AN Ω(D log(n/d)) LOWER BOUND FOR BROADCAST IN RADIO NETWORKS

Transcription

1 SIAM J. COMPUT. c 1998 Society for Industria and Appied Mathematics Vo. 27, No. 3, pp , June AN Ω(D og(n/d)) LOWER BOUND FOR BROADCAST IN RADIO NETWORKS EYAL KUSHILEVITZ AND YISHAY MANSOUR Abstract. We show that for any randomized broadcast protoco for radio networks, there exists a network in which the expected time to broadcast a message is Ω(D og(n/d)), where D is the diameter of the network and N is the number of nodes. This impies a tight ower bound of Ω(D og N) for any D N 1 ε, where ε>0 is any constant. Key words. radio networks, broadcast, ower bounds AMS subject cassifications. 68Q22, 68M10 PII. S Introduction. Traditionay, radio networks have received considerabe attention due to their miitary significance. The growing interest in ceuar teephones and wireess communication networks has reinforced the interest in radio networks. The basic feature of radio networks, which distinguishes them from other networks, is that a processor can receive a message ony from a singe neighbor at a certain time. If two (or more) neighbors of a processor transmit concurrenty, then the processor woud not receive either messages. In many appications, the users of the radio network are mobie, and therefore the topoogy is unstabe. For this reason, it is desirabe for radio-networks agorithms to refrain from making assumptions about the network topoogy, or about the information that processors have concerning the topoogy. In this work we assume that none of the processors initiay have any topoogica information, except for the size of the network and its diameter. 1 See [Tan81, Ga85, BGI92, BGI91] for a discussion on this mode and reated modes. We study broadcast protocos; those protocos are initiated by a singe processor (the originator) that has a message M it wishes to propagate to a the other processors in the network. In many of the radio-networks appications (e.g., ceuar phones) broadcast is a centra primitive which is frequenty used, for exampe, to perform a network-wide search for a user. Bar-Yehuda, Godreich, and Itai [BGI92] present a randomized broadcast agorithm, that runs in expected O(D og N + og 2 N) time sots, where N is the number of processors in the network and D is its diameter. In contrast, they show that for any deterministic broadcast agorithm there are networks of constant diameter on which the agorithm needs Ω(N) time sots. Received by the editors September 8, 1994; accepted for pubication (in revised form) Apri 3, An eary version of this paper appeared in Proc. 12th ACM Symp. on Principes of Distributed Computing, ACM, New York, 1993, pp Aiken Computation Lab., Harvard University, Cambridge, MA Current address: Dept. of Computer Science, Technion, Haifa, 32000, Israe (eyak@cs.technion.ac.i). This research was supported by research contracts ONR-N J-1981 and NSF-CCR Computer Science Dept., Te-Aviv University, Ramat Aviv, Te Aviv 69978, Israe and IBM T. J. Watson Research Center (mansour@cs.tau.ac.i). 1 Usuay, when the topoogy is unstabe, the diameter is unknown to the processors and ony a bound on the size of the network is avaiabe. However, since we are proving a ower bound, this assumption ony makes the resut stronger. 702

2 LOWER BOUND FOR BROADCAST IN RADIO NETWORKS 703 Aon et a. [ABLP91] made the first step towards proving the optimaity of the upper bound of [BGI92]. Their resut can be viewed as a graph-theoretic resut; they show that there exist networks of diameter D = 3 on which any schedue needs at east Ω(og 2 N) time sots. This ower bound shows that there are networks on which broadcast requires this many time sots, and it matches the known upper bounds [BGI92, CW87], in the case of constant-diameter networks. In this work we compete the picture by proving an Ω(D og(n/d)) ower bound. Our resut is of a different nature; we show that for any randomized broadcast agorithm and parameters N and D, there is an ordering of the N processors in a network of diameter D such that the expected number of time sots, used by the agorithm, is Ω(D og(n/d)). For D N 1 ε this gives an Ω(D og N) ower bound. Hence, it proves the tightness of the upper bound of [BGI92] for a N and D N 1 ε. Moreover, the ower bound hods even if each of the N processors is aowed to use a different program (e.g., the processors can use their IDs). In a recent work, Gaber and Mansour [GM95] have shown that for every network, there exists a schedue whose time is O(D + og 5 N). The scheduer there needs to get the topoogy of the network in advance, in order to buid the schedue. The resut of [GM95] shows that the ower bound