JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 23, 1073-1086 (2007) Analysis on a Localize Pruning Metho for Connecte Dominating Sets JOHN SUM 1, JIE WU 2 AND KEVIN HO 3 1 Department of Information Management Chung-Shan Meical University Taichung, 402 Taiwan E-mail: pfsum@csmu.eu.tw 2 Department of Computer Science an Engineering Floria Atlantic University Boca Raton, Floria 33431, U.S.A. E-mail: jie@cse.fau.eu 3 Department of Computer Science an Communication Engineering Provience University Taichung, 433 Taiwan E-mail: ho@pu.eu.tw While restricte rule-k has been succeee in generating a connecte ominating set (CDS) of small size, not much theoretical analysis on the size has been one. In this paper, an analysis on the expecte size of a CDS generate by such algorithm an its relation to ifferent noe ensity is presente. Assume N noes are eploye uniformly an ranomly in a square of size L N L N (where N an L N ); three results are obtaine. (1) It is prove that the noe egree istribution of such a network follows a Poisson istribution. (2) The expecte size of a CDS that is erive by the restricte pruning rule-k is a ecreasing function with respect to the noe ensity n ˆ. For ˆn 30. it is foun that the expecte size is close to N / n ˆ. (3) It is prove that the lower boun on the expecte size of a CDS for a Poissonian network of noe ensity ˆn is given by 1 nˆ { exp( ( nˆ 1))} N. The secon result is of paramount importance for practinˆ 1 nˆ 1 tioners. It provies the information about the expecte size of a CDS when the noe ensity ˆn is between 6 an 30. The ata (expecte CDS size) for this range can harly be provie by simulations. Keywors: connecte ominating set (CDS), expecte size, lower boun, restricte pruning rule, wireless mobile a hoc network 1. INTRODUCTION In wireless a hoc networks, the selecting of a subset of noes (i.e., construction of a virtual backbone) for efficient message routing is always a crucial problem. In the last ecae, much research has been conucte in orer to evelop a simple an yet efficient algorithm for the construction of such a virtual backbone. Amongst them, istribute algorithms base on the iea of connecte ominating set (CDS) [4] have been propose an have succeee in generating a virtual backbone of small size [2, 10-13]. In these algorithms, a CDS is constructe by going through two processes, namely the marking process an the pruning process. In the marking process, a noe will mark itself true if it Receive September 15, 2006; accepte February 6, 2007. Communicate by Ten H. Lai, Chung-Ta King an Jehn-Ruey Jiang. 1073
1074 JOHN SUM, JIE WU AND KEVIN HO has two unconnecte neighbors. Otherwise, it will mark itself false. Once the marking process has finishe, each true noe will check if its local conition fulfills the conitions specifie by a pruning rule-k. With respect to the pruning rule-k, a marke noe unmarks itself if there exists a set of connecte noes whose coverage can cover all its neighbors an, at the same time, the ID of the marke noe is smaller than the IDs of the connecte noes. If the connecte noes are restricte to irect neighbors of the marke noe, the pruning rule is calle restricte rule-k. 1 Otherwise, it is calle unrestricte rule. Although the Wu & Li ecentralize algorithm is simple an efficient in terms of computational complexity, little theoretical work has been one concerning the size of the CDS being generate. Only Dai & Wu in [2] an Hansen et al. in [5] have provie analytical stuies on this issue. Let N be the total number of noes an N, Dai & Wu showe that the size of a pruning rule k CDS is of constant-times-larger than the minimal CDS. Hansen et al. consiere the situation that the size of the square (say L 2 N ) grows linearly with N. The expecte size of a CDS erive by the restricte pruning 2 rule-k is in an orer linear to L N an lower boune by L 2 N / π. As observe from the simulate results presente in [2], this lower boun oes not fit the cases when the noe ensity is low. In this regar, an in-epth investigation about the expecte size of a CDS generate by restricte pruning rule-k is inevitable. In particular, we woul like to investigate how the size changes with the noe ensity, an when it reaches the lower boun as erive in [5]. To o so, we nee to erive the expecte size of a CDS in terms of the noe egree istribution, an the probability that a true noe will turn out to be marke false uring the pruning process. We call the latter probability the unmarke probability. Here, noe egree is efine as the total number of neighbors a noe has. Unfortunately, we will state later in the text that this unmarke probability cannot be obtaine analytically. The ranom sampling technique is neee to etermine these values numerically. Therefore, the contributions of this paper are as follows. 