Sparse power-efficient topologies for wireless ad hoc sensor networks

Sparse p o w er-efficien t top o l ogies for wireless ad ho c sensor net w ork s ∗ Amitabha Bagc hi Dept of Computer Scie nce and Engg Indian Institute of T ec hnology Hauz Khas, New Delhi 110016, India. bagc hi@cse.iitd.ernet.in. No v em b er 18, 2018 Abstract W e study the problem of pow er-efficient routing for multihop wireless ad hoc sensor netw orks. The guid- ing insight of our w ork is that unlike an ad ho c wireless netw ork, a wireless ad hoc sensor netw ork do es not require full connectivity among the nodes. As long as the sensing region is well co vered by connected nod es, the netw ork can perform its task. W e consider t wo kinds of geometric random graphs as base inter- connection structures: u nit disk graphs UDG(2 , λ ) and k -nearest-neighbor graphs NN(2 , k ) bu il t on points generated by a Pois son p oint pro cess of density λ in R 2 . W e provide sub gra ph constructions for these tw o mod els UD G-SENS (2 , λ ) and NN-SENS(2 , k ) and show that there are values λ s and k s above w hich these constructions ha ve the f ollow ing goo d properties: (i) they are sparse; (ii) they are pow er-efficient in the sense that the graph d istance is no more than a constan t times the Euclidean distance b et ween any pair of p oin ts; (iii) they co ver the space well; (iv) the subgraphs can b e set up easily using local info rmation at each node. Our analyses pro ceed b y coupling th e random graph constru ctions in R 2 with a site percolation pro cess in Z 2 and u si ng the prop erties of the latter to d eriv e prop erties of the former. An imp ortan t consequence of our constructions is that they provide new upp er b ounds for the critical v alues of the parameters λ and k for the mod els UD G (2 , λ ) and NN(2 , k ). W e also describ e a simple local al gorithm requiring only location informa- tion (from a GPS for example) and communication with immediate neighbors for setting u p the su b net works UDG-SENS(2 , λ ) and NN(2 , k ) and for routing p ackets on them. 1 In tro duction Multihop sensor netw orks, wher e no des act not only to sense but also to relay information, hav e prov en adv antages in terms o f energy efficiency ov er single hop se nsor net works [11]. Not o nly is sensor -to-sensor comm unication useful for necessa ry tas k s like time synchronization [6], for cer tain k inds of collab orative sensing functions like target tracking [23] senso r-to-sensor co mm unication is essential. The question of how to achieve connectivity arises here just as it do es in a d ho c wireless networks with one crucial difference: It is not necessar y tha t every sensor b e part of a co nnected net work. It is o nly necessary that the density of connected sensors is high enough to pe rform the sensing function. In other w ords, even if some sensors are wasted in the sense tha t the data they sens e cannot b e relay ed, it does not matter as lo ng as the area b eing sensed is well c overed with useful s ensors whic h a r e part of multihop net work that can relay data. The difference from other ad hoc wireless net works is that ea c h node exp e cts connectivity as a service provided to it, while in the W ASN individual no des are not important, the ov erall task is. In this pap er we follow this critica l insight to prop ose sparse easy - to-compute power-efficien t constructions for multihop W ASNs. W e co nsider t wo differe n t t yp es of geometr ic random graphs as the base interconnection s tructures. The no des are mo delled b y a P oiss on p oint pro cess with densit y λ on the plane, R 2 . The in terconnections b et ween ∗ This work has b een submitted to the IEEE for p ossible publication. Copyrigh t m a y be transferred without notice, after whic h this v ersi on m a y no l on ger b e accessible. 1 these points are modeled in t wo ways: 1) Unit Disk Gr aphs in which there is an edge b et ween tw o nodes if the Euclidean distance betw e e n them is at mo st 1 and 2) k -ne ar est neighb or gr aphs in whic h each no de establishes (undirected) edges to the k po in ts nearest to it. W e will refer to the former model as UDG(2 , λ ) and the latter as NN(2 , k ) (following t he notation in tro duced in [8]). The 2 in bo th these terms denotes the dimension of the space. Both these mo dels display a critica l phenomeno n. F or UDG(2 , λ ) it is k no wn that there is a v alue λ (2) c such that if λ > λ (2) c the g raph contains an infinite comp onent [16]. F or NN(2 , k ) the density is not relev ant, instead Hag gstr¨ om and Meester sho w that k is the cr itica l pa rameter (se e [8]) i.e. there is a k c (2) such that for all k > k c (2), NN(2 , k ) ha s a n infinite component. W e s how that as long as the cr itical parameters take at least certain v alues (whic h are higher than the critical v alues) it is po s sible in both cases to construct a subgr aph of the infinite co mponent with the following prop erties: P1. ( Sp arsity ) The subgraph has a maxim um degree of 4. P2. ( Constant str etch ) The distance b et ween any tw o po in ts in the subgr aph is at mo st a co ns tan t factor