presented here must rey heaviy on the ack of topoogica knowedge at the processors. Broadcast in radio networks has received considerabe attention in previous works. [CW87] present a deterministic sequentia agorithm that, given the network, finds in poynomia time a ega schedue that requires at most O(D og 2 N) time sots. Broadcast that is based on using a spanning tree was suggested in [CK85a, CK87]. In [BII93] it is shown how to reduce the amortized cost per broadcast by using a breadthfirst-search (BFS) tree. Simuation of point to point networks on radio networks is found in [CK85b, ABLP92, BGI91]. An important issue in the study of radio networks is whether coisions can be detected; namey, whether a istener can distinguish between the case when none of its neighbors transmit and the case when two or more of them transmit. In our mode it is assumed that the istener cannot distinguish between the two cases (say, it hears noise in both cases). There is another common mode in which it is assumed that the two cases are distinguishabe (say, if no neighbor transmits, the istener hears sience, whie if two or more neighbors transmit, the istener hears noise). A discussion justifying both modes can be found in [Ga85, BGI92]. Wiard [Wi86] studies a broadcast probem in a singe mutiaccess channe under this second mode (i.e., when coision detection is avaiabe). He shows matching upper and ower bounds of Θ(og og n) expected time sots 2 in this mode. Our main emma impies an Ω(og n) ower bound for the same probem in our mode. Again, this ower bound hods even if the processors use different programs. Hence, we demonstrate a provabe exponentia gap between these two modes. The rest of this paper is organized as foows: section 2 contains some necessary definitions. Section 3 contains the proof of the main emma in the uniform case, where a the processors use the same program. Section 4 contains the proof of the main emma in the nonuniform case, where processors may use different programs. The 2 Wiard shows an Ω(og og n) ower bound in the singe mutiaccess channe mode. Athough this bound appies to a different mode, it shoud be noted that his bound is aso significanty restricted by the types of agorithms for which it appies. In particuar, he requires independence between the decision whether to transmit in a certain time sot and the decisions made in previous time sots. In our case such a restriction is unacceptabe, as the upper bound of [BGI92] has such dependencies. Aso, he does not hande the case where each processor uses a different program.

3 704 EYAL KUSHILEVITZ AND YISHAY MANSOUR proof for this case is based on a probabiistic reduction to the uniform case. Finay, in section 5, we prove the main theorem. The proof invoves constructing a difficut network in a probabiistic way. 2. Preiminaries. A radio network is described by an undirected graph G(V,E), 3 where N = V and D is the diameter of the graph. The nodes of the graph represent processors of the network, and an edge between nodes v and u impies that v can send messages to u (and vice-versa). The neighborhood ofanodeu incudes a the nodes v such that there is an edge (u, v) ine. The time is viewed as divided into sots (or rounds). In any given sot, a node (processor) can either transmit some message (a string in {0, 1} ) or not (i.e., remain sient). A radio network has the property that if two or more nodes in the neighborhood of a node u transmit at the same time sot, then none of the messages is received at u. More formay, we can define the set of possibe transmissions as W = {0, 1} {sient}. If exacty one of the node s neighbors transmits at time t and the message that this neighbor transmits is some m {0, 1}, then m is received by the node. In any other case (i.e., if either none of the neighbors transmits or more than one neighbor transmits) this node hears sient. The history of ength ofanode is a vector in W which consists of its view of the first rounds. Each processor P i in the radio network uses a probabiistic program. This program defines whether the processor wi transmit at the next time sot j or not. As we are not concerned with the computationa power of the processors we can simpy view this program as a probabiity distribution, which may depend on the history. More formay, for each processor P i and step j there is a probabiistic function Γ j i : W j 1 W that, based on the history, determines the action of P i in step j (i.e., whether it remains sient, or ese the vaue of the message it sends). The program of P i is a coection Γ i =(Γ 1 i, Γ2 i,...) that defines the actions of P i in each step. A protoco P N,D is simpy a