On the longest edge of Gabriel graphs in wireless ad hoc networks

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(1)IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,. VOL. 18,. NO. 1,. JANUARY 2007. 111. On the Longest Edge of Gabriel Graphs in Wireless Ad Hoc Networks Peng-Jun Wan and Chih-Wei Yi, Member, IEEE Abstract—In wireless ad hoc networks, without fixed infrastructures, virtual backbones are constructed and maintained to efficiently operate such networks. The Gabriel graph (GG) is one of widely used geometric structures for topology control in wireless ad hoc networks. If all nodes have the same maximal transmission radii, the length of the longest edge of the GG is the critical transmission radius such that the GG can be constructed by localized and distributed algorithms using only 1-hop neighbor information. In this paper, we assume a wireless ad hoc network is represented by a Poisson point process with mean n on a unit-area disk, and nodes have the same maximal transmission radii. We give three results on the length of the longest edge of the GG. First, we qffiffiffiffiffiasymptotic ffi almost surely equal to 2. Next, we show that for any , show that the ratio of the length of the longest edge to lnnn is asymptotically qffiffiffiffiffiffiffiffiffiffi. the expected number of GG edges whose lengths are at least 2. ln nþ n. is asymptotically equal to 2e . This implies that ! 1 is an. asymptotically almost sure sufficient condition for constructing the GG by 1-hop information. Last, we prove that the number of long edges q isffiffiffiffiffiffiffiffiffi asymptotically Poisson with mean 2e . Therefore, the probability of the event that the length of the longest edge is less ffi ln nþ . than 2 n is asymptotically equal to exp 2e Index Terms—Wireless ad hoc network, Gabriel graph, asymptotic probability distribution, the longest edge, poisson point process, topology control.. Ç 1. INTRODUCTION. A. wireless ad hoc network is a collection of wireless devices (transceivers) distributed over a geographic region. Each node is equipped with an omnidirectional antenna and has limited transmission power. A communication session is established either through a single-hop radio transmission if the communication parties are close enough, or through relaying by intermediate devices otherwise. Since they have no need for a fixed infrastructure, wireless ad hoc networks can be flexibly deployed at low cost for varying missions such as decision making in the battlefield, emergency disaster relief, and environmental monitoring. In wireless ad hoc networks, each node is associated with a maximal transmission radius. The network topology of a wireless ad hoc network is a graph in which two nodes have an edge between them if they are within each other’s transmission range. A spanner is a subset of the network topology in which the total cost, e.g., distance or energy consumption, between any pair of nodes is only a constant fact larger than in the original network topology. Hence, spanners are good candidates of virtual backbones. The topics about how to construct and maintain spanners are called topology control. Geometric structures, including Euclidean minimal spanning trees (EMST), relative neighbor . P.-J. Wan is with the Department of Computer Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong. E-mail: pwan@cs.cityu.edu.hk . C.-W. Yi is with the Department of Computer Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu City, Hsinchu 300, Taiwan. E-mail: yi@cs.nctu.edu.tw. Manuscript received 15 Feb. 2005; revised 1 Sept. 2005; accepted 2 Sept. 2005; published online 28 Nov. 2006. Recommended for acceptance by I. Stojmenovic. For information on obtaining reprints of this article, please send e-mail to: tpds@computer.org, and reference IEEECS Log Number TPDS-0100-0205. 1045-9219/07/$20.00 ß 2007 IEEE. graphs (RNG), Gabriel graphs (GG), Delauney triangulations (DT), and Yao’s graphs (YG), are widely used ingredients for constructing spanners [1], [2], [3]. A topology control algorithm is localized if each node only needs to collect information from few hops neighbors. In this paper, we study the critical transmission radius for Gabriel graphs. In the GG, two nodes have an edge between them if and only if there is no other node on the disk using the segment of these two nodes as its diameter. Assume all nodes have the same maximal transmission radius r. Then, the induced network topology is exactly the r-disk graph over the set of nodes V , denoted by Gr ðV Þ. To construct the GG only by 1-hop neighbor information, the transmission radius r should be large enough such that the GG is a subgraph of the r-disk graph. Thus, the transmission radius should be not less than the length of the longest edge of the GG. On the other hand, for each node, if it can gather the information of nodes that are not farther than its farthest neighbor in the GG, it can decide all GG edges incident to it. Therefore, the length of the longest edge of the GG is called the critical transmission radius for GGs. For modeling radio networks, Gilbert [4] proposed a random geometric graph model in which devices are represented by an infinite random point process over the entire plane and two devices are joined by an edge if and only if their distance is at most r. For modeling wireless ad hoc networks which consist of finite radio nodes in a bounded geographic region, a bounded (or finite) variant of the Gilbert’s model has been used by Gupta and Kumar [5] and others. In this variant, instead of an infinite random point process, the ad hoc device is typically presented by a uniform point process or Poisson point process over a disk or a square by proper scaling. The largest nearest-neighbor link problem Published by the IEEE Computer Society.

(2) 112. IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,. has been studied by Dette and Henze [6], and the longest edge of the EMST that is related to the connectivity problem has been studied by Penrose [7]. Based on their results, the probability of the eventqthat ffiffiffiffiffiffiffiffiffithe ffi length of the longest edge of the EMST is less than. ln nþ n. for some constant is equal to. expðe Þ asymptotically. Recently, Kozma et al. [8] proved that the maximal length of an edge ffiffiffiffiffiffithe qin DT of a uniform 3 ln n n-point process in a unit disk is O n . In what follows, kxk is the Euclidean norm of a point x 2 IR2 . jAj is shorthand for 2-dimensional Lebesgue measure (or area) of a measurable set A IR2 or the cardinality of a countable set A. All integrals considered will be Lebesgue integrals. The topological boundary of a set A IR2 is denoted by @A. The disk of radius r centered at x is denoted by Bðx; rÞ. The special unit-area disk centered at the origin is denoted by D D. For any set S and positive integer k, the k-fold Cartesian product of S is denoted by S k . An event is said to be asymptotically almost sure (abbreviated by a.a.s.) if it occurs with a probability that converges to one as n ! 1. An event is said to be asymptotically almost rare (abbreviated by a.a.r.) if it occurs with a probability that converges to zero as n ! 1. The symbols O, o, always refer to the limit n ! 1. To avoid trivialities, we tacitly assume n to be sufficiently large if necessary. For simplicity of notation, the dependence of sets and random variables on n will be frequently suppressed. The remainder of this paper is organized as follows: In Section 2, we give a brief review of our main results. In Section 3, we present several useful geometric results and integrals. In Section 4, we derive the asymptotic length of the longest edge. In Section 5, we drive the asymptotic expected number of long edges. In Section 6, we drive the asymptotic distribution of the length of the longest edge. We summarize this paper in Section 7.. 2. MAIN RESULTS. In this paper, we assume a wireless ad hoc network is represented by a Poisson point process over a unit-area disk with mean n, denoted by P n , and all nodes have the same maximal transmission radius rn which is a function of n. We use GðP n Þ to denote the Gabriel graph over P n . For simplicity, the edges of GGs are called Gabriel edges. If G is a geometric graph, we use ðGÞ to denote the maximal length of an edge of G and NðG; lÞ to denote the number of edges of G whose length is at least l. Our first main result is the next theorem. Theorem 1. For any constant " > 0, we have " rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi# ln n ln n ¼ 1: ðGðP n ÞÞ ð1 þ "Þ2 lim Pr ð1 "Þ2 n!1 n n According to Theorem 1, q if ffiffiffiffiffi each ffi node sets its maximal ln n transmission radius to rn ¼ n for some constant , then the rn -disk graph over P n a.a.s. contains the GG if > 2, and. VOL. 18,. NO. 1,. JANUARY 2007. qffiffiffiffiffiffiffiffiffiffi nþ Now, we assume rn ¼ 2 lnn for some constant . For a given , we call the edge whose length is not less than rn a long edge. The next theorem gives us the asymptotic expectation of the number of long Gabriel edges. Theorem 2. For the expectation of the number of long Gabriel edges, we have " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# ln n þ ¼ 2e : lim E N GðP n Þ; 2 n!1 n Since Pr½X ¼ 0 ¼ 1 Pr½X 1 1 E½X for any nonnegative integer value RV X, " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ln n þ Pr ðGðP n ÞÞ < 2 n " # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ln n þ ¼0 ¼ Pr N GðP n Þ; 2 n " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# ln n þ 1 E N GðP n Þ; 2 n 1 2e : Therefore, ". rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ln n þ ¼ 1; lim lim Pr ðGðP n ÞÞ < 2 !1 n!1 n and ! ffi 1 is an a.a.s. sufficient condition for ðGðP n ÞÞ < qffiffiffiffiffiffiffiffiffi ln nþ 2 n . In the next theorem, we give the asymptotic probability distribution of the number of long Gabriel edges, and that implies the asymptotic probability distribution of the length of the longest edge. Theorem 3. For any constant ,qthe total ffiffiffiffiffiffiffiffiffi ffi number of Gabriel edges nþ is asymptotically Poisson whose lengths are at least 2 lnn with mean 2e . Since " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ln n þ Pr ðGðP n ÞÞ < 2 n " # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ln n þ ¼0 ; ¼ Pr N GðP n Þ; 2 n according to Theorem 3, we have " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ln n þ ¼ expð2e Þ: lim Pr ðGðP n ÞÞ < 2 n!1 n. 3. PRELIMINARIES. on the contrary, the rn -disk graph a.a.r. contains the GG if. In this section, we shall give some definitions and lemmas that will be used to prove our main results.. < 2. Therefore, ¼ 2 is the threshold for constructing the. 3.1. GG by 1-hop information. For reference, we remark that ¼ 1. The results in this section are purely geometric, with no probabilistic content. Let D D denote a unit-area disk, R0 ¼ p1ffiffi. is the threshold for the rn -disk graph being connected [5], [9].. Geometry Preliminaries.