greater than the Euclidea n dista nce betw een the points. P3. ( Cover age ) The subgraph is infinite and the proba bility of a squar e reg ion of R 2 not cont aining any points of the subgr aph deca ys exp onentially with the size of the region. P4. ( L o c al c omputability ) E ac h no de can determine if it is part o f the subgraph by us ing its lo cation informatio n and by co mm unicating with its immediate neighbours. The subgraph constr uc ted for UDG(2 , λ ) is called UDG-SENS(2 , λ ) and that constructed for NN(2 , k ) is called NN-SENS(2 , k ). W e will show that there is a v alue λ s (and k s ) such that for all λ ≥ λ s (resp. k ≥ k s ), UDG-SENS(2 , λ ) (resp. NN-SENS(2 , k )) has prop erties (P1)-(P4). These prop erties align w ell with the prop- erties o f p o wer-efficient spanner s s tudied in the context of ad ho c wireless netw o r ks (see [18, pp 177 -178].) The difference b eing that not every p oin t of the po in t pro cess is require d to b e pa rt of the netw or k as long as the sensing function is satisfied (which the cov erage prop ert y (P 3) ens ur es.). Prop erty (P2) is o f ma jo r consequence to the p o wer consumption of the netw ork. This follows from the r e la- tionship b et ween the stretch in distance betw een t wo p oin ts and the consequent incr ease in the power co nsumed in comm unicating b et ween th em. F ormally , if w e consider a wir eless netw ork G formed on a set o f no des V in R 2 , the d istanc e str etch , δ , of a subgraph H ⊆ G is defined as δ = max u,v ∈ V d H ( u, v ) d G ( u, v ) , where d G ( u, v ) is the graph dista nce betw een u a nd v in G i.e. distance betw een u a nd v using the edge s graph G and d H ( u, v ) is the graph dista nc e in H . Li, W an and W ang [14, L emma 2 ] show ed that giv en a connection net work G a nd a subgr aph H with distance stretch δ , the pow er taken to comm unicate b et ween any tw o nodes is at mo st δ β where β is a pa rameter v arying betw een 2 a nd 5 . Clearly a net work with prop erty (P2) achiev es a co nstan t p ow er stretch since the E uclidean distance betw een tw o po ints is a lower b ound on the distance betw een them in b oth UDG(2 , λ ) and NN(2 , k ). Hence we c la im tha t our constructions are pow er - efficien t up to a co nstan t factor. Additionally we hav e prop erty (P 3 ) that guar an tees cov erag e o f the regio n being s e ns ed. W e show that the pro babilit y that a region does not co ntain a p oint o f UDG-SENS(2 , λ ) (or NN-SENS(2 , k )) decays exp onen tially with the size of the regio n when λ ≥ λ s (resp ectiv ely k ≥ k s ). In both cases the decay is shar per if a larger v alue of λ is chosen. This allows us to achiev e a target coverage by incr easing the densit y to a hig h enough level. The basic idea b ehind our constr uctions is to couple the rando m g r aph in R 2 with a discr ete site p e rcolation pro cess in Z 2 . The subg r aph we co nstruct mimics the no des of a p ercolated mesh. One imp o rtan t bypro duct of our constructions is that they g iv e the b est kno wn upp er b ounds on the critical v alues for b oth our setting. W e improv e the be s t known b o und o f 2 13 (due to T eng and Y ao [21]) for the cr itical v alue of k for NN(2 , k ) to 18 8. W e also improv e the bound for the critical v alue of λ in UDG(2 , λ ) to 1 . 568. The construction is easy to r ealize using lo cation information (which can be obtained using a GPS) and lo cal computation, hence sa tis fying proper ty (P4). The algor ithm for ro uting on o ur subgr aph constr uc tio ns is based on a simple distributed algorithm for routing on the p ercolated lattice g iv en by Ang el et. al. [1]. In the rest of this sec tio n w e in tro duce so me notation a nd definitions tha t will be required. W e als o discuss the v ar ious strands of r esearch relev ant to our paper . In Section 2 w e describ e our co nstructions UDG-SENS(2 , λ ) 2 and NN(2 , k ) and give lower b o unds on the v a lue s λ s and k s ab o ve which prop erties (P1)-(P 4 ) hold. The results regar ding stretch and cov erag e a re detailed in 3. In Section 4 we discuss the alg orithmic is sues inv olved in forming our subgr aphs fro m the underlying structure and also de s cribe how to route pack ets once the struc tur es are made. 1.1 Preliminaries W e will use the notation d ( x, y ) to denote the Euclidea n distance be tw een tw o points x, y ∈ R 2 . In general w e will denote the g raph distance (i.e. the sho rtest path along the edges of the graph) of tw o vertices u, v o f a graph G by d G ( u, v ). F or the r a ndom graphs NN(2 , k ) and UDG(2 , λ ) we will use the notation d n ( u, v ) and d u ( u, v ) to denote the gr aph distance be tween them. P oisso n p oin t process es Our rando m p oint sets ar e g enerated by homog e nous Poisson p oin t pro cesses of int ensity λ in R 2 . Under this mo del the num b er o f p oin ts in a regio n is a random v ariable that dep ends o nly on its d -dimensional v olume i.e. the n umber of points in a bo unded, measur able set A is Poisson distributed with mean λV ( A ) where V ( A ) is the d -dimensional volume of A. F ur ther , the rando m v ar iables asso