that st have x = 1 decreases by at east one. Snce each commttee ntay contans at most k nodes, after k rounds a nodes n a commttees have x = 0, and a output 0. Our strategy for sovng the countng probem s as foows: for k = 1, 2,4, 8,..., sove the k-commttee eecton probem, then execute the k-verfcaton protoco. If k n, termnate and output the count; ese, contnue to the next vaue of k. Here we use the fact that our mode s amenabe to sequenta composton. The strategy outned above requres a nodes to begn the k- verfcaton protoco n the same round. Our protoco for sovng k-commttee eecton ensures that ths occurs. The protoco aso has the usefu property that f k n, every node knows the UIDs of a other nodes n the graph at the end of the protoco. Thus, when k n, nodes can determne the exact count. 5. A k-committee PROTOCOL FOR 1-INTERVAL CONNECTED GRAPHS To sove k-commttee eecton n 1-nterva connected graphs, we magne that there s a unque eader n the network, and ths eader nvtes k nodes to jon ts commttee. Of course we do not truy have a pre-eected eader n the network; we w soon show how to get around ths probem. The protoco proceeds n k cyces, each consstng of two phases. Pong phase: For k 1 rounds, a nodes n the network propagate the UID of the smaest node they have heard about that has not yet joned a commttee. Intay each node broadcasts ts own UID f t has not joned a commttee, or f t has; n each round nodes remember the smaest vaue they have sent or receved so far n the executon, and broadcast that vaue n the next round. Invtaton phase: The eader seects the smaest UID t heard durng the pong phase, and ssues a message nvtng that node to jon ts commttee. The message carres the UID of the eader and of the nvted node. The nvtaton s propagated by a nodes for k 1 rounds. At the end of the nvtaton phase, a node that receved an nvtaton jons the eader s commttee. At the end of the k cyces, nodes that have joned the eader s commttee output the eader s UID as ther commttee ID. Any node that has not been nvted to jon a commttee jons ts own commttee, usng ts UID as the commttee ID. Because we do not ntay have a unque eader n the network, a nodes start out thnkng they are the eader, and contnue to pay the roe of a eader unt they hear a UID smaer than ther own. At that pont they swtch to payng the roe of a non-eader. However, once nodes jon a commttee they do not change ther mnds. THEOREM 5.1. The protoco sketched above soves k-commttee eecton n O(k 2 ) rounds. PROOF SKETCH. The frst condton of k-commttee eecton requres each commttee to be of sze at most k. Ths condton s satsfed because no node ever nvtes more than k nodes to jon ts commttee (each node ssues at most one nvtaton per cyce). For the second condton we must show that f k n then a nodes jon the same commttee. Thus, suppose that k n. The pong phase of the frst cyces asts for k 1 n 1 rounds, and from Coroary 3.3, ths s suffcent for a nodes to hear the UID of the smaest node n the network. Thus, after the frst pong phase there s ony one eader, and no other node ever ssues an nvtaton. Usng Coroary 3.3 we see that the k 1 rounds of each pong phase are suffcent for the eader to successfuy dentfy the smaest node that has not yet joned ts commttee. Smary, the nvtaton phase s ong enough for that node to receve the eader s nvtaton, so n every cyce one node jons the eader s commttee. Snce there are k n cyces, a nodes jon the eader s commttee, and a output the eader s UID as ther commttee ID. We remark that when k n, the k-commttee eecton protoco can aso be used to sove a-to-a token dssemnaton. To do so we smpy have nodes attach ther token to ther UID n every message they send. Each node s snged out for k 1 n 1 rounds durng whch t s nvted to jon the eader s commttee, and the nvtaton reaches a nodes n the graph. Thus, nodes can coect a the tokens by recordng the tokens attached to a nvtatons they hear. In partcuar, f node UIDs are used as tokens, nodes can coect a the UIDs n the network. COROLLARY 5.2. When used together wth the k-verfcaton protoco from Secton 4, the k-commttee eecton protoco yeds an O(n 2 )-round protoco for countng or a-to-a token dssemnaton. 