Managing Random Sensor Networks by means of Grid Emulation

Transcription

1 Maagig Radom Sesor Networks by meas of Grid Emulatio Zvi Lotker 1 ad Alfredo Navarra 2 1 Cetrum voor Wiskude e Iformatica Kruislaa 413, NL-1098 SJ Amsterdam, Netherlads lotker@cwi.l 2 Computer Sciece Departmet, Uiversity of L Aquila., Via Vetoio I L Aquila, Italy avarra@di.uivaq.it Abstract. A commo assumptio i sesor etworks is that the sesors are located accordig to a uiform radom distributio. I this paper we show that uiform radom poits o the two dimesioal uit square are almost a grid. I particular, for a sychroous geographic sesor etwork we show how to emulate ay grid protocol o radom sesor etworks, with high probability. This suggests the followig framework. I order to solve a problem o a radom sesor etwork we solve the same problem o the grid. The we use our emulatio to make the obtaied solutio suitable for radom sesor etwork. We aalyze the cost of the emulatio i terms of cosumed eergy ad time. Fially we provide three examples that illustrate our method. Keywords: Routig, Schedulig, MAC-layer, Collisios, Grid. 1 Itroductio A sesor etwork is usually modeled as a radio etwork where the sesors are spread out at radom over a give area accordig to a uiform distributio. The structure of sesor etworks is complex ad presets may challeges. This is due to its radom characteristic ad its iduced physical limitatios (i.e., eergy cosumptio, trasmissio rage ad ope medium access costraits). I a radom sesor etwork usually each sesor does ot have ay kowledge about the etwork i which it is workig, uless some local iformatio is obtaied by exchagig cotrol messages with its eighbors. Moreover, sice the sesors are placed at radom, a first glace might suggest a total lack of structure. This is ot ecessarily the case. Dealig with radomess is always a problem. Oe way of dealig with it is by simulatios. This solutio is time ad effort cosumig ad its accuracy is usually hard to evaluate. Aother way of The research was partially fuded by the Europea projects COST Actio 293, Graphs ad Algorithms i Commuicatio Networks (GRAAL) ad COST Actio 295, Dyamic Commuicatio Networks (DYNAMO). F. Boavida et al. (Eds.): NETWORKING 2006, LNCS 3976, pp , c IFIP Iteratioal Federatio for Iformatio Processig 2006

2 Maagig Radom Sesor Networks by meas of Grid Emulatio 857 approachig this problem is by applyig sophisticated stochastic geometry tools. This approach is agai time costly ad it is ot always simple. Uderstadig the structure of radom sesor etworks is a quitessetial problem i the field of sesor etworks. Clearly a uderstadig of this structure ca lead to a major improvemet i eergy cosumptio ad i the overall performace of the radom etwork. A stadard ad elegat techique whe dealig with complex structures is to fid a simpler structure that is close eough to the complex oe, ad yet simple eough to uderstad (see for istace [8]). This is our mai goal i this work. Our cotributio is a grid protocol emulatio for radom sesor etworks. I order to achieve this we develop optimal schedulig schemes that avoid collisios. More precisely we propose a geeral framework that is capable of emulatig ay protocol based o a grid structure for radom sesors. I this way, we break the problem ito 2 steps. The first step is to solve the problem o the grid. Sice the grid is a well kow ad well researched structure, a textbook solutio there probably already exists. The secod step is to emulate the solutio o the radom sesor etwork usig our grid emulatio protocol. The advatages of this approach are evidet. First, there are may problems that are already optimally solved o grids. Secod, usually it is much easier to solve a problem o grids tha o a radom set of poits. Moreover we are goig to show that the cost of the grid emulatio i terms of cosumed eergy ad time is ot too high. I particular, we use our method to solve the Broadcast, the Gossipig ad the Leafy Tree problems o radom sesor etworks, obtaiig satisfactory solutios. Last, the grid emulatio ca also be used as a rule of thumb to evaluate the correctess of simulatios. I order to achieve the grid emulatio we develop a collisio-free schedulig scheme. Usig this schedulig scheme we developed a collisio-free routig algorithm that ca be easily applied i order to perform ay desired commuicatio o ay sesed area of iterest. Our scheme is completely idepedet of the routig protocol amog the locatio-aware oes [1, 11]. It is worth otig that the combiatio of the routig protocol with the schedulig scheme is the mai key for the coservatio of eergy i ay commuicatio. While the routig scheme, i fact, miimizes the eergy eeded to perform a desired commuicatio, the schedulig prevets cases where commuicatios must be repeated several times before succeedig. This cocept was iitiated by [7] where the authors dealt with radom ad determiistic schedulig fuctios. The mai differeces reside i their mai assumptios for which each sesor is aware about the positio of ay other oe ad moreover each virtual grid square is assumed to be ot empty. They also assume three basic states for the sesors. Active, whe a sesor ca trasmit, Passive, whe a sesor ca receive ad Sleep whe a sesor is switched off i order to save eergy. Cocerig collisios, those are caused by superpositios of the trasmissio rages of the sesors as i [3] but i [7] also by a extra rage, called iterferece rage (R p ). For the sake of clarity we do ot cope with such a extra rage but everythig is easily scalable.