(3) WAN AND YI: ON THE LONGEST EDGE OF GABRIEL GRAPHS IN WIRELESS AD HOC NETWORKS. Fig. 1. The partition of the unit-area disk D D.. 113. Fig. 2. If x 2 D Dr ð1Þ, then ðx; rÞ ¼ 2ffaxb.. denote the radius of D D, and o denote the origin. Without loss of generality, we assume D D is centered at the origin. For. Furthermore, the following lemma proved in [10] gives a tighter lower bound for r ðxÞ.. a given transmission radius r, the unit-area disk D D is. Lemma 4. For any x 2 D Dr ð1Þ,. partitioned into D Dr ð0Þ, D Dr ð1Þ, and D Dr ð2Þ as shown in Fig. 1: D Dr ð0Þ is the disk of radius p1ffiffi r centered at the origin, qffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 centered at D Dr ð1Þ is the annulus of radii pffiffi r and qrffiffiffiffiffiffiffiffiffiffiffiffi the origin, and D Dr ð2Þ is the annulus of radii 1 r2 and p1ffiffi . centered at the origin. Then, pffiffiffi 2 1 Dr ð1Þj ¼ 2r pffiffiffi r ; jD Dr ð0Þj ¼ ð1 rÞ ; jD and jD Dr ð2Þj ¼ r2 : If R is a positive number, for any finite set of nodes V ¼ fx1 ; ; xk g, we use GR ðx1 ; ; xk Þ or GR ðV Þ to denote the R-disk graph over fx1 ; ; xk g in which there is an edge between two nodes if and only if their Euclidean distance is at most R. For any positive integers k and m with 1 m k and any positive number R, let Ckm ðRÞ denote the set of Dk satisfying that G2R ðx1 ; ; xk Þ has exactly ðx1 ; ; xk Þ 2 D m connected components. For any two points u and v, let Duv denote the disk with the segment uv as a diameter, i.e., kuvk . We have ; Duv ¼ B uþv 2 2 1 jDuv j ¼ ku vk2 : 4 For any r ¼ ðr1 ; ; rk Þ 2 ð0; R0 Þk and x ¼ ðx1 ; ; xk Þ 2 k D D , Bðx1 ; r1 Þ; ; Bðxk ; rk Þ are called feasible if r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi. kxi xj k r2i r2j for any i 6¼ j and i ok ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qkx. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R20 r2i for any i. We remark. R20 r2i if and only if xi 2 D Dri ð0Þ [ D Dri ð1Þ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 and kxi xj k jri rj j if and only if both disks Bðxi ; ri Þ. that kxi ok . and Bðxj ; rj Þ do not contain each other’s diameter. Let 1rx be an indicator such that 1rx ¼ 1 if Bðx1 ; r1 Þ; ; Bðxk ; rk Þ are feasible, and 1rx ¼ 0 otherwise. In what follows, we only consider feasible disk sets. We use r ðxÞ to denote the area S D, and sometimes by slightly abusing the of ki¼1 Bðxi ; ri Þ \ D notation, to denote the union region itself. If k ¼ 1, for Dr ð1Þ, r ðxÞ 12 r2 . x2D Dr ð0Þ, r ðxÞ ¼ r2 and, for x 2 D. 1 1 r ðxÞ r2 þ pffiffiffi kxk r: 2 The next lemma gives a lower bound for the area of the r-neighborhood of more than one nodes. 1 Lemma 5. Let R 100 R0 , c ¼ 0:03, x ¼ ðx1 ; ; xk Þ 2 D Dk , and. 1 k r ¼ ðr1 ; ; rk Þ 2 2 R; R . Assume x1 has the largest norm among x1 ; ; xk , and kxi xj k 2R if and only if ji jj 1. If 1rx ¼ 1, then. r ðxÞ r1 ðx1 Þ þ cR. k1 X. kxIþ1 xi k:. I¼1. The proof of Lemma 5 is given in the Appendix. 1 R0 and c ¼ 0:03. If x ¼ ðx1 ; ; Corollary 6. Assume R 100. k xk Þ 2 Ck1 ðRÞ, r ¼ ðr1 ; ; rk Þ 2 12 R; R , 1rx ¼ 1, and x1 has the largest norm among x1 ; ; xk , then. r ðxÞ r1 ðx1 Þ þ cR max kxi x1 k: 2ik. Proof. Without loss of generality, we assume that kxk x1 k achieves max kxi x1 k. Let P be a min-hop path between 2ik. x1 and xk in G2R ðx1 ; x2 ; ; xk Þ and t be the total length of P . Then, every pair of nodes in P that are not adjacent nodes in P are separated by a distance of more than R. Thus, by applying Lemma 5 to the nodes in P , we obtain ðfri jxi 2P gÞ ðfxi j xi 2 P gÞ r1 ðx1 Þ þ cRt: Since r ðxÞ ðfri jxi 2P gÞ ðfxi jxi 2 P gÞ and t kxk x1 k, the corollary follows. u t For x 2 D D and r 2 IR, let ðx; rÞ denote the (total) central angle corresponding to the portion of @Bðx; rÞ in which, if a diameter of Bðx; rÞ has endpoints, it is fully contained in D D. For example, in Fig. 2, x 2 D Dr ð1Þ, b and c are intersection points of @Bðx; rÞ and @D D, and segment ac is a diameter of Bðx; rÞ. Then, ðx; rÞ ¼ 2ffaxb. If x 2 D Dr ð1Þ, we use tðxÞ to denote the distance between x and @D D. We have.

(4) 114. IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,. VOL. 18,. NO. 1,. JANUARY 2007.

(5). In Pr min Yi > 0 ¼ Pr½X ¼ 0 ¼ 1 Pr½X > 0 i¼1. 1 E½X ¼ 1 . In X. E½Xi ¼ 1 . i¼1. In X. Pr½Yi ¼ 0:. i¼1. Therefore, if c and i ln n,

(6). In In X X In Pr½Yi ¼ 0 ¼ 1 ei Pr min Yi > 0 1 i¼1. 1. i¼1. In X. e ln n. i¼1. . ¼1. nc lnc n. i¼1. n c ¼1 e ln n ln n. 1:. If Y1 ; Y2 ; ; YIn are independent, we have Fig. 3. A polyquadrate is a collection of grids that intersect with a polygon.. ðx; rÞ ¼ 2; if x 2 D Dr ð0Þ; tðxÞ tðxÞ 4 ; if x 2 D Dr ð1Þ; ðx; rÞ 4 arcsin r 2 r ðx; rÞ ¼ 0; if x 2 D Dr ð2Þ:. i¼1. ð1Þ. An "-tessellation is to divide the plane by vertical and horizontal lines into a grid in which each cell is with width ". Without loss of generality, we assume the origin is a corner in the grid. A collection of grid cells intersecting with a polygon or a convex compact set is called a polyquadrate. For example, in Fig. 3, the shaded grid cells form a polyquadrate. The horizontal span of a polyquadrate is the horizontal distance measured by the number of grid cells from the left to the right. The vertical span of a polyquadrate is with similar definite, but for the vertical distance. Lemma 7. If S consists of m grid cells and n is a constant, the number of polyquadrates with span less than n and intersecting with S is ðmÞ. Proof. Since n is a constant, the number of polyquadrates that have spans less than n and contain a specified grid cell is also constant. Since S consists of m grid cells, the lemma follows. u t. 3.2 Extremes of a Collection of Poisson RVs The next lemma gives an a.a.s. upper bound and lower bound of a collection of Poisson RVs. Lemma 8. Let Yi be a Poisson RV with rate i for i ¼ 1; ; In and let c > 0 and > 0 be constants. Suppose In ¼ ððlnnnÞc Þ. If c and i ln n, we have

(7). In lim Pr min Yi > 0 ¼ 1: n!1. i¼1. If Y1 ; Y2 ; ; YIn are independent, 2 ð0; cÞ, and i ln n, we have

(8). In lim Pr min Yi ¼ 0 ¼ 1: n!1. i¼1. Proof. Let Xi ¼ 1fYi ¼0g and X ¼. PIn. i¼1. Xi . We have.

(9).

(10). In Y In In Pr½Yi > 0 Pr min Yi ¼ 0 ¼ 1Pr min Yi > 0 ¼ 1 ¼ 1. i¼1. 1¼i. In Y. In Y. 1¼i. 1¼i. ð1Pr½Yi ¼ 0Þ 1. e Pr½Yi ¼0 ¼ 1e. PIn i¼1. Pr½Yi ¼0. :. Therefore, if 2 ð0; cÞ and i ln n, PIn PIn i Pr min Yi ¼ 0 1 e i¼1 PrðYi ¼0Þ ¼ 1 e i¼1 e 1iIn. 1 e. PIn i¼1. e ln n. c. ¼ 1 eððln nÞ Þe n. ln n. ¼ 1 e. c n lnc n. 1: t u. 3.3 Palm Theory and Brun’s Sieve Here, we state the Palm theory [11], [12] on the Poisson point process. Theorem 9. Let n > 0. Suppose k is a positive integer and hðY; X Þ is a bounded measurable function defined on all pairs of the form ðY; X Þ with X IR2 being a finite subset and Y being a subset of X satisfying hðY; X Þ ¼ 0 except when Y has k elements. Then, " # X nk hðY; P n Þ ¼ E½hðX k ; X k [P n Þ; E k! Y P n. where the sum on the left-hand side is over all subsets Y of the random Poisson point set P n and, on the right-hand side, the set X k is a binomial process with k nodes, independent of P n . The next theorem is called Brun’s sieve, which is a traditional approach to the Poisson process and will be used to prove Theorem 3. Theorem 10. Assume mðnÞ is a nonnegative integer random variable. Let B1 ; ; BmðnÞ be events, Y be the number of Bi that holds, and X S ðjÞ ¼ Pr½Bi1 ^ ^ Bij : fi1 ;;ij g f1;;qmðnÞg. Suppose there is a constant such that, for every fixed k, E½S ðjÞ . 1 j : j!. Then, Y is also asymptotically Poisson with mean ..