1. For a network of N noes that are uniformly an ranomly generate in a square of size L N L N, the noe egree istribution follows a Poisson istribution when L N, N. 2. A proceure base on the iea of ranom sampling is propose an the unmarke probabilities against ifferent noe egrees are obtaine. 3. It is foun that the expecte size of a CDS is almost a ecreasing function with respect to the noe ensity. The size of the CDS reaches its lower boun when the noe ensity is greater than or equal to 30. 4. The lower boun on the expecte size of a CDS for a Poissonian network of noe ensity ˆn is given by { 1 nˆ exp( ( nˆ 1))} N. nˆ 1 nˆ 1 The thir result is of paramount importance for practitioners. It provies the information about the expecte size of a CDS when the noe ensity ˆn is between 6 an 30. The expecte CDS size for this range can harly be provie by simulations. The rest of the paper is organize in four sections. In the next section, the algorithm of the marking process an the restricte pruning rule will be presente. The noe egree istribution of a network of ranomly eploye noes will be analyze an presente in 1 In this paper, the terms restricte rule-k an restricte pruning rule-k are use interchangeably.
LOCAL PRUNING METHOD FOR CDS 1075 section 3. An empirical proceure to obtain the unmarke probability an the analysis on the expecte size will be eluciate in the same section. Finally, the conclusion is presente in section 4. 2. RESTRICTED PRUNING RULE-k Consier a mobile a hoc network of N noes that are ranomly an uniformly eploye within a two-imensional square of area L L. Because of the transmission power of a raio signal, two noes can communicate with each other if their istance apart is less than an allowable transmission range, say r (r << L). In other wors, two noes are neighbors if the istance between them is less than r. Once a noe has been eploye, (i) it generates a uniformly ranom ID for itself an broacasts to other noes nearby (if any) about its ID. Then, (ii) it waits an listens to the signals from nearby noes about their IDs an the IDs of their neighbors. In accorance with the receive list of IDs, the noe can check whether its ID is unique. If the ID alreay exists, the noe will generate another ranom ID an then repeat steps (i) an (ii). (iii) As long as the IDs have been receive, it upates the list of the IDs of its neighbors an broacasts this neighbor information to its neighbors. The listen-upate-broacast cycle is then repeate a few more times until there are no more upates on the neighbor list. The resultant network graph is enote by V. When a complete list of neighbor information has been obtaine, each noe carries out the following algorithm to etermine whether it is a gateway noe for message routing. Let i(x) an N(x) be the ID an the set of neighbor noes of a noe locate at x. The marker of x is enote by M(x). Wu-Li Marking Process [13]: A noe locate at x sets its marker to True, i.e. M(x) = T, if there exists two unconnecte neighbor noes. Dai-Wu Restricte Pruning Rule k [2]: A marke noe unmarks itself if its neighbor noes can be covere by a set of connecte neighbor noes whose IDs are larger than noe x. That is to say, M(x) = F, if there exists x 1, x 2,, x k N(x) such that (i) M(x) = T. (ii) i(x) < i(x i ) for all j = 1, 2,, k. (iii) N(x) N(x 1 ) N(x 2 ) N(x k ). (iv) x 1, x 2,, x k form a connecte graph. To realize step (ii) in practice, each marke noe first sorts the IDs of its neighbors in ascening orer. Then, all noes with IDs larger than i(x) will be selecte. The selecte neighbor noes are further checke with respect to their coverage (step (iii)) an connectivity (step (iv)). Finally, M(x) changes to F if the selecte noes form a connecte graph an can cover all the neighbors of x. The beauty of this pruning rule is that the algorithm is completely istribute. Only irect neighbor information is neee. No global information is require. Each noe can perform the pruning locally. Plus, the computational complexity is small. For a noe of
