Trap Coverage: Allowing Coverage Holes of Bounded Diameter in Wireless Sensor Networks


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1 Tra Coverage: Allowing Coverage Holes of Boune Diameter in Wireless Sensor Networks Paul Balister Zizhan Zheng Santosh Kumar Prasun Sinha University of Memhis The Ohio State University Abstract Tracking of movements such as that of eole, animals, vehicles, or of henomena such as fire, can be achieve by eloying a wireless sensor network. So far only rototye systems have been eloye an hence the issue of scale has not become critical. Reallife eloyments, however, will be at large scale an achieving this scale will become rohibitively exensive if we require every oint in the region to be covere (i.e., full coverage), as has been the case in rototye eloyments. In this aer we therefore roose a new moel of coverage, calle Tra Coverage, that scales well with large eloyment regions. A sensor network roviing Tra Coverage guarantees that any moving object or henomena can move at most a (known) islacement before it is guarantee to be etecte by the network, for any trajectory an see. Alications asie, tra coverage generalizes the efacto moel of full coverage by allowing holes of a given maximum iameter. From a robabilistic analysis ersective, the tra coverage moel exlains the continuum between ercolation (when coverage holes become finite) an full coverage (when coverage holes cease to exist). We take first stes towar establishing a strong founation for this new moel of coverage. We erive reliable, exlicit estimates for the ensity neee to achieve tra coverage with a given iameter when sensors are eloye ranomly. Our ensity estimates are more accurate than those obtaine using asymtotic critical conitions. We show by simulation that our analytical reictions of ensity are quite accurate even for small networks. We then roose olynomialtime algorithms to etermine the level of tra coverage achieve once sensors are eloye on the groun. Finally, we oint out several new research roblems that arise by the introuction of the tra coverage moel. I. INTRODUCTION Several romising alications of wireless sensor networks with a high otential to imact human society involve etection an tracking of movements. Movements may be of ersons, animals, an vehicles, or of henomena such as fire. Examles inclue tracking of thieves fleeing with stolen objects in a city, tracking of intruers crossing a secure erimeter, tracking of enemy movements in a battlefiel, tracking of animals in forests, tracking the srea of forest fire, an monitoring the srea of cro isease. So far only rototye systems have been eloye an hence the issue of scale has not become critical. Reallife eloyments, however, will be at large scale, an achieving this scale will become rohibitively exensive if we require every oint in the region to be covere (i.e., full coverage or blanket coverage [8]), as has been the case in rototye eloyments [3], [6], [2]. The requirement of full coverage will soon become a bottleneck as we begin to see reallife eloyments. In this aer, we therefore roose a new moel of coverage, calle Tra Coverage, that scales well with large eloyment regions. We efine a Coverage Hole in a target region of eloyment A to be a connecte comonent of the set of uncovere oints of A. A sensor network is sai to rovie Tra Coverage with iameter to A if the iameter of any Coverage Hole in A is at most. For every eloyment that rovies tra coverage with iameter of, the sensor network guarantees that every moving object or henomena of interest will surely be etecte for every islacement that it travels in A. At any instant, we can either in oint the location of a moving object recisely, or can oint to a coverage hole of iameter at most in which it is trae. With this moel, the ensity of sensors can be ajuste to meet the esire quality of tracking while economizing on the number of sensors neee. Large scale sensor eloyments for tracking thus become economically feasible with this new moel of coverage. Figure shows an examle eloyment region where the size of the largest uncovere region is. Hole Hole iameter = Hole Fig.. In this eloyment, is the iameter of the largest hole. Notice that although the iameter line intersects a covere section, it still reresents the largest islacement that a moving object can travel within the target region without being etecte. Tra Coverage Generalizes Full Coverage: If the value of is set to 0, then tra coverage is equivalent to full coverage. By relaxing the requirement of having every oint covere, tra coverage generalizes the moel of full coverage. Traitionally, the fraction of target region that is covere has been use as an inicator of the quality of coverage [3], [25]. Notice that even if a large fraction of region is covere, the iameter of the largest hole may be arbitrarily large. Therefore, tra coverage may better inicate the Quality of Full Coverage as it rovies a eterministic guarantee in the worst case. Here connecte refers to the connectivity of a set of oints in the real lane that comrise the target region.