(11) WAN AND YI: ON THE LONGEST EDGE OF GABRIEL GRAPHS IN WIRELESS AD HOC NETWORKS. Proof. To prove this, we need to show that, for any k,. Pr½Y ¼ k !. k e : k!. For any i j 1, let X. ðjÞ. Si ¼. Pr½Bi1 ^ ^ Bij j mðnÞ ¼ i:. fi1 ;;ij g f1;;mðnÞg. ð0Þ. For convenience, let S ð0Þ ¼ 1 and Si note that, for any j 1, E½S ðjÞ ¼. 1 X. ¼ 1. In addition,. ðjÞ. Si Pr½mðnÞ ¼ i:. i¼j. According to the inclusion-exclusion principle, we have Pr½Y ¼ 0 ¼ 1 S ð1Þ þ S ð2Þ ! 1 i X X j ðjÞ Pr½mðnÞ ¼ i ¼ ð1Þ Si i¼0. ¼ ¼. j¼0. !! 1 1 X X ðjÞ j ð1Þ Si Pr½mðnÞ ¼ i. j¼0 1 X. i¼j. 115. . @ðx1 ; ; xk1 ; xk Þ @ðx1 þ pðx1 Þ; ; xk1 þ pðxk1 Þ; xk Þ. ¼. @ðz ; ; z ; r; Þ . @ðz1 ; ; zk1 ; r; Þ 1 k1 @ x1 þpðx1 Þ ; ; xk1 þpðxk1 Þ ; x k . 2 2. ¼ 4k1 . @ðz1 ; ; zk1 ; r; Þ. @ðz1 ; ; zk1 ; xk zk1 Þ. ¼ 4k1 @ðz1 ; ; zk1 ; r; Þ . I2 0 0. . . . . .. . . ... ... k1 . ¼ 4 0 I. 0 2. 0 0 cos r sin . sin r cos ¼ 4k1 r: Now, we give several lemmas about the limits of integrals, but leave their proofs in the Appendix. qffiffiffiffiffiffiffiffiffiffi nþ Lemma 11. Let r ¼ 2 lnn for some constant , and either qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffi n Rn ¼ 3 lnnn or Rn ¼ 2 ln nþ n with n ¼ oðln nÞ and n ! 1. Then, n2 2. Z. Rn 2 r. r¼ 2. Z z2D D. enr ðzÞ 4rðz; rÞdzdr 2e :. ð1Þj E½S ðjÞ :. j¼0. In the general form, we have k ðkÞ k þ 1 ðkþ1Þ S S þ k k ! 1 i X X j ðjÞ jk S ¼ Pr½mðnÞ ¼ i ð1Þ jk i i¼k j¼k !! X 1 1 X j ðjÞ jk ¼ ð1Þ Si Pr½mðnÞ ¼ i jk i¼j j¼k 1 X j E½S ðjÞ : ð1Þjk ¼ j k j¼k. Pr½Y ¼ k ¼. Then, following the proof outline given in [13], Chapter 8, the theorem can be proved. u t Here, we remark that Theorem 10, in which the number of events is a random variables is an extension to the traditional Brun’s sieve described in [13] which is an implication of the Bonferroni inequalities.. 3.4 Integral Preliminaries First, we introduce a technique to obtain the Jacobian determinant in the change of variables. Assume a tree topology is fixed over x1 ; x2 ; ; xk 2 IR2 . Without loss of generality, we may assume ðxk1 ; xk Þ is one of edges. Let zk1 ¼ 12 ðxk1 þ xk Þ, r ¼ 12 kxk xk1 k, and be the slope of xk1 xk . For 1 i k 2, we use pðxi Þ to denote xi ’s parent in the tree rooted at xk , and let zi ¼ 12 ðxi þ pðxi ÞÞ. Then,. In the remainder of this section, we always assume r ¼ qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi nþ n for some constant , and Rn ¼ 2 ln nþ with n ¼ 2 lnn n oðln nÞ and n ! 1. Lemma 12. For any fixed integer k 2, 2 k Z Z k Y n r nr ðzÞ 1 e 4ri ðzi ; ri Þdzi dri ¼ o ð1Þ: z r 2 r2½ 2 ;R2n k z2Ck1 ðR2n Þ i¼1. Lemma 13. For any fixed integers 2 m < k, 2 k Z Z k Y n 1rz enr ðzÞ 4ri ðzi ; ri Þdzi dri r R k 2 r2½ 2 ; 2n z2Ckm ðR2n Þ i¼1 ¼ o ð1Þ: Lemma 14. For any fixed integer k 2, 2 k Z Z k Y n enr ðzÞ 4ri ðzi ; ri Þdzi dri r R k 2 r2½ 2 ; 2n z2Ckk ðR2n Þ i¼1 ð2e Þk : Lemma 15. For any constant m 3 and m2 t i 1, let S ij denote the set of ðz1 ; z2 ; ; zmt Þ 2 CðmtÞ1 ðR2n Þ satisfying that zi is the one with largest norm and zj is the one with smallest norm among z1 ; ; zk . Then,.

(12) 116. IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,. nm. Z. Z. r2½ m t Y. r R t n 2; 2. dzk. !. enðri ðzi ÞþcRn kzi zj kÞ. z2S ij. t Y. VOL. 18,. NO. 1,. JANUARY 2007. ! rk ðzk ; rk Þdrk. k¼1. o ð1Þ:. k¼1. 4. ASYMPTOTIC LENGTH. OF THE. LONGEST EDGE. This section is dedicated to the proof of Theorem 1.. 4.1 Lower Bounds for the Longest Edge Length In this section, we are going to prove the following lemma that gives lower bounds for the length of the longest Gabriel edge. Lemma 16. For any constant 2 ð0; 2Þ, " rffiffiffiffiffiffiffiffi# ln n ¼ 1: lim Pr ðGðP n ÞÞ n!1 n Assume 1 and 2 are positive constants, and R1 and R2 are given by nR21 ¼ 1 ln n and nR22 ¼ 2 ln n, respectively. Choose 1 ; 2 such that maxð1; 2 Þ < 1 < 2 < 4 and 2 R 1 Lemma 5. It follows that ffi qffiffiffiffiffi c2 1 R2 < 1. Here, c is given by 1 ln n 3 n -tessellation and the 2 R2 R1 R2 . Consider a grid cells within D D. Let In denote the number of cells fully contained in D D. Here, In ¼ O lnnn . For eachqcell ffiffiffiffiffiffi fully contained in D D, we draw a disk with radius 12 lnnn at the center of the cell. For 1 i In , let Ei be the event that there exist Gabriel edges whose midpoints are on the ith disk and whose lengths are between R1 and R2 . Hence, " rffiffiffiffiffiffiffiffi# ln n Pr ½at least one Ei occurs: Pr ðGðP n ÞÞ n We have that E1 ; ; EIi are identical. If oi denotes the center of the ith disk, u and v are two points such that their midpoint is on the ith disk, and the distance between them is between R1 and R2 , then, for any point w 2 Duv , we have 1 þ oi 1 ðu þ vÞ ðu þ vÞ w kw oi k 2 2 rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi 1 1 ln n 3 ln n Rþ < : 2 2 n 2 n Therefore, u, v, and Duv are contained in the ith cell. (See Fig. 4.) Hence, E1 ; ; EIi are independent. Then, In. Pr ½none of Ei occurs ¼ ð1 Pr½E1 Þ ¼ e. In lnð1Pr½E1 Þ. eIn PrðE1 Þ : If In PrðE1 Þ ! 1, we may have " rffiffiffiffiffiffiffiffi# ln n Pr ðGðP n ÞÞ ! 1: n In the following, we shall prove that In PrðE1 Þ ! 1. Let A denote the first disk. Assume V is a point set and Y V . Let h1 ðY ; V Þ denote a function such that h1 ðY ¼ fx1 ; x2 g; V Þ ¼ 1 only if 12 ðx1 þ x2 Þ 2 A, R1 kx1 x2 k R2 ,. Fig. 4. Duv is fully contained in the cell.. and there is no other node of V in the disk area Dx1 x2 ; otherwise, h1 ðY ; V Þ ¼ 0. Hence, E1 is the event that there exist two nodes X; Y 2 P n such that h1 ðfX; Y g; P n Þ ¼ 1. In the remainder of this section, we use X1 , X2 , X3 , and X4 to denote independent random points with uniform distribution over D D and independent of P n . Let F1 be the event that h1 ðfX1 ; X2 g; fX1 ; X2 g [ P n Þ ¼ 1; F2 be the event that h1 ðfX1 ; X2 g; fX1 ; X2 ; X3 g [ P n Þ h1 ðfX1 ; X3 g; fX1 ; X2 ; X3 g [ P n Þ ¼ 1; and F3 be the event that h1 ðfX1 ; X2 g; fX1 ; X2 ; X3 ; X4 g [ P n Þ h1 ðfX3 ; X4 g; fX1 ; X2 ; X3 ; X4 g [ P n Þ ¼ 1: We claim that Pr½E1 . n2 n3 n4 Pr½F1 Pr½F2 Pr½F3 : 2! 2 8. ð2Þ. We shall prove this claim by the Palm theory and Boole’s inequalities. For clarity, we use X10 , X20 , X30 , and X40 to denote elements of P n . For any fx1 ; x2 ; x3 g V , let h2 ðfx1 ; x2 ; x3 g; V Þ ¼ h1 ðfx1 ; x2 g; V Þ h1 ðfx1 ; x3 g; V Þ þ h1 ðfx2 ; x1 g; V Þ h1 ðfx2 ; x3 g; V Þ þ h1 ðfx3 ; x1 g; V Þ h1 ðfx3 ; x2 g; V Þ: For any fx1 ; x2 ; x3 ; x4 g V , let h3 ðfx1 ; x2 ; x3 ; x4 g; V Þ ¼ h1 ðfx1 ; x2 g; V Þ h1 ðfx3 ; x4 g; V Þ þ h1 ðfx1 ; x3 g; V Þ h1 ðfx2 ; x4 g; V Þ þ h1 ðfx1 ; x4 g; V Þ h1 ðfx2 ; x3 g; V Þ: Let F10 ðfX10 ; X20 gÞ be the event that h1 ðfX10 ; X20 g; P n Þ ¼ 1; F20 ðfX10 ; X20 ; X30 gÞ be the event that h2 ðfX10 ; X20 ; X30 g; P n Þ ¼ 1;.