1076 JOHN SUM, JIE WU AND KEVIN HO egree, its complexity is in an orer of O( 2 ) 2. Consier a graph of finite-mean-noe egree (say μ) an variance (say σ 2 ). It can be shown by the Chebyshev Inequality that 1 P( μ mσ). m2 In other wors, for a finite m, pruning (1 m -2 ) noes has a complexity of just O(m 2 σ 2 ). The cost pai for conucting the marking an pruning process is low. 3. ANALYSIS Assume a network of N noes, i.e. V = N. Let P() be the noe egree istribution of V. P(M(x) = F eg(x) = ) is the probability that a noe of egree is unmarke after the pruning step. During the marking process, a noe will be marke if there are two neighbor noes of x that are not neighbors to each other. As noes are eploye uniformly an ranomly, the probability that a noe of egree will be marke in the marking process will be given by P(Noe x is marke eg(x) = ) = 1 β (-1)/2, (1) where β is the probability that the istance of any two ranom noes within a unit circle is less than or equal to the raius. By conucting a computer simulation that generates 10,000 points uniformly an ranomly in a circle of raius r, an then counts the percentage of pairs of noes whose separation is less than r, it fins that β is equal to 0.5852. Then P(Noe x is marke eg(x) = ) = 0.995 for = 5. Fig. 1 shows the probability that a noe of egree is not marke uring the marking process. Clearly, one can assume that this probability vanishes when > 6. 10 0 10 2 10 4 10 6 10 8 10 10 10 12 0 2 4 6 8 10 Fig. 1. The probability that a noe of egree is not marke uring the marking process, i.e. β (-1)/2 versus. 2 Theorem 4 in [2].
LOCAL PRUNING METHOD FOR CDS 1077 Suppose a network graph is of Poisson noe egree istribution with large mean noe egree. The percentage of noes of small noe egree will be very small. One can thus assume that all noes are marke after the initial marking process has been performe. The expecte size of the CDS can be given by E Vcs = N(1 P(M(x) = F eg(x) = )P(eg(x) = )). (2) x As P(eg(x) = ) is homogenous for all x V, the expecte size can simply be expresse as follows: E Vcs = N(1 P(M() = F )P()), (3) where the factor P(M() = F ) correspons with to the probability that a noe of egree is unmarke. 3.1 Noe Degree Distribution P() Suppose the noes are ranomly an uniformly istribute, an let ˆn be the average number of noes within a unit isk. The average noe egree λ will thus be nˆ 1. The noe egree istribution of V follows a Poisson istribution. Theorem 1 For a mobile a hoc network V, in which the mobile noes are ranomly an uniformly istribute, the noe egree istribution P() is given by λ P ( ) = exp( λ), λ = nˆ 1, (4)! where ˆn is the average noe ensity. Proof: Let N be the total number of noes of V, an the area of eployment is much larger than a unit isk. The number of noes eploye within a unit isk will then follow a binomial istribution, N! P{Exactly n noes in a unit isk} (1 ) n!( N n)! δ δ = where Area of unit isk δ =. Deployment Area For large N, ˆn = Nδ an n ( N n) nˆ P{Exactly n noes in a unit isk} = exp( nˆ ). n! Therefore, the probability of a noe having noe egree (i.e., the number of neighbor noes) is given by a Poisson istribution with average noe egree λ = nˆ 1. n
1078 JOHN SUM, JIE WU AND KEVIN HO 0.14 0.12 λ = 19 λ = 9 0.1 0.08 0.06 0.04 0.02 0 0 10 20 30 40 50 Fig. 2. The noe egree istributions of V for which the noe ensities are 20 (λ = 19) an 10 (λ = 9) respectively. The y-axis correspons to value of P(), while the x-axis correspons to the noe egree. For illustration, Fig. 2 shows two examples in which ˆn equals 20 an 10 respectively. 3.2 Unmarke Probability P(M() = F ) that Recall that a marke noe x unmarks itself if there exists x 1, x 2,, x k N(x) such (i) M(x) = T. (ii) i(x) < i(x j ) for all j = 1, 2,, k. (iii) N(x) N(x 1 ) N(x 2 ) N(x k ). (iv) x 1, x 2,, x k form a connecte graph. Consier the conition (i). As we have assume that all the noes are marke, P(M(x) = T) = 1, x V. Consier the conition (ii). For a noe of egree, it might have 1 neighbor noe, 2 neighbor noes, 3 neighbor noes an so on, up to neighbor noes that have IDs larger than itself. Since all noe IDs are uniformly an ranomly generate in a constant range, say [0, 1], the probability that i(x) < i(y) for all y N(x) is given by 1 1 P(i(x) < i(y) y N(x)) = (1 uu ) =. 0 2 As a result, the probability that exactly k (k ) neighbor noes that have larger IDs is given by 1 P(Exactly k out of neighbors having larger IDs) =, k 2 for all k = 0, 1, 2,,. (5)