2 2 II. KEY CONTRIBUTIONS AND ROADMAP In aition to introucing a new moel that generalizes the traitional full coverage moel, we make several contributions in this aer, some of which may be of ineenent interest. First, we erive a reliable estimate of the ensity (similar as in [3]) neee to achieve tra coverage with a esire iameter when sensors are eloye ranomly. Roughly seaking, the critical ensity conition is of the form λ(2r + πr 2 ) log n, () where λ is the execte ensity of sensors er unit area, r is the sensing range, an n = λ A is the execte total number of sensors in the target region A. In other wors, we exect that having, on average, log n sensors in the r neighborhoo of a thin long hole of iameter will suffice for achieving tra coverage with a iameter of. We also show how our estimate for the ensity can be aate to a nonisk moel of sensing region, by using ellises of ranom orientation as an examle. (Section IV) Secon, the moel of tra coverage exlains the ga that has long existe between the ercolation threshol (when holes become finite an isolate) an the critical ensity for achieving full coverage (when holes cease to exist). Looking at (), we can observe that if r is constant w.r.t. n, which is the case for ercolation to occur, is of the orer of log n, matching the known behavior that for fixe λr 2 above the ercolation threshol, the maximum hole iameter is on average of orer log n. On the other han, if is a constant, an 0 in articular, then λr 2 is of the orer of π log n, matching the known behavior for achieving full coverage [8]. Thus, the tra coverage moel not only generalizes the moel of full coverage, it also hels exlain the robabilistic behavior of coverage between the ercolation threshol an critical ensity for full coverage. (See Figure 6 for an illustration.) Once sensors have been eloye on the groun (either ranomly or eterministically), it may be necessary to etermine the level of tra coverage that they rovie, since some may fail at or after the eloyment for unforeseen reasons. Our thir contribution, therefore, is olynomial time algorithms to etermine the level of tra coverage that an arbitrary eloye sensor network rovies. Our algorithms not only works for nonconvex moels of sensing regions, but also when sensing regions are uncertain (e.g., robabilistic sensing moels). Further, they take into consieration the comlications that may arise ue to the bounary of the eloyment region (see Figure 8 for an examle). (Section V) III. RELATED WORK Most work on robabilistic ensity estimates for coverage assume the full coverage moel [8], [24], [30]. As we show in Section IV, the naïve aroach of increasing the sensing range by an then eriving the conitions for full coverage will lea to overeloyment, no matter how small the value of >0 is. For larger, overeloyment will be orers of magnitue more than neee in our estimates. Work on full coverage that oes consier holes focuses on the fraction of region that is (un)covere, see [24], [30]. They attemt to asymtotically minimize the area of vacant region an o not rovie any simle exression for the ensity neee in a ranom eloyment to achieve a esire fraction of uncovere region. Even if there existe such an exression, it coul not be use to reaily erive an estimate of ensity neee for bouning the iameter of coverage holes. This is because holes of large iameter ten to be long an thin, an their area is not tyically large (even close to zero). Perhas, the work closest to tra coverage are [8], [] that allow holes for surveillance alications. Here the quality of surveillance metric is base on the istance that a moving target, starting at a ranom location, moving in a ranom irection can travel in a straight line before it is etecte by a sensor. In [8], istance to etection by a giant connecte comonent is also stuie. There are several issues with such a metric. For one, they o not rovie any worst case guarantee on how far a target can move before being etecte, unlike tra coverage. For examle, if the ensity chosen is just large enough that a giant comonent exists almost surely, as in [8], the hole iameters are not boune by any constant; they grow as a function of log n where n is the number of sensors eloye. Further, even though the average istance may be boune, even close to zero, the worst case istance coul be arbitrarily large (as show in Figure 2). As shown in a tyical eloyment (Figure 4), holes that have larger iameters are usually thin an long, so the average istance measure is quite likely to be misleaing. Therefore, neither of these metric can be use to erive a ensity estimate for tra coverage. In summary, there oes not exist any work that can be use to erive estimates of ensity (or even critical conitions) neee in a ranom eloyment to achieve tra coverage of a given iameter, a mathematically challenging roblem that we aress comrehensively in this aer. We ostone iscussing existing work relate to algorithmic etermination of the status of tra coverage to Section VA. IV. ESTIMATING THE DENSITY FOR RANDOM DEPLOYMENTS In this section, we erive a reliable estimate for ensity that will ensure tra coverage of a given iameter. We take a rogressive aroach in eriving our estimate for simlicity of exosition. We first consier a isk moel of sensing. For this moel, we first erive a crue but rigorous boun that may aeal to intuition. We then show that large holes occur with a Poisson istribution. In Section IVA, we estimate the intensity of this Poisson istribution. Once we have an accurate estimate of the intensity with which large holes occur, we can accurately etermine the ensity neee to achieve tra coverage of a given iameter with any given robability (such as with robability ). We show in Section IVB that our ensity estimate is accurate even for small eloyment regions, a significant imrovement over asymtotic critical ensities that work only for large eloyments. Finally, we show in Section IVC, how our erivations can be aate