(13) WAN AND YI: ON THE LONGEST EDGE OF GABRIEL GRAPHS IN WIRELESS AD HOC NETWORKS. and F30 ðfX10 ; X20 ; X30 ; X40 gÞ be the event that. 117. We have. h3 ðfX10 ; X20 ; X30 ; X40 g; P n Þ ¼ 1:. Z Z. Pr½F1 ¼. Pr½F1 j X1 ¼ x1 ; X2 ¼ x2 dx1 dx2. S1. According to the Palm theory (Theorem 9), we have X Pr½F10 ðfX10 ; X20 gÞ fX10 ;X20 g P n. 2. ¼ E4. Let z ¼. Pr½F1 ¼. n ¼ E½h1 ðfX1 ; X2 g; fX1 ; X2 g [ P n Þ 2! n2 ¼ Pr½F1 ; 2. h2 ðfX10 ; X20 ; X30 g; P n Þ5 ð4Þ. 3. n E½h2 ðfX1 ; X2 ; X3 g; fX1 ; X2 ; X3 g [ P n Þ 3! n3 n3 ¼ 3 Pr½F2 ¼ Pr½F2 ; 3! 2. ¼. 2. ¼ E4. ð5Þ. n3 E½h3 ðfX1 ; X2 ; X3 ; X4 g; fX1 ; X2 ; X3 ; X4 g [ P n Þ 3! n4 n4 ¼ 3 Pr½F2 ¼ Pr½F3 : 4! 8 ¼. Let z1 ¼. Pr½F2 16 16. Z z1 2A. Pr½F20 ðfX10 ; X20 ; X30 gÞ. Z. r1 ¼. Z. R1 2. R2 2. r1 ¼. Z. Z. R2 2. enðr1 þcR2 kz1 z2 kÞ 2r1 dr1 dz1 dz2 2. z2 2A 2. R1 2. R2 2. enr1 2r1 dr1 dz1. Pr½F30 ðfX10 ; X20 ; X30 ; X40 gÞ. n2 n3 n4 Pr½F1 Pr½F2 Pr½F3 : 2 2 8. Z. ecnR2 kz1 z2 k dz2. z2 2A. 2 enr1 d r21 dz1. R r1 ¼ 21 R2 2 2. Z. 0 1. 16. AjAj 2 ¼ @ enr . n ðcnR2 Þ2 R r¼ 21 1 2 8 ¼ 2 2 3 n 4 n 4 ln n: c nR2 n. fX10 ;X20 ;X30 ;X40 g P n. 1 ðx1 ; x2 Þj ðx1 þ x2 Þ 2 A; R1 kx1 x2 k R2 : 2. Z z1 2A. 16. Hence, our claim is true. In the next, we derive the probabilities of F1 , F2 , and F3 . Let S1 denote the set. Z z1 2A. fX10 ;X20 ;X30 g P n. ¼. 3 , r1 ¼ 12 kx1 x2 k, z2 ¼ x1 þx 2 , and ¼ kz1 z2 k.. x1 þx2 2. Then,. fX10 ;X20 g P n. X. 2. enr dðr2 Þdz. Let S2 denote the set 8 9 x1 þx2 = Dx1 x3 ; = 2 2 A; R1 kx1 x2 k R2 ; x1 ; x2 2 <. : ðx ; x ; x Þ : 1 2 3 x1 þx3 ; 2 A; R kx x k R ; x ; x 2 = D 1 1 3 2 1 3 x x 1 2 2. Applying Boole’s inequalities and (3), (4), and (5), we have X Pr½F10 ðfX10 ; X20 gÞ Pr½E1 . . z2A. R r¼ 21. S2. h3 ðfX10 ; X20 ; X30 ; X40 g; P n Þ5. fX10 ;X20 ;X30 ;X40 g P n. . 8rdrdz ¼ 4. R2 2. Therefore, Z Z Z Pr½F2 ¼ Pr½F2 j X1 ¼ x1 ; X2 ¼ x2 ; X3 ¼ x3 dx1 dx2 dx3 S Z Z Z2 2 x1 þx2 x1 þx3 1 en ð2kx1 x2 kÞ þcR2 k 2 2 k dx1 dx2 dx3 :. 3. X. e. Z. Z. Pr ½F2 j X1 ¼ x1 ; X2 ¼ x2 ; X3 ¼ x3 enjDx1 x2 [Dx1 x3 j 2 x1 þx2 x1 þx3 1 en ð2kx1 x2 kÞ þcR2 k 2 2 k :. Pr½F30 ðfX10 ; X20 ; X30 ; X40 gÞ X. R r¼ 21. nr2. Applying Lemma 5, if ðx1 ; x2 ; x3 Þ 2 S2 , we have. and. fX10 ;X20 ;X30 ;X40 g P n. R2 2. ð6Þ. fX10 ;X20 ;X30 g P n. X. Z. Z. ! R 2 4 nr2 22 1 1 e ¼ R1 jAj ¼ 2 n 4 n 4 ln n: r¼ 2 n n. 3. X. 1. and r ¼ 12 kx1 x2 k. Then,. z2A. Pr½F20 ðfX10 ; X20 ; X30 gÞ. ¼ E4. x1 þx2 2. ð3Þ. fX10 ;X20 g P n. 2. 2. enð2kx1 x2 kÞ dx1 dx2 :. S1. h1 ðfX10 ; X20 g; P n Þ5. fX10 ;X20 ;X30 g P n. Z Z. ¼. 2. X. enjDx1 x2 j dx1 dx2. S1. 3. X. Z Z. ¼. 1. ecnR2 2 d. ¼0. ð7Þ Let S3 denote the set. ) x1 þx2 = Dx3 x4 ; 2 2 A; R1 kx1 x2 k R2 ; x1 ; x2 2 : ðx1 ; x2 ; x3 ; x4 Þ x3 þx4 = Dx1 x2 2 2 A; R1 kx3 x4 k R2 ; x3 ; x4 2. (.

(14) 118. IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,. Applying Lemma 5, if ðx1 ; x2 ; x3 ; x4 Þ 2 S3 , we have. VOL. 18,. NO. 1,. JANUARY 2007. Lemma 17. For any constant > 2, we have " rffiffiffiffiffiffiffiffi# ln n ¼ 0: lim Pr ðGðP n ÞÞ n!1 n qffiffiffiffiffiffi Proof. Let d ¼ lnnn and r ¼ d2 . Pick a constant 1 2 ð2; Þ, qffiffiffiffiffiffi 0 0 Þ pffiffi . Let Cr be the and let d0 ¼ 1 lnnn, r0 ¼ d2 and " ¼ ðrr 2. Pr ½F3 j X1 ¼ x1 ; X2 ¼ x2 ; X3 ¼ x3 ; X4 ¼ x4 2 x1 þx2 x3 þx4 1 enkDx1 x2 [Dx3 x4 j en ð2kx1 x2 kÞ þcR2 k 2 2 k : Therefore,. 3 X 1 ¼ x1. 6 X ¼ x 7 27 6 2 Pr6F3 Pr½F3 ¼ 7dx1 dx2 dx3 dx4. 4 X ¼ x S3 35 3 x ¼x 4 4 Z Z Z Z 2 x þx x þx n ð12kx1 x2 kÞ þcR2 k 1 2 2 3 2 4 k e 2. collection of all feasible r-disks, whose diameters are. Z Z Z Z. contained in D D. If xy is a Gabriel of the GG, there is no node on Dxy . Therefore, ðGðP n ÞÞ d implies that there exists a disk of Cr on which there is no node of P n . Therefore,. S3.

(15). Pr½ðGðP n ÞÞ d Pr min jC \ P n j ¼ 0 :. dx1 dx2 dx3 dx4 :. C2Cr. Let z1 ¼. x1 þx2 2. , r1 ¼. 1 2 kx1. x2 k, z2 ¼. x3 þx4 2. , r2 ¼. 1 2 kx3. x4 k,. and ¼ kz1 z2 k. Then, Pr½F3 . Z. Z z1 2A. Z. R2 2. r1 ¼. R1 2. Z. z2 2A. R2 2. r2 ¼. enðr1 þcR2 kz1 z2 kÞ 2. R1 2. ð8r1 dr1 dz1 Þð8r2 dr2 dz2 Þ ! Z R2 Z 2 nr21 4 e 2dr1 dz1 R z1 2A. . r1 ¼. 1 2. Z R2 R2 R1 ecnR2 kz1 z2 k dz2 2 2 2 z2 2A ! Z R2 Z 2 2 nr21 e d r1 dz1 4 R . 8. . z1 2A. . r1 ¼. ð8Þ. 1 2. Z 1 R2 R2 R1 ecnR2 2 d 2 2 2 ¼0 ! ln n 1 42 42 4 n n R2 ðR2 R1 Þ ¼ n2 ðcnR2 Þ2 2 42 R1 1 n 4 n 4 ln n: ¼ 2 4 1 c n R2 . i¼1. 8. Combining (2), (6), (7), and (8), we have ! 1 2 1 4 2 R1 2 2 2 1 n 4 n 4 ln n Pr½E1 2 c nR2 2c R2 2 1 2 1 R1 1 2 1 n 4 n 4 ln n: 2 c R2 ln n 2 1 Since c2 1 R R2 < 1 and In ¼ n , we have 1 2 Pr½E1 ¼ n 4 n 4 ln n and 1 In Pr½E1 ¼ n1 4 ! 1: This complete the proof of Lemma 16.. Note that Yi is a Poisson RV with rate njPi j. Assume that E is the collection of polyquadrates induced by r0 -disks with center on D Dr0 ð0Þ and F is the collection of polyquadrates induced by feasible r0 -disks with their center on D DnD Dr0 ð0Þ. Since E [ F ¼ fP1 ; ; PIn g,

(16).

(17). In Pr min Yi ¼ 0 ¼ Pr min Yi ¼ 0 or min Yi ¼ 0 i¼1 Pi 2E Pi 2F

(18).

(19). Pr min Yi ¼ 0 þ Pr min Yi ¼ 0 : Pi 2E. u t. Pi 2F. For any Pi 2 E, we have 1 1 njPi j n ð2r0 Þ2 ¼ 12 ln n > ln n: 4 4 Applying Lemma 7, we also have jEj ¼ "12 ¼ lnnn . Therefore, by Lemma 8,

(20).

(21). Pr min Yi ¼ 0 ¼ 1 Pr min Yi > 0 0: Pi 2E. 4.2 Upper Bounds for the Longest Edge Length In this section, we are going to give upper bounds for the length of the longest Gabriel edge.. Divide the plane into an "-tessellation. The distance of any two points within a grid cell is at most ðr r0 Þ. If A and B, respectively, are r-disk and r0 -disk with the same center, the grid cells intersected with B are contained in A since any point of B is apart from @A by at least ðr r0 Þ. So, the polyquadrate induced by B is contained in A. Let fP1 ; ; PIn g denote the set of polyquadrates induced by r0 -disks of Cr0 , and Yi denote the number of nodes of P n on Pi . Since any r-disk of Cr contains at least one Pi , we have

(22).

