AALBORG UNIVERSITY. The distribution of communication cost for a mobile service scenario. Jesper Møller and Man Lung Yiu. R June 2009

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1 AALBORG UNIVERSITY The distribution of communication cost for a mobie service scenario by Jesper Møer and Man Lung Yiu R June 29 Department of Mathematica Sciences Aaborg University Fredrik Bajers Vej 7 G DK Aaborg Øst Denmark Phone: Teefax: URL: e ISSN On-ine version ISSN

2 The distribution of communication cost for a mobie service scenario June 13, 29 Jesper Møer, Department of Mathematica Sciences, Aaborg University, Fredrik Bajers Vej 7G, DK-922 Aaborg Ø, Denmark. Emai: jm@math.aau.dk. Man Lung Yiu, Department of Computer Science, Aaborg University, Sema Lageröfs Vej 3, DK-922 Aaborg, Denmark. Emai: my@cs.aau.dk. Abstract: We consider the foowing mathematica mode for a mobie service scenario. Consider a panar stationary Poisson process, with its points radiay ordered with respect to the origin (the anchor); these points may correspond to ocations of e.g. restaurants. A user, with a ocation different from the origin, asks for the ocation of the first Poisson point and keeps asking for the ocation of the next Poisson point unti the first time that he can be competey certain that he knows which Poisson point is his nearest neighbour. The distribution of this waiting time, caed the communication cost, is anaysed in detai. In particuar the expected communication cost and the asymptotic behaviour as the distance between the user and the anchor increases are of interest. Keywords: nearest-neighbour search; Poisson process; radia simuation agorithm; waiting time. 1 Introduction The mobie Internet offers services that e.g. receive the ocation of the nearest point of interest such as a store, restaurant, or tourist attraction, see Yiu et a. (28, 29) and the references therein. Toos from appied probabiity 1

3 and in particuar stochastic geometry shoud be usefu for an anaysis of the performance of such services. We iustrate this by considering a user ocated at a given point q R 2 and data points p 1, p 2,... R 2 corresponding to the ocations of e.g. stores. In order to preserve some privacy, the user queries a server for nearby points but he reports not his correct ocation q but another ocation q R 2 referred to as the anchor. An incrementa query processing on the server is used so that the data points are ordered in increasing distance to the anchor. The user then stops to queries the server as soon as possibe, i.e., when the nearest data point with respect to q can be determined. The waiting time for this to happen is caed the communication cost and is denoted M; further detais are given in Section 2. The purpose of this note is to anayse the distribution of M assuming that the data points foow a stationary Poisson process Φ = {p 1, p 2,...} with intensity ρ > (see, e.g., Kingman, 1993, or Chapter 3 in Møer and Waagepetersen, 24). In particuar the expected communication cost and the asymptotic behaviour as the distance between q and q increases are of interest. Section 3 discusses our resuts. 2 The communication cost We use the foowing notation. By stationarity and isotropy of Φ, we can without oss of generaity take q = (, ) and q = (, ), where > is the distance between the anchor and the user ocation. Let (R i, θ i ) [, ) [, 2π) be the poar coordinates of p i with respect to the anchor (the origin), and order the data points such that the R i are increasing, R := R 1 R Let p = p i if p i be the first NN (nearest-neighbour) to q among p 1, p 2,.... For i = 1, 2,..., et q i denote the first NN to q among the i first data points p 1,..., p i, and et R i = q q i denote the distance from q to q i. Moreover, et B(x, r) = {y R 2 : y x r} denote the cosed disc with centre x R 2 and radius r. Then the communication cost M is defined as the first time m such that the user can be competey ensured that p = q m when he has ony received p 1,..., p m from the server, i.e., no matter where the points p m+1, p m+2,... potentiay coud be ocated in R 2 \ B(q, R m ). In other words, if for m N, we define the demand space D m and the suppy space S m by D m = disc(q, R m ), S m = disc(q, R m ), then M = inf{m N : D m S m } (1) 2