coection of N such programs, one per processor. A protoco is uniform if a processors use the same program. Otherwise, if each processor has a different program, the protoco is nonuniform. The above definition aows the protocos to use the vaues of N and D. On the other hand, the protoco does not know the topoogy of the graph, meaning that the same protoco must work for a graphs of N nodes and diameter D. A broadcast protoco is a protoco that is initiated by a singe processor, caed originator, that hods a message M. Any other processor is inactive (i.e., it remains sient) unti receiving a message for the first time. The aim of the protoco is that each processor in the network wi receive a copy of the message M. 3. Uniform processors. In this section we prove the main emma for the uniform case, where a processors use the same program. It shows that if there are n processors 4 arranged in a cique, then there exists a t (2 t n) such that if t processors wish to transmit (we ca these t processors the participants), then the expected number of rounds (time sots) unti a round in which exacty one of them transmits is Ω(og n). In fact, we show that this is the case for most of the t s of the form t =2 i. Note that the assumption that the topoogy is not known to the processors, in the context of this emma, means that t, the number of processors that 3 None of the resuts presented in this work wi be changed if the network is a directed one. However, it is common in this area to assume that the network is undirected. 4 Note that we use here n (and not N) as the number of processors. This wi be convenient whie using the emma in the proof of the theorem.

4 LOWER BOUND FOR BROADCAST IN RADIO NETWORKS 705 are trying to transmit, is not known to any processor. We can view the scenario as having a famiy of networks with n+1 nodes, composed from a cique of size n and an originator which is connected to t of the nodes in the cique. The (unknown) topoogy is chosen to be one of these networks. For a broadcast protoco Π, we ca a round successfu if exacty one processor transmits. Let E(T Π ) denote the expected number of rounds unti the first successfu round, given that the number of participants is 2 (the expectation is taken over the probabiistic choices of the processors). Lemma 1. Let Π be a broadcast protoco, et the network be as above, and et n be an upper bound on the number of participants. Then, E [E(T Π )] = Ω(og n), where E denotes the expectation when is chosen uniformy from the range 1 og n. Proof. The first observation that we make is that the emma deas ony with the first success. This, in a sense, aows us to get rid of the dependency in the history we can assume that the (probabiistic) decision as to which rounds a processor tries to transmit is made at the beginning of the protoco. This is done by etting each of the 2 processors choose whether to transmit in round s or not in the same way as it chooses in the origina protoco, when a previous rounds were unsuccessfu. Ceary, as far as the first success is concerned, this modification has no effect on the protoco. Aso, as ony the first success is considered, it does not matter what the vaues of the messages that the processors try to transmit are. Hence, the decision of a processor on whether to transmit in round s may depend on the round number, s, and the probabiistic choices of the processor in the first s 1 rounds, but it does not depend on choices made by other processors. 5 Therefore, we can think about the processors as if they choose in advance, for every round s =1, 2,..., whether they wi try to transmit. 6 For simpicity of notation, we assume that n is a power of 2. Define p s, = Pr(faiure in rounds 1,...,s 1 and success in round s 2 participants). As the events described in the definition are disjoint (for fixed and different s s), and assuming that the protoco succeeds with probabiity 1 (no matter what is), we have for a (1) p s, =1. s=1 At some point in the proof beow, it wi be inconvenient if p s, depends on events that happen in previous rounds. However, we can get rid of this dependency simpy by writing (2) p s, Pr( success in round s 2 participants). 5 The message M that the processors need to broadcast aso infuences their decisions. However, it can be thought of as part of the program used by the processors. 6 To avoid measurabiity concerns, it is convenient to assume that the protoco is such that s is in the range 1,...,F, for some finite F. If this is not the case, we can aways choose F such that the probabiity of choosing ony in the range 1,...,F is arbitrariy cose to 1. This wi cause minor changes in our proof.