ciated with the nu mber of points in disjoint sets ar e independent. Unit disk graphs The random gr a ph mo del UDG(2 , λ ) is defined as follows: Given a set of p oin ts S g enerated by a Poisson point pro cess in R 2 with density λ , th er e is an edge b et ween p oints x ∈ S and y ∈ S if d ( x, y ) ≤ 1. k -nearest neig h b or graphs The r andom graph model NN(2 , k ) defined as follows: Given a set of points S generated b y a P oisso n point proce s s in R 2 there is an (undirected) edge betw een p oin ts x ∈ S the k p oin ts in S \ { x } that a r e closest to x . Note that the even t that t wo p oin ts ha ve exactly the same distance from a point is a measure 0 ev ent, but for prac tical purp oses we c an use any tie-breaking mechanism w e deem fit. Site p ercolation Consider an infinite graph defined on the vertex set Z d with edges b et ween p oin ts x and y such that k x − y k 1 = 1. Site p ercolation is a probabilistic proces s on this graph. Each point o f Z d is taken to be op en with pro babilit y p and close d with probabilit y 1 − p . Hence w e hav e a sample space Ω = Q x ∈ Z d { 0 , 1 } , individual elements of which are c onfi gur ations ω = ( ω ( x ) : x ∈ Z d ). The pro duct of all the measures for individual points forms a measure for the spa c e of p ossible configurations. An edge betw een tw o open vertices is co ns idered open. All other edges are cons idered closed. A comp onent in which open vertices ar e c o nnected through paths o f op en edges is known as an open cluster. I t is known that there is a v alue p c such that for all p > p c the gra ph o btained has an infinite op en cluster. This v alue is known as the critical proba bilit y . When p > p c then ea c h p oin t of Z d has s o me no n-zero probability of b eing pa rt of an infinite cluster . The rea der is referred to [7] for a full trea tmen t of p ercolation and to [4] for a recent up date on some new directio ns in this area. When the lattice undergo es p ercolatio n, the pa th b et ween tw o connected vertices might b ecome long a nd tortuous. W e intro duce so me notation for this setting. The distance b etw e e n tw o lattice p oin ts x, y ∈ Z d will be denoted D ( x, y ). When the la ttice has b een pe r coalted with pr obabilit y p , the dista nce will b e deno ted D p ( x, y ). An tal and Pisztora studied this setting and prov ed a powerful theor em which we state here as a lemma [2, Theorems 1.1 a nd 1.2]. W e adopt the restatemen t of Angel et. al. [1, Lemma 8]. Lemma 1. 1 [2, 1] F or any p > p c and any x, y c onne cte d thr ough an op en p ath in a cub e M d of the infin ite lattic e. F or some ρ, c 2 > 0 dep ending only on the dimension and p and for any a > ρ · D ( x, y ) pr ( D p ( x, y ) > a )) < e − c 2 a . 1.2 Related w ork Wireless netw o rk s T op ology control in wireless netw ork s has b een studied extensively (see e.g. the surveys b y Santi [18] and Ra jaraman [17].) Tw o impo rtan t go a ls of the resea r c h in this a r ea have b een ensuring co nnec tivity of all no des and energy-efficiency . 3 The approach has been to take the underlying top ology as a unit disk gra ph [19] or a pro ximity graph (for which sev er al prop osals exist c.f. surveys cited ab ov e) on a point set and then construct some kind of spa nning subgraph of this po in t set with low deg ree, constant stretc h and the prop ert y that e ac h no de can compute its connections using lo c al information. Although this line of research has the s a me flav our a s our work, it is different in a fundamen tal way - w e do not requir e all no des to be c o nnected - and so we do not sur vey the litera ture in detail instead referring the reader to g eneral sur v eys on top ology control by and and a sp e cific survey on spanners by Li and W ang [15]. Geometric random graphs and p ercolation The study of ra ndom g raphs o btained b y applying connection rules on stationar y po in t pro cesses is known as co n tin uum pe rcolation. Meester and Roy’s mono graph on the sub ject provides an excellen t view of the deep theory that has been developed around this genera l se ttin g [16]. UDG(2 , λ ) is studied in [16], where the existence and non- trivialit y of the cr itical density is demonstr ated. Kong and Zeh [12] show a low er b ound of 0.76 98 o n λ c . An upper b ound of 3.372 w as earlier shown by Hall [9 ]. Hall’s pap er states an upp er bound 0.84 3 for a mo del of intersecting spheres which scales by a facto r of 4 using scaling prop ert y of co nin umm p ercolation mo dels [16]. The NN( d, k ) mo del was intro duced b y H¨ agg str¨ om and Meester [8]. T hey show ed tha t there w as a finite critical v alue, k c ( d ) for all d ≥ 2 such tha t an infinite cluster exists in this mo del. They prov ed that the infinite cluster was unique and that there was a v alue d 0 such that k c ( d ) = 2 for a ll d > d 0 . T eng and Y ao gav e an upp er bo und of 213 for k c ( d ) [21]. k -nearest neig h bo r gr a phs on random p oin t sets contained inside a finite r e g ion hav e be e n extensively studied. The ma jor concern, differen t from ours, has b een to ensure that al l the points within the regio n are connected within the same cluster. Ballister , Bo llob´ as, Sark ar and W alters [3] show ed that the smalle st v alue of k that will ensure connectivit y lies b et ween 0 . 