6. COUNTING AND TOKEN DISSEMINA- TION IN MORE STABLE GRAPHS In ths secton we show that n T -nterva connected graphs the computaton can be sped up by a factor of T. To do ths we empoy a neat ppenng effect, usng the temporary stabe subgraphs that T -nterva connectvty guarantees; ths aows us to dssemnate nformaton more qucky. For convenence we assume that the graph s 2T -nterva connected for some T Fast T -Token Dssemnaton n 2T -Interva Connected Graphs Procedure dssemnate gves an agorthm for exchangng at east T peces of nformaton n n rounds when the dynamc graph s 2T -nterva connected. The procedure takes three arguments: a set of tokens A, the parameter T, and a guess k for the sze of the graph. If k n, each node s guaranteed to earn the T smaest tokens that appeared n the nput to a the nodes. The executon of procedure dssemnate s dvded nto k/t phases, each consstng of 2T rounds. Durng each phase, each node mantans the set A of tokens t has aready earned and a set S of tokens t has aready broadcast n the current phase (ntay empty). In each round of the phase, the node broadcasts the smaest token t has not yet broadcast n the current phase, then adds that token to S. S for = 0,..., k/t 1 do for r = 0,..., 2T 1 do f S A then t mn (A \ S) broadcast t S S {t} receve t 1,..., t s from neghbors A A {t 1,..., t s} S return A Procedure dssemnate(a, T, k)

6 Because the graph s 2T -nterva connected, n each phase there s a stabe connected subgraph G that perssts throughout the phase. We use A u(r),s u(r) for the vaues of node u s oca varabes A, S at the begnnng of round r of phase. We say that u knows token t whenever t A u. Let K (t) denote the set of nodes that know t at the begnnng of phase, and et tdst (u, t) denote the mnma dstance n G between node u and any node n K (t). Correctness hnges on the foowng property. LEMMA 6.1. For any node u V, token t S v V Av(0) and round r such that tdst (u, t) r 2T, ether t Su(r+1) or S u(r + 1) ncudes at east (r tdst (u, t)) tokens that are smaer than t. The ntuton behnd Lemma 6.1 s that f r tdst (u, t), then r rounds are enough tme for u to receve t. If u has not receved t and sent t on, the path between u and the nearest node that knows t must have been bocked by smaer tokens, whch node u receved and sent on. Usng Lemma 6.1 we can show: LEMMA 6.2. If k n, at the end of procedure dssemnate the set A u of each node u contans the T smaest tokens. PROOF SKETCH. Let N d (t) := {u V tdst (u, t) d} denote the set of nodes at dstance at most d from some node that knows t at the begnnng of phase, and et t be one of the T smaest tokens. From Lemma 6.1, for each node u N T (t), ether t S u(2t+ 1) or S u(2t + 1) contans at east 2T T = T tokens that are smaer than t. But t s one of the T smaest tokens, so the second case s mpossbe. Therefore a nodes n N T (t) know token t at the end of phase. Because G s connected we have N T (t) mn {n K (t), T }; that s, n each phase T new nodes earn t, unt a the nodes know t. Snce there are no more than k nodes and we have k/t phases, at the end of the ast phase a nodes know t. Remark 1. If each stabe subgraph G enjoys good expanson then dssemnate requres fewer than n phases. For exampe, f G s aways f-connected for some parameter f, then each token s earned by f T new nodes n each phase unt a nodes know t, and we ony requre n/f phases. Smary, f G s aways a vertex expander we ony requre O(og n) phases. 