5 706 EYAL KUSHILEVITZ AND YISHAY MANSOUR The next caim gives a bound on the sum of the success probabiities in a given round. Intuitivey it says that you cannot have high probabiity of success in (a fixed) round s for more than a few vaues of 2. This woud impy that since the number of participants is unknown, Ω(og n) rounds woud be required to reach a success for a numbers of participants. Formay, we make the foowing caim. Caim 2. For any s, og =1 Pr(success in round s 2 participants) < 2. Proof. Fix s. As aready discussed, we assume that the processors make a their choices in advance. The history of choices of a processor is a string in {0, 1} s 1, where the vaue of the ith bit means trying ( 1 ) or not trying ( 0 ). Define q(s) = Pr(trying in round s) = history h Pr(h) Pr(trying in round s h). Note that q(s) does not depend on. We assume, without oss of generaity, that q(s) > 0 (rounds with q(s) = 0 can be omitted from the protoco). Reca that a successfu round is one in which exacty one processor is trying to transmit. Therefore, We get og =1 Pr(success in round s 2 participants) = 2 q(s) (1 q(s)) 2 1. Pr(success in round s 2 participants) = og =1 2 q(s)(1 q(s)) 2 1 og = q(s) 2 (1 q(s)) 2 1 =1 n 1 2 q(s) (1 q(s)) j 1 (1 q(s))n =2 q(s) < 2 q(s) which competes the proof of the caim. Let k be a parameter (to be fixed ater). We are interested in k s=1 p s,, which is intuitivey the probabiity that, given that there are 2 participants, the agorithm succeeds in one of the first k rounds. Using equation (2) and Caim 2, we get (3) og =1 s=1 By definition, k p s, k og s=1 =1 E [E(T Π )] = og =1 j=1 Pr(success in round s 2 participants) < 2k. 1 og n p s, s s=1 k og og n =1 s=k p s,.

6 LOWER BOUND FOR BROADCAST IN RADIO NETWORKS 707 By equation (1), this equas k og n og =1 ( ) k 1 1 p s,, s=1 which, by equation (3), is greater than By choosing k = 1 4 og n, we have that k (og n 2(k 1)). og n E [E(T Π )] 1 8 og n + 1 2, which competes the proof of the emma Nonuniform processors. In this section we prove the main emma for the nonuniform case, where the n processors may use different programs. The main idea of the proof is to reduce the nonuniform case to the uniform one, and use the resut of the previous section (Lemma 1). Lemma 3. Let Π be a protoco for n distinct processors P 1,...,P n that run (possiby) different programs. Let E(T Π ) denote the expected number of rounds unti the first successfu round, given that a random set of 2 processors participates (the expectation is taken over the choice of the set and the probabiistic choices made by the processors). Then E [E(T Π )] = Ω(og n), where is chosen uniformy from the range 1 og n. Proof. As argued in the previous section, as ony the first successfu round is considered, each program can be thought of as a schedue a choice of a subset of rounds in which the processor wi transmit. Processor P i chooses its schedue from a distribution µ i. We now define, based on the (possiby different) programs used by P 1,...,P n,a new program that wi be used by each of L uniform processors Q 1,...,Q L : processor Q j chooses (uniformy) at random 1 i n and simuates the program of processor P i. Namey, it chooses a schedue s with probabiity n 1 n i=1 µ i(s), where µ i (s) is the probabiity that processor P i chooses the schedue s. We denote by c(q j ) the processor P i that Q j chose to simuate. We emphasize that a the Q j s run the same program (i.e., they are uniform), and that different Q j s may choose to simuate the same processor P i (we wi choose L sma enough so that this wi happen ony with a sma probabiity). The foowing caim says that, given that a the c(q j ) s are distinct for Q 1,...,Q 2, then the probabiity distribution of the schedues chosen by the Q j s is the same as that of a random set of 2 processors P i. Caim 4. Let Q = {Q 1,...,Q 2 }. For every Q j Q, et c(q j ) be a random processor P i. If j 1 j 2 : c(q j1 ) c(q j2 ), then P = {c(q j ) Q j Q} is a random 7 In the origina version of this paper [KM93], we proved a sighty better ower bound of 1 og n; 4 however, the proof here is simper.