3 3 R L Fig. 2. Region R an line L in roof of lower boun on P(h m ). L is uncovere an so forms a long thin hole rovie R is voi of any sensors. q q q γ to nonisk sensing moels. We rovie the erivation for ranomly oriente ellises as an examle. We consier a Poisson eloyment with intensity λ in a eloyment region A that inclues a large target region A of area A. Write n = λ A for the execte number of sensors within the target region, an h m for the maximum hole iameter. Before we obtain a boun on the robability that h m, we make some remarks on the effect of the bounary. Generally seaking, if the eloyment region A is the same as the target region A, then coverage is more likely to fail at the bounary than in the interior (see [3]). Thus a similar result woul be execte to occur for tra coverage, at least when /r is small. One simle way of avoiing roblems at the bounary is to enlarge A so that it inclues all oints within istance r of A. (We shall assume in the following that the bounary of A is small, i.e., A (r +) A. Thus enlarging the eloyment region as above will not increase its area much, i.e., A / A.) This makes coverage of oints on the bounary of A as likely as oints in the interior, an large holes are no more likely to aear at the bounary than in the interior (in fact less likely since there is less area near the bounary than the interior, an holes are confine to lie insie A). In the following analysis we shall assume that the eloyment region has been enlarge in this manner. We first erive a lower boun on P(h m ). LetL be a straight line of length insie A. If there is no sensor within istance r of L then L lies in the interior of a hole, which then must have iameter at least. LetR be the set of oints within istance r of L. Then R consists of a 2r rectangle with two semicircular cas of raius r attache to each en (see Figure 2). The robability that R contains no sensor is e λ R where R = 2r + πr 2. We can lace R insie a 2r ( +2r) rectangle which has area less than 2 R. Thus if A is large enough an of a reasonable shae (in articular, if it has small bounary as mentione above), we can ack at least A /(2 R ) =n/(2λ R ) isjoint coies of R into A.The event that one coy of R is evoi of sensors is ineenent of any of the other coies, so the robability that the maximum hole iameter is at least is boune below by the robability that at least one of the coies of R is emty. Thus P(h m ) ( e λ R ) n/(2λ R ) e I A, where I = (2(2r + πr 2 )) e λ(2r+πr2). (2) (Here we have use the fact that x e x. The quantity I is essentially a boun on the average number of holes of iameter er unit area.) If we write λ(2r + πr 2 )=λ R = log n log log n t, (3) Fig. 3. Left: calculation of the area of R γ(s). Right: Examle of selfoverlaing R γ(s) with s = r. R is lightly shae region, R 2 is heavily shae region. If γ aroaches within 2( 3 )r >rof itself, then one can shorten γ by cutting across along ashe line q. e t log n 2(log n log log n t) = then for t = t(n) =o(log n), I A = (.5+o())e t.ift as n we have I A an thus P(h m ). Now, we give an uer boun on P(h m ), which is more involve. Suose a hole H of iameter h m exists. Suose x, y H are oints with x y = an let γ be the shortest ath from x to y insie the hole H. We may assume that x lies at a crossing oint of the bounaries of the sensing regions of two sensors (see Lemma 5. below). Note that γ consists of straight line segments ossibly joine together with arcs of circles of raius r. In articular, the raius of curvature of γ at any oint is never less than r. Lemma 4.: Suose 0 <s r. Then the set R γ (s) of oints that lie within istance s of γ has area at least s( γ + )+πs 2, where γ is the arc length of the curve γ. Proof: Suose first that R γ (s) oes not wra aroun on itself, i.e., no oint on R γ (s) is istance s from more than one oint of γ (see Figure 3). Then the area of R γ (s) is exactly 2s γ + πs 2. To see this, cut γ into small segments each of (aroximately) constant raius of curvature, an make corresoning cuts in R γ (s) orthogonally to γ at the laces where γ is cut. Suose one segment of γ has raius of curvature R an subtens an angle δθ. The length of this segment is Rδθ, while the area of the corresoning slice of R γ (s) is 2 (R +s)2 δθ 2 (R s)2 δθ =2sRδθ (the ifference between sectors of two isks). Aing u these areas for each segment of γ givesanareaof2s γ, an aing the two halfisks centere at the enoints of γ gives the result. Now assume R γ (s) selfintersects. Then the above argument will overestimate the area. However, istant arts of γ cannot aroach too closely. Inee, suose there are two oints an q on γ such that q an the istance between an q is a local minimum for oints on γ. Then there are sensors at, q with, q lying on the segment q an γ following the bounaries of the sensor regions of an q (see Figure 3). No sensor on the oosite sie of γ to an q can have a sensor region intersecting the sensor regions of or q,butif q < 2 3r this imlies no sensor region intersects the line segment q. Thus if q < 2( 3 )r the line segment from to q is uncovere by any sensor an γ can be shortene by joining across from to q, contraicting the assumtion that γ was the shortest ath from x to y. A similar argument shows that no oint can lie in a trile self