6.2 Countng and Token Dssemnaton To sove countng and token dssemnaton wth up to n tokens, we use Procedure dssemnate to speed up the k-commttee eecton protoco from Secton 5. Instead of nvtng one node n each cyce, we can use dssemnate to have the eader earn the UIDs of the T smaest nodes n the pong phase, and use procedure dssemnate agan to extend nvtatons to a T smaest nodes n the seecton phase. Thus, n O(k + T) rounds we can ncrease the sze of the commttee by T. THEOREM 6.3. It s possbe to sove k-commttee eecton n O(k + k 2 /T) rounds n T -nterva connected graphs. When used n conjuncton wth the k-verfcaton protoco, ths approach yeds an O(n+n 2 /T)-round countng a-to-a token dssemnaton protoco. 6.3 Unknown Interva Connectvty The protoco sketched above assumes that a nodes know the degree of nterva connectvty present n the communcaton graph; f the graph s not 2T -nterva connected, nvtatons may not reach ther destnaton, and the commttees formed may contan ess than k nodes even when k n. However, even when the graph s not 2T -nterva connected, no commttee ever contans more than k nodes, smpy because no node ever ssues more than k nvtatons. Thus, f nodes guess a vaue for T and use the protoco to check f k n, ther error s one-sded: f ther guess for T s too arge they may fasey concude that k < n when n fact k n, but they w never concude that k n when k < n. Ths one-sded error aows us to try dfferent vaues for k and T wthout fear of mstakes. We can count n O(nog n+n 2 og n/t) tme n graphs where T s unknown by teratng over varous combnatons of k and T unt we reach a par (k, T) such that k n and the graph s T -nterva connected. In the worst case, the graph s 1-nterva connected, and we need to try a the vaues T = 1, 2,4,..., k for each k; we pay a og n factor n the round compexty. Ths ony mproves upon the orgna O(n 2 ) agorthm when the graph s ω(og n)-nterva connected. However, we can execute the orgna agorthm n parae wth the adaptve one, and termnate when the frst of the two termnates. In ths way we can sove countng or token dssemnaton n O(mn n 2, n og n + n 2 og n/t ) rounds when T s unknown. Usng smar deas we can aso adapt to unknown expanson of the graph, e.g., we mght guess that t s aways f-connected for some nta vaue of f, and decrease f unt we fnd the rght vaue. 7. APPROXIMATE COUNTING In ths secton we show that under certan restrctons on the dynamc-graph adversary, t s possbe to use randomzaton to compute an approxmate count n amost-near tme, even when the dynamc graph s ony 1-nterva connected. The technques we use are based on a gosspng protoco from [36]. We assume that nodes know some (potentay oose) upper bound N on the sze n of the network; ths upper bound determnes the message sze. For any ε > 0, the agorthm computes a (1 + ε)-approxmaton of the number of nodes n. There are two varants of the agorthm: the frst termnates n O(n) tme wth hgh probabty (n N) and uses messages of sze O(og N (og og N + og(1/ε)/ε 2 )); the second requres O(n (og og N+og(1/ε)/ε 2 )) rounds wth hgh probabty, but uses messages of sze ony O(og N). Both versons of the agorthm assume that the dynamc graph s generated by an obvous adversary, whch determnes the compete sequence of graphs before the executon begns. In partcuar, the adversary s not prvy to the resuts of the nodes con tosses n prevous rounds, and t aso cannot see ther states and ther messages, whch revea the resuts of those con tosses. For smpcty, we descrbe here ony the agorthm that runs n O(n) rounds w.h.p. but uses sghty arger messages. The agorthm rees on the foowng emma from [36], whch shows how the sze of the network can be estmated by computng the mnmum of exponenta random varabes (and repeatng ths procedure to decrease the error probabty). LEMMA 7.1 ([36]). Let S be a setn of -tupes of ndependent exponenta varabes wth rate 1: S = Y (1) 1,..., Y (1),..., o Y (m) 1,..., Y (m). Defne ˆn(S) := P =1 mn. 1 j S Y (j)