(23). In Pr min jC \ P n j ¼ 0 Pr min Yi ¼ 0 i¼1 C2Cr

(24). In ¼ 1 Pr min Yi > 0 :. Pi 2E. For any Pi 2 F , we have 1 1 1 2 njPi j n ð2r0 Þ > ln n: 2 4 2 0 Applying Lemma 7, we also have jF j ¼ "r2 ¼ pffiffiffiffiffiffi n ln n . Therefore, by Lemma 8,.

(25) WAN AND YI: ON THE LONGEST EDGE OF GABRIEL GRAPHS IN WIRELESS AD HOC NETWORKS.

(26).

(27). 6. Pr min Yi ¼ 0 ¼ 1 Pr min Yi > 0 0: Pi 2F. Pi 2F. Put all together, and the lemma is proved.. u t. Lemmas 16 and 17, respectively, give lower and upper bounds for the length of the longest Gabriel edge. Hence, Theorem 1 is now an immediate consequence of Lemmas 16 and 17.. 5. EXPECTED NUMBER. OF. LONG EDGES. In the previous section, we provedqthat ffiffiffiffiffiffi the ratio of the ln n length of the longest Gabriel edge to n is a.a.s. equal to 2. In this section, we are going to prove Theorem 2, which gives the expectation of the number of long Gabriel edges. Proof of Theorem 2. Assume Y and V are point sets and Y V . Let hr ðY ; V Þ denote a function such that hr ðY ¼ fx1 ; x2 g; V Þ ¼ 1 only if kx1 x2 k r and there is no other node of V in the disk area Dx1 x2 ; otherwise, hr ðY ; V Þ ¼ 0. Let X1 and X2 denote independent random points with uniform distribution over D D and independent of P n . According to the Palm theory, 2 3 X 6 7 E½NðGðP n Þ; rÞ ¼ E4 hr X10 ; X20 ; P n 5 0 0 fX1 ;X2 g P n n2 ¼ E½hr ðfX1 ; X2 g; fX1 ; X2 g [ P n Þ: 2! Let F ðrÞ be the probability of the event that X1 X2 is a Gabriel edge and kX1 X2 k r. Then, F ðrÞ ¼ E ½hr ðfX1 ; X2 fX1 ; X2 g[P n Þ RR and F ðrÞ ¼ x1 ;x2 2D D kx1 x2 kr.

(28). X1 ¼ x1. Pr Xa X2 is a Gabriel edge dx1 dx2 X2 ¼ x2 Z Z x1 þx2 ¼ envkx1 x2 k=2 ð 2 Þ dx1 dx2 ; x1 ;x2 2D D kx x kr Z 1 Z1 2 ¼ env ðzÞ 4rðz; Þdzdr: ¼2r z2D D 2 and ¼ 12 kx1 x2 k. In the last equality, we let z ¼ x1 þx 2 According to Theorem 1, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rffiffiffiffiffiffiffiffi! ln n þ ln n þ ln n F 2 F 3 F 2 n n n p ffiffiffiffi Z 3 ln n Z 2 n ¼ env ðzÞ 4rðz; Þdzd : pffiffiffiffiffiffiffi nþ ¼ lnn z2D D. Hence, applying Lemma 11, " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ln n þ n2 ln n þ ¼ F 2 2e : E N GðP n Þ; 2 n n 2! t u. 119. ASYMPTOTIC PROBABILITY DISTRIBUTION MAXIMAL LENGTH. OF THE. qffiffiffiffiffiffiffiffiffiffiffiffi n for some sequence n such that n ! 1, Let Rn ¼ 2 ln nþ n n ¼ oðln nÞ, and mðnÞ ¼ nðn 1Þ=2. For applying Theorem 10 (Brun’s sieve) to prove Theorem 3, let Bi be the event that the edge between the ith pair of nodes is a Gabriel edge whose length is at least r but less than Rn for 1 i mðP oðnÞÞ, and Y be the number of Bi that holds. Then, Y is exactly the q number ffiffiffiffiffiffiffiffiffiffi of Gabriel edges whose nþ but less than Rn . According lengths are at least r ¼ 2 lnn to Theorem 2, Y is a.a.s. equal to the number of Gabriel edges whose length is at least r . For any fixed integer k, let K1 denote the collection of set fi1 ; ; ik g f1; ; mðP oðnÞÞg such that ei1 ; ; eik are not incident to the same nodes, and K2 be the collection of set fi1 ; ; ik g f1; ; mðP oðnÞÞg such that ei1 ; ; eik have some common endpoints. Then, X. S ðkÞ ¼. Pr½Bi1 ^ ^ Bik . fi1 ;;ik g f1;;mðP oðnÞÞg. X. ¼. Pr½Bi1 ^ ^ Bik . fi1 ;;ik g2K1. X. þ. Pr½Bi1 ^ ^ Bik :. fi1 ;;ik g2K2. In Lemma 18, we shall prove that the expectation of the sum over fi1 ; ; ik g 2 K1 are asymptotically equal to ð2e Þk . In Lemma 19, we shall prove that the expectation of the sum over fi1 ; ; ik g 2 K2 are asymptotically equal to zero. Therefore, Theorem 3 follows Lemmas 18 and 19 by applying Theorem 10. In the proofs of Lemmas 18 and 19, for applying the Palm theory, let X l ¼ fX1 ; ; Xl g denote a uniform l-points process over D D and independent of P n , and Bi be the event that the edge between the ith pair of nodes of X l is a Gabriel edge over X l [ P n whose length is at least r but less than Rn for 1 i mðlÞ. In addition, let ei denote the edge of X2i1 X2i (or x2i1 x2i ), zi ¼ x2i12þx2i be the midpoint of edge ei , and ri ¼ k x2i12x2i k be the radius of Dx2i1 x2i . kei k is shorthand for kx2i1 x2i k, i.e., the length of edge ei . Lemma 18. For any fixed k, 2 3 X E4 Pr ½Bi1 ^ ^ Bik 5 ð2e Þk : fi1 ;;ik g2K1. Proof. Applying the Palm theory and due to the identity property, we have 2 3 X E4 Pr ½Bi1 ^ ^ Bik 5 fi1 ;;ik g2K1. ! 2k2k2 2 2 n2k 2 2 Pr ½B1 ^ ^ Bk ¼ ð2kÞ! k! k 1 n2 Pr ½B1 ^ ^ Bk : ¼ k! 2.

(29) 120. IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,. VOL. 18,. NO. 1,. JANUARY 2007. So, the asymptotic equality (9) is also true for any fixed k 2. Hence, Lemma 18 is proved. u t. It is sufficient to show that, for any fixed k, 2 k n Pr ½B1 ^ ^ Bk ð2e Þk : 2. ð9Þ Lemma 19. For any fixed k, 2 3 X Pr ½Bi1 ^ ^ Bik 5 o ð1Þ: E4. For k ¼ 1, we have x1 þx2 2. Pr ½B1 j X1 ¼ x1 ; X2 ¼ x2 ¼ enkx1 x2 k=2 ð. Þ:. fi1 ;;ik g2K2. Hence, 2. 2. n n Pr ½B1 ¼ 2 2 ¼. n2 2. n2 ¼ 2. Z Z D D. Z Z. x1 ;x2 2 r kx1 x2 kRn. Pr ½B1 j X1 ¼ x1 ; X2 ¼ x2 dx2 dx1 x1 þx2 2. D D. x1 ;x2 2 r kx1 x2 kRn. Z. Z. Rn 2 r. r1 ¼ 2. enkx1 x2 k=2 ð. fi1 ;;ik g2K2. r1 ðz1 Þ. z 1 2D D. Þ dx dx 2 1. e. . 4r1 ðz1 ; r1 Þdz1 dr1 2e :. ¼. 2X k1. 2 k n Pr ½B1 ^ ^ Bk 2. " # 2 k Z Z for any 1 i 2k; n. ¼ Pr B1 ^ ^ Bk . 2 X i ¼ xi x1 ;;x2k 2D D r ke1 kRn .. . r kek kRn. Pr ½Bi1 ^ ^ Bik :. For any fi1 ; ; ik g 2 K2m , ei1 ; ; eik contain a subgraph that is a forest incident to m nodes and with at least one tree composed of more than two nodes. Let m denote the collection of k-edges forest topologies that are incident to exactly m nodes and contain at least one tree component composed of more than two nodes. Applying the Palm theory, we have 2 3 X E4 Pr ½Bi1 ^ ^ Bik 5 fi1 ;;ik g2K2. !. dxi :. . i¼1. 2k1 X. X. m¼3 2 m. Let z ¼ ðz1 ; ; zk Þ and r ¼ ðr1 ; ; rk Þ. We have nr ðzÞ. Pr ½B1 ^ ^ Bk j for any 1 i 2k; Xi ¼ xi e In addition, if z 2 Ckk. X. m¼3 fi1 ;;ik g2K2m. The last asymptotic equality is given by Lemma 11. So, the asymptotic equality (9) is true for k ¼ 1. Now, fix k 2. We have. 2k Y. Proof. For 3 m 2k 1, let K2m be the collection of set fi1 ; ; ik g 2 K2 such that ei1 ; ; eik have some common endpoints and the total number of nodes incident to S m ei1 ; ; eik is m. Then, K2 ¼ 2k1 m¼3 K2 and X Pr ½Bi1 ^ ^ Bik . Rn 2. : ð10Þ. h^ i O ðnm Þ Pr B : i e 2. i. Note that, for a fixed k, we have j m j ¼ Oð1Þ. Therefore, it is enough to prove that, for any forest topology 2 m , h^ i nm Pr B o ð1Þ: i e 2. i. , nr ðzÞ. Pr ½B1 ^ ^ Bk j for any 1 i 2k; Xi ¼ xi ¼ e. :. ð11Þ From (10) and Lemmas 12 and 13, 2 k

(30) n Rn k Pr B1 ^ ^ Bk and z 2 D D nCkk 2 2 2 k Z Z k Y n 1rz enr ðzÞ 4ri ðzi ; ri Þdzi dri r R k k R n n 2 r2½ 2 ; 2 z2D D nCkk ð 2 Þ i¼1 ¼ o ð1Þ: From (11) and Lemma 14, 2 k

(31) n Rn Pr B1 ^ ^ Bk and z 2 Ckk 2 2 2 k Z Z k Y n ¼ enr ðzÞ 4ri ðzi ; ri Þdzi dri r R k Rn n 2 r2½ 2 ; 2 z2Ckk ð 2 Þ i¼1 k 2e :. Let ¼ fT1 ; ; Tt g denote a forest composed of e1 ; ; emt incident to X1 ; ; Xm such that at least one tree contains more than two nodes. Let E be the event B1 ^ ^ Bmt . Without loss of generality, we assume ei is contained in Ti for 1 i t. Let i denote the collection of all i-partition of f1; 2; ; tg. For any $ ¼ fP1 ; P2 ; ; Pi g P S 2 i , let m$ ðjÞ ¼ k2Pj jV ðTk Þj, V$ ðjÞ ¼ k2Pj V ðTk Þ, $ ðjÞ Dmt such that ¼ fT : k 2 Pj g, and Sð$Þ be the set of z 2 D Rn S k zi 2V$ ðjÞ B zi ; 2 forms a connected component. For a fixed topology , let S ij ¼ (. ) for any 1 k m 1; r kx x k R k1 k2 n ; x2D D . and kzi k kzk k kzj k m. S ij ¼ Rn z 2 CðmtÞ1 j kzi k kzj k for any 1 j m t : 2 First, consider t ¼ 1, i.e., X1 ; ; Xm form a tree by e1 ; ; em1 . We have.