LOCAL PRUNING METHOD FOR CDS 1079 The final question left behin is this: If there are k neighbor noes with larger IDs, will these noes form a connecte graph, an simultaneously will the rest of the other k noes be neighbors of these noes? Unfortunately, it is not an easy question to answer. It all epens on the locations of these neighbor noes. Take a look at the illustrative examples shown in Fig. 3. In both cases, k = 6. Even though both sets of neighbor noes can cover the whole circle, one is connecte (Fig. 3 (a) an the other is isconnecte (Fig. 3 (b)). r r r r x x x x (a) (b) (c) () Fig. 3. (a) an (b) show two illustrative examples in which the neighbor noes of x can cover the whole circle. (a) Six neighbor noes are locate evenly on the circumference of the circle. They form a connecte graph. (b) Three neighbor noes are locate evenly on the circumference an three other noes are locate at the lower half of the circle. The graph being inuce from these neighbor noes is a isconnecte graph. (c) an () show two illustrative examples in which the neighbor noes of x cannot cover the whole circle. Again, one forms a connecte graph (c) an the other oes not (). Let Ω(x) be the unit circle centere at location x. Let X = (x 1, x 2,, x k ) Ω(x) k be an augmente ranom vector, in which x 1, x 2,, x k Ω(x). The graph inuce by X is enote by G X. Furthermore, we let I(X) be an inicator function efine as follows: 1 if GX is connecte, I( X) = (6) 0 if G is not connecte. X The coverage of X is enote by Cov(X), where k Area covere by j= 1 ( Ω( xj) Ω( x)) Cov( X ) =. Area covere by Ω( x) (7) Therefore, the probability that ( k) ranom noes in Ω(x) can be covere by the other k ranom noes in Ω(x) will be given by Pk (, ) I ( XCovX k = ) ( ) X. (8) k X Ω( x) The probability that a noe of egree will be unmarke will thus be given by 1 P(M() = F ) = Pk (, ) k = 1 k (9) 2
1080 JOHN SUM, JIE WU AND KEVIN HO an the expecte size of CDS is given by exp( λ)( λ/ 2) E Vcs = N 1 P( k, ). k= 1 k!( k)! (10) Unfortunately, there is no simple close-form solution for the probability P(k, ), Eq. (8). We obtaine the values empirically by a ranom sampling proceure, as epicte in Fig. 4. The iea of the proceure can be sketche as follows. In the first step, we generate Z ranom noes, x 1, x 2,, x Z, within a unit isk centere at the origin (step 1). In the secon step, for each value of noe egree, say k, we generate another k ranom noes, y 1, y 2,, y k, within the same unit isk (step 2.1.1). Then, we count the fractional number of x i s being covere by the y 1, y 2,, y k an store the value in the array OL (steps 2.1.2 an 2.1.3). Next, the connectivity of the graph inuce by y 1, y 2,, y k is checke (step 2.1.4). Finally the fractional number counte in the step 2.1.3 will be store in an array PC if the graph is connecte. Otherwise, the store value will be set to zero. The secon step is repeate M times. In our simulation, Z is set to 10,000. The value of k ranges from 1 to 25, an M is set to 10,000. So, before the simulation, we initialize three 2D arrays (OL, CN an PC) of imension 25 10,000. Their kjth elements, where k = 1, 2,, 25 an j = 1, 2,, 10,000, correspon to the intermeiate results obtaine in the jth simulation for noe egree k. Step 0: Initialize OL, CN, PC R 25 10,000. Step 1: Let Ω 0 be the unit isk centere at (0, 0) an then uniformly ranomly generate x 1, x 2,, x 10,000 insie Ω 0. Step 2: For k = 1, 2,, 25, 2.1 For j = 1, 2,, 10,000 2.1.1 Uniformly ranomly generate y 1, y 2,, y k insie Ω 0, k Ω κ = 1 2.1.2 Set NI equals the number of x i s that are locate insie ( y ) Ω κ 0. 2.1.3 Set OL kj equals NI/10,000. 2.1.4 Set CN kj equals 1 if y 1,, y k form a connecte graph. 2.1.5 Set PC kj equals OL kj CN kj. Fig. 4. Ranom sampling proceure for obtaining the probability P(k, ), Eq. (8). Since the kjth element in the array PC is the value I(X)Cov(X) of the jth set of ranom k noes, the value P(k, ) can then be obtaine empirically by M 1 k Pk (, ) = PCkj (11) M j= 1 for all k. The unmarke probability of a noe of egree can be obtaine. Fig. 5 shows the unmarke probability P(M() = F ) against noe egree. It is clear that the minimum unmarke probability is attaine when equals 5, where the minimum unmarke probability is equal to 0.3722. (This is ue to the fact that there is a small chance for a 5-noe inuce graph to form a connecte inuce graph.)
