With this preamble out of the way, we are now in a position to spell out the details of our leader election protocol. Protocol Nonuniform-election initially all the stations are active; Phase 1: for i ǟ 0 to ϱ Sieve(i); exit for-loop if the status of the channel is NULL; Phase 2: t ǟ i – 1; for i ǟ t downto 0 Sieve(i); Phase 3: repeat Sieve(0); forever We now turn to the task of evaluating the number of time slots it takes the protocol to terminate. In Phase 1, once the status of the channel is NULL the protocol exits the for loop. Thus, there must exist an integer t such that the status of the channel is: ț SINGLE or COLLISION in Sieve(0), Sieve(1), Sieve(2), , Sieve(t – 1) ț NULL in Sieve(t) Let f Ն 1 be an arbitrary real number. Write s = log log (4nf ) (10.13) Equation (10.13) guarantees that 2 2 s Ն 4nf. Assume that Sieve(0), Sieve(1), , Sieve(s) are performed in Phase 1 and let X be the random variable denoting the number of stations that transmitted in Sieve(s). Suppose that we have at most n active stations, and Sieve(s) is performed. Let X denote the number of stations that transmits in Sieve(s). Clearly, the expected value E[X] of X is E[X] ՅՅ = (10.14) Using the Markov inequality (10.4) and (10.14), we can write Pr[X Ն 1] Յ Pr[X Ն 4fE[X]] Յ (10.15) Equation (15) guarantees that with probability at least 1 – 1/4f , the status of the channel in Sieve(s) is NULL. In particular, this means that 1 ᎏ 4f 1 ᎏ 4f n ᎏ 4nf n ᎏ 2 2 s 10.5 NONUNIFORM LEADER ELECTION PROTOCOL 235 t Յ s holds with probability at least 1 – (10.16) and, therefore, Phase 1 terminates in t + 1 Յ s + 1 = log log (4nf ) + 1 = log log n + O(log log f ) time slots. In turn, this implies that Phase 2 also terminates in log log n + O(log log f ) time slots. Thus, we have the following result. Lemma 5.1 With probability exceeding 1 – 1/4f , Phase 1 and Phase 2 combined take at most 2 log log n + O(log log f ) time slots. Recall that Phase 2 involves t calls, namely Sieve(t – 1), Sieve(t – 2), , Sieve(0). For convenience of the analysis, we regard the last call, Sieve(t), of Phase 1 as the first call of Phase 2. For every i (0 Յ i Յ t 2 ) let N i denote the number of active sta- tions just before the call Sieve(i) is executed in Phase 2. We say that Sieve(i) is in fail- ure if N i > 2 2 i ln(4f (s + 1)) and the status of the channel is NULL in Sieve(i) and, otherwise, successful. Let us evaluate the probability of the event F i that Sieve(i) is failure. From [1 – (1/n)] n Յ (1/e) we have Pr[F i ] = ΂ 1 – ΃ N i < e –N i /2 2 i < e –ln[4f (s 2 +1)] = In other words, Sieve(i) is successful with probability exceeding 1 – [1/4f (s + 1)]. Let F be the event that all t calls to Sieve in Phase 2 are successful. Clearly, F = FF ෆ 0 ෆ ʝ F ෆ 1 ෆ ʝ ··· ʝ F ෆ t ෆ = F ෆ 0 ෆ ෆ ʜ ෆ ෆ F ෆ 1 ෆ ෆ ʜ ෆ ෆ · ෆෆ · ෆෆ · ෆෆ ʜ ෆ ෆ F ෆ t ෆ and, therefore, we can write Pr[F] = Pr[F ෆ 0 ෆ ෆ ʜ ෆ ෆ F ෆ 1 ෆ ෆ ʜ ෆ ෆ · ෆෆ · ෆෆ · ෆෆ ʜ ෆ ෆ F ෆ t ෆ ] > 1 – Α t i=0 Ն 1 – (10.17) Thus, the probability that all the t 2 calls in Phase 2 are successful exceeds 1/4f, provided that t Յ s. Recall, that by (10.16), t Յ s holds with probability at least 1 – 1/4f . Thus, we conclude that with probability exceeding 1 – 1/2f all the calls to Sieve in Phase 2 are successful. Assume that all the calls to Sieve in Phase 2 are successful and let t Ј (0 Յ tЈՅt) be the smallest integer for which the status of the channel is NULL in Sieve(tЈ). We note that since, by the definition of t, the status of the channel in NULL in Sieve(t), such an 1 ᎏ 4f 1 ᎏ 4f(s + 1) 1 ᎏ 4f(s 2 + 1) 1 ᎏ 2 2 i 1 ᎏ 4f 236 LEADER ELECTION PROTOCOLS FOR RADIO NETWORKS integer tЈ always exists. Our choice of tЈ guarantees that the status of the channel must be COLLISION in each of the calls Sieve( j), with 0 Յ j Յ tЈ – 1. Now, since we assumed that all the calls to Sieve in Phase 2 are successful, it must be the case that N t Ј Յ 2 2 t Ј ln[4f (s + 1)] (10.18) Let Y be the random variable