4 4 intersection of R γ (s). Inee, if w is such a oint an, 2, 3 are istinct locally closest oints on γ, then there are sensors at i, where i lies on the segment w i an γ follows the bounary of the sensor region of i near i.ifanyofthe istances i j, i j, are less than 2 3r, then γ may be shortene. But if all i j 2 3r then their sensor regions o not intersect, an so w oes not exist. Thus of the area R γ (s), no art can be more than ouble counte by the estimate 2s γ +πs 2 above. In other wors, we can write R γ (s) as the union of two regions R an R 2, with R +2 R 2 =2s γ + πs 2. Now any line L erenicular to xy between x an y must intersect R in line segments of total length at least 2s since no oint on L before the first oint of γ or after the last oint of γ can be in a selfintersection of R γ (s). AlsoR contains two halfisks at x an y. Thus R 2s + πs 2 an R γ (s) = R + R 2 = R /2+( R +2 R 2 )/2 s( γ + )+πs 2 as require. Now aroximate γ with a ath γ that is mae u from a sequence of arcs of circles, each of raius r/2 an length rε (so they curve by an angle of 2ε). Each arc curves either to the left or the right. One can show that γ can be chosen so that it starts at x, the angle that γ makes with the horizontal at x is a multile of ε, an all oints of γ are within istance Crε 2 of γ, where C is some absolute constant. Hence there is no sensor within istance r( Cε 2 ) of γ. Given x, there are (2π/ε)2 k choices for γ when γ consists of k segments. Given γ, one knows γ to within istance Crε 2, so icking any γ consistent with γ, we know R γ (r( 2Cε 2 )) contains no sensors. Since the length of γ an γ agree to within a factor of +O(ε 2 ),anyγ gives us a region of area (r 2 kε + r + πr 2 )( C ε 2 ) evoi of sensors, so the robability of some such γ existing starting from x is at most k /rε (2π/ε)2k e λ(r2 kε+r+πr 2 )( C ε 2 ) 2π )( C ε 2 )+(/rε)log2 ε( 2e λr2ε/2 ) e λ(2r+πr2 Setting ε =(λr 2 ) 2/3 an assuming λr 2, thisisatmost C (λr 2 ) 2/3 e λ(2r+πr2 )( O((λr 2 ) 2/3). (4) The execte number of intersection oints in A we can choose for x is 4λπr 2 n, so we obtain P(h m ) C (λr 2 ) 5/3 ne λ(2r+πr2 )( O((λr 2 ) 2/3 ) (5) for some constant C.Forλr 2 = O(log n), this tens to 0 when λ(2r + πr 2 )( O((λr 2 ) 2/3 )) log n + O(log log n). Combining this with the lower boun (3) above, we see that the maximum hole size h m = tyically occurs when λ(2r + πr 2 )( O((λr 2 ) 2/3 )) = log n, (6) (the O((λr 2 ) 2/3 ) error term swallowing the log log n terms in both cases). We observe that (from both the lower an uer bouns above) the holes with the largest iameter are long an thin, basically being obtaine by insisting that Fig. 4. Examle of Poisson eloyment. Rectangle enotes target region. Notice that holes of larger iameters are tyically long an thin, although this nee not be true for smaller iameter holes. an almost straight ath γ of length is not covere by any sensing region. We show in Figure 4, a reresentative Poisson eloyment for which some holes exist. Note that although the holes are of various shaes, the holes with the largest iameters are usually long an thin, confirming our analytical conclusion. Comarison with an obvious extension of the full coverage moel. Note that our estimate is significantly better than the naïve boun obtaine by increasing r by an then emaning that this rovies full coverage. Inee, our boun (assuming λr 2 ) isoftheform λ(2r + πr 2 ) log n, (7) while if we require full coverage with sensing range r + we woul nee (relacing by 0 an r by r + in (7)) λπ(r + ) 2 = λ(π 2 +2πr + πr 2 ) log n. Even for small we woul unerestimate by a factor of π (2πr vs. 2r), an for large the iscreancy tens to ( c log n vs. cr log n for fixe λ). Note that enlarging the sensor range by /2 is not sufficient in general to eliminate all holes of iameter, but even if it were, the (incorrect) boun obtaine on woul still always be worse than our result. The reason for the iscreancy between our estimate an the naïve boun however becomes clear when we observe that a long thin hole can be covere with just a small increase in r, rather than increasing it by. Estimating the Probability Distribution of Large Holes. Large holes, when they exist, shoul be well searate, so one woul exect the istribution of the number of holes with iameter to follow an aroximately Poisson istribution. This is inee true for large λr 2. To show this, suose H is a coverage hole. Then H eens on the Poisson rocess within aregionh consisting of all oints at istance r from H. To show the number of holes is aroximately Poisson, one can use the SteinChen metho (see []). In our case, it reuces to showing (a) that the execte number of airs of holes H an H 2 for which H an H 2 intersect is o(), an (b) that this woul also be true if the H i were truly ineenent. Conition (b) is easy to show since the H i are much smaller than A. Conition (a) hols since conitione of the state of the Poisson rocess in H, it is unlikely there is a hole close by. (Effectively this reuces to showing holes are rarely near