LOCAL PRUNING METHOD FOR CDS 1081 1 0.9 Unmarke Probability 0.8 0.7 0.6 0.5 0.4 0.3 0 5 10 15 20 25 Noe Degree () Fig. 5. Unmarke probability. The unmarke probability reaches 0.9661 when = 25. It can be further note from the figure that the value of P(M() = F ) increases as increases, an then approaches 1 when is large. 3.3 Expecte Size of CDS In accorance with the formulae erive earlier for the average number of marke noes (Eq. (3)) an the theorem about the nature of noe egree istribution (Theorem 1), the expecte size of a CDS-erive restricte pruning rule can be expresse as follows: E Vcs λ = 1 exp( λ) P( ( ) = F ), N M! (12) where λ correspons to the average noe egree. Then, the expecte size of a CDS erive is evaluate by putting the values of P(M() = F ) as shown in Fig. 5 an ifferent values of λ into the Eq. (12). Fig. 6 shows the expecte size of CDS against λ. (For convenience, we simply let P(M() = F ) = 0.9661 for > 25.) The soli line with squares correspons to the lower boun (λ + 1) -1, while the soli line with circles correspons to our results. The soli line with squares correspons to the lower boun (λ + 1) -1. (Please refer to Appenix A for the erivation of this lower boun.) It is observe that the size is about 0.55% of the original network size when λ = 6. The factor matches the result obtaine in [2] for the same λ an N = 200. (Please refer to Appenix B for the reason why the comparison is only mae for λ = 6, not for other values of λ in their paper.) In accorance with Fig. 6, one can also see that the size of a CDS rops as the λ increases. Eventually, it rops to its lower boun when λ is close to 30. 3.4 A Tighter Lower Boun A tighter lower boun for Poissonian P() can inee be erive from the Eq. (12). Consier a marke noe of egree. One conition that a marke noe will be staying marke, after the pruning process, is when its ID is larger than all its neighbors. This probability is given by ( + 1) -1 for a marke noe with neighbors. Hence,
1082 JOHN SUM, JIE WU AND KEVIN HO 0.7 0.6 0.5 Expecte Size 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 Mean Noe Degree (λ) Fig. 6. Expecte size of a CDS erive by restricte pruning rule. The soli line with squares correspons to the lower boun (λ + 1) -1, while the soli line with circles correspons to our results. 0.2 0.18 Hansen et al Our lower boun 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 10 20 30 40 50 λ Fig. 7. Comaprison between Hansen et al. an our lower bouns. The y-axis correspons to the value E[ V cs ]/N, while the x-axis correspons to noe ensity of V. E Vcs λ exp( λ). N 1 ( + 1)! Since λ exp( λ) 1 λ =, ( + 1)! λ 1 λ + 1 E Vcs exp( λ ) N. λ λ 1 In terms of noe ensity n ˆ, 1 nˆ E Vcs exp( nˆ + 1) N. nˆ 1 nˆ 1 (13) It is equal to N /( n ˆ 1) when ˆn is large an this boun is tighter than N / n ˆ for all
LOCAL PRUNING METHOD FOR CDS 1083 ˆn 4 (i.e. λ 3). Fig. 7 compares the ifference between the Hansen et al. lower boun an our lower boun. It is clear that there is no significant ifference when λ is large. 4. CONCLUSION In this paper, we have provie an analysis on the size of a CDS erive by the restricte pruning rule-k algorithm. For a network of N noes that are uniformly an ranomly generate in a square of size L N L N, we have shown that the noe egree istribution follows a Poisson istribution when L N, N. To argue that the noe egree istribution of the network oes not change much after the marking process has been performe; we have iscusse the probability of a noe being marke in the marking process an shown that such a probability tens to vanish when the noe ensity is high. After that, we have erive an equation to evaluate the expecte size of a CDS, in terms of the network noe egree istribution an the unmarke probabilities. As there is no close-form solution for the connecte probability an the coverage of a graph inuce by ranom noes within a circle, a