3043 log n and 0 . 5 139 log n , impro ving earlier res ults of Xue and Kuma r [22]. Ballister et. al. also studied the problem of c overing the r egion with the discs containing the k -nearest neighbour s of the p oints. W e r e fer the reader to [3] fo r an interesting discussion relating this setting to earlier work b y Penrose and others. 2 The subgraph constructions In this sectio n we desc r ibe our constr uctions and prove some imp ortant prop erties. W e be g in by giv ing a general ov erview of our technique, then move on to the specifics of the t wo s ettings. F or b oth co ns tructions w e pro ceed by viewing R 2 as a union of a coun tably infinite set of square tiles. Inside each tile we lo ok for a t wo kinds of po in ts. The first kind is what we call a r epr esentative p oint. Repres en tative po in ts lie a t the cen tre of the tile, roughly sp eaking. W e a lso lo ok for re lay p oints , whic h help connect represen tative points to neigh b ouring tiles. Both these kinds of po in ts ha ve precise definitions that diff er for UDG-SENS(2 , λ ) and NN-SENS(2 , k ), w e will discuss those in Section 2.1 and 2.2 resp ectiv ely . A tile in which we find b oth kinds of points w e call a go o d tile, other tiles are b ad . W e co nnect r epresen tative p oin ts to four relay p oin ts, one for each neighbouring tile. Sev er al po in ts within and o utside go od tiles may be left unconnec ted. See Figure 1 for a pictorial depiction. Note that representative p o in ts hav e degr ee 4 and relay p oin ts hav e deg ree 2. The subg r aph drawn by connecting re pr esen tative p oin ts thro ugh relay p o in ts will b e the netw ork we will use for sensing. In order to prove prop erties (P1 )-(P4) we will couple the tiling with a site p ercolation pro cess in Z 2 . W e a ssocia te each tile in R 2 with a p oin t in Z 2 . W e declare a site in Z 2 op en o nly if the tile co rrespo nding to it in R 2 is go o d. Hence the proba bility that a s ite is op en is equal to the probabilit y tha t its cor responding tile is go od. O ur de finitio ns of r epresen tative and rela y po ints will ensur e that if t wo neighbour ing tiles are g oo d then their representativ e points are co nnec ted throug h their relay p oints as shown in Figur e 2. This cor responds to the edge b et ween tw o op en sites b eing op en in Z 2 (see Figure 2 .) Since paths in Z 2 corres p onds to paths betw een po in ts in R 2 it follows that if p ercolation o ccurs in the Z 2 then a n infinite comp onen t must e x ist in the ge o metric random g raph mo del as w ell. Hence we can c o nclude that if the probability of a tile being go o d exceeds the critical probability for s ite pe r colation, the geo metr ic random graph mo del als o has an infinite comp onent in it almo st surely . Let us now tak e a more sp ecific loo k at the constr uctions for UDG(2 , λ ) and NN(2 , k ). 4 Representatives Relays Unconnected points Figure 1: A p ortion of the tiling of R 2 . Figure 2: The part of Z 2 corres p onding to the tiles sho wn in Figur e 1 . 2.1 Unit-disk graphs The internal structure of a tile for the cons truction of UDG-SENS(2 , λ ) is shown in Figure 3. W e co nsider squar e tiles of side 4/ 3. F or the sake of exp osition let us assume that the tile shown in Figure 3 is centred at (0,0 ) a nd it’s low er left corner is (-2 /3, - 2 /3). Within ea ch tile we cons ider fiv e disjoint reg ions, the r epresen tative reg ion C 0 ( t ), a nd the relay reg io ns E l ( t ) , E r ( t ) , E t ( t ) a nd E b ( t ). C 0 ( t ) is a cir cle of ra dius 1/ 2 centred at the origin. The regions E i ( t ) , i ∈ { l , r, t, b } ar e intuitiv e to understand but slightly tricky to describ e formally . W e describ e one o f them, E r ( t ). In o rder to do so, let us denote by t r the tile immediately to the right of the tile t i.e. the tile c en tred at (4/3,0) with b ottom left c orner (2/3,-2 /3). It’s leftmost relay region is E l ( t r ). Now we define E r ( t ) as the part of the int er section lying wholly within t of all circles of unit radius centred a t po in ts in C 0 ( t ) a nd E l ( t r ). F rom this set we remove all the p oin ts o f C 0 ( t ). In the figure the region is depicted by an ellipse, but clearly it is a less r egular sha p e. W e call a tile t go o d if each of C 0 ( t ) a nd E i ( t ) , t ∈ { l , r, b, t } co n tains at le ast one p oin t o f the p oin t pr o cess. One of the po in ts contained in C 0 ( t ) will b e the repr esen tative point for this tile, denoted rep( t ). F our other po in ts, one fro m each of the regio ns E i ( t ) , t ∈ { l , r , b, t } will b e the relay p o in ts for this go o d tile. If a r egion has more tha n one point, the tie has to be broken. This will be done in a distributed fashion. W e p ostp one the discussion of this aspect to Section 4. Note that s ome of the relay r egions o verlap a nd hence it may b e the case that one p oint fulfils tw o rela y functions. According to the pro gram described ea rlier we create a bijection, φ , b et ween the tiles in R 2 and p oint s in Z 2 such that neighbouring tiles in R 2 corres p ond to neighbouring points in Z 2 . W e couple UDG(2 , λ ) to a site per colation pro cess in Z 2 by saying that a given p oint x in Z 2 is open only if the tile t = φ − 1 ( x ) is go o d. Now we can claim that the e xistence of an edge in Z 2 implies the existence of a path from the representativ e po in ts of the t wo tiles corresp onding to the t wo end p oin ts of the edge. W e state this formally , including an observ ation ab out the distance str etc h b et ween the t wo representativ e p oin ts. Claim 2. 