5 5 the bounary of a eloyment region R 2 \ H, which hols since the bounary of H is tyically not large.) We refer the reaer to [4] for more etails of these calculations. As a result, for sufficiently large λπr 2 P(h m ) e I A, where (8) I = λe λ(2r+πr2 )( O((λr 2 ) 2/3), I being the execte number of holes of iameter at least er unit area (i.e., the intensity of the Poisson rocess for the occurrence of holes of iameter ). Once again the O() error term in I swallows the olynomial factors in front of the exonentials in the uer an lower bouns given above. We shall refine this estimate in the next section. A. Refining the Estimate In this section we shall give a much more accurate estimate for the robability of occurrence of holes of iameter. We only rovie an outline of our erivation here an efer the etaile roofs to [4]. To obtain an imrove estimate, we comare the tra coverage moel with that of barrier coverage, where sensors are eloye in a long (but 2 imensional) horizontal rectangular stri S h of height h, an one asks whether there are coverage holes crossing the stri (see [3] for etails). We shall count the number of holes that cut across this stri in two ifferent ways, leaing to a comarison between barrier coverage an tra coverage. First let I tra be the number of holes of iameter at least er unit area an assume u, v are enoints of such a hole with u lying below v. Then since the holes are tyically long an thin, this hole will cut across S h rovie u an v lie on oosite sies of S h.letθ be the angle uv makes with the vertical, an x the istance of u below the bottom of S h (see Figure 5). Then we nee u v (x + h)/ cos θ. The intensity I of such holes er unit length along S h is therefore given aroximately by π/2 I I tra π (x+h)/ cos θ xθ. π/2 0 To relate this to I tra at a articular value of, we note that by our simle estimates in the revious section that I tra ecays exonentially with, I tra +ε Itra e 2λrε. Using this aroximation (an evaluating the xintegral) gives I I tra h 2πλr π/2 π/2 e 2λrh(/ cos θ ) cos θθ I tra h (4πλ2 r 2 (λrh )) /2, where the last aroximation is vali for large λrh. Now we evaluate I by comarison with barrier coverage. A hole across S h results in a break as efine in [3], however when efining barrier coverage one assumes eloyment only insie the stri S h. Thus for a break to efine a hole crossing S h, we also nee that sensors outsie of S h o not estroy the break. From the results in [3] we know that most breaks are aroximately rectangular an thin cutting erenicularly h v S h x θ u Fig. 5. Left: hole with iameter uv crossing stri S h. Right: aitional vacant semicircular areas allow break to form hole. across S h. Using this it follows that for this break to make a hole, one nees at least one oint on the to bounary of S h insie the break to be uncovere by sensors outsie of S h, an similarly at least one oint on the bottom bounary of S h to be uncovere (see Figure 5). One can show that the robability of some oint on the to bounary of S h in a fixe interval of length W to be uncovere by sensors above S h is aroximately ( + λrw )e πλr2 /2. One may assume the to an bottom bounaries are ineenent for large h (in fact λh 3 r is enough), so this gives I Ih barrier ( + λre(w )) 2 e πλr2, where E(W ) is the execte with of the uncovere interval on the bounary of S h that occurs at a break, an Ih barrier is the average number of breaks er unit istance along S h. One can show using the techniques of [3] that E(W ) cλ 2/3 r /3 with c Also [3] gives the following estimate for Ih barrier. Ih barrier λ 2/3 (2r) /3 e 2λr( α(4λr2 ) 2/3 )+β. where α.2794 an β.056. (Note that the value of r in [3] is twice the sensor raius.) Putting these together gives the following aroximation for I tra. I tra C 0 λ(λr 2 ) 2/3 ( + c(λr 2 ) /3 ) 2 (λr )/2 e 2λr( α(4λr2 ) 2/3 ) πλr 2, (9) where C 0 = π /2 2 4/3 e β.56, α.2794, c As in [3], this estimate shoul be vali for λ 3 r, an λr 2, which in our context means not too close to either full coverage πλr 2 log n or the ercolation threshol λr 2 constant. Since coverage holes of iameter follow Poisson istribution (using the same SteinChen argument as in the revious section), we have P(h m ) e A Itra (0) when I tra is small. B. Simulation to Valiate Our Density Estimates In this section, we resent some simulation results to suort our analytical results. We consier a eloyment region A of size 256r 256r, where we lace oints accoring to Poisson rocess of intensity N. WevaryN from 0 to 500, 000 an track the maximum coverage hole iameter. We reeat our exeriment 0, 000 times for each value of N for statistical accuracy. We also ran simulations with smaller A, obtaining very similar results even own to a 8r 8r region. We have two istinct goals in our simulation.
6 6 6 A = [0,256r] μ λr2 6 λr A = [0,8r] μ λr2 Fig. 6. Mean size of largest hole (μ, left han scale) together with estimate base on (9) an (0) (otte line). Probability (right han scale) that hole size becomes finite ( 0), i.e., ercolation occurs, an robability that holes cease to exist ( ), i.e., full coverage occurs..) Valiating the accuracy of our analytical estimates. We show results of our simulation in Figures 6 an 7. We first exlain our rationale for icking the various axis before exlaining the results. For xaxis in Figure 6, we use λr 2,a imensionless arameter which inicates the level of coverage. (Each oint is covere by an average of πλr 2 sensing regions.) We have two arameters for the yaxis. On the left scale, we use λr, a imensionless quantity to measure the hole iameter, which also haens to be the xaxis in Figure 7. Since ecreases with an increase in λ or r, using this unit allows us to resent the entire sectrum of variation in the hole iameter in one grah. The right scale of yaxis in Figures 6 an the left scale in Figure 7 are robabilities. Note that the only quantity fixe in Figure 6 is the size of A relative to r. We observe that the mean value of the maximum hole iameter observe in simulation (soli line) is mostly inistinguishable from our analytical estimates (otte line) for 256r 256r region an quite close even for the 8r 8r region, which is smaller than many reallife eloyments. In Figure 7, we show the entire robability istribution for hole iameters for some ensities, which rovies significantly more information than the mean values of iameter. This confirms that our estimate of the robability istribution of hole iameters (to Poisson) an our estimation of the arameter of this istribution are highly accurate, making it quite useful in reallife eloyments. 