denoting the number of stations that are transmitting in Sieve(0) of Phase 2. To get a handle on Y, observe that for a given station to transmit in Sieve(0) it must have transmitted in each call Sieve( j) with 0 Յ j Յ tЈ – 1. Put differ- ently, for a given station the probability that it is transmitting in Sieve(0) is at most = = Therefore, we have E[Y] ՅՅ = 2 ln[4f (s + 1)] (10.19) Select the value ␦ > 0 such that (1 + ␦ )E[Y] = 7 ln[4f (s + 1)] (10.20) Notice that by (19) and (20) combined, we have 1 + ␦ = Ն = In addition, by using the Chernoff bound (1) we bound the tail of Y, that is, Pr[Y > 7 ln[4f(s + 1)]] = Pr[Y > (1 + ␦ )E[Y]] as follows: Pr[Y > (1 + ␦ ) E[Y]] < ΂΃ (1+ ␦ )E[Y] = ΂΃ 7ln[4f (s+1)] < e –ln[4f (s+1)] < We just proved that, as long as all the calls to Sieve are successful, with probability ex- ceeding 1 – 1/4f , at the end of Phase 2 no more than 7 ln[4f (s + 1)] stations remain active. Recalling that all the calls to Sieve are successful with probability at least 1 – 1/2f , we have the following result. Lemma 5.2 With probability exceeding 1 – 3/4f, the number of remaining active sta- tions at the end of Phase 2 does not exceed 7 ln[4f (s + 1)]. 1 ᎏ 4f 2e ᎏ 7 e ᎏ 1 + ␦ 7 ᎏ 2 7 ln[4f (s + 1)] ᎏᎏ 2 ln[4f (s + 1)] 7 ln[4f (s + 1)] ᎏᎏ E[Y] 2·2 2 t Ј ln[4f (s 2 + 1)] ᎏᎏᎏ 2 2 t Ј 2N t Ј ᎏ 2 2 t Ј 2 ᎏ 2 2 t Ј 1 ᎏ 2 2 t Ј –1 1 ᎏᎏ 2 2 t Ј –1 2 2 t Ј –2 ··· 2 2 0 10.5 NONUNIFORM LEADER ELECTION PROTOCOL 237 Let N be the number of remaining active stations at the beginning of Phase 3 and as- sume that N Յ 7 ln[4f (s + 1)]. Recall that Phase 3 repeats Sieve(0) until, eventually, the status of channel becomes SINGLE. For a particular call Sieve(0) in Phase 3, we let NЈ, (NЈՆ2), be the number of active stations just before the call. We say that Sieve(0) is successful if ț Either the status of the channels is SINGLE in Sieve(0), or ț At most NЈ/2 stations remain active after the call. The reader should have no difficulty confirm that the following inequality holds for all NЈ Ն 2 ΂΃ + ΂΃ + ··· + ΂΃ Ն 2 NЈ It follows that a call is successful with probability at least 1 – 2 . Since N stations are active at the beginning of Phase 3, log N successful calls suffice to elect a leader. Let Z be the random variable denoting the number of successes in a number ␣ of inde- pendent Bernoulli trials, each succeeding with probability 1 – 2 . Clearly, E[Z] = ␣ /2. Our goal is to determine the values of ⑀ and ␣ in such a way that equation (10.3) yields Pr[Z < log N] = Pr[Z < (1 – ⑀ )E[Z]] < e –( ⑀ 2 /2)E[Z] = (10.21) It is easy to verify that (21) holds whenever Ά (10.22) hold true. Write A = Solving for ⑀ and E[Z] in (22) we obtain: 0 < ⑀ = < 1 and E[Z] = ln(4f )[2A + 1 + ͙ 4 ෆ A ෆ + ෆ 1 ෆ ] < ln(4f )(6A + 2) = 3log N + 2 ln(4f ). 2 ᎏᎏ 1 + ͙ 4 ෆ A ෆ + ෆ 1 ෆ log N ᎏ 2 ln(4f ) (1 – ⑀ )E[Z] = log N ᎏ ⑀ 2 2 ᎏ E[Z] = ln(4f ) 1 ᎏ 4f 1 ᎏ 2 NЈ  ᎏ N 2 Ј ᎏ  NЈ 2 NЈ 1 238 LEADER ELECTION PROTOCOLS FOR RADIO NETWORKS If we assume, as we did before, that N Յ 7 ln[4f(s + 1)], it follows that log N Յ 3 + log ln(4f(s + 1)) = O(log log log n + log log f ) Thus, we can write ␣ = 2E[Z] = 4 ln f + O(log log log log n + log log f ) Therefore, if N Յ 7 ln[4f(s + 1)] then Phase 3 takes 4 ln f + O[log log log log n + log log f ] time slots with probability at least 1 – 1/4f . Noting that N Յ 7 ln[4f (s + 1)] holds with probability at least 1 – 3/4f , we have obtained the following result. Lemma 5.3 With probability at least 1 – 1/f , Phase 3 terminates in at most 4 ln f + O(log log log log n + log f ) time slots. Now Lemmas 5.1 and 5.3 combined imply that with probability exceeding 1 – 3/4f – 1/4f = 1 – 1/f the protocol Nonuniform-election terminates in 2 log log n + O(log log f ) + 4 ln f + O(log log log log n + log log f ) = 2 log log n + 4 ln f + o(log log n + log f ) < 2 log log n + 2.78 log f + o(log log n + log f ) time slots. Thus, we have Lemma 5.4 Protocol Leader-election terminates, with probability exceeding 1 – 1/f , in 2 log log n + 2.78 log f + o(log log n + log f ) time slots for every f Ն 1. 