1 If an e dge exists in the p er c olate d mesh Z 2 b etwe en two p oints x and y then 5 E b E t a = 4 / 3 E r E l C 0 Figure 3: A tile t and it’s 5 relev ant regio ns. Note that the reg io n E r is the part of the intersection lying wholly within t of all unit disc s centred at p oints in C 0 and in the r e g ion E l (i.e. the left relay regio n) of the neighbour tile t r . 1. Ther e is a p ath b etwe en the re pr esentative p oints r ep ( φ − 1 ( x )) and r ep ( φ − 1 ( y )) of the tiles c orr esp onding t o x and y in UDG (2 , λ ) and 2. ther e is a c onstant c u ≤ 3 such that d k ( r ep ( φ − 1 ( x )) , r ep ( φ − 1 ( y ))) ≤ c u · d ( r ep ( φ − 1 ( x )) , r ep ( φ − 1 ( y ))) . (1) Figure 4: A path be tw een the representativ e po in ts of neig h bo ring g oo d tiles. Pro of. Clearly if tw o neighbouring tiles t and t ′ are go od, by the g oo dness condition there will b e an edg e from the representativ e po in t of one of them to a rela y po in t in the direction of its neighbor. This relay p oint will subsequently connect to the relay p oin t of that neighbor clos est to it, whic h will in turn b e connected to the representative p o in t of the neighbour (see Figure 4). Clear ly eac h o f the three edg es on the pa th from rep( t ) to rep( t ′ ) is at mo st 1 unit in length so c u ≤ 3. ⊓ ⊔ The larges t connected c omponent fo rmed by the represent ative p oin ts and relay p oin ts is UDG-SENS(2 , λ ). F rom Cla im 2.1, it is e asy to deduce that if an infinite comp onen t exis ts in the site per colation s etting, then an infinite comp onen t exists in UDG(2 , λ ). Hence w e need to determine for what v alues o f λ the site per colation pro cess is supercr itica l. The critical probability for site pe rcolation lies b etw e e n 0.592 and 0.593 (see e.g. [13]). Numerical calcula tio ns show ed that the smallest v alue of λ for whic h the probability of a tile being go o d exceeds 0.593 is λ s = 1 . 568. Hence for λ larg er than this v alue UDG-SENS(2 , λ ) is infinite. Since this improves the b est known upp e r bo und of 3.372 [12], we state it here as a theor em. 6 Theorem 2.2 F or UDG (2 , λ ) λ (2) c < 1 . 568 . In Section 3 we will show tha t UDG-SENS(2 , λ ) for λ ≥ λ s has co nstan t str etc h. F or now we mov e on to the construction of NN-SE NS(2 , k ). 2.2 Nearest-neigh b or graphs The internal structure of a tile for the cons truction of NN-SENS(2 , k ) is sho wn in Figure 5. C x C b C l E l C 0 E b z E r C r C t E t x C z Figure 5: A tile t a nd it’s 9 r e le v ant regions. Note that the re g ion E r lies wholly within all discs of the form C x and C z centred a t p oin ts on the b oundary of the discs C 0 and C r . Let us say that the tile in Figur e 5 is centred at (0 , 0) with bo tto m left co rner ( − 5 a, − 5 a ) and top rig h t corner (5 a, 5 a ). F or conv enience we will refer to the tiles sur r ounding the tile t as, couunterclockwise starting from the right t r , t t , t l and t b . W e consider five circles o f radius a : C 0 centred at (0 , 0), C l centred at ( − 4 a , 0), C r centred at (4 a, 0 ), C t centred at (0 , 4 a ) and C b centred at (0 , − 4 a ). There are four o ther region which are named E l , E r , E t and E b in the figure. E r is defined as follows. Cons ide r the largest circle centred at a n y p oin t in C 0 or C r that lies wholly within the tw o tiles t and t r . Two such cir cles, C x and C z , are depicted in Figure 5 . E r is the loc us of the p oints cont aine d in all s uc h circles. The regions E l , E t and E b are defined similary by C 0 alongwith C l , C t and C b resp ectiv ely and th e tiles t l , t t and t b resp ectiv ely . Now, we call tile t go o d if 1. the n umber of p oint s inside t is a t mos t k / 2 and 2. the nine r egions C 0 , C r , C t , C l , C b , E r , E t , E l and E b contain a t least one p oin t eac h. One p oint contained in C 0 will be the re pr esen tative p oint of the tile t , deno ted rep( t ). A p oin t from each o f the o ther 8 r egions will b e r ela y p oin ts. If these regions con tain multiple p oints w e will hav e to select one from each and disca rd the rest. As in the case of UDG-SENS(2 , λ ) we p ostpo ne the dis cussion of how to select this one p oin t ea c h to Section 4. According to the pro gram described ea rlier we create a bijection, φ , b et ween the tiles in R 2 and p oint s in Z 2 such that neig h bo uring tiles in R 2 corres p ond to neighbour ing p oin ts in Z 2 . W e couple NN(2 , k ) to a site per colation pro cess is Z 2 by saying tha t a given p oin t x in Z 2 is open only if the tile t = φ − 1 ( x ) is g oo d. Now we can claim that the e xistence of an edge in Z 2 implies the existence of a path from the representativ e po in ts of the t wo tiles corresp onding to the t wo end p oin ts of the edge. W e state this formally , including an observ ation ab out the distance str etc h b et ween the t wo representativ e p oin ts. Claim 2. 