2.) Grahically emonstrating the new continuum from ercolation to full coverage. Figure 6 illustrates how the moel of tra coverage fills the continuum between ercolation an full coverage. The curve labele 0 eicts the robability of ercolation, i.e., largest hole iameters becoming smaller than the eloyment region. As the ensity increases, hole iameter ecreases. The curve labele eicts the robability of full coverage. As this curve aroaches, the execte largest hole iameter aroaches zero. Note that the value of λr 2 corresoning to 0 reresents ercolation threshol, while that corresoning to reresents critical conitions for full coverage. Until this result of ours, the behavior in between these two imortant values of λr 2 was unknown. The introuction of the tra coverage moel in this aer now exlains the continuum 5 4 A = [0,256r] λr Fig. 7. Cumulative robability istribution, P(h m ), oflargest hole size for λr 2 =,...,6, together with estimate base on equation (9) an (0) (otte line). For examle, if λ =,anr =2 (so λr 2 =4), then from Figure 6, left sie, we have λr 2 on average (so ), however it can range between about an 6 ( =0.5 to 3) with a robability istribution as shown here. between these two imortant curves comrehensively, with the curve for the tra coverage iameter. C. Extening to Nonisk Sensing Regions The above analysis assumes that the sensing regions are isks. However, it is clear from the lower boun argument for P(h m ) that we can generalize this to other shaes of sensing region. To recall, for the lower boun we require that no sensing region intersects a line L. The robability that this occurs can be calculate for any require (even robabilistic) moel of the sensing region. The fact that the uer boun for isks is close to the lower boun suggests that this will also hol for most reasonably isklike sensing regions. As an examle, we consier the case of ranomly oriente ellises (to moel biase gain along a ranomly oriente axis). Lemma 4.2: Suose the sensing regions are ellises, each with maximum an minimum raii r an αr resectively, an with orientation that is ranom an uniform. Then the execte number of sensor regions meeting a fixe line L of length is given exactly by λ(πr 2 α +2r 2 π E( α2 )), () where E(m) = π/2 ( m sin 2 θ) /2 is an ellitic integral. 0 Proof: Consier the sensors whose smaller raius lies in some small angle [θ, θ + θ] from the irection of the line L. These sensors occur as a Poisson rocess of intensity λ 2π θ. If we scale the lane by stretching by a factor /α in the irection of the smaller raius, the sensor regions become circular with raius r, while the ensity of sensors is now θ. The line L is now also stretche, an has a new length = (α 2 cos 2 θ +sin 2 θ) /2. The execte number of these sensors meeting L is therefore equal to α λ 2π (πr 2 +2r )α λ 2π θ = λ 2π (πr2 α +2r( ( α 2 )sin 2 θ) /2 )θ. The result follows by integrating this from θ =0to 2π. Note that since we are assuming Poisson eloyment, the hysical location of the sensor within the ellise is irrelevant (as long as it is ineenent of the location an orientation 2
7 7 of the ellise), so we may for examle assume the sensor is at the center, or at a focal oint, or at one en of the ellise. The results will be ientical in all cases. The lower boun argument for P(h m ) follows exactly as before, using () in lace of the exression λ R = λ(πr 2 +2r). Similarly, the uer boun argument also follows, excet that the raii of curvature of the ath γ may nee to be reuce, leaing to worse constants in the O() term in (5) when α is small. Similar results can be shown for robabilistic sensing regions. For examle, if the raii r varie ranomly then one obtains the same results with λ R relace with Eλ R = λ(πe(r 2 )+2E(r)) (for the isk moel), rovie the ranom raii r is is boune, r <r<r 2, an with the error terms eening on r an r 2. V. COMPUTING THE TRAP COVERAGE DIAMETER Even though we rovie an accurate robabilistic estimate of the ensity neee to achieve tra coverage of a given iameter when eloying sensors ranomly, it may be useful to ascertain eterministically whether a target hole iameter has been achieve after eloyment, esecially in the face of unanticiate an unknown eloyment failures [5]. In orer to etermine whether a eloye network continues to rovie tra coverage over time, efficient algorithms are neee to etermine the largest hole iameter. In this section, we roose such algorithms. Figure 8 shows a target region with several sensing coverage holes. Although the sensors are lotte as isks in the figure, we are not assuming a isk sensing moel. Further, the sensing regions of ifferent sensors may be ifferent. Excet in Section VD, where sensing regions are assume to be star convex, the only assumtions we make are: ) Two sensor noes are within the transmission range of each other if their sensing regions overla; 2) The accurate ositions of noes can be etermine; 3) The bounary A of the target region A is a simle olygon an is known. To etermine the largest iameter of coverage holes, the following two stes are alie. First, the bounary of each hole is foun. Secon, the iameters of these holes are comute base on their bounaries to obtain the largest iameter. The goo news is that several ieas from existing work on iscovering exact hole bounaries [6], [4], [22], [25], [28] can be alie here. However, the following challenges, which are critical to the tra coverage moel, are not aresse there. ) The bounary of a coverage hole may involve art of A, such as hole H 7 in Figure 8, so that it is har to iscover the entire bounary. 2) In a realistic sensing moel, the bounary of a coverage hole may have an arbitrary shae, which makes the comutation of the accurate iameter nontrivial. 