7 Then «2 Pr ˆn(S) S > 3 ǫ S 2e ε2/27. For parameters ε (0,1/2) and c > 0, we defne := (2 + 2c) 27 n(n)/ε 2. The scheme for approxmate countng s gven n Ag 2. The man dea s as foows. Intay, each node v V computes ndependent exponenta random varabes Y (v) 1,..., Y (v) wth rate 1. The objectve of a nodes s to compute ˆn(V ), whch s a good estmate for n wth hgh probabty. To do ths they must compute mn v V Y (v) for each []. From Proposton 3.4 we know that nodes can fnd mn v V Y (v) by propagatng the smaest vaue they have heard so far for n 1 rounds. However, n s not known to the nodes (we coud wat N 1 rounds, but N may be a very oose upper bound). We use a combnaton of Lemma 3.2 and Lemma 7.1 to decde when to stop. Let C u(r) := {v V : (v,0) (u, r)} be the set of nodes whose vaue Y (v) has reached u by round r. In round r node u s abe to compute mn v Cu(r) Y (v), but t cannot see vaues Y (v) for v C u(r). Therefore we want node u to hat ony when C u(r) = V. From Lemma 3.2 we know that C u(r) r + 1 for a r n 1. Because we assume an obvous adversary, the set C u(r) s chosen before the nodes choose ther random varabes. We can use Lemma 7.1 to show that wth hgh probabty, f (1 ε)r > ˆn u(c u(r)), then C u(r) = V. We use ths crteron to know when to termnate. (Reca that n Proposton 3.5 we used a determnstc verson of ths test: we hated exacty when r > C u(r).) Sendng exact vaues for Y (v) woud requre nodes to send rea numbers, whch cannot be represented usng a bounded number of bts. Instead nodes send rounded and range-restrcted approxmatons Ỹ (v) for Y (v) ; we omt the technca detas here. Each vaue can be represented usng O(og og N + og(1/ε)) bts. Ỹ (v) Z (u) (Ỹ (u) 1,..., Ỹ (u) ) for r = 1, 2,... do broadcast Z (u) receve Z (v1),..., Z (vs) from neghbors for = 1,..., do Z (u) mn n Z (u) o, Z (v 1),..., Z (vs) ñ u(r) / P =1 Z(u) f (1 ε)r > ñ u(r) then termnate and output ñ u(r) Agorthm 2: Randomzed approxmate countng n near tme (code for node u) For ack of space, the foowng theorem s gven wthout proof. THEOREM 7.2. For ε (0,1/2) and c > 0, wth probabty at east 1 1/N c, (a) every node n the graph computes the same vaue ñ v(r) =: ñ, and furthermore, (b) ñ n εn. 8. LOWER BOUNDS ON TOKEN DISSEMINATION Our agorthms for token dssemnaton do not combne tokens or ater them n anyway, ony store and forward them. We ca ths stye of agorthm a token-forwardng agorthm. Formay, et A u(r) denote the set of messages node u has receved by the begnnng of round r, pus node u s nput I(u). A token-forwardng agorthm satsfes: (a) for a u V and r 0, the message sent by u n round r s a member of A u(r) { }, where denotes the empty message; and (b) node u cannot hat n round r uness A u(r) = S v V I(v), that s, node u has receved a the tokens ether n messages from other nodes or n ts nput. In ths secton we gve two ower bounds on token dssemnaton wth token-forwardng agorthms. 8.1 Lower Bound on Centrazed Token Dssemnaton For ths ower bound we assume that n each round r, some centra authorty provdes each node u wth a vaue t u(r) A u(r) to broadcast n that round. The centrazed agorthm can see the state and hstory of the entre network, but t does not know whch edges w be schedued n the current round. Centrazed agorthms are more powerfu than dstrbuted ones, snce they have access to more nformaton. To smpfy, we begn wth each of the k tokens known to exacty one node (ths restrcton s not essenta). We observe that whe the nodes ony know a sma number of tokens, t s easy for the agorthm to make progress; for exampe, n the frst round of the agorthm at east k nodes earn a new token, because connectvty guarantees that k nodes receve a token that was not n ther nput. However, as nodes earn more tokens, t becomes harder for the agorthm to provde them wth tokens they do not aready know. Accordngy, our strategy s to charge a cost of 1/(k ) for the -th token earned by each node: the frst token each node earns comes at a cheap 1/k, and the ast token earned costs deary (a charge of 1). Formay, the potenta of the system n round r s gven by Φ(r) := X u V A u(r) 1 X =0 1 k. In the frst round we have Φ(0) = 1, because k nodes know one token each. If the