7 708 EYAL KUSHILEVITZ AND YISHAY MANSOUR set of 2 processors (in P 1,...,P n ), and the foowing hods: for every choice of 2 schedues s 2 =(s 1,...,s 2 ), Pr[ s 2 processors Q run ]=Pr[ s 2 processors P run ]. The foowing caim is the main too in the reduction from the nonuniform case to the uniform case. Caim 5. Let Q be as above and et Q = {Q 1,...,Q 2 } be a set of 2 processors. Each processor Q j runs the program of Q j at the odd steps and the [BGI92] program at the even steps. (Note that the [BGI92] program is aso a uniform protoco, and therefore, so is the program run by the processors Q.) Let β be the probabiity that j 1 j 2, c(q j1 ) c(q j2 ). Let T Q be the random variabe indicating the time of first success when the 2 identica programs in Q run, and reca that T Π is the random variabe indicating the time of first success when a random subset of 2 distinct programs P i1,...,p i2 run. Then, E[T Q ] 2β E[T Π ] + 8(1 β ) og n. In the above caim we mixed the given (unknown) protoco with the [BGI92] protoco. This is because we have no guarantee about the running time of the simuation, in the case when some Q j s choose to simuate the same P i. For exampe, a protoco that ets processor P i transmit at time sot i woud not terminate if a the Q j simuate the same processor P i. Proof. Let unique be the event that Q j1,q j2 Q,c(Q j1 ) c(q j2 ). Then, E[T Q ]=E[T Q unique] Pr[unique]+ E[T Q not unique] Pr[not unique]. By definition, Pr[unique] =β. By Caim 4, E[T Q unique] 2E[T Π ], where the additiona factor of 2 is due to the intereaving of the two protocos. In the case when the choices of c(q j ) are not unique, we cannot use the properties of the origina protoco. However, we can use the fact that the [BGI92] protoco has the expected time unti the first success of at most 4 og n. Therefore, E[T Q not unique] 8 og n, which competes the proof of the caim. The next caim says that with high probabiity the choices c(q j ) are unique. Caim 6. Let β be the probabiity that j 1 j 2, c(q j1 ) c(q j2 ), and assume that 2 n 1/4. Then, β > 1 1. n Proof. Note that Pr[j 1 j 2 and c(q j1 )=c(q j2 )] = 1 n.