(32) WAN AND YI: ON THE LONGEST EDGE OF GABRIEL GRAPHS IN WIRELESS AD HOC NETWORKS. 121. Fig. 5. Two intersecting domains.. X. m. n Pr ½E . n. m. 1i6¼jm1. X. Oð1Þ. nm. Z. r. ri ¼ 2. 1i6¼jm1 m1 Y. Rn 2. !. # for 1 i m Y m. Pr E dxi Xi ¼ xi x2S ij i¼1 Z enðri ðzi ÞþcRn kzi zj kÞ ri ðzi ; ri Þ ". Z. z2S ij. dzk dri. k¼1. 0: The last equality is given by Lemma 15. Now, we consider t > 1. For Pr½E ^ fZ 2 CðmtÞ1 ðR2n Þg, we have

(33) Rn m n Pr E ^ Z 2 CðmtÞ1 2 Z m X Y nm Pr ½E j Xi ¼ xi for 1 i m dxi x2S ij. 1i6¼jmt. Oð1Þ. X. nm. r. !. rk ðzk ; rk Þdrk. k¼1. i¼1. Z t. r2½ 2 ;R2n . 1i6¼jm1 t Y. Z. m t Y. enðri ðzi ÞþcRn kzi zj kÞ. z2S ij. !. APPENDIX. dzk. k¼1. 0: The last equality is given by Lemma 15. For Pr½E ^ fZ 2 S$ g, applying the same argument used in Lemma 13, we have nm Pr ðE ^ fZ 2 S$ gÞ O ð1Þ. i Y. nm Pr ½E ^ fZ 2 S$ g. i¼1 $2i. O ð1Þ. t X Y i X i¼1 $2i. n. m$ ðjÞ. 7. ! o ð1Þ: Pr E $ ðjÞ. j¼1. Therefore, the lemma is proved.. Lemma 20. If the boundaries of S and T at x and y can be expressed by continue functions along the axis parallel to uv, then. Proof. For convenience, assume v is on T . After shifting T apart from S by 4t, x, y, and v on T are moved to x0 , y0 and v0 respectively. See Fig. 5. Then, jS [ T 0 j jS [ T j is equal to the area of the shaded region between xy and x0 y0 . Since the boundaries can be expressed by continue functions, the height of the shaded region can be estimated by kx yk þ oð1Þ. The distance between xy and x0 y0 is equal to 4t. So, the area of the shaded region can be expressed by ðkx yk þ oð1ÞÞ4t. u t. Putting it all together, we have t X X. First, we give two technical lemmas. Assume S and T are two convex compact sets, and @S and @T intersect only at two points x and y. Let u and v be two points to the different side of xy such that uv and xy are perpendicular. T 0 is obtained by shifting T away from S along uv by distance 4t. Then, we have the following lemma:. jS [ T 0 j jS [ T j ¼ ðkx yk þ o ð1ÞÞ 4t:. nm$ ðjÞ Pr E $ ðjÞ o ð1Þ:. j¼1. nm Pr ½E ¼. and can be constructed by distributed and localized algorithms. If all nodes have the same transmission radii, the maximal length of Gabriel edges is the smallest transmission radius for constructing the GG by only 1-hop neighbor information. In this paper, we assume a wireless ad hoc network is represented by a Poisson point process with mean n on a unit-area disk. We first showed the ratio of the ffi qffiffiffiffiffithat ln n maximal length of Gabriel edges to n is a.a.s. equal to 2. Next, we proved that, for any constant , the expected ffiffiffiffiffiffiffiffiffiffi qnumber nþ of long Gabriel edges, whose lengths are at least 2 lnn , is a.a.s. equal to 2e . This implies that, if ! 1, it is a.a.s. that qffiffiffiffiffiffiffiffiffiffi nþ the maximal length is less than 2 lnn . Last, we proved that the number of long Gabriel edges is asymptotically Poisson with mean 2e . Therefore, the probability of the event that ffi qffiffiffiffiffiffiffiffiffi ln nþ the maximal length of Gabriel edges is less than 2 n is asymptotically equal to expð2e Þ.. u t. CONCLUSION. The Gabriel graph is one of the widely used geometric structures in topology control of wireless ad hoc networks. Lemma 21. Assume R is ffia constant, 12 R r2 r1 R, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r1 r22 . Let t ¼ kx2 x1 k and fðtÞ ¼ kx2 x1 k jBðx2 ; r2 Þ n Bðx1 ; r1 Þj. Then, fðtÞ 0:16Rt: Proof. Assume r1 is fixed. If t < r1 þ r2 , let y1 y2 be the common chord of @Bðx1 ; r1 Þ and @Bðx2 ; r2 Þ. See Fig. 6a..

(34) 122. IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,. VOL. 18,. NO. 1,. JANUARY 2007. Fig. 6. The area of two intersecting disks.. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi r21 12 R . For any t 2 ½0; t0 , the minimum of pffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðtÞ occurs as r2 ¼ r21 t2 . So, without loss of generalpffiffiffiffiffiffiffiffiffiffiffiffiffiffi ity, we assume r2 ¼ r21 t2 . Let a (respectively, b) be. Let t0 ¼. the intersection point of the ray x1 x2 and circle @Bðx1 ; r1 Þ (respectively, @Bðx2 ; r2 Þ). See Fig. 6b. Then, 1 fðtÞ 2j4aby1 j ¼ 2 ðr2 þ t r1 Þr2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21 t2 ðr1 tÞ2 1 1 r21 t2 ðr1 tÞ ¼ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 2 r21 t2 þ ðr1 tÞ 1 2tðr1 tÞ 1 1 pffiffiffi Rt > 0:16Rt: ¼ Rt qffiffiffiffiffiffiffi ¼ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 þt 2 4 þ 3 r21 t2 þ ðr1 tÞ þ1 r1 t. qffiffiffiffiffiffiffi r1 þt Note that the maximum of r1 t occurs as r1 ¼ R, pffiffi 3 1 r2 ¼ 2 R, and t ¼ 2 R. For any fixed t 2 ½t0 ; 2R, the minimum of fðtÞ occurs as r2 ¼ 12 R. If t < r1 þ r2 , according to Lemma 20, we have f 0 ðtÞ ¼ ky1 y2 k. If t ¼ t0 , y1 y2 is the diameter of Bðx2 ; r2 Þ. Therefore, for t 2 ½t0 ; 2R, f 0 ðtÞ is decreasing, and fðtÞ is concave. Since fð2RÞ 0:16Rt0 ; ¼ R > 0:16R ¼ 2R 8 t0. 2 1 1 r ðxÞ r1 ðx1 Þ ¼ r2 ðx2 Þ R ¼ Rð2RÞ 0:03Rt: 3 2 24 Now, we only need to consider 0 t < r1 þ r2 . If Dr1 ð0Þ, then r ðx1 ; x2 Þ r1 ðx1 Þ is exactly fðtÞ and, x1 2 D thus, the lemma follows immediately from fðtÞ 0:16Rt. =D Dð0Þ. Note that, for the same So, we assume that x1 2 distance t, r ðx1 ; x2 Þ r1 ðx1 Þ achieves its minimum D. It is sufficient to prove when both x1 and x2 are in @D D. Let ‘1 be the line perpendithe lemma for x1 ; x2 2 @D cular to ox1 and through x1 . Furthermore, if x2 moves out of D D, r ðxÞ becomes smaller. So, consider that x2 is on ‘1 . We use d1 (respectively, d2 ) to denote the intersection D) far from x1 . point of @Bðx2 ; r2 Þ and ‘1 (respectively, @D Let a1 be the intersection point of ‘1 and y1 y2 , and b1 be the intersection point of ‘1 and @Bðx1 ; r1 Þ near to x2 . We use c1 to denote the point on ‘1 and in Bðx2 ; r2 Þ such that kc1 d1 k ¼ ka1 b1 k. Let z1 and z2 be the two points in @Bðx2 ; r2 Þ such that z1 z2 is perpendicular to ‘1 and through c1 . Let ‘2 denote the line perpendicular to ox1 and through d2 . We use y3 (respectively, a2 and c2 ) to denote the intersection point of ‘2 and ox1 (respectively, y1 y2 and z1 z2 ), and b2 to denote the intersection point of ‘2 and @Bðx1 ; r1 Þ near x2 . (See Fig. 6c.) We have ka1 c1 k ¼ ka1 x2 k þ kx2 c1 k ka1 x2 k þ kx1 a1 k ¼ t; and if R 15 R20 ,. we have fð2RÞ 0:16Rt0 0:16Rt0 ¼ 0:16R: 2R t0 t0 Let 4t ¼ t t0 . Then, fð2RÞ 0:16Rt0 4t 0:16Rt0 þ 0:16R4t fðtÞ 0:16Rt0 þ 2R t0 ¼ 0:16Rðt0 þ 4tÞ ¼ 0:16Rt: Note that the final inequality does not depend on r1 . Therefore, the lemma is proved. t u In the remainder of the Appendix, we give the proofs of Lemmas 5 , 11, 12, 13, 14, and 15. Proof of Lemma 5. We prove the lemma by induction on k. We begin with k ¼ 2. Let t ¼ kx2 x1 k and fðtÞ ¼ jBðx2 ; r2 Þ n Bðx1 ; r1 Þj. Since jBðx2 ; r2 Þ n Bðx1 ; r1 Þj > jBðx1 ; r1 Þ n Bðx2 ; r2 Þj if r2 > r1 , without loss of generality, we may assume r1 r2 . According to Lemma 6, we have fðtÞ 0:16Rt. If r1 þ r2 t 2R, since Bðx2 ; r2 Þ and Bðx1 ; r1 Þ are disjoint, we have. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ka1 a2 k ¼ kx1 ok kd2 ok2 kd2 y3 k2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 R20 ð3RÞ2 ¼. 9R2 9R2 5R2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : 4 R0 R0 þ R20 ð3RÞ2 R0 þ 5R0. The area of a1 a2 b2 b1 , surrounded by the segments a1 a2 , a1 b1 , a2 b2 , and arc b1 b2 , is larger than the area of c1 c2 d2 d1 , surrounded by the segments c1 c2 , c1 d1 , c2 d2 , and arc d1 d2 . 1 R0 , we have Therefore, if R 100 1 r ðxÞ r1 ðx1 Þ fðtÞ ka1 c2 k ka1 a2 k 2 5R 0:08Rt Rt ¼ 0:03Rt R0 and, thereby, the lemma for k ¼ 2 follows. Next, we assume the lemma is true for at most k 1 nodes and we shall show that the lemma is true for k nodes. If k ¼ 3, then.