3 If an e dge exists in the p er c olate d mesh Z 2 b etwe en two p oints x and y then 7 1. Ther e is a p ath b etwe en the re pr esentative p oints r ep ( φ − 1 ( x )) and r ep ( φ − 1 ( y )) of the tiles c orr esp onding t o x and y in NN (2 , k ) and 2. ther e is a c onstant c k such tha t d k ( r ep ( φ − 1 ( x )) , r ep ( φ − 1 ( y ))) ≤ c k · d ( r ep ( φ − 1 ( x )) , r ep ( φ − 1 ( y ))) . (2) Figure 6: A path b et ween the re presen tative p oint s o f t wo neig h bo ring go od tiles. Pro of. The pro of of th e claim is depicted in Figure 6 Clea rly a ny circle drawn fro m rep( t ) that sta ys within t contains all of E r in it b y the definition o f E r . Since there ar e at most k / 2 p oin ts in every go od tile, hence there is an edge from rep( t ) to the p oint gua ran teed to be con tained in E r , let’s call it x r . W e do not make a n y claims o n where the edg es established b y x r to its neighbours lie, o bserving only that any p oin t that lies in C r m ust hav e an edg e to x r , again b y the definition o f E r . How ever, any disc cent re d at a point in C r that remains within t and t r m ust contain the left disc o f its neighbor ing tile. Hence, if t and t r are both g oo d then a path from rep( t ) to re p( t r ) o ccurs. The s econd par t o f the claim is obviously true. The constant c k can easily b e calculated using c a lculus. ⊓ ⊔ W e define NN-SENS(2 , k ) as the lar gest connec ted comp onent of the graph built o n repres en tative and relay po in ts. Note tha t unlike UDG-SENS(2 , λ ) there are 8 relay points within eac h tile her e a nd the path betw een t wo represe ntative p oint s contains 4 relay po ints. Also note that only the regio ns E l , E r , E t or E b can s hare relay po in ts. The regions C l , C r , C t and C b do not intersect in any way . Each of the regions may contain mo re than one p oint of the p oin t pro cess. In this c ase o ne p oin t has to be b e chosen as a repr e s en tative or relay as the ca se may b e. This can b e ea sily a chieved by running a simple leader election alg orithm [20] b et ween the no des in the region which are all co nnected to each other in an y case. W e will discuss this issue further in Sec tion 4 . F rom Cla im 2.3, it is e asy to deduce that if an infinite comp onen t exis ts in the site per colation s etting, then an infinite comp onen t exists in NN(2 , k ). Hence we need to determine for wha t s ettings of our par ameters a and, more imp ortantly , k , the s ite p ercolation pr ocess is sup ercritical. The critica l proba bilit y for site per colation lies betw een 0.59 2 and 0.593 (see e.g . [13]). Numerica l calculations show ed that the smallest v alue o f k for which the proba bilit y of a tile being goo d exceeds 0.59 3 is 1 88, and the v alue o f a for which this happ ens is 0.8 93. Like H¨ a ggstr¨ om and Meester’s pro of for the existence of a critical v alue [8] a nd T eng and Y ao’s pro of for the weak er of their tw o upp er b ounds on k c (2) [21] our pro of of Theorem 2.4 pro ceeds by constructing a coupling with a s ite p e rcolation pro cess on Z 2 . How ever, our cons truction gives a b etter upp er b o und than T eng and Y ao’s improvemen t o f their own r e sult (also in [21]) to k c (2) ≥ 213 whic h uses a coupling to a mixed per c o lation pro cess. Hence we state this r e sult as a theorem: Theorem 2.4 F or NN (2 , k ) , k c (2) ≤ 188 . Having de s cribed the constructions of UDG-SENS(2 , λ ) a nd NN-SENS(2 , k ) and having shown that there are v alues of the critical parameter s for whic h these co nstructions ex ist, let us no w pro ceed to show that these constructions indeed have c o nstan t stretch. 8 3 Stretc h and co v erage In this section we prov e that our constructions have co nstan t stretch with high probability when the critica l parameters have hig h enough v alues. W e also show that the coverage of our constructions is v er y g oo d in a probabilistic se ns e. 3.1 Constan t stretch The argument for co ns tan t stretch of b oth NN-SENS(2 , k ) , k ≥ k s and UDG-SENS(2 , λ ) , λ ≥ λ s follow s imila r lines so w e present them toge ther . F or the purpo ses of this section we denote the tile lengths chosen for the tw o constructions as a u (= 4 / 3) a nd a k (= 0 .8 93). In wha t fo llows, all a rgumen ts hold for b oth settings exce pt wher e explicitly noted o therwise. Also in the follo wing we implicitly ass ume th at k ≥ k s and λ ≥ λ s . Let us co nsider any t wo tiles t 1 and t 2 whose representativ e p oin ts rep( t 1 ) and rep( t 2 ) lie UD G-SE NS (2 , λ ) or NN-SENS(2 , k ). Fir st we relate the distance in the (unp ercolated) lattice to the euclidean distance betw een these tw o p oin ts b y observing a simple fact. F act 3.1 Given the c onstants c u define d in Claim 2. 