3) When the shaes of sensing regions are unknown or uncertain (as in robabilistic sensing moels), the bounaries of iniviual holes may not be accurately etermine. We escribe in Sections VB an VC a moification to existing algorithms that comutes an accurate iameter for H H 2 H 3 H 4 H 5 H 6 H 7 H 8 Fig. 8. An instance of eloyment with eight coverage holes, H to H 8. The rectangle shows the bounary of the target region. Note that only the holes within the target region are counte. The small isks are sensing regions. convex sensing regions an aroximate iameter for nonconvex but known sensing regions. In Section VD, we escribe an outline of a simler algorithm that comutes an aroximate iameter for both known an unknown (uncertain) sensing regions. We first review existing work in this area before escribing our algorithms. A. Relate Work Tools from both algebraic toology an comutational geometry have been use for etecting coverage holes. Most focus on coverage verification an bounary noe etection without comuting the exact hole bounaries [6], [0], [4], [23], [25], [28], an several of them assume a isk sensing moel an an oen target region [6], [0], [23], [25], [28]. In toology base aroaches, certain criteria to etect holes or verify coverage [0], [23] are erive from the toology of the covere region without using the ositions of noes. However, these criteria are comute in a centralize way an the comlexity is not well stuie yet. In contrast, geometry base aroaches assume the ositions of noes are known [4], [25], [28] or at least the accurate istances among neighboring noes are known [6] an use certain locally comutable geometric objects to etect noes on a coverage bounary. The first localize aroach is roose in [4] where every noe can locally etermine whether it is on the bounary of a kcoverage hole by counting the coverage levels of its sensing erimeter, which is simlifie in the case of coverage in [29]. The location free version of [4] is roose in [6]. Another geometric aroach uses Voronoi iagrams [9], [25], [28], which is not alicable to nonconvex or heterogeneous sensing regions. Base on [4], [22] rooses an algorithm to etermine exact bounaries of coverage holes. However, it can only fin those bounaries with at most one iece from A, such as H 5 an H 6 in Figure 8, an it assumes a isk sensing moel. An algorithm to fin the bounaries of routing holes is roose in [9], an [27] rooses a metho to etermine the bounaries of communication holes using only the connectivity grah an a general sensing moel. However, A is not consiere in either aroach. B. Discovering Hole Bounary In this an the next section, we assume that each noe knows the shae of its sensing region (not necessarily convex). The imact of sensing uncertainty is iscusse in Section VD.
8 8 Our algorithm first alies the erimeter coverage base aroach [4] to etect noes on the bounaries of coverage holes. The iea is that the sensing erimeter of one noe is ivie into one or more ieces by the sensing erimeters of the neighboring noes. Every such iece is calle a sensing segment. A noe is on the bounary of a coverage hole iff it has a sensing segment that is not covere by other noes. The bounaries of coverage holes neee for iameter comutation are then erive base on the following observations, which can be verifie in Figure 8. First, the bounary of a coverage hole is comose of one or more close curves, but its iameter is only etermine by the outermost one, calle the hole loo. For instance, H 3 in Figure 8 has two bounary curves, but the inner one the erimeter of the two overlae sensing regions can be safely ignore. Secon, if a hole is comletely containe in another hole, it can be ignore, such as H 8 in Figure 8. Thir, each curve is comose of sensing segments an (ossibly) arts of A. If it is comose of only sensing segments, the entire curve can be foun by traversing the noes on it. Otherwise, each iece that is comose of only sensing segments on the curve can be foun. Once all the ieces of hole bounaries are known, a olygon clier algorithm [20] can be extene to fin the hole loos by also taking A into account. We efer the etails to [4]. C. Diameter Comutation Let H enote a hole loo, an X H enote the set of crossings on that loo, where a crossing is efine as an intersection oint of either two sensing erimeters, or a sensing erimeter with A, or a vertex of the simle olygon A. The following lemma states that X H is inee a goo aroximation of H in terms of the iameters, even if sensing region is not convex. Lemma 5.: iam X H iam H iam X H +2D, where D is the maximum iameter of all sensing regions. Moreover, if the sensing regions are convex, then iam X H = iam H. Proof: iam X H iam H follows since X H H. Let x an y be two oints on H with x y = iam H, where enotes the Eucliean istance. Let x be the crossing on H closest to x, an y the crossing closest to y. Then x y x y + x x + y y x y +2D. As x y iam X H, iam H = x y iam X H +2D. If the sensing regions are convex, then H is containe within the convex hull of X H. Since a oint set an its convex hull have the same iameter, the result follows. Accoring to Lemma 5., when the sensing regions are all convex, it suffices to maintain the set of crossings on each hole loo instea of their accurate shaes in orer to fin the largest iameter D. For arbitrary sensing regions, this also gives a goo aroximation when D 2D. D. Coing with Sensing Region Uncertainty Sensing regions show irregularity ue to harware calibration an obstacles an therefore are har to characterize eterministically [5]. A more realistic way to characterize sensing regions is to use a samling base aroach, where the sensing region