8 LOWER BOUND FOR BROADCAST IN RADIO NETWORKS 709 Therefore, β = Pr[ j 1 j 2 ( ) 2 1 : c(q j1 ) c(q j2 )] 1 2 n. Since 2 n 1/4 the emma foows. Let L = n 1/4. By Caims 5 and 6, E[T Q ] 2β E[T Π ]+(1 β )8 og n 2E[T Π ]+ 8 og n n or E[T Π ] 1 2 Q E[T ] 4 og n. n We now take the expectation over a vaues 1 og L and get By Lemma 1, which impies that E [E[T Π ]] 1 2 E [E[T Q ]] 4 og n. n E [E[T Q ]] = Ω(og L) = Ω(og n), E [E[T Π ]] = Ω(og n), as desired. 5. Main theorem. In this section we prove the main theorem. We show that for every broadcast agorithm that does not know the topoogy of the network, for every N and every D, there exist networks of N processors and diameter D such that the expected running time of the agorithm (unti a processors receive the message) is Ω(D og(n/d)). This impies a simiar ower bound for the worst case running time, when a sma probabiity of error is aowed (which is the scenario in which the upper bound of [BGI92] is described). Given an agorithm and the vaues N and D, we construct a network as foows. Let n = N/D, and assume for simpicity that n is a power of 2. We construct a compete ayered network of D + 2 ayers. The first ayer (ayer 0) contains one node, s, which wi be the originator of the broadcast. Each of the next D ayers (ayers 1, 2,...,D) consists of n i =2 i n nodes, where i is chosen uniformy (and independenty for each ayer i) in the range 1,...,og n. The ast ayer contains a the other nodes (so that the tota number of nodes wi be N). Eachnodeinayeri is connected to a nodes in ayers i 1 and i + 1. (See Figure 1.) Reca that the topoogy of the network is not known to the processors. (If the topoogy was known, then an efficient uniform protoco woud be to et a processor at ayer i broadcast with probabiity 1/n i, with expected time O(D). A nonuniform protoco that knows the topoogy simpy ets one node in each ayer transmit.) The agorithm can depend, however, on other information that the processors have, in particuar, the history, the number of steps, etc. (As mentioned, other information which is independent of the graph, such as the message M to be broadcast, the

9 710 EYAL KUSHILEVITZ AND YISHAY MANSOUR Fig. 1. Structure of the network. processors IDs, or the vaue of a cock, can be thought of as encoded into the programs of the processors.) We discuss the uniform case, in the sense that a the processors at ayer i have the same protoco. The extension to the nonuniform case empoys the techniques of the previous section, and the proof is the same but the notation becomes cumbersome. (In particuar, in the nonuniform case, at each ayer i we wi choose not ony n i but aso a random set of n i processors.) The main property of this construction is the foowing. For a i and a runs of the protoco, a the processors in ayer i have the same view; every message received at one of these processors is received by a other processors at the same time. Therefore, the broadcast progresses in a ayer-by-ayer fashion. Moreover, this impies that a the processors in ayer i choose schedues according to the same distribution µ (the choice of µ depends on the history, but a the processors of ayer i share the same history), which aows us to use Lemma 1. Finay, before going into the detais, we make one more assumption that makes our argument simper. We give the processors of ayer i, at the time they get the first message from a processor in ayer i 1, a the other messages they wi get from ayer i 1 in the future, as we as the actua vaues of 1,..., i 1. As this extra information can ony hep the processors to make the broadcast faster, we are aowed to make this assumption. Let t i be the random variabe indicating the number of rounds from the time the processors of ayer i get the message (and become active) unti their success (the first time that a singe processor in ayer i transmits). We need to show that for some choice of 1,..., D we get E Π ( D i=1 t i)=ω(d og(n/d)), where the expectation is taken over the random choices of the agorithm Π. Certainy, it is enough to show that E 1,..., D,Π( D i=1 t i)=ω(d og(n/d)). By inearity of expectation, we get E 1,..., D,Π ( D ) t i = i=1 D E 1,..., D,Π(t i ). So a we have to bound now is E 1,..., D,Π(t i ). Ceary, the choice of i+1,..., D has i=1