3 858 Z. Lotker ad A. Navarra The paper is orgaized as follows. I the ext sectio we describe the model ad motivatios that led to the assumptios made i the paper. I Sectio 3 we take care of the MAC-layer i order to avoid collisios i the commuicatios. We also provide aalysis i order to estimate the eeded time for a sourcedestiatio commuicatio. I Sectio 4 we show how the combiatio of a routig protocol with our schedulig scheme ca be applied i order to emulate grid structures hece implyig a virtual ifrastructure o the etwork (see for istace [14]). Fially, i Sectio 5 we discuss some coclusive remarks. 2 Model As assumed i the large majority of the papers we cosider radom istaces of sesor etworks i the two dimesioal space (see [1, 11] for a survey o sesor etworks routig protocols). The radomess of the spread sesors is usually motivated by the applicatios. The area of iterest, i fact, where the sesig must be computed, ca be a impervious, eve dagerous area so that the sesors caot be suitably set up. Without loss of geerality we cosider a square area usig a uiform distributio. Each sesor kows its ow locatio iside the cosidered area. Positioig iformatio ca be obtaied through GPS systems, but also by cheaper meas such as services like Ad-Hoc Positioig System (APS) [15] or the GPS-less low-cost outdoor localizatio for very small devices proposed i [4]. Sesors are assumed to be sychroized. As for the locatio awareess, the sychroizatio ca be accomplished either by some strog assumptio like a cetral clock to which each sesor refers (a GPS device ca be also used for this purpose) or by meas of cheaper strategies like the oe preseted i [16]. About the eergy cosumptio cocerig the sesor commuicatios we refer to the most commo power atteuatio model [17] by which the sigal power P s of a sesor s decreases as a fuctio of the distace i such a way that ay statio s at distace s, s from s ca receive a message from s if P s O( s, s 2 ). If a sesor is reached simultaeously by more tha oe trasmissio, a collisio occurs ad the received messages are assumed to be ureadable. Note that, i what follows, with high probability we mea a probability of 1 1 N with N = beig the umber of cosidered sesors. 3 MAC-Layer I this sectio we describe a determiistic MAC-layer schedule based o the locatios of the trasmitters. For simplicity we assume that the sesors lie o a regular 2-dimesioal grid G of N = vertices V. We will remove this assumptio i Sectio 4. For the sake of geerality, we assume that some of the grid poits are free from sesors ad that some of them have more tha oe. The secod case ca be simplified just by cosiderig oe sesor i such grid poits, sice sesors i the same locatio ca check the presece of overlappig oes without loosig too much eergy ad time. Moreover, we assume that each sesor kows its positio but they do ot kow aythig about the topology