1 and c k define d in Claim 2. 1 , t hen for two tiles t 1 , t 2 D ( φ ( r ep ( t 1 )) , φ ( r ep ( t 2 ))) ≤ √ 2 a · d ( r ep ( t 1 ) , r ep ( t 2 )) c wher e c ∈ { c u , c k } and a ∈ { a u , a k } r esp e ctively. F act 3.1 along with Le mma 1.1 gives us the following theorem: Theorem 3.2 1. F or UDG-SENS (2 , λ ) , with λ ≥ 1 . 56 8 ther e ar e c onstants α and c 1 dep ending only on λ such tha t P ( d u ( x, y ) > α · D ( x, y )) < e − c 1 · D ( x,y ) . 2. F or NN-SENS (2 , k ) , with k ≥ 188 ther e ar e c ons t ants β and c 2 dep ending only on k such that P ( d k ( x, y ) > β · D ( x, y )) < e − c 2 · D ( x,y ) . Theorem 3.2 is an existential result. In Section 4 we will show how to actually find the co nstan t str etc h paths in a dis tributed way with b ounded overhead. F or now we pro ceed to show that our co nstructions have go o d cov erag e. 3.2 Co v erage Let us co nsider a s quare r egion o f size ℓ × ℓ . Let us call this B ( ℓ ). W e will arg ue that the probability that B ( ℓ ) contains no p oin t of UDG-SENS (2 , λ ) (or NN-SENS(2 , k )) decays exp onen tially with ℓ . As b efore the a rgumen ts here also apply to both the mo dels. W e claim the follo wing theorem: Theorem 3.3 1. F or λ ≥ 1 . 56 8 t her e ar e c onstant s c 3 , c 4 dep ending only on λ such that P [ | B ( ℓ ) ∩ UDG-SENS (2 , λ ) | = 0] ≤ c 4 · ℓ 2 · e − c 3 · ℓ . 2. F or k ≥ 188 ther e ar e c onstants c 5 , c 6 dep ending only on k such that P [ | B ( ℓ ) ∩ NN-SENS (2 , k ) | = 0] ≤ c 5 · ℓ 2 · e − c 6 · ℓ . Let us deno te by T B ( ℓ ) the s et of tiles fully or partia lly co n tained in the B ( ℓ ). Let us co nsider the set φ ( T B ( ℓ )) i.e. the s et of all p oin ts in Z 2 which are ima ges of the tiles in T B ( ℓ ) under the mapping defined ear lier. F or B ( ℓ ) ∩ UDG-SENS(2 , λ ) to be empt y , each p oint of φ ( T B ( ℓ )) must be o utside the infinite c lus ter of the sup e rcritical per colation pro cess. With this insight we now refer the r eader to Theo rems 8.18 and 8.21 of [7] dealing with the radius of finite cluster s in the sup ercritical phas e. A slight mo dification o f the pro of of Theo r em 8.21 of (due to [5]) will yield the the pro of o f Theorem 3.3. The details ar e tedious and do not add an ything to the pro of describ ed in [7] so we omit them here. Theorem 3.3 yields the following simple corollary 9 Corollary 3. 4 1. Ther e is a c onstant c 7 such tha t for ℓ ≥ c 7 log n P [ | B ( ℓ ) ∩ UDG-SENS (2 , λ ) | = 0] < 1 n . 2. Ther e is a c onstant c 8 such that fo r ℓ ≥ c 8 log n P [ | B ( ℓ ) ∩ NN-SENS (2 , k ) | = 0] < 1 n . The constants c 7 and c 8 may b e la rger than what the netw ork r e q uires for its sensing function. It seems int uitive that adding more no des s ho uld decr ease the v a lues of thes e co ns tan ts, but the s ta temen t of Theo rem 3 .3 do es no t s eem to provide this. How ever we claim this int uition is indeed sa tis fie d for UDG-SENS(2 , λ ) and NN-SENS(2 , k ). In order to use Theor em 3.3 to provide a guara n tee of desir ed coverage, we argue that the exp onen tial decay indeed grows sharpe r when the densit y λ increa ses. This is true for bo th UDG-SENS(2 , λ ) and NN-SENS(2 , k ). F or the case of UDG-SENS(2 , λ ) it is not hard to see, since an increase in λ directly leads to an incr ease in the probability of a tile being go o d. This, through th e coupling, increa ses the probabilit y of the cor r esponding site in Z 2 being o p en. In the site p ercolation setting the probabilit y of a site b eing part of t he infinite cluster, θ ( p ) is known to inc r ease montonically with the probability p o f sites b eing op en. Our cla im follows b ecause the increase in θ ( p ) is centrally involv ed in the (omitted) pro of of Theorem 3.3. The claim has a sligh tly subtler prov enance for NN-SENS(2 , k ). Essent ially the a rgumen t is that if we fix some v alue k ≥ k s , increa sing the density λ allows us to use tiles of s maller side length and still achiev e the desired proba bilit y ( > p c ) o f a tile being goo d. This implies that the num b er of tiles within a region increases, and hence the exp onen tial decrease bec omes sharp er. 4 Algorithmic issues W e now fo cus on the a lgorithmic issues inv olved in building UDG-SENS(2 , λ ) and NN(2 , k ) and routing packets in them o nce they are built. 4.1 F orming t he net works After the no des are laid out in their p ositions they ha ve to undertake four ba s ic steps. Firstly , they have to identify which tile they belo ng to. This involv es using their lo cation information (assumed to b e of the for m ( x, y ) ∈ R 2 ) and the v alue of the tile width (deno ted a u for UDG-SENS(2 , λ ) and a k for NN-SENS(2 , k )) programmed in to the no des. In the seco nd step e ac h no de determines whether it belongs to one of the specia l regions within the tile as describ ed in Section 2. In the third step all the no des within a region communicate to elect a leader who is then designated as the representativ e point o f the tile o r a relay point, as applicable. In the fourth step the elected p oin ts of each reg ion form co nnections with the