10 LOWER BOUND FOR BROADCAST IN RADIO NETWORKS 711 no infuence on the expectation of t i ; i.e., E 1,..., D,Π(t i )=E 1,..., i,π(t i ). Aso, by the discussion above, with every history (which depends on the random choices made in the first i 1 ayers, incuding the choice of 1,..., i 1 )wecan associate a probabiity distribution µ used by the processors in ayer i to choose their schedues. (Note that since we assume that the processors of ayer i get a the future information with the first message, they can make a their random choices at this time.) Therefore, we can write E 1,..., i,π(t i )= E i,π(t i 1 = b 1,..., i 1 = b i 1 ) Pr[ 1 = b 1,..., i 1 = b i 1 ]. b 1,...,b i 1 (4) It remains to bound the expression E i,π(t i 1 = b 1,..., i 1 = b i 1 ). As mentioned, we aow the processors at ayer i to have access to b 1,...,b i 1 (the actua vaues of 1,..., i 1 ). Therefore, we need to evauate E i,π i (t i ), where Π i is the protoco at ayer i, with the additiona information about the ower ayers. By Lemma 1, for each such Π i, E i,π i (t i ) c og n for some constant c. Therefore, for every b 1,...,b i 1,wehave which by (4), impies This impies E i,π(t i 1 = b 1,..., i 1 = b i 1 ) c og n, E 1,..., D,Π E 1,..., i,π(t i ) c og n. ( D ) t i =Ω(Dog n) =Ω(Dog(N/D)), i=1 which competes the proof of our main theorem. Theorem 7. For any nonuniform broadcast protoco, for every number of processors N and every diameter D, there exists a network in which the expected time to compete a broadcast is Ω(D og(n/d)). When D N 1 ε, the above proof shows a ower bound of Ω(D og N). Combining our resut with the resuts of Aon et a. [ABLP91] and Bar-Yehuda, Godreich, and Itai [BGI92], we have the foowing tight resut. Coroary 8. For any nonuniform broadcast protoco, for every number of processors N and every diameter D, there exists a network in which the time to compete a broadcast is Ω(og 2 N + D og(n/d)). Furthermore, there is a (uniform) protoco that requires ony O(og 2 N + D og N) expected time (which is tight for D N 1 ε ). Note that unike [ABLP91] we show that for any protoco there exists a network for which the ower bound hods, whie they prove that there exists a network on which any protoco requires the ower bound.

11 712 EYAL KUSHILEVITZ AND YISHAY MANSOUR Acknowedgments. We wish to thank Oded Godreich and the anonymous referees for their very usefu comments. REFERENCES [ABLP91] N. Aon, A. Bar-Noy, N. Linia, and D. Peeg, A ower bound for radio broadcast, J. Comput. System Sci., 43 (1991), pp [ABLP92] N. Aon, A. Bar-Noy, N. Linia, and D. Peeg, Singe round simuation on radio networks, J. Agorithms, 13 (1992), pp [BGI91] R. Bar-Yehuda, O. Godreich, and A. Itai, Efficient emuation of singe-hop radio network with coision detection on muti-hop radio network with no coision detection, Distrib. Comput., 5 (1991), pp [BGI92] R. Bar-Yehuda, O. Godreich, and A. Itai, On the time-compexity of broadcast in muti-hop radio networks: An exponentia gap between determinism and randomization, J. Comput. System Sci., 45 (1992), pp [BII93] R. Bar-Yehuda, A. Israei, and A. Itai, Mutipe communication in muti-hop radio networks, SIAM J. Comput., 22 (1993), pp [CK85a] I. Chamtac and S. Kutten, On broadcasting in radio networks probem anaysis and protoco design, IEEE Trans. Comm., COM-33 (1985), pp [CK85b] I. Chamtac and S. Kutten, A spatia reuse TDMA/FDMA for mobie muti-hop radio networks, in INFOCOM, 1985, pp [CK87] I. Chamtac and S. Kutten, Tree-based broadcasting in mutihop radio networks, IEEECOM, C-36 (1987), pp [CW87] I. Chamtac and O. Weinstein, The wave expansion approach to broadcasting in mutihop radio networks, in INFOCOM, pp , [Ga85] R. Gaager, A perspective on mutiaccess channes, IEEE Trans. Inform. Theory, 31 (1985), pp [GM95] I. Gaber and Y. Mansour, Broadcast in radio networks, in Proc. 6th ACM-SIAM Symposium on Discrete Agorithms, SIAM, 1995, pp [KM93] E. Kushievitz and Y. Mansour, An Ω(D og(n/d)) ower bound for broadcast in radio networks, in Proc. 12th ACM Symp. on Principes of Distributed Computing, 1993, pp [Tan81] A. S. Tanenbaum, Computer Networks, Prentice-Ha, Engewood Ciffs, NJ, [Wi86] D. E. Wiard, Log-ogarithmic seection resoution protocos in a mutipe access channe, SIAM J. Comput., 15 (1986), pp