10.5.2 Nonuniform Leader Election in log log n Time Slots In this subsection, we modify Nonuniform-election to run in log log n + O(log f ) + o(log log n) time slots with probability at least 1 – 1/f . The idea is to modify the protocol such that Phase 1 runs in o(log log n) time slots as follows. In Phase 1 the calls Sieve(0 2 ), Sieve(1 2 ), Sieve(2 2 ), , Sieve(t 2 ) are performed until, for the first time, the status of the channel is NULL in Sieve(t 2 ). At this point Phase 2 begins. In Phase 2 we perform the calls Sieve(t 2 – 1), Sieve(t 2 – 2), , Sieve(0). In Phase 3 repeats Sieve(0) in the same way. Similarly to subsection 10.4.2 we can evaluate the running time slot of the modified Nonuniform-election as follows. Let f Ն 1 be any real number and write s =  ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ ( ෆ 4 ෆ n ෆ f) ෆ . (10.23) The reader should have no difficulty to confirm that t Յ s holds with probability at least 1 – (10.24) 1 ᎏ 4f 10.5 NONUNIFORM LEADER ELECTION PROTOCOL 239 Therefore, Phase 1 terminates in t + 1 Յ s + 1 =  ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ ( ෆ 4 ෆ n ෆ f) ෆ  + 1 = O( ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ n ෆ + ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ f ෆ ) time slots. In turn, this implies that Phase 2 terminates in at most t 2 Յ s 2 < ( ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ ( ෆ 4 ෆ n ෆ f) ෆ + 1) 2 Յ log log n + log log f + O( ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ n ෆ + ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ f ෆ ) time slots. Thus, we have the following result. Lemma 5.5 With probability exceeding 1 – 1/4f , Phase 1 and Phase 2 combined take at most log log n + log log f + O( ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ n ෆ + ͙ lo ෆ g ෆ l ෆ o ෆ g ෆ f ෆ ) time slots. Also, it is easy to prove the following lemma in the same way. Lemma 5.6 With probability exceeding 1 – 3/4f , the number of remaining active sta- tions at the end of Phase 2 does not exceed 7 ln[4f (s 2 + 1)]. Since Phase 3 is the same as Nonuniform-election, we have the following theorem. Theorem 5.7 There exists a nonuniform leader election protocol terminating in log log n + 2.78 log log f + o(log log n + log f ) time slots with probability at least 1 – 1/f for any f Ն 1. 10.6 CONCLUDING REMARKS AND OPEN PROBLEMS A radio network is a distributed system with no central arbiter, consisting of n radio trans- ceivers, referred to as stations. The main goal of this chapter was to survey a number of re- cent leader election protocols for single-channel, single-hop radio networks. Throughout the chapter we assumed that the stations are identical and cannot be distin- guished by serial or manufacturing number. In this set-up, the leader election problem asks to designate one of the stations as leader. In each time slot, the stations transmit on the channel with some probability until, eventually, one of the stations is declared leader. The history of a station up to time slot t is captured by the status of the channel and the transmission activity of the station in each of the t time slots. From the perspective of how much of the history information is used, we identified three types of leader election protocols for single-channel, single-hop radio networks: oblivious if no history information is used, uniform if only the history of the status of the channel is used, and nonuniform if the stations use both the status of channel and the transmission activity. We noted that by extending the leader election protocols for single-hop radio networks discussed in this chapter, one can obtain clustering protocols for multihop radio networks, in which every cluster consists of one local leader and a number of stations that are one 240 LEADER ELECTION PROTOCOLS FOR RADIO NETWORKS