(35) WAN AND YI: ON THE LONGEST EDGE OF GABRIEL GRAPHS IN WIRELESS AD HOC NETWORKS. r23 r1 ðx1 Þ þ R2 12 3 2 X r1 ðx1 Þþ0:03R 4R r1 ðx1 Þ þ 0:03R kxiþ1 xi k:. r ðxÞ r1 ðx1 Þ þ r3 ðx3 Þ r1 ðx1 Þ þ. 123. Therefore, n2 2. i¼1. Z. Z. Rn 2 r. z2D D. r¼ 2. enr ðzÞ 4rðz; rÞdzdr 2e : t u. If k > 3, then by the induction hypothesis,. Proof of Lemma 12. Let S denote the set of ðz1 ; z2 ; ; zk Þ 2 Ck1 ðR2n Þ satisfying that z1 is the one with largest norm among z1 ; ; zk and z2 is the one with longest distance from z1 among z2 ; ; zk . Then,. r ðxÞ ðr1 ;;rk2 Þ ðx1 ; ; xk2 Þ þ rk ðxk Þ r1 ðx1 Þ þ 0:03R. k3 X. kxiþ1 xi k þ. i¼1. r1 ðx1 Þ þ 0:03R. k3 X. r2k 3. 2 k Z Z k Y n r nr ðzÞ 1 e 4ri ðzi ; ri Þdzi dri z k r 2 r2½ 2 ;R2n z2Ck1 ðR2n Þ i¼1 2 k Z Z k Y n r nr ðzÞ kðk 1Þ 1 e 4ri ðzi ; ri Þdzi dri : z k r 2 r2½ 2 ;R2n z2S i¼1. kxiþ1 xi k þ 0:03R 4R. i¼1. r1 ðx1 Þ þ 0:03R. k1 X. kxiþ1 xi k:. i¼1. u t. Therefore, the lemma is true by induction.. So, it suffices to prove. Proof of Lemma 11. First, we calculate the integration over D Dr ð0Þ. 2Z. n 2. r r¼ 2. z2D Dr ð0Þ. Z. n2 2. ¼. r. ¼ 2n2. Rn 2. enr 2rdr ¼ 2n2. r. Z. Z. Rn 2. Rn 2 r. enr. 2. r. r¼ 2. z2D Dr ð1Þ. O ð1Þn2. Z. Rn 2. zi 2 B ðz1 ; kz2 z1 kÞ; for 3 i k;. r¼ 2. r. r¼ 2. r. enð2r 1. þc1 rtÞ. O ð1Þn e. Z. r. Z. Z. ¼ O ð1Þn e. Z. Rn 2. r¼ 2 Rn 2. r r¼ 2. 12ðln nþÞ. Z. Rn 2 r. r¼ 2. Z. ec1 nrt tdtdr. r. r1 ¼ 2. 2 12ðln nþÞ. ðnrÞ dr O ð1Þn e. 2. ðnr Þ Rn. rffiffiffiffiffiffiffiffi!1 ln n ¼ O ð1Þðln nÞ1=2 ¼ o ð1Þ: n. z2D Dr ð2Þ. encRn kz2 z1 k dz2. 2r2 dr2 z2 2Bðz1 ;ðk1ÞRn Þ. Z. Rn 2. ri ¼ 2. 2ri dri. dzi zi 2Bðz1 ;kz2 z1 kÞ. enr ðzÞ 4rðz; rÞdzdr ¼ 0:. Z. Rn 2. enr1 ðz1 Þ r1 ðz1 ; r1 Þdz1 dr1. r. r1 ¼ 2 z1 2D D. encRn kz2 z1 k kz2 z1 k2ðk2Þ dz2. Oð1Þn2k ðRn ðRn r Þk1 r1 ðz1 ; r1 Þdz1 dr1. Z. 1. Z. Z. Rn 2 r. z 1 2D D. r1 ¼ 2. enr1 ðz1 Þ. encRn 2k3 d. 0. ¼ Oð1Þ. ri ðzi ; ri Þdzi dri. enr1 ðz1 Þ r1 ðz1 ; r1 Þdz1 dr1. Z. Rn 2. k Y i¼1. z2 2Bðz1 ;ðk1ÞRn Þ. t¼0. Z. z1 2D D. Z. ec1 nrt tdtdr. 2. z2S. Z. Rn 2. ri ðzi ; ri Þdzi dri. enðr1 ðz1 ÞþcRn kz2 z1 kÞ. Z Oð1Þn2k ðRn ðRn r ÞÞk1. r. 1. k. k Y i¼1. Z. r2½ 2 ;R2n . r. t¼0. Z. 1rz enr ðzÞ. z2S. r. i¼3. Now, we calculate the integration over D Dr ð2Þ. Since ðz; rÞ ¼ 0 for any z 2 D Dr ð2Þ, n2 2. n. k. Z. k Z Y. t¼0. r. 2 12ðln nþÞ. 2k. Z. tdtdr. ec1 nrt tdtdr. Z. 1. r. r2½ 2 ;R2n . n. r. r r¼ 2 Rn 2. O ð1Þn2 e2ðln nþÞ. Z. r2 ¼ 2. Z. r¼ 2 1. Z. r. Rn 2. ¼ O ð1Þn2 e2ðln nþÞ. n2k. 2k. t¼0. 2 18nr2. O ð1Þe. 2. z2 2 B ðz1 ; ðk 1ÞRn Þ:. Thus,. enr ðzÞ 4rðz; rÞdzdr. Z. ri ðzi ; ri Þdzi dri ¼ o ð1Þ:. i¼1. for some constant c by Corollary 6, otherwise, 1rz ¼ 0, and. d r2. Rn 1 2 2 enr r 2e : n r¼ 2. Z. z2S. 1 r1 ðz1 Þ þ cRn kz2 z1 k r ðzÞ k R2n 4. Next, we calculate the integration over D Dr ð1Þ. Let t denote the distance from z to @D D. According to Lemma 4 and (1), there exist constants c1 and c2 such that r ðzÞ 1 2 2 r þ c1 rt and rðz; rÞ c2 t. Then, n2 2. k Y. 1rz enr ðzÞ. Note that, for any ðz1 ; z2 ; ; zk Þ 2 S, if 1rz ¼ 1,. enr 8rdzdr. 2. r¼ 2. . r. 2. z2D Dr ð0Þ. r¼ 2. 2n2. enr ðzÞ 4rðz; rÞdzdr. Z. Rn 2. Z. Z. r2½ 2 ;R2n k. Z. Rn 2. Z. n2k. n2k ðRn ðRn r ÞÞk1. Z. Z. Rn 2. ðnRn Þ2ðk1Þ r1 ¼ 2 z1 2D D r1 ðz1 ; r1 Þdz1 dr1 Rn r k1 ¼ Oð1Þ ð2e Þ ¼ oð1Þ: Rn r. enr1 ðz1 Þ.