leader s of their neighbouring regions. See Figure 7 for a formal statement of the a lgorithm for building UDG-SENS(2 , λ ). The alg orithm for NN-SENS(2 , k ) is very similar. The function electLe ader ca n b e realized us ing any distributed leader election a lgorithm on a c o mplete g raph top ology since a ll the nodes within a region can talk to ea c h other (see e.g. [20]. T he fu nction na med co nnect is simply a handshake b et ween the tw o no des mentioned. O nce the calls to this function are ov er the set up phas e is completed. Note that the alg orithm of Figur e 7 will not ju st form the largest co mp onent but will also for m other s ma ll comp onen ts. It is p ossible to detect how lar ge a giv en comp onen t is by attempting to send pack ets to distant no des. The no des o f a s mall comp onen t can then turn themselv es off if they rea lize they are not par t of UDG-SENS(2 , λ ). Detecting connectivity is an area of research in itself so we do not address the iss ue s here, refering the reader to some recent w or k in this area [10]. 4.2 The r outing algorit hm F or routing purp oses the represe n tative points of a tile act as if they ar e open lattice p oin ts in Z 2 . They use relay points to send pack ets to the representative points of their neighbo uring go od tiles (see Figur e 8) hence 10 Algorithm c onstruct (UDG-SENS(2 , λ ) , a ) 1. At each no de v do (a) Determine ( locatio n v ( x ) , lo cation v ( y )). (b) co mpute id v ( x ) = lo cation v ( x ) /a u (c) compute id v ( y ) = lo cation v ( y ) /a u (d) co mpute regio n v 2. F or each tile t and eac h reg ion r ∈ { C 0 ( t ) , E r ( t ) , E l ( t ) , E t ( t ) , E b ( t ) } do (a) Build S ( r , t ) = { v ∈ t | region v = r } . (b) r ep ( t ) ← electLeader ( S ( C 0 , t )) (c) F or r ∈ { E r ( t ) , E l ( t ) , E t ( t ) , E b ( t ) } relay ( t, r ) ← electLeader ( S ( r, t )) 3. F or each tile t with neigh b ours t l , t r , t t , t b do (a) F or each r ∈ { E r ( t ) , E l ( t ) , E t ( t ) , E b ( t ) } connect (rep( t ) , relay( t, r )). (b) co nnect (relay ( t, E r ( t )) , relay( t, E l ( t r )). (c) conne ct (rela y( t, E l ( t )) , relay( t, E r ( t l )). (d) co nnect (relay ( t, E t ( t )) , relay( t, E b ( t t )). (e) conne ct (rela y( t, E b ( t )) , relay( t, E t ( t b )). Figure 7: Building UDG-SENS(2 , λ ). realizing o pen edg es in Z 2 . With this simple idea in pla ce, we can just plug in any algorithm which per forms routing in the perc olated mesh. W e r ely on the a lgorithm for efficient distributed routing in the gia n t comp onen t of a p ercolated mesh given by Angel et. al. [1]. T heir algorithm pro ceeds by trying to follo w a shortest path from source to destination. If at a ny p oin t the pa th is broken (i.e. one of the no des is clos ed) they try to find the next no de along the path that is open by performing a distributed BFS from the curr en t location of the pac ket. F or our purpo s es we assume that the cano nical shortest path betw een an y t wo no des ( x 1 , y 1 ) , ( x 2 , y 2 ) ∈ Z 2 is the x − y path: the path that pro ceeds to first fix the x co ordinate then the y co ordinate i.e. ( x 1 , y 1 ) → ( x 2 , y 1 ) → ( x 2 , y 2 ). W e describ e the routing alg orithm in Figure 9. The function c omputeNext mentioned in Figure 9 finds the next no de along the x − y path. The function isOp en inv olves chec king if the next tile along the pa th is go o d o r not. This can b e done by ask ing the relev a nt relay if it has a neighbour in the ta rget tile. The function sendT o uses the r ela ys to pass the pack et to the representative node of the next tile whic h then contin ues the routing pro cess. This is the simple par t of the algorithm. If the target tile is not g oo d a BFS is launched to find the next go o d tile along the x − y path. This is a distributed algor ithm that re quires no des to b e prob ed as the sea rc h pro ceeds. Finally when it finds the destination it has to also rep ort the path ba c k to th e no de that launc hed the search. Once this path is kno wn the function s endT o Node sends the pack et along this pa th to the discovered no de. W e refer the reader to Angel et. al. [1] for a pro of that the exp ected num b er of prob es re quired for this algor ithm is at most a co nstan t times the length of the shortest path. 5 Conclusion In this pap er we hav e shown that it is p ossible to construct s parse p ower-efficien t wireless ad ho c sens or net works with go od coverage if we are willing to accept a certain level of redundancy in the system. Ideas from p ercolatio n 11 Figure 8: The path b et ween tw o representativ e p oints mimics the path in Z 2 using relay p oin ts to realize edges. theory have b een used to demonstrate that the infinite clus ter of tw o kinds of geometric ra ndom gra phs contains a go o d subgra ph with the prop erties tha t we seek a nd that this subgr aph can b e built e fficie ntly using o nly lo cal information. It is our conjecture that the subg raphs w e build should exis t whenev er a n infinite cluster exists in the geo metric rando m gr aphs we study . 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