hop away from the leader. Thus, every cluster is a two-hop subnetwork [18]. We note that a number of issues are still open. For example, it is highly desirable to elect as a leader of a cluster a station that is “optimal” in some sense. One optimality criterion would be a central position within the cluster. Yet another nontrivial and very important such criterion is to elect as local leader a station that has the largest remaining power level. ACKNOWLEDGMENTS Work was supported, in part, by the NSF grant CCR-9522093, by ONR grant N00014-97- 1-0526, and by Grant-in-Aid for Encouragement of Young Scientists (12780213) from the Ministry of Education, Science, Sports, and Culture of Japan. REFERENCES 1. H. Abu-Amara, Fault-tolerant distributed algorithms for election in complete networks, IEEE Transactions on Computers, C-37, 449–453, 1988. 2. Y. Afek and E. Gafni, Time and message bounds for election in synchronous and asynchronous complete networks, SIAM Journal on Computing, 20, 376–394, 1991. 3. R. Bar-Yehuda, O. Goldreich, and A. Itai, Efficient emulation of single-hop radio network with collision detection on multi-hop radio network with no collision detection, Distributed Comput- ing, 5, 67–71, 1991. 4. J. Bentley and A. Yao, An almost optimal algorithm for unbounded search, Information Process- ing Letters, 5, 82–87, 1976. 5. D. Bertzekas and R. 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Boggs, Ethernet: distributed packet switching for local computer net- works, Communications of the ACM, 19, 395–404, 1976. 13. R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge: Cambridge University Press, 1995. 14. K. Nakano and S. Olariu, Randomized O(log log n)-round leader election protocols in radio net- works, Proceedings of International Symposium on Algorithms and Computation (LNCS 1533), 209–218, 1998. 15. K. Nakano and S. Olariu, Randomized leader election protocols for ad-hoc networks, Proceed- ings of Sirocco 7, June 2000, 253–267. REFERENCES 241 16. K. Nakano and S. Olariu, Randomized leader election protocols in radio networks with no colli- sion detection, Proceedings of International Symposium on Algorithms and Computation, 362–373, 2000. 17. K. Nakano and S. Olariu, Uniform leader election protocols for radio networks, unpublished manuscript. 18. M. Joa-Ng and I T. 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Willard, Log-logarithmic selection resolution protocols in a multiple access channel, SIAM Journal on Computing, 15, 468–477, 1986. 242 LEADER ELECTION PROTOCOLS FOR RADIO NETWORKS CHAPTER 11 Data Broadcast JIANLIANG XU and DIK-LUN LEE Department of Computer Science, Hong Kong University of Science and Technology QINGLONG HU IBM Silicon Valley Laboratory, San Jose, California WANG-CHIEN LEE Verizon Laboratories, Waltham, Massachusetts 11.1 INTRODUCTION We have been witnessing in the past few years the rapid growth of wireless data applica- tions in the commercial market thanks to the advent of wireless devices, wireless high- speed networks, and supporting software technologies. We envisage that in the near future, a large number of mobile users carrying portable devices (e.g., palmtops, laptops, PDAs, WAP phones, etc.) will be able to access a variety of information from anywhere and at any time. The types of information that may become accessible wirelessly are boundless and include news, stock quotes, airline schedules, and weather and traffic information, to name but a few. There are two fundamental information delivery methods for wireless data applica- tions: point-to-point access and broadcast. In point-to-point access, a logical channel is es- tablished between the client and the server. Queries are submitted to the server and results are returned to the client in much the same way as in a wired network. In broadcast, data are sent simultaneously to all users residing in the broadcast area. It is up to the client to select