(36) 124. IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS,. Here, ¼ kz2 z1 k, the second to last equality follows from Lemma 11, and the last equality is based on Rn r u t Rn ¼ oð1Þ. Proof of Lemma 13. For any m-partition ¼ fK1 ; K2 ; ; Km g of f1; 2; ; kg, let D Dk ðÞ denote the set of ðz1 ; z2 ; ; zk Þ 2 D Dk such that, for any 1 j m, the S Rn i2Kj Bðzi ; 2 Þ forms a connected component. Then, Ckm ðR2n Þ is the union of D Dk ðÞ over all m-partitions of f1; 2; ; kg. So, it is sufficient to show that, for any m-partition of f1; 2; ; kg,. n2k. Z. Z r. k. z2D D ðÞ. r2½ 2 ;R2n k. 1rz enr ðzÞ. k Y. ri ðzi ; ri Þdzi dri ¼ oð1Þ:. i¼1. VOL. 18,. NO. 1,. JANUARY 2007. Thus, 2 k Z Z k Y n enr ðzÞ 4ri ðzi ; ri Þdzi dri r R k 2 r2½ 2 ; 2n z2Ckk ðR2n Þ i¼1 2 k Z Z Pk n n ðz Þ i¼1 ri i e ¼ k r 2 r2½ 2 ;R2n z2Ckk ðR2n Þ k Y 4ri ðzi ; ri Þdzi dri i¼1. 2 k Z Z Pk Y k n n ðz Þ i¼1 ri i ¼ e 4ri ðzi ; ri Þdzi dri k r 2 r2½ 2 ;R2n z2D Dk i¼1 2 k Z Z Pk n n ðz Þ i¼1 ri i e k r 2 r2½ 2 ;R2n z2D Dk nCkk ðR2n Þ k Y 4ri ðzi ; ri Þdzi dri : i¼1. Now, fix an m-partition ¼ fK1 ; K2 ; ; Km g of f1; 2; ; kg. For 1 j m, let lj ¼ jKj j, and rðjÞ and let zðjÞ , respectively, denote the subsequence of r and z corresponding to Kj . Then,. j¼1. Rn ; 2. m X. ðjÞ. m Y. k. D D ðÞ. Clj 1. and for any z 2 D Dk ðÞ,. r ðzÞ ¼. rðjÞ ðz Þ:. Thus, n2k. Z r. r2½ 2 ;R2n . ¼ n2k ¼ n2k ¼ n2k. Z. ¼. k. z2D D ðÞ. Z r2½. Z. r R k n 2; 2. . r. r2½ 2 ;R2n . k. z inD D ðÞ. Z k. z2D D ðÞ. n. j¼1. lj. r2½ 2 ;R2n . Z. m Y. r. lj. Pm j¼1. rðjÞ ðzðjÞ Þ. ðjÞ enrðjÞ ðz Þ. k Y. k Y. j¼1. z2Clj 1 ðR2n Þ. Z. r2½ 2 ;R2n . ri ðzi ; ri Þdzi dri. z2Clj 1 ðR2n Þ. lj Y i¼1. 1rz enr ðzÞ. lj Y. ri ðzi ; ri Þdzi dri. ri ðzi ; ri Þdzi dri. ri ðzi ; ri Þdzi dri !. ri ðzi ; ri Þdzi dri. 2 k Z Z k Pk Y n en i¼1 ri ðzi Þ 4ri ðzi ; ri Þdzi dri r R k k 2 r2½ 2 ; 2n z2D D nCkk ðR2n Þ i¼1 Z k1 2 kZ k Pk Y X n en i¼1 ri ðzi Þ 4ri ðzi ; ri Þdzi dri r R k 2 r2½ 2 ; 2n z2Ckm ðR2n Þ m¼1 i¼1 ¼ oð1Þ; where the last equality can be proved by the arguments used in Lemma 12 and Lemma 13. u t. i¼1. ¼ oð1Þ; where the last equality follows from Lemma 12 and the u t fact that at least one lj 2. Rn Proof of Lemma 14. For any z 2 Ckk 2 , r ðzÞ ¼. 1 ri ðzi Þ k R2n : 4. Note that this is similar to Corollary 6, but with no need of the feasible condition. Thus,. i¼1. 1rz enr ðzÞ. k X i¼1. i¼1. Z r. 2lj. 1rz e. 1rz. r1 ðz1 þ cRn kz1 z2 k . i¼1. n. k. m Z Y j¼1. m Y. k. 1rz enr ðzÞ. k Y. 2 k Z Z Pk Y k n n ðz tÞ i¼1 ri i e 4ri ðzi ; ri Þdzi dri k r 2 r2½ 2 ;R2n z2D Dk i¼1 2 k Z Z k Y n ¼ enri ðzi Þ 4ri ðzi ; ri Þdzi dri r R k 2 r2½ 2 ; 2n z2D D i¼1 ! Z Rn Z k Y k n2 2 nr ðzÞ ¼ 4rðz; rÞdzdr 2e ; e r 2 r¼ 2 z2D D i¼1 where the last equality follows from Lemma 11. If ðz1 ; z2 ; ; zk Þ 2 Ck1 R2n satisfying that z1 is the one with largest norm among z1 ; ; zk and z2 is the one with longest distance from z1 among z2 ; ; zk , it can be proved that. j¼1. Z. We shall show that the first term is asymptotically equal to ð2e Þk , and the second term is asymptotically negligible. Indeed,. k X i¼1. ri ðzi Þ:. Proof of Lemma 15. Without loss of generality, assume i ¼ 1. In addition, assume j ¼ 2. (If 1 j t, the following proof still works. If t þ 1 j m t, the following proof works after some minor modifications.) Note that zi 2 Bðz1 ; kz2 z1 kÞ; for 3 i m t; z2 2 Bðz1 ; ðm t 1ÞRn Þ:.

(37) WAN AND YI: ON THE LONGEST EDGE OF GABRIEL GRAPHS IN WIRELESS AD HOC NETWORKS. [2]. (In the following, we assume t 2.) Thus, nm. Z. Z. r2½. r R t n 2; 2. m t Y. . !. enðri ðzi ÞþcRn kzi zj kÞ. t Y. z2S ij. ! rk ðzk ; rk Þdrk. k¼1. dzk. k¼1. Z. Oð1Þnm. !. Z. Rn 2 r. r1 ¼ 2. z 1 2D D. Z. enr1 ðz1 Þ r1 ðz1 ; r1 Þdz1 dr1. e t Z Y. !. Rn 2 r. ri ¼ 2. dz2 !. m t Z Y. ri dri. i¼3. Z. Z. Rn 2 r. z1 2D D. r1 ¼ 2. z2 2Bðz1 ;ðk1ÞRn Þ. Oð1Þn. t1 Rn ðRn r Þ. r1 ðz1 ; r1 Þdz1 dr1. Z. 1. Z. Z. Rn 2 r. r1 ¼ 2. z1 2D D. enr1 ðz1 Þ. [8]. t1 Z n Rn ðRn r Þ m. ¼ Oð1Þ. enr1 ðz1 Þ. encRn 2ðmtÞ3 d. 0. ðnRn Þ2ðmt1Þ. ½r ;R3 . Z z1 2D D. [9]. [10]. [11] [12]. enr1 ðz1 Þ. r1 ðz1 ; r1 Þdz1 dr1 t1 Rn ðn Þ nm2 ðR r Þn n ¼ Oð1Þ mt1 nmt1 nR2n ! Z Rn Z 2 2 nr1 ðz1 Þ n r1 ðz1 ; r1 Þdz1 dr1 e r r1 ¼ 2 z1 2D D ! ðn Þt1 ¼ Oð1Þ ðe Þ ¼ oð1Þ; lnmt1 n where the second to last equality follows from Lemma 11 and the last equality is based on n ¼ oðln nÞ and m t 1 1. u t. ACKNOWLEDGMENTS The work of P.-J. Wang is supported in part by the US National Science Foundation (NSF) under Grant 557904 and CityU of Hong Kong under Grant 7200031. The work of C.W. Yi was supported in part by the NSC under Grant NSC94-2218-E-009-030 and NSC95-2221-E-009-059-MY3, by MOE ATU Program, and by Intel JRP, and was partially done when the author visited the CityU of Hong Kong.. REFERENCES [1]. [6] [7]. r1 ðz1 ; r1 Þdz1 dr1 Z encRn kz2 z1 k kz2 z1 k2ðmt2Þ dz2 . [5]. dzi. zi 2Bðz1 ;kz2 z1 kÞ. Oð1Þnm ðRn ðRn r ÞÞt1. m. [4]. !. ncRn kz2 z1 k. z2 2Bðz1 ;ðmt1ÞRn Þ. i¼2. [3]. N. Li, J.C. Hou, and L. Sha, “Design and Analysis of a MST-Based Distributed Topology Control Algorithm for Wireless Ad-Hoc Networks,” Proc. 22nd Ann. Joint Conf. IEEE Computer and Comm. Soc. (INFOCOM ’03), vol. 3, pp. 1702-1712, Apr. 2003.. [13]. 125. Y. Wang and X.-Y. Li, “Localized Construction of Bounded Degree and Planar Spanner for Wireless Ad Hoc Networks,” Proc. 2003 Joint Workshop Foundations of Mobile Computing (DIALM-POMC ’03), pp. 59-68, Sept. 2003. J. Cartigny, F. Ingelrest, D. Simplot-Ryl, and I. Stojmenovic, “Localized LMST and RNG Based Minimum-Energy Broadcast Protocols in Ad Hoc Networks,” Ad Hoc Networks, vol. 3, no. 1, pp. 1-16, 2004. E.N. Gilbert, “Random Plane Networks,” J. Soc. for Industrial and Applied Math., vol. 9, no. 4, pp. 533-543, Dec. 1961. P. Gupta and P.R. Kumar, “Critical Power for Asymptotic Connectivity in Wireless Networks,” Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W.M. McEneaney, G. Yin, and Q. Zhang, eds., pp. 547-566, Birkhauser, Mar. 1998. H. Dette and N. Henze, “The Limit Distribution of the Largest Nearest-Neighbour Link in the Unit d-Cube,” J. Applied Probability, vol. 26, pp. 67-80, 1989. M.D. Penrose, “The Longest Edge of the Random Minimal Spanning Tree,” The Annals of Applied Probability, vol. 7, no. 2, pp. 340-361, 1997. G. Kozma, Z. Lotker, M. Sharir, and G. Stupp, “Geometrically Aware Communication in Random Wireless Networks,” Proc. 23rd Ann. ACM Symp. Principles of Distributed Computing, pp. 310319, July 2004. P.-J. Wan and C.-W. Yi, “Asymptotic Critical Transmission Ranges for Connectivity in Wireless Ad Hoc Networks With Bernoulli Nodes,” Proc. IEEE Wireless Comm. and Networking Conf. (WCNC ’05), Mar. 2005. C.-W. Yi, P.-J. Wan, X.-Y. Li, and O. Frieder, “Asymptotic Distribution of the Number of Isolated Nodes in Wireless Ad Hoc Networks with Bernoulli Nodes,” Proc. IEEE Wireless Comm. and Networking Conf. (WCNC ’03), Mar. 2003. M. Penrose, Random Geometric Graphs. Oxford Univ. Press, 2003. H. Zhang and J.C. Hou, “On the Critical Total Power for Asymptotic k-Connectivity in Wireless Networks,” Proc. 24th Ann. Joint Conf. IEEE Computer and Comm. Soc. (INFOCOM ’05), Mar. 2005. N. Alon and J.H. Spencer, The Probabilistic Method, second ed. Wiley, Mar. 2000. Peng-Jun Wan received the PhD degree from the University of Minnesota, the MS degree from The Chinese Academy of Science, and the BS degree from Tsinghua University. He is currently an associate professor of computer science at the Illinois Institute of Technology and the City University of Hong Kong. His research interests include wireless networks, optical networks, and algorithm design and analysis.. Chih-Wei Yi received the PhD degree from the Illinois Institute of Technology, and the MS and BS degrees from National Taiwan University. He is currently an assistant professor in the Department of Computer Science, National Chiao Tung University. His research focuses on wireless ad hoc networks. He is a member of the IEEE.. . For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib..

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