agorthm termnates n round r then we must have Φ(r) = n H k = Θ(nog k), because a n nodes must know a k tokens. We construct an executon n whch the potenta ncrease s bounded by a constant n every round; ths gves us an Ω(n og k) bound on the number of rounds requred. THEOREM 8.1. Any determnstc centrazed agorthm for k- token dssemnaton n 1-nterva connected graphs requres at east Ω(n og k) rounds to compete n the worst case. PROOF. We construct the communcaton graph for each round r n three stages (ndependenty of prevous or future rounds). Stage I: addng the free edges. An edge {u, v} s sad to be free f t u(r) A v(r) and t v(r) A u(r); that s, f when we connect u and v, nether node earns anythng new. Let F(r) denote the set of free edges n round r; we add a of them to the graph. Let C 1,..., C denote the connected components of the graph (V, F(r)). Observe that any two nodes n dfferent components must send dfferent vaues, otherwse they woud be n the same component. We choose representatves v 1 C 1,..., v C from each component arbtrary. Our task now s to construct a connected subgraph over v 1,..., v and pay ony a constant cost. We assume that 12, otherwse we can connect the nodes arbtrary for a constant cost. Let mssng(u) := k A u(r) denote the number of tokens node u does not know at the begnnng of round r.

8 Stage II. We spt the nodes nto two sets, Top and Bottom, accordng to the number of tokens they know, wth nodes that know many tokens on top : Top := {v mssng(v ) /6} and Bottom := {v mssng(v ) > /6}. Snce top nodes know many tokens, connectng to them coud be expensve. We w choose our edges n such a way that no top node w earn a new token. Bottom nodes are cheaper, but st not free; we w ensure that each bottom node w earn at most three new tokens (see Fg. 1). We begn by boundng the sze of Top. To that end, notce that ` Top Pu Top mssng(u) : 2 for a, j such that u, v Top, ether t u(r) A v(r) or t v(r) A u(r), otherwse {u, v} woud be a free edge and u, v woud be n the same component. Therefore each par u, v Top contrbutes at east one mssng token to the sum, and P ` Top u Top mssng(u). 2 On the other hand, snce each node n Top s mssng at most /6 tokens, t foows that P u Top mssng(u) Top (/6). Puttng the two facts together we obtan Top /3 + 1, and consequenty aso Bottom Top 2 Top ( 12) 6. Next we show that because there are many more bottom nodes than top nodes, we have enough fexbty to use ony cheap edges to connect to top nodes. Stage III: Connectng the nodes. The bottom nodes are reatvey cheap to connect to, so we connect them n an arbtrary ne (see Fg. 1). In addton we want to connect each top node to a bottom node, such that no top node earns somethng new, and no bottom node s connected to more than one top node. That s, we are ookng for a matchng between Top and Bottom, usng ony edges n P = {{u, v} : u Top, v Bottom and t v A u(r)}. Snce each top node s mssng at most /6 tokens, and each bottom node broadcasts a dfferent vaue from a other bottom nodes, for each top node there are at east Bottom /6 edges n P to choose from. To construct the matchng, we go through the top nodes n arbtrary order v 0,..., v p Top, and choose for each v some unmatched bottom node u such that {v, u } P and u u j for a j <. Before each step the number of unmatched bottom nodes s at east Bottom > Bottom Top /6. We aready saw that each top node s connected to a but /6 bottom nodes n P, so there s aways some unmatched P -neghbor of v to choose n step. What s the tota cost of the graph? Top nodes earn no tokens, and bottom nodes earn at most two tokens from other bottom nodes and at most one token from a top node. Thus, the tota cost s bounded by X u Bottom Bottom 6 6 mn{3,mssng(u)} X =1 36 = mssng(u) ( 1) 8.2 Lower Bound on Token Dssemnaton wth Knowedge-Based Agorthms A token-forwardng randomzed agorthm for k-token dssemnaton s sad to be knowedge-based f the dstrbuton