of a noe is aroximate by the iscrete oints a b c e 2 s s 2 Fig. 9. The aroximation of covere region by a lanar grah. The ashe lines show art of A. The ashe curves show the real sensing erimeters (unknown) of noes s an s 2. e an e 2 are two events etecte by both of them. a, c, an are oints on the eges of A intersecting the two sensing regions, an b is a vertex of A. Three faces, s abcs, s e 2s 2e s,ans cs 2e 2s are shown. corresoning to the events etecte by the noe [5]. In this section, we consier how to comute the largest iameter of coverage holes if only a limite number of samles are known. To this en, we first construct a lanar grah base on the samles observe. This grah is use to aroximate the real covere region, that is, the union of all the sensing regions. We then show that uner certain assumtions, the largest iameter of coverage holes can by estimate by the largest iameter of the faces of this grah. Let B s enote the sensing region of noe s. Wealsouses to enote the osition of noe s an e to enote the osition where event e haene. We make the following assumtions. ) The ositions of noes an events observe are known. 2) Each B s is a star convex subset of R 2 with resect to s, that is, any line segment joining s to a oint t in B s, enote as st, lies in B s. Figure 9 shows an examle of two overlae starconvex sensing regions. 3) For every connecte comonent C i of B s B s2,s s 2, there is at least one event etecte in each C i, i.e., there is a oint e i C i known such that s e i lies in B s an s 2 e i lies in B s2. For instance, the two sensing regions in Figure 9 intersect at two connecte subregions, with one common event etecte in each. 4) For each noe s, it is known whether B s is comletely insie of A, or is comletely outsie of A, or intersects A. In the last case, the set of eges of A that intersect B s is known. Let S enote the set of noes whose sensing regions are within or intersect A, an E enote the set of events observe by noes in S.LetA enote the set of vertices of A. For each noe s San each ege of A that intersects B s, ick an arbitrary oint on that ege that is within B s, such as oints a, c, an in Figure 9. Name the set of such oints I. We construct a geometric grah G(V,E), where V = S E A I, an each ege in E corresons to either a line segment joining a noe s an an event e etecte by s, ora line segment joining a noe s an a oint a I on an ege of A intersecting B s, or a line segment on A joining oints in A an I. See Figure 9 for reference. Notice that, the eges of G may intersect at oints other than vertices. We make G lanar by treating these intersections as vertices as well. We then observe that G is a lanar grah without oen faces. Let D e
9 9 an D enote the largest iameter of coverage holes an that of the faces of G, resectively. Then uner the assumtions mae above, we can observe that D D D +2D, where D is the maximum sensing iameter. We efer the roof of this statement to [4]. Notice that, the above aroximation can also be alie to the case where all the sensing regions are known. It is not as accurate as the aroach sketche in Section VB, but more efficient since the faces of G an their iameters can be easily comute. If 2D, the aroximation may be esirable. In aition, if more events than require are etecte, they can be use to imrove the accuracy of the aroximation. VI. OPEN PROBLEMS Although we have aresse the roblems of ranom eloyment an algorithmic etermination of the status of tra coverage, introuction of this new moel of coverage oens u an oortunity to revisit several funamental eloyment an toology control roblems afresh. First the roblem of otimal eterministic eloyment for various ranges of an r remains oen. Secon, the roblem of joint coverage an connectivity (both from a eterministic eloyment ersective [2] an from an algorithmic ersective [2], [26]) remain oen. Thir is the roblem of coverage restoration uon sensor failures [7]. Finally, the roblem of sleewakeu [7], [3], [8], [29] which has traitionally assume full coverage moel or the barrier coverage moel [9], also nees to be reinvestigate for this new moel. VII. CONCLUSION This aer generalizes the traitional moel of full coverage by allowing systematic holes of boune iameter. With this new moel, eterministic guarantees on etection, articularly tracking can be maintaine even if not all oints in the region are covere, whether ue to failure of eloye sensors or ue to the exense of eloying sensors to cover every oint in a large region. Tra coverage thus makes sensor eloyment scalable. Of ineenent interest is also the fact that the tra coverage moel briges the longstaning ga between the threshols for ercolation an for full coverage. ACKNOWLEDGMENT This work was sonsore artly by NSF Grants CNS , CNS07287, CCF , NIH Grant U0DA02382 from National Institute for Drug Abuse (NIDA), an FIT at the University of Memhis. The content is solely the resonsibility of the authors an oes not necessarily reresent the official views of the sonsors. REFERENCES [] R. Arratia, L. Golstein, an L. Goron, Two Moments Suffice for Poisson Aroximations: The ChenStein Metho, Annals of Probability, vol. 7,. 9 25, 989. [2] X. Bai, S. Kumar, D. Xuan, Z. Yun, an T. H. Lai, Deloying Wireless Sensors to Achieve Both Coverage an Connectivity, in ACM MobiHoc, [3] P. Balister, B. Bollobás, A. Sarkar, an S. Kumar, Reliable Density Estimates for Coverage an Connectivity in Thin Stris of Finite Length, in ACM MobiCom, [4] P. Balister, Z. Zheng, S. Kumar, an P. Sinha, Tra Coverage: Allowing Holes of Boune Diameter in Wireless Sensor Networks, University of Memhis, CS0900, Tech. Re., [5] S. Baat, V. Kulathumani, an A. 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