4 (setting inf = ). Furthermore, p = q M (2) is returned as the NN to q, provided of course M <. See Figure 1. p 3 q p 1 p 2 q Figure 1: An exampe with M = 3, showing the corresponding demand space D 3 and suppy space S 3. 3 Resuts Since M = 1 impies that p 1 ies on the hafine H with endpoint q and direction q q, and this happens with probabiity zero, we have amost surey that M 2 and R M. Simiary, with probabiity one, for m {2, 3,...}, M = m impies that p m cannot be the NN to q among p 1,..., p m, because this woud impy that p m H. Moreover, with probabiity one, the sequence R 1, R 2,... is stricty increasing to infinity, and so the sequence of suppy spaces S 1 S 2... tends to R 2. On the other hand, the sequence of demand spaces decreases. Consequenty, by (1) and (2), with probabiity one, 2 M <, D 1 D 2... D M, and p = q M = q M 1, RM = R M 1, R M. (3) The distribution of M may be estimated by Monte Caro methods using a radia simuation agorithm due to Quine and Watson (1984). The radia simuation agorithm simpy utiizes the fact that the squared radia coordinates R1, 2 R2, 2... form a homogeneous Poisson process with intensity πρ on the positive hafine, so Ri 2 Ri 1, 2 = 1, 2,..., are independent and exponentiay distributed with mean 1/(πρ), and they are independent of the 3

5 anguar coordinates θ 1, θ 2,..., which in turn are independent and uniformy distributed on [, 2π). The foowing Lemma 1 foows straightforwardy from these properties. In Lemma 1, when we consider the first m data points and ignore their radia ordering, we write {x 1,..., x m } for the corresponding point configuration. Moreover, Unif([, 2π)) denotes the uniform distribution on [, 2π), and Γ(α, β) the gamma distribution with shape parameter α and inverse scae parameter β. Lemma 1 For any m N, we have the foowing. (i) R 2 m Γ(m, πρ) is independent of θ m Unif([, 2π)). (ii) Conditiona on R m+1 = r, the configuration of the first m points {x 1,..., x m } is a binomia point process on B(q, r) (i.e., the x i are independent and uniformy distributed on B(q, r)). (iii) Conditiona on R m+1 = r, we have that θ m Unif([, 2π)) is independent of R m, where R m /r B(1, m 1). If m 2 and we condition on both R m+1 = r and (R m, θ m ) with R m = t, then {x 1,..., x m 1 } is a binomia point process on B(q, t). We now turn to exact resuts for the distribution of the communication cost M, where its distribution function turns out to be more tractabe than its probabiity density function, and we can obtain an expression for the expectation EM. We need the foowing notation. Let f m+1 (r) = 2 (πρ)m+1 r 2m+1 exp ( πρr 2), r >, (4) m! denote the density function of R m+1, cf. (i) in Lemma 1. ϕ < 2π, et For r and v(ϕ, r) = ( r 2 2 sin 2 ϕ ) 1/2 cos ϕ. (5) Finay, for r s v(ϕ, r), define where g(r, s) = B(q, r) \ B(q, s) B(q, r) = 1 h(r, s)/ ( πr 2) (6) h(r, s) = B(q, r) B(q, s) (7) ( ) = r r 2 s 2 arccos 2 + r 2 s 2 [ 4 2 r 2 ( 2 + r 2 s 2) ] 2 1/2 2r 4 ( ) 2 + s s 2 r 2 arccos 2 + s 2 r 2 [ 4 2 s 2 ( 2 + s 2 r 2) ] 2 1/2. 2s 4 2 4

6 Here denotes area (Lebesgue measure), and we suppress in the notation that the functions in (5)-(7) depend on >. Proposition 1 For m = 2, 3,..., we have that and P(M > m) = EM = 2 + πρ 2 is finite. m f m+1 (r) dr+ 2π v(ϕ,r) 2π v(ϕ,r) r r s πr 2 g(r, s)m 1 f m+1 (r) ds dϕ dr (8) [ 1 + πρr 2 ] exp ( πρr 2) dr+ (9) 2πρ 2 sr [ exp( ρh(r, s)) exp ( πρr 2)] ds dϕ dr Proof Using (1) and (3), considering each of the cases R m+1 < and R m+1, we see that the probabiity that M > m is equa to the sum of two terms, viz the term and the term P(R m ) = f m+1 (r) dr m P(x i is the NN to q among x 1,..., x m and i=1 By (ii) in Lemma 1, B(q, x i q ) B(q, r) R m+1 = r)f m+1 (r) dr. P(x i is the NN to q among x 1,..., x m and (1) B(q, x i q ) B(q, r) R m+1 = r) ( ) B(q m 1, r) \ B(q, x q ) 1 = dx. (11) B(q, r) B(q, r) B(q,r)\B(q,r ) Hence, by making a shift of coordinates from x = q + s(cos ϕ, sin ϕ) 5