that determnes whch token s broadcast by node u n round r s a functon of the UID of u, the sequence A u(0),..., A u(r 1), where A Top (nodes mssng at most /6 tokens) Bottom (nodes mssng more than /6 tokens) Fgure 1: Iustraton for the proof of Theorem 8.1 s the set of tokens receved by u by the begnnng of round (ncudng ts nput), and the sequence of u s con tosses up to round r (ncusve). Knowedge-based agorthms can base ther decsons on the set of tokens currenty known, the order n whch tokens were acqured, and even the round n whch each token was acqured; however, they cannot rey on other factors, such as the number of tmes a partcuar token was heard, or whch tokens were receved n the prevous round. Nevertheess, the cass of knowedge-based agorthms ncudes many natura strateges for sovng the token dssemnaton probem, and t ncudes the agorthms n ths paper. (Other knowedge-based strateges ncude round-robn over the known tokens, choosng a token to broadcast unformy at random, and choosng each token wth a probabty that depends on how ong ago that token was acqured.) Knowedge-based agorthms have the property that once a node earns a the tokens, the dstrbuton of tokens broadcast n future rounds s fxed and does not depend on the dynamc graph. We use ths property to show the foowng ower bound. THEOREM 8.2. Any knowedge-based agorthm for k-token dssemnaton n T -nterva connected graphs requres Ω(n + nk/t) rounds to succeed wth probabty 1/2. Further, f the sze of the namespace for UIDs U = Ω(n 2 k/t), then determnstc agorthms requre Ω(n + nk/t) rounds even when each node starts wth exacty one token. PROOF SKETCH. An Ω(n) ower bound s trvay demonstrated n a statc ne graph where some token starts at one end of the ne. Thus we assume that k > 1. For smpcty, we choose an nput assgnment n whch some node u knows a the tokens, and the other nodes have no tokens. Let r 0 = (n 1)(k 1)/(4T) 2 = Θ(nk/T). We say that a token t s nfrequent n a gven executon f node u broadcasts t ess than (n 1)/(2T) tmes n rounds 0,..., r 0 of the executon. Snce node u knows a the tokens, ts behavor s determned: regardess of the dynamc graph we choose n rounds 0,..., r 0, the dstrbuton of tokens broadcast by node u n these rounds s fxed. In partcuar, snce r 0 < (n 1)(k 1)/(4T) 1, the nearty of expectaton and Markov s nequaty show that there s some token t such that n any dynamc graph, token t w be nfrequent wth probabty at east 1/2. We w construct a specfc dynamc graph G n whch whenever t s nfrequent, the agorthm does not termnate by round r 0. Thus, n the graph we construct, the agorthm requres Ω(nk/T) rounds w.p. at east 1/2. Intay there are n 1 nodes that do not know t (a nodes but u). Our goa n constructng G s to ensure that every tme node u broadcasts t, at most 2T new nodes earn t. Reca that t s sad to be nfrequent when t s broadcast ess than (n 1)/(2T) tmes by round r 0. Hence, whenever t s nfrequent, some node n G has st not earned t by round r 0 and agorthm cannot termnate. We construct the dynamc graph n phases of two types. When

9 u has not broadcast t for a whe (T rounds to be precse), the network s n a quet phase. A quet phase extends unt the frst tme u broadcasts t (ncudng that round). Durng quet phases the communcaton graph remans statc and comprses two components (see Fg. 2(a)): component U (for unaware, shown as whte nodes n Fg. 2(a)) contans nodes that are guaranteed not to know t, arranged n a ne v 1,..., v. The frst node n the ne, v 1, s connected to node u. (Note that u U, because u knows t from the start). The other component, K (for knowedgeabe ), contans the remanng nodes. These nodes may or may not know t, and we connect them to each other arbtrary. Intay, K = {u} and U = V \ {u}, wth the nodes n U ordered arbtrary. A quet phase ends mmedatey after u broadcasts token t. At ths pont v 1 U knows t; f we eave the network statc, the nodes n U may forward t to each other, unt