4 Maagig Radom Sesor Networks by meas of Grid Emulatio 859 Fig. 1. Schedule scheme for 1-uit square grid trasmissios. It eeds 5 time slots to perform all the commuicatios at distace 1. The white odes are the trasmittig oe, the grey are the receivers ad the black are iactive i order to avoid collisios. of the etwork except that all the sesors are o some grid poits. I order to save eergy, collisios should be avoided. We ow describe a algorithm to perform commuicatios without collisio. Sice a sesor does ot have iformatio about the other sesors, we have to assig slots of commuicatio to each pair of the etwork to esure commuicatio. A time slot is just a widow of time durig which some sesors are allowed to termiate oe trasmissio operatio. Its duratio is depedet by the techology of the used sesors ad without loss of geerality we ca cosider oe time slot as oe uit of time (see for istace Figure 1). Idepedetly of the grid structure we eed ( ) ( N 2 = 2) 2 time slots (oe for each possible pair). Ideed we ca parallelize some of the trasmissios i order to reduce the time eeded to perform evetual commuicatios. Let D = {D(x, r) :x V,r IR} be the set of disks of radius r cetered at ode x. Aschedule S :IN 2 D is a fuctio from time step to a subset of disks. Next we defie two properties of determiistic schedule. Defiitio 1. Let S be a determiistic schedule, 1 S has o collisios if ay two odes trasmittig at the same time caot reach a commo ode, i.e., c IN, S(c) is a subset of disjoit disks. 2 S is uiversal if ay source x V destiatio y V pair x, y ca commuicate ifiitely may times, i.e., x, y V ad t IN t >t: D(x, r) S(t ) with r x y. Let S(x, y, k) be the umber of slots i the schedule S that the ode x eeds to wait i order to commuicate with ode y for the k-th times. Defiitio 2. Let S be a schedule, the fairess of S is φ(s) = max {S(x, y, k) S(x, y, k 1)}. x,y V,k IN Note that, without ay iformatio about the topology of the etwork, φ represets the time eeded i the worst case to perform ay commuicatio. Lemma 1. For ay uiversal schedule S without collisios φ(s) =Θ( 4 ).

5 860 Z. Lotker ad A. Navarra Proof. Assume, by cotradictio, that φ(s) < This meas that cosiderig ay iterval of time equal to 4 64 we must fid i S all the source-destiatio pairs. Sice the umber of pairs at distace more tha 4 2 is bigger tha 16 ad that without collisios we ca parallelize at most 4 of them, at least 4 64 time slots are eeded. The claim the holds by rememberig that the umber of all the source-destiatio pairs is ( ) 2. 2 Sice we are iterested i radom poits with uiform distributio, a atural questio is whether we ca improve the expectatio of the commuicatio time. Depedig o the desired commuicatios, i may cases a good idea for a routig algorithm may be to prefer short hop istead of log oes. This is due to the fact that short trasmissios are less expesive i terms of cosumed eergy ad moreover they ca be parallelized much more tha the log oes. Let us divide all the source-destiatio pairs accordig to their Euclidea distace. Let P = {π 1,π 2,...,π d } be such a partitio where π i = {(x, y) :x, y V (G) adi 1 x y <i} is the set of all the pairs at distace i o the grid ad d = 2 is the diameter of G. Followig the previous ideas we wat to perform all the commuicatios of each π i i the best way. Sice the disks close to the boudary of the grid are ot full, we defie b(r) = max x V (G) D(x, r) to be the maximal umber of grid poits cotaied i a ball of radius r. Letopt i be the optimal umber of time slots eeded to perform the commuicatios defied by π i. Note that there is a big differece i the umber of possible disjoit disks used by opt betweethecaseofradiusr< 4 ad the case of r> 4 hece we describe two differet procedures. I the first case, we cosider a dese maximal disjoit packig P 1 (r) of the grid poits by disks of the radius r. Sicesucha packig leaves holes betwee the circles, we eed aother shifted oe, P 2 (r) to cover them (see for istace Figure 2). Sice for each disk i P 1 (r), (resp. P 2 (r)) there are oly 9 discs at distace less tha 2r (see figure 3), we partitio P 1 (r), (resp. P 2 (r)) ito 9 subparts i a way that all the distaces betwee discs i each subpart is bigger tha 2r Fig. 2. The coverage of the whole grid by the two described complemetary packigs P 1(r) (empty circles) ad P 2(r) (shaded circles)