the data it wants. Later we will see that in a special kind of broadcast system, name- ly on-demand broadcast, the client can also submit queries to the server so that the data it wants are guaranteed to be broadcast. Compared with point-to-point access, broadcast is a more attractive method for several reasons: ț A single broadcast of a data item can satisfy all the outstanding requests for that item simultaneously. As such, broadcast can scale up to an arbitrary number of users. ț Mobile wireless environments are characterized by asymmetric communication, i.e., the downlink communication capacity is much greater than the uplink communica- tion capacity. Data broadcast can take advantage of the large downlink capacity when delivering data to clients. 243 Handbook of Wireless Networks and Mobile Computing, Edited by Ivan Stojmenovic´ Copyright © 2002 John Wiley & Sons, Inc. ISBNs: 0-471-41902-8 (Paper); 0-471-22456-1 (Electronic) ț A wireless communication system essentially employs a broadcast component to deliver information. Thus, data broadcast can be implemented without introducing any additional cost. Although point-to-point and broadcast systems share many concerns, such as the need to improve response time while conserving power and bandwidth consumption, this chapter focuses on broadcast systems only. Access efficiency and power conservation are two critical issues in any wireless data system. Access efficiency concerns how fast a request is satisfied, and power conservation concerns how to reduce a mobile client’s power consumption when it is accessing the data it wants. The second issue is important because of the limited battery power on mobile clients, which ranges from only a few hours to about half a day under continuous use. Moreover, only a modest improvement in battery capacity of 20–30% can be expected over the next few years [30]. In the literature, two basic performance metrics, namely ac- cess time and tune-in time, are used to measure access efficiency and power conservation for a broadcast system, respectively: ț Access time is the time elapsed between the moment when a query is issued and the moment when it is satisfied. ț Tune-in time is the time a mobile client stays active to receive the requested data items. Obviously, broadcasting irrelevant data items increases client access time and, hence, deteriorates the efficiency of a broadcast system. A broadcast schedule, which determines what is to be broadcast by the server and when, should be carefully designed. There are three kinds of broadcast models, namely push-based broadcast, on-demand (or pull-based) broadcast, and hybrid broadcast. In push-based broadcast [1, 12], the server disseminates information using a periodic/aperiodic broadcast program (generally without any inter- vention of clients); in on-demand broadcast [5, 6], the server disseminates information based on the outstanding requests submitted by clients; in hybrid broadcast [4, 16, 21], push-based broadcast and on-demand data deliveries are combined to complement each other. Consequently, there are three kinds of data scheduling methods (i.e., push-based scheduling, on-demand scheduling, and hybrid scheduling) corresponding to these three data broadcast models. In data broadcast, to retrieve a data item, a mobile client has to continuously monitor the broadcast until the data item of interest arrives. This will consume a lot of battery pow- er since the client has to remain active during its waiting time. A solution to this problem is air indexing. The basic idea is that by including auxiliary information about the arrival times of data items on the broadcast channel, mobile clients are able to predict the arrivals of their desired data. Thus, they can stay in the power saving mode and tune into the broadcast channel only when the data items of interest to them arrive. The drawback of this solution is that broadcast cycles are lengthened due to additional indexing informa- tion. As such, there is a trade-off between access time and tune-in time. Several indexing techniques for wireless data broadcast have been introduced to conserve battery power while maintaining short access latency. Among these techniques, index tree [18] and sig- nature [22] are two representative methods for indexing broadcast channels. 