n U 1 rounds a nodes n U know t. Reca that we want to ensure that at most 2T nodes earn t after every tme u broadcasts t. To contan the propagaton of t, we begn an actve phase. If we wanted to satsfy ony 1-nterva connectvty, we woud smpy move v 1 from U to K and connect u to v 2 nstead. Ths woud prevent v 1 from spreadng t to other nodes n U, but t voates T -nterva connectvty for T > 1. In order to move nodes from U to K we need more edges, so that we can remove some wthout breakng connectvty. Thus, at the begnnng of an actve phase we connect u to v, cosng the ne to form a rng (see Fg. 2(b)). Then we wat for T rounds. Fnay, we remove edge v 2T, v 2T+1 (see Fg. 2(c)). Ths ends the actve phase. Note that T -nterva connectvty s preserved aong the ne v 2T+1, v 2T+2,..., v, u, v 1,..., v 2T (shown n bod nes n the fgures). Ths s why we use a rng nstead of a ne. At the begnnng of an actve phase, token t may be known ony by nodes v 1,..., v T n the rng. (If the precedng phase was quet, ony node v 1 knows t; f the precedng phase was actve more nodes may know t, see beow.) An actve phase asts T rounds. Durng ths tme, token t propagates n one of two ways: (1) Nodes v 1,..., v T may forward t to nodes v T+1,..., v 2T. (2) Node u may broadcast t agan durng the phase, n whch case nodes v,..., v T (ndcated n cross-hatchng n Fg. 2(c)) may aso earn t. At the end of the phase we remove the nk v 2T, v 2T+1, cuttng off the propagaton of t aong that sde of the rng, and set U U \ {v 1,..., v 2T }. Notce that now the ony node n U to whch u s connected s v. To retan consstency n notaton we reverse the ne, renamng as foows: 2T, 1, 2 1,..., 2T+1. If u dd not broadcast t durng the phase (that s, case (2) above dd not occur), then no remanng node n U knows t, and we begn another quet phase. If u dd broadcast t, at most v 1,..., v T (whch were abeed v,..., v T before the renamng) know t, and we begn another actve phase. The constructon above aows us to charge at most 2T nodes to each tme u broadcasts t: an actve phase s ony trggered when u broadcasts t, and each actve phase ends wth the remova of 2T nodes from U. If t s nfrequent n an executon, then there are ess than (n 1)/(2T) actve phases by round r 0 of the executon; snce ntay U = n 1, by round r 0 = Ω(nk/T) there s st some node n U whch does not know t, and the agorthm s not done. Snce t s nfrequent w.p. at east 1/2, ths shows that any knowedge-based agorthm for k-token dssemnaton requres Ω(nk/T) rounds w.p. at east 1/2. K U v 1 v 2 u v (a) The network n a quet phase. v 1 v 2 u v v T (b) The begnnng of an actve phase. (c) The end of an actve phase. u u v 1 v 2 v 2T v (d) The begnnng of the next actve phase. Fgure 2: The constructon for Theorem 8.2, wth T = 2. Nodes that do not know t are shown n whte, nodes that may know t are shown n grey. Edges aong whch T -nterva connectvty s preserved are shown n bod. 9. CONCLUSION In ths work we consder a mode for dynamc networks whch makes very few assumptons about the network. The mode can serve as an abstracton for wreess or mobe networks, to reason about the fundamenta unpredctabty of communcaton n ths type of system. We do not restrct the mobty of the nodes except for retanng connectvty, and we do not assume that geographca nformaton or neghbor dscovery are avaabe. Nevertheess, we show that one can effcenty compute any computabe functon n our mode, takng advantage of stabty f t exsts n the network. We beeve that the T -nterva connectvty property provdes a natura and genera way to reason about dynamc networks. It s easy to see that wthout any connectvty assumpton no non-trva functon can be computed, except possby n the sense of computaton n the mt (as n [3]). However, our connectvty assumpton s easy weakened to ony requre connectvty once every constant number of rounds, or to ony requre eventua connectvty n the stye of Prop. 3.1, wth a known bound on the number of rounds. U