6 Maagig Radom Sesor Networks by meas of Grid Emulatio Fig. 3. The subpartitio of P j 1, j =1,..., 9. The umbers i the figure determie to which subpartitio the ode belogs. All the grey areas show the total area that ca be covered by odes from P1 9. The dark grey areas show the odes that receive the trasmissio from the cetral ode i P1 9. Note that i this case b(3) = 13, the total time it takes to get all the commuicatios of π 3 is = 234. For the sake of clarity the grid is show just i the left part of the figure. Deote P j i to be the j =1,..., 9 subparts of the packig i =1, 2. We schedule the poits covered by P j 1 (r) to trasmit before the oes covered oly by P j 2 (r). Let g 0,0 be the poit at the ceter of the Grid ad let us cosider the circle cetered o it. We label the cotaied grid poits from 1 to b(r) isucha way that each ode get a uique label. We use the same umberig process for each circle of both P 1 (r) adp 2 (r). This umberig represets the trasmittig sequece i which every ode of P 1 (r) (resp.p 2 (r)) with the same label ca simultaeously trasmit. procedure S(T,P 1(r),P 2(r)) 1: for j =1to 9 do 2: Let v i be a ode covered by P j 1 ad let i be its label. 3: for i =1to b(r) do 4: every ode labelled as i is allowed to trasmit at radius 2r i the (T + i + b(r)(j 1))-th time slot 5: ed for 6: Let v i be a ode covered by P 2(r) adlet i be its label. 7: for i =1to b(r) do 8: every ode labelled as i is allowed to trasmit at radius 2r i the (T + i + b(r)j)-th time slot 9: ed for 10: ed for I the secod case, that is, whe r> 4, if our schedule uses oe disk i a time slot it is still ok sice the optimal solutio caot parallelize too may of such commuicatios, i.e., o more tha 9. I this way we just loose a costat factor. Lemma 2. The schedule S(T,P 1 (r),p 2 (r)) performs all the commuicatios of π i i O(opt i ) time slots without ay collisio.

7 862 Z. Lotker ad A. Navarra Proof. Let us first provide a lower boud for opt i i the case of i 4.Cosider the disk D(i) ofradiusi placed at g 0,0. Such a disk cotais exactly b(i) odes. The disks cetered i those poits have a overlappig i g 0,0. This meas that, i order to avoid the collisio i the cetral ode, all those odes have to trasmit i differet time slots. Note that the Schedule algorithm S(T,P 1 (r),p 2 (r)) eeds 18b(i) time steps for all commuicatios of π i. This meas that i this case we obtai a 18-approximatio o the umber of time slots eeded to perform all the commuicatios. If i> 4 usig packig argumets, opt i caot trasmit with more tha 9 disks at the same time, hece a 18-approximatio holds. To see that S(T,P 1 (r),p 2 (r)) is collisio free we use the fact that the distace of the discs i each P j i is bigger tha 2r. Sice each sesor trasmits at radius 2r, the sesors that trasmit simultaeously do ot iterfere with each other, ad the lemma follows. 4 Grid Emulatio I this sectio we apply the previous results i order to achieve a geeral techique for emulatig ay grid protocol with radom sesors. The idea is to move the poits to a grid structure. The movemets (Log or Short) are performed by icreasig the radius of trasmissios to esure that all the eighbors of the grid structure ca commuicate. The differece betwee the Log ad the Short movemets cocers the size of the grid structure ad the techique to calculate the relative locatios of the poits. More precisely, sometimes we use global or local iformatio. Aother importat issue is the graularity of the cosidered grid. I what follows we also estimate the eeded overhead for the cosumed eergy iduced by our emulatio strategy. Note that, sice our schedulig scheme is placed at the MAC-layer, our results ca be achieved with ay locatio-aware routig protocol. Let us assume a protocol A performed o grid etworks. Actually for each ode (x, y) ofthegridaprotocola defies the istructio A x,y (t) ithasto compute at time t. LetΓ be a mappig from the set of radom poits P to the grid odes. Note that Γ chages accordig to the size of the chose grid. I order to perform the emulatio we accumulate several time steps ito oe phase. Each phase ca be cosidered as oe basic time step i the protocol A. Theumber of time steps that defies oe phase is the output of the schedule we use i order to perform oe sigle commuicatio i the grid. Let μ be the maximal distace betwee ay pair (x, y) ad its image Γ (x, y). From lemma 2 it follows that S(T,P 1 (μ +1),P 2 (μ + 1)) has o collisio. Moreover the real distace betwee two sesors that are eighbors o the grid is less tha or equal to 2μ +1. It follows that two sesors that are eighbors o the grid ca commuicate with each other. Log Movemet. To achieve a oe to oe mappig betwee the grid poits G, ad the 2 sesors we use the results of [18]. By allowig each sesor to move at most O(log 3 4 ) we achieve such a matchig Γ with high probability.