7 to the poar coordinates (s, ϕ), we obtain (8) after a straightforward cacuation, noticing the foowing facts: in (11) we can set q = (, ) and q = (, ); for fixed r > and ϕ [, 2π), since x B(q, r) \ B(q, r ), we obtain that s ranges from r to v(ϕ, r); here the atter bound foows from the equation (, ) + s(cos ϕ, sin ϕ) = r, which has ony one soution because s >. See aso Figure 2. q' r x i s ϕ q Figure 2: Iustration of the notation used in the proof of Proposition 1. The function v(ϕ, r) is the upper bound on s considering a case with i data points such that x i = q i is the nearest-neighbour to q, B(q, r) is the corresponding suppy space, and B(q, s) is the corresponding demand space. Since M 2 is a non-negative discrete random variabe, we have that EM = 2 + m=2 P(M > m). Combining this with (8), we obtain EM = 2 + f m+1 (r) dr + m=2 2π k(ϕ,r) r s πr 2 mg(r, s) m 1 f m+1 (r) ds dϕ dr. m=2 From this and (4), we easiy obtain (9). Since k(ϕ, r) r +, it foows from (9) that EM is finite if for some number r >, I := r+ r r sr [ exp( ρh(r, s)) exp ( πρr 2)] ds dr is finite. This is indeed the case, since for any ɛ >, if r > +ɛ is sufficienty arge, then h(r, s) π(r ɛ) 2 whenever r r and r s r +, and so I 2 r r 2 exp ( πρ(r ɛ) 2) dr 2 6 r r 2 exp ( πρr 2) dr

8 where both integras are finite. The integras in (8) and (9) can be evauated by numerica methods. In (9), as increases towards infinity, EM is dominated by the term πρ 2, meaning that the dominating part of the expected communication cost is proportiona to ρ and 2. It is worth noticing that πρ 2 is the expected number of points in the disc b(q, ). Note that the distribution of M depends ony on (ρ, ) through ρ. Figure 3 visuaizes the difference between EM and its dominant term πρ 2 (shown as O ), where for different vaues of and with ρ fixed to 1, EM has been estimated both by using (9) and a numerica method (shown as ) and by the Monte Caro method based on 1 independent simuations obtained by the radia simuation agorithm (shown as ). The axes in Figure 3 are shown in the og-scae, the estimates of EM obtained by the two methods are rather cose, and the dominating term of EM is seen to converge to EM as. The figure aso shows the Monte Caro estimates of the 5% and 95% quanties (shown as + ). Figures 4 and 5 show further Monte Caro estimates with ρ = 1 and based on 1 independent simuations obtained by the radia simuation agorithm for each tested vaue of. Figure 4 suggests an approximate oginear reation between VarM/EM and. Considering vaues of 2, the regression ine in Figure 4 was estimated by the method of east squares to y =.5524x.9147, where (x, y) corresponds to (og 1, og 1 VarM/EM) (og 1 denotes the ogarithm function with base 1). Figure 5 shows four histograms of the distribution of M corresponding to the simuations with =.1, 1, 1, 1. For the arger vaues of, the distribution of M seems to be we approximated by a Gaussian distribution. Acknowedgements: Supported by the Danish Natura Science Research Counci, grant , Point process modeing and statistica inference. References [1] J. F. C. Kingman. Poisson Processes. Carendon Press, Oxford, [2] J. Møer and R. P. Waagepetersen. Statistica Inference and Simuation for Spatia Point Processes. Chapman and Ha/CRC, Boca Raton, 24. 7

9 [3] M. P. Quine and D. F. Watson. Radia simuation of n-dimensiona Poisson processes. Journa of Appied Probabiity, 21: , [4] M. L. Yiu, C. S. Jensen, X. Huang, and H. Lu. SpaceTwist: Managing the trade-offs among ocation privacy, query performance, and query accuracy in mobie services. In M. Casteanos, A. P. Buchmann, and K. Ramamritham, editors, Proceedings of the 24th IEEE Internationa Conference on Data Engineering, pages IEEE Computer Society, Los Aamitos, CA, 28. [5] M. L. Yiu, C. S. Jensen, J. Møer, and H. Lu. Design and anaysis of an incrementa approach to ocation privacy for ocation-based services. Submitted for pubication, 29. Number of received points % and 95% quanties of simuated cost.1 Average of simuated cost Numeric vaue of E(M) Dominating term of E(M) = dist(q,q') Figure 3: Estimates of EM vs. when ρ = 1 and EM is estimated by the resut in Proposition 1 using a numerica method (shown as ) and by the Monte Caro method (shown as ), together with the dominant term πρ 2 (shown as O ) and Monte Caro estimates of the 5% and 95% quanties (shown as + ). 8

10 sqrt(varm)/em Figure 4: Monte Caro estimates of VarM/EM vs. when ρ = 1. 9

11 = = = Figure 5: Histograms of the distribution of M corresponding to 1 independent simuations with ρ = 1 and =.1, 1, 1, 1. =1 1