244 DATA BROADCAST [...]... Hu, and D L Lee, A study of channel allocation methods for data dissemination in mobile computing environments, ACM/Baltzer Journal of Mobile Networks and Applications (MONET), 4(2): 117–129, 1999 W.-C Lee and D L Lee, Using signature techniques for information filtering in wireless and mobile environments, Journal of Distributed and Parallel Databases (DPDB), 4(3): 2 05 227, July 1996 W.-C Lee and. .. information broadcast and filtering in mobile environments, ACM/Baltzer Journal of Wireless Networks (WINET), 5( 1): 57 –67, 1999 C W Lin and D L Lee, Adaptive data delivery in wireless communication environments, in Proceedings of the 20th IEEE International Conference on Distributed Computing Systems (ICDCS’2000), pp 444– 452 , Taipei, Taiwan, April 2000 S.-C Lo and A L P Chen, Optimal index and data allocation... Helal, and R Alonso, Bit-sequences: A new cache invalidation method in mobile environments, ACM/Baltzer Journal of Mobile Networks and Applications (MONET), 2(2): 1 15 127, 1997 K C K Lee, H V Leong, and A Si, A semantic broadcast scheme for a mobile environment based on dynamic chunking, in Proceedings of the 20th IEEE International Conference on Distributed Computing Systems (ICDCS’2000), pp 52 2 52 9,... beginning of every replicated index to direct clients to a proper branch in the index tree This additional index information for nav- 255 c2 3 c1 0 b1 6 c3 9 c4 12 c5 b2 a1 15 c6 18 c7 21 c8 b3 b5 b6 b7 b8 a3 b9 24 27 30 33 36 42 45 48 51 54 57 Figure 11.3 A full index tree 39 60 63 66 69 72 75 Replicated Part Non-Replicated Part 78 c9 c10 c11 c12 c13 c14 c 15 c16 c17 c18 c19 c20 c21 c22 c23 c24 c 25 c26... probabilities and moves them to one level lower so as to minimize the average index search cost Figure 11.7 shows an index tree built using the I a1 (1, 28) (2, 2) a2 (3, 2) (4, 2) a3 (5, 04) (6, 04) a4 (7, 02) (8, 0 05) (9, 0 05) (10, 0 05) (11, 0 05) Figure 11.6 Index tree of a fixed fan-out of three 11.3 AIR INDEXING 259 I a1 a2 a3 (1, 28) (2, 2) (3, 2) (4, 2) a4 a5 (5, 04) (6, 04) (7, 02) (8, 0 05) (9, 0 05) (10,... evaluation of a wireless hierarchical data dissemination system, in Proceedings of the 5th Annual ACM/IEEE International Conference on Mobile Computing and Networking (MobiCom’99), pp 163–173, Seattle, WA, USA, August 1999 14 Q L Hu, W.-C Lee, and D L Lee, A hybrid index technique for power efficient data broadcast, Journal of Distributed and Parallel Databases (DPDB), 9(2), 151 –177, 2001 15 Q L Hu,... distribution of information items with unequal length, in Proceedings of the 31st Conference on Information Science and Systems (CISS’97), March 1997 34 C J Su, L Tassiulas, and V J Tsotras, Broadcast scheduling for information distribution, ACM/Baltzer Journal of Wireless Networks (WINET), 5( 2): 137–147, 1999 35 K L Tan and J X Yu, Energy efficient filtering of nonuniform broadcast, in Proceedings of the... 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Data broadcast can take advantage of the large downlink capacity when delivering data to clients. 243 Handbook of Wireless Networks and Mobile Computing, Edited by Ivan Stojmenovic´ Copyright. years the rapid growth of wireless data applica- tions in the commercial market thanks to the advent of wireless devices, wireless high- speed networks, and supporting software technologies. We. 1998. 15. K. Nakano and S. Olariu, Randomized leader election protocols for ad-hoc networks, Proceed- ings of Sirocco 7, June 2000, 253 –267. REFERENCES 241 16. K. Nakano and S. Olariu, Randomized