8 Maagig Radom Sesor Networks by meas of Grid Emulatio 863 Therefore we have μ O(log 3 4 ). Without loss of geerality let μ IN. The schedule will be S(T,P 1 (μ +1),P 2 (μ + 1)) accordig to the procedure described i Sectio 3. By Lemma 2, the time eeded to perform it is the O(μ 2 ). Moreover, sice it is possible that several (roughly log()) sesors will be i the same grid square, we must multiply by a factor of O(log ) i order to eable all the possible commuicatios give by the emulated protocol A. I this case we eed global iformatio to compute Γ, i.e., each sesor has to kow its associated grid ode. I order to perform every local commuicatio roud o the grid, usig the schedulig algorithm of Sectio 3, we eed time Θ(log 3 2 ) ad also the eergy must be multiplied by the same factor. This meas that up to a poly-log factor we achieve a upper boud for the eergy ad time eeded for radom poits i the plae to emulate the protocol A. More precisely, usig the log movemet strategy we get the followig theorem. Theorem 1. Ay protocol A over a grid etwork G, ca be emulated with high probability o a set of 2 radom poits with stretch factors of O(log 5 2 ) i time ad O(log 3 2 ) i eergy. Note that, cosiderig oe source-destiatio pair, the previous method is the fastest oe i terms of time (schedulig steps), o the other had each sesor trasmits for log distace, i.e., O(log() 3 4 ). This is expasive i terms of eergy cosumptio. This suggests to cosider a suitable routig scheme i order to maage a good trade-off betwee the miimum delivery time ad the miimum eergy cosumptio. This remais a challegig issue accordig to the actual desired patters of commuicatio. Short Movemet. I this case we cosider a grid G O( log ),O( log ) but still with 2 sesors, therefore there is still a average of log odes that belog to the same grid square. We associate to the left bottom grid ode of each grid square oe sesor lyig i that square. This is accomplished by meas of stadard local leader electio strategies [13], which costs log log time steps with high probability. We summarize the Short movemet performace i the ext theorem. Theorem 2. Ay protocol A over a grid etwork G 8, ca be emulated log 8 log with probability 1 1 o a set of 2 radom poits with costat stretch factors 6 i time ad i eergy. Proof (sketch). I order to emulate A we eed to have a sesor i each grid square. Note that the size of each square is 64 log, ad so the expected umber of sesor i each grid square is 64 log. Formally let 0 <i,j< 8.Deote log the umber of sesors i the grid square i, j by X i,j. Usig Cheroff we get log() 64λ2 Pr[X i,j 64(1 λ)log()] e 2

9 864 Z. Lotker ad A. Navarra Takig λ =1/2itfollows that Pr[X i,j 32 log()] 1. To boud the 8 probability that oe of the 2 64 log grid squares i G 8, is empty we log 8 log use uio boud. Sice the umber of grid squares is less tha 2 it follows that: Pr[Mi{X i,j :0<i,j< 8 2 } 32 log()] < log 8 = 1 6 I order to perform every local commuicatio roud o the grid, usig the schedulig algorithm of Sectio 3 it follows that we eed a costat stretch factor i time ad i eergy. The costat time factor follows from the fact that we use π 1 schedulig. The eergy costat follows from locality, i.e., two eighbors o the grid are at distace 8 log (o the grid), ad their real distace o the plae is less tha 38 log. Moreover i the short movemet, the grid eighborhood of each ode coicides with the real eighborhood o the plae. Therefore usig the π 1 schedulig we ca perform all the desired commuicatios. This is accomplished by the trick of throwig a umber of sesors ( 2 ) that is bigger tha the grid dimesio ( 2 log ). The Short Movemet performs much better tha the Log oe both i time ad i eergy cosumptio. This is due to the fact that spreadig more sesors tha the umber of grid odes substatially icreases the probability that some sesor is close to a grid ode. I fact, each grid square may cotai several poits (usually log poits). Therefore, the short movemet has a overhead geerated i the iitial stage due to the leader electio which ca be doe i O(log log()). This leader will be the ode that emulates the grid odes. Moreover we ca permute the leader role amog all the sesors that belog to the same grid square. By doig this we balace the eergy cosumptio amog all the sesors, ot oly amog the leaders. By doig this we prolog the lifetime of the etwork. The time eeded to achieve this permutatio is O log() log log()). Sice such a procedure is very local it is also ot expesive i eergy. By meas of such movemets we are fially able to remove the assumptio of Sectio 3 for which the sesors were placed o the grid odes. 4.1 Applicatios I order to demostrate the stregth of our results, we ow describe some importat applicatio problems o radom sesor etworks easily solvable by meas of our techique with high probability related to the locatio of the sesors. We focus o the upper bouds obtaied by such a method. Roughly speakig the idea is to cosider a geeric protocol A ad perform it by the Log or the Short movemet. Broadcast. Oe of the most importat protocols i ay kid of commuicatio etwork is give by the Broadcast protocol (see for istace [2, 5, 9, 10]). I [5] the authors describe the optimal algorithm for grid structure roughly showig that it eeds D + 2 hops where D is the diameter of the grid. By applyig the Short Movemet we ca achieve the Broadcast i O(D) timeado( 2 )eergy