computer simulate proceure base on the iea of ranom sampling has been evelope to obtain such probabilities. The probability that a noe of egree will be unmarke is obtaine an hence the expecte size of a CDS can be obtaine. Finally, the expecte size of a CDS erive by the restricte pruning rule-k is analyze with respect to ifferent noe ensities. It is foun that the size is almost a ecreasing function with respect to the noe ensity. The size reaches its lower boun when the noe ensity is equal to or greater than 30. That is to say, the CDS erive by the restricte pruning rule-k algorithm in a high noe ensity situation is a minimal CDS. The results are consistent with the existing results previously obtaine in [2] an [5]. More important, our results have fille in the gap, 6 λ 30, that has not been investigate before. By applying a similar technique, analysis on other istribute methos, such as the extene works presente in [10, 14, 15], for constructing CDS might also be possible. REFERENCES 1. B. N. Clark, C. J. Colbourn, an D. S. Johnson, Unit isk graphs, Discrete Mathematics, Vol. 86, 1990, pp. 165-177. 2. F. Dai an J. Wu, An extene localize algorithm for connecte ominating set formation in a hoc wireless networks, IEEE Transactions on Parallel an Distribute Systems, Vol. 15, 2004, pp. 1-13. 3. B. Das, R. Sivakumar, an V. Bhargavan, Routing in a hoc networks using a spine, in Proceeings of the 6th IEEE International Conference on Computers Communication an Networks, 1997, pp. 34-39. 4. S. Guha an S. Khuller, Approximation algorithms for connecte ominating sets, Algorithmica, Vol. 20, 1998, pp. 374-387. 5. F. Hansen, E. Schmutz, an L. Sheng, The expecte size of the rule k ominating set, Algorithmica, Vol. 46, 2006, pp. 409-418. 6. F. Ingelrest, D. Simplot-Ryl, an I. Stojmenovic, A ominating sets an target raius base localize activity scheuling an minimum energy broacast protocol for
1084 JOHN SUM, JIE WU AND KEVIN HO a hoc an sensor networks, in Proceeings of the Meiterranean A Hoc Networking Workshop, 2004, pp. 351-359. 7. C. R. Lin an M. Gerla, Aaptive clustering for mobile wireless networks, IEEE Journal on Selecte Areas in Communication, Vol. 15, 1996, pp. 1265-1275. 8. H. Liu, Y. Pan, an J. Cao, An improve istribute algorithm for connecte ominating sets in wireless a hoc networks, in Proceeings of the International Symposium on Parallel an Distribute Processing an Applications, 2004, pp. 340-351. 9. P. J. Wan, K. Alzoubi, an O. Frieer, Distribute construction of connecte ominating set in wireless a hoc networks, in Proceeings of IEEE INFOCOM, Vol. 3, 2002, pp. 1597-1604. 10. J. Wu, Extene ominating-set-base routing in a hoc wireless networks with uniirectional links, IEEE Transactions on Parallel & Distribute Systems, Vol. 13, 2002, pp. 866-881. 11. J. Wu an F. Dai, Mobility control an its applications in mobile a hoc networks, IEEE Network, Vol. 18, 2004, pp. 30-35. 12. J. Wu an F. Dai, Efficient broacasting with guarantee coverage in mobile a hoc networks, IEEE Transactions on Mobile Computing, Vol. 4, 2005, pp. 259-270. 13. J. Wu an H. Li, On calculating connecte ominating set for efficient routing in ahoc wireless networks, in Proceeings of the 3r International Workshop on Discrete Algorithms an Methos for Mobile Computing an Communication, 1999, pp. 7-14. 14. J. Wu, W. Lou, an F. Dai, Extene multipoint relays to etermine connecte ominating sets in MANETs, IEEE Transactions on Computers, Vol. 55, 2006, pp. 334-347. 15. J. Wu, B. Wu, an I. Stojmenovic, Power-aware broacasting an activity scheuling in a hoc wireless networks using connecte ominating sets, Wireless Communications an Mobile Computing, Vol. 4, 2003, pp. 425-438. APPENDIX A. Hansen et al. LOWER BOUND [5] Instea of running extensive computer simulations, Hansen et al. have presente a theoretical analysis on the size of a CDS erive by the restricte pruning rule in [5]. In one of their theorems (Theorem 5 in [5]), they show that the size of a CDS is lowerboune by l 2 N / π for ln. Here l N is the length of the square where the mobile noes are eploye. For an a hoc network consisting of N noes, l 2 N / π is equal to the total number of noes over the noe ensity. As noe ensity is equal to the average noe egree plus 1 (i.e. ˆn = λ + 1), the lower boun of the expecte size of a CDS erive by the restricte pruning rule epens on the average noe egree of the Poissonian noe egree istribution: E V cs N. λ + 1 (14)