10 Maagig Radom Sesor Networks by meas of Grid Emulatio 865 hece solvig the problem almost optimally. Note that, for the specific broadcast applicatio it is useless to apply the Log Movemet wastig much more time ad eergy. Corollary 1. A Broadcast over a grid etwork G O( log ),O( log ) ca be emulated with high probability o a set of 2 radom poits with O( log ) time ad O( 2 ) eergy. Gossipig. Aother importat basic protocol i commuicatio tasks is the gossipig. Each ode participatig i the protocol is assumed to have a value which should be trasmitted to all the other oes. A trivial solutio is the give by performig 2 broadcastig commuicatio, that is, oe per ode. I [19] a O( 3 ) determiistic Gossipig algorithm for radio etworks of 2 odes is preseted without ay kowledge about the ode locatios. Restrictig the attetio to G, as show i [6] the umber of commuicatios has a upper boud of O( 2 ) ad the eeded time has a upper boud of O(). Corollary 2. A Gossipig over a grid etwork G, ca be emulated with high probability o a set of 2 radom poits with O( log 5 3 ) time ad O( 2 log 3 2 ) eergy. I order to apply the Short Movemet we have to pay attetio to the values that belog to the odes that are ot actively participatig i the protocol. We divide the protocol ito two phases. I the first oe each elected leader, represetative of every grid ode, has to collect all the values belogig to the surroudig sesors that are physically associated to the same grid ode (at most O(log )). This phase costs O(log log log ). I the secod phase the real gossipig starts betwee the grid odes. Corollary 3. A Gossipig over a grid etwork G O( log ),O( log ) ca be emulated with high probability o a set of 2 radom poits with O( log ) time ad O( 2 log ) eergy. Leafy Trees. Give a graph G =(V,E) the problem is to fid a spaig tree with a maximal umber of leaves [12]. Such a problem is very iterestig i the field of sesor etworks sice icreasig the umber of leaves reduces the umber of eeded trasmissios ad hops. Usually the uderlyig graph that models a sesor etwork is complete so the leafy tree ca be trivially solved by oe ode coected to all the other oes hece obtaiig 2 1 leaves. Actually such a solutio is practically ot feasible due to the limited resources of the sesors, moreover, we are iterested i the emulatio of grid structures. O the full grid the maximal umber of leaves is approximatively (see Figure 4). Usig the Short Movemet we obtai a umber of leaves proportioal to 2 3 of the grid odes plus all the odes associated to the same grid ode but oe, hece obtaiig,

11 866 Z. Lotker ad A. Navarra Fig. 4. The Leafy Tree for a grid etwork of 81 odes. It cotais 45 leaves. Corollary 4. A Leafy T ree over a grid etwork G O( log ),O( log ) ca be emulated with high probability o a set of 2 radom poits obtaiig roughly log + 2 (1 1 log ) leaves. 5 Coclusio I this paper we have show that the combiatio of routig ad the MAC-layer ca be efficiet i a sesor etwork i terms of eergy cosumptio ad delivery time. We have proposed a schedulig scheme that perfectly matches with ay locatio-aware routig protocol, hece obtaiig a fully fuctioal protocol for sesor etworks. We have show that a simple algorithm ca avoid ay collisio whe the sesors kow their ow locatio ad whe they are sychroized. Actually we have proposed a powerful framework able to emulate ay protocol based o grid structures for radom istaces of sesors. This ca be used as a rule of thumb, that is, istead of solvig problems o radom sesors, we solve the problems o grid etworks ad adapt the obtaied solutios to the radom istaces. We have also show the stregth of the proposed framework by solvig basic problems like Broadcast, Gossipig ad Leafy Trees. Refereces 1. Al-Karaki, J. N., ad Kamal, A. E. Routig techiques i wireless sesor etworks: a survey. IEEE Wireless Commuicatios 11, 6 (2004), Alo, N., Bar-Noy, A., Liial, N., ad Peleg, D. A lower boud for radio broadcast. Joural o Computer ad System Scieces 43, 2 (1991), Barriere, L., Fraigiaud, P., ad Narayaa, L. Robust positio-based routig i wireless ad hoc etworks with ustable trasmissio rages. I Proc. of the 5th It. Workshop o Discrete algorithms ad methods for mobile computig ad commuicatios (DIALM) (2001), ACM Press, pp Bulusu, N., Heidema, J., ad Estri, D. GPS-less low-cost outdoor localizatio for very small devices. IEEE Persoal Commuicatios 5 (2000). 5. Farley, A. M., ad Hedetiem, S. T. Broadcastig i grid graphs. I Proc. of the 9th S-E Cof. combiatorics, graph theory, ad computig (1978), pp

12 Maagig Radom Sesor Networks by meas of Grid Emulatio Farley, A. M., ad Proskurowski, A. Gossipig i grid graphs. Joural of Combiatorics, Iformatio ad System Sciece 5, 2 (1980), Friedma, R., ad Korlad, G. Timed grid routig (tigr) bites off eergy. I Proc. of the 16th ACM It. Symp. o Mobile ad hoc etworkig ad computig (MobiHoc) (2005), pp Klasig, R., Lotker, Z., Navarra, A., ad Perees, S. From Balls ad Bis to Poits ad Vertices. I Proc. of the The 16th Aual It. Symp. o Algorithms ad Computatio (ISAAC) (2005), vol of Lecture Notes i Computer Sciece, Spriger-Verlag, pp Kortsarz, G., ad Peleg, D. Approximatio algorithms for miimum time broadcast. I Proc. of the Symp. o Theory of computig ad systems (ISTCS) (1992), Spriger-Verlag, pp Kowalski, D. R., ad Pelc, A. Determiistic broadcastig time i radio etworks of ukow topology. I Proc. of the 43rd Symp. o Foudatios of Computer Sciece (FOCS) (2002), IEEE Computer Society, pp Kozma, G., Lotker, Z., Sharir, M., ad Stupp, G. Geometrically aware commuicatio i radom wireless etworks. I Proc. of the 23rd Aual ACM Symp. o Priciples of distributed computig (PODC) (2004), pp Lu, H., ad Ravi, R. The power of local optimizatio: Approximatio algorithms for maximum-leaf spaig tree. I Proc. of the 30th Aual Allerto Cof. o Commuicatio, Cotrol ad Computig (1992), pp Malpai, N., Welch, J. L., ad Vaidya, N. H. Leader electio algorithms for mobile ad hoc etworks. I Proc. of ACM Joit Workshop o Foudatios of Mobile Computig (DIALM-POMC) (2000). 14. McCa, J. A., Navarra, A., ad Papadopoulos, A. A. Coectioless Probabilistic (CoP) routig: a efficiet protocol for Mobile Wireless Ad-Hoc Sesor Networks. I Proc. of the 24th It. Performace Computig ad Commuicatios Cof. (IPCCC) (2005), pp Niculescu, D., ad Nath, B. Ad Hoc Positioig System (APS). I Proc. of the 44th IEEE Global Telecommuicatios Cof. (GLOBECOM) (2001). 16. PalChaudhuri, S., Saha, A. K., ad Johso, D. B. Adaptive clock sychroizatio i sesor etworks. I Proc. of the 3rd iteratioal Symp. o Iformatio processig i sesor etworks(ipsn) (2004), ACM Press, pp Rappaport, T. S. Wireless commuicatios: priciples ad practice. Pretice- Hall, Eglewood Cliffs, NY, Shor,P.W.,adYukich,J.E. Miimax Grid Matchig ad Empirical Measures. The Aals of Probability 19, 3 (1991), Xu, Y. A O( 1.5 ) determiistic gossipig algorithm for radio etworks. Algorithmica 36, 1 (2003),