LOCAL PRUNING METHOD FOR CDS 1085 B. Dai &Wu RESULT [2] In our analysis, we assume the network is of Poisson noe egree istribution. For a network of N noes eploye in a square of size L L, an each noe has transmission range r, the Poisson noe egree istribution happens when r << L an r = λ + 1 L π N. This conition is equivalent to λ << N for when L is finite. Therefore, the noe egree istribution is close to a Poisson istribution only when λ is small. The simulate results in [2] for the expecte size of a CDS at λ = 6 is consistent the results obtaine in this paper. On the contrary, the noe egree istribution of a large λ network coul harly follow a Poisson istribution. A comparison between their results an the results presente here coul not be mae. John Sum ( ) receive the B.Eng. in Electronic Engineering from Hong Kong Polytechnic University in 1992. Then, he receive his M.Phil. an Ph.D. in Computer Science an Engineering from Chinese University of Hong Kong in 1995 an 1998 respectively. Before joining the Chung Shan Meical University (Taiwan), John was teaching in several universities in Hong Kong, incluing the Hong Kong Baptist University, the Open University of Hong Kong an the Hong Kong Polytechnic University. Currently, he is an assistant professor of the Department of Information Management in the Chung Shan Meical University, Taichung, R.O.C. His current research interests inclue neural computation, mobile sensor networks an scale-free network. John Sum is a senior member of IEEE an an associate eitor of the International Journal of Computers an Applications. Jie Wu ( ) is a Distinguishe Professor at the Department of Computer Science an Engineering, Floria Atlantic University. He has publishe over 350 papers in various journals an conference proceeings. His research interests are in the areas of wireless networks an mobile computing, routing protocols, fault-tolerant computing, an interconnection networks. Dr. Wu was on the eitorial boar of IEEE Transactions on Parallel an Distribute Systems an was a co-guest-eitor of IEEE Computer an Journal of Parallel an Distribute Computing. He serve as the program co-chair for MASS 2004, program vice-chair for ICDCS 2001, an program vice-chair for ICPP 2000. He was also general co-chair for MASS 2006 an is general vice-chair for IPDPS 2007. He is the author of the text Distribute System Design publishe by the CRC press. He was also the recipient of the 1996-97 an 2001-2002 Researcher of the Year Awar at Floria Atlantic University. Dr. Wu has serve as an IEEE Computer Society Distinguishe Visitor an is the Chairman of IEEE Technical Committee on Distribute Processing (TCDP). He is a Member of ACM an a Senior Member of IEEE.
1086 JOHN SUM, JIE WU AND KEVIN HO Kevin I-J Ho ( ) receive the B.S. in Computer Engineering from National Chiao Tung University (Taiwan) in 1983. From 1985 to 1987, he was an assistant engineer of Institute of Information Inustry, Taiwan. Then, he receive his M.S. an Ph.D. in Computer Science from University of Texas at Dallas in 1990 an 1992, respectively. Currently, he is an associate professor of Department of Computer Science an Communication Engineering, Provience University, Taiwan. His current research interests inclue image processing, algorithm esign an analysis, scheuling theory, an computer networks.