Separability and Topology Control of Quasi Unit Disk Graphs

Transcription

1 Sepaability and Topology Contol of Quasi Unit Disk Gaphs Jiane Chen, Anxiao(Andew) Jiang, Iyad A. Kanj, Ge Xia, and Fenghui Zhang Dept. of Compute Science, Texas A&M Univ. College Station, TX {chen, ajiang, School of CTI, DePaul Univesity, 4 S. Wabash Avenue, Chicago, IL 664. ikanj@cs.depaul.edu. Depatment of Compute Science, Lafayette College, Easton, PA 184. gexia@cs.lafayette.edu. Abstact A deep undestanding of the stuctual popeties of wieless netwoks is citical fo evaluating the pefomance of netwok potocols and impoving thei designs. Many potocols fo wieless netwoks outing, topology contol, infomation stoage/etieval and numeous othe applications have been based on the idealized unit-disk gaph (UDG) netwok model. The significant deviation of the UDG model fom many eal wieless netwoks is substantially limiting the applicability of such potocols. A moe geneal netwok model, the quasi unitdisk gaph (quasi-udg) model, captues much bette the chaacteistics of wieless netwoks. Howeve, the undestanding on the popeties of geneal quasi-udgs has been vey limited, which is impeding the designs of key netwok potocols and algoithms. In this pape, we pesent esults on two impotant popeties of quasi-udgs: sepaability and the existence of powe efficient spannes. Netwok sepaability is a fundamental popety leading to efficient netwok algoithms and fast paallel computation. We pove that evey quasi-udg has a coesponding gid gaph with small balanced sepaatos that captues its connectivity popeties. We also study the constuction of wieless backbones though topology contol fo efficient communication and pesent a distibuted localized algoithm that builds a nealy plana backbone in any quasi-udg with low constant stetch facto and degee. We demonstate the excellent pefomance of these popeties though simulations and show, among many applications, thei application in efficient outing. I. INTRODUCTION The connectivity stuctues of wieless netwoks exhibit stong coelations with the physical envionment due to the signal tansmission model of wieless nodes. A deep undestanding of the stuctual popeties of wieless netwoks is citical fo evaluating the pefomance of netwok potocols and impoving thei designs. So fa, many potocols have been based on the idealized unit-disk gaph (UDG) netwok model, whee two wieless nodes can diectly communicate if and only if thei physical distance is within a fixed paamete R. Examples of these potocols include outing [], [9], topology contol [1], distibuted infomation stoage/etieval [4] and a geat vaiety of othe applications. In pactice, howeve, the UDG model significantly deviates fom many eal wieless netwoks, due to easons including multi-path fading [6], [1], antenna design issues, inaccuate node position estimation, etc. It is not uncommon to obseve stable links that ae five times longe than unstable shot links [1]. The significant deviation of the UDG model fom pactice is substantially limiting the applicability of potocols based on UDGs. To combat the poblem, a much moe geneal netwok model, the quasi unit-disk gaph (quasi-udg) model, has been poposed in ecent yeas to captue the non-unifomity chaacteistic of most wieless netwoks. Fomally, it is defined as follows. Definition 1: A quasi-udg model is chaacteized by two positive paametes R and (R ). Fo any two nodes u, v in a quasi-udg netwok deployed in a plane, let d(u, v) denote thei Euclidean distance. Then, if d(u, v), an edge (link) exists between u and v; if d(u, v) > R, the edge does not exist; if < d(u, v) R, the edge may o may not exist. The undestanding on the popeties of geneal quasi-udgs, howeve, has been vey limited. That is in shap contast to UDG, whose popeties have been well undestood [1], [9]. Among the limited knowledge about quasi-udg, a notable esult is the link-cossing popety discoveed fo quasi- UDGs whee R []. The seious lack of undestanding on the popeties of geneal quasi-udgs is impeding the designs of key netwok potocols and algoithms. In this pape, we pesent esults on two impotant popeties of quasi-udgs: sepaability and the existence of powe efficient spannes. Netwok sepaability is a fundamental popety leading to efficient netwok algoithms and fast paallel computation [11]. A (vetex) sepaato of a gaph G is a set of vetices whose emoval splits the gaph into two non-adjacent pats of simila sizes. We call a gaph G well sepaable if any subgaph of G has elatively small sepaatos. A well sepaable gaph has stong locality popeties. As a esult, the pefomance of potocols fo outing, infomation etieval, netwok monitoing, etc., can be significantly impoved fo such gaphs. We fist constuct a gid gaph that is an abstaction of the given quasi-udg G and show that the gid gaph is well sepaable. The sepaato we obtain is of size O( N) and can split the gaph into two pats of size oughly N, whee N is the numbe of nodes of the gid gaph. In addition, both the degees of the gid nodes and the numbe of edges cossing any edge ae uppe bounded by constants. Among many applications of the sepaatos, we pesent, as an example, a compact outing potocol based on the gid gaph constuction and distance labelling. We pove that the outing table size of each node in ou potocol is bounded by O( N log N), which is much bette than the tight bound poved fo geneal gaphs and close to the lowe bound of Ω( N) fo degee bounded gaphs in [7]. The atio of the outing path length to the shotest path length is uppe bounded by + ǫ whee ǫ is a small constant. Moe extensions of the

2 esults ae also included. In the second pat of the pape we study the existence and the constuction of enegy efficient backbones fo quasi-udgs. A backbone is a spanning subgaph of the wieless netwok fo efficient communication, obtained though puning a set of edges. By using only those edges in the backbone fo communication, signal intefeence, outing table size and powe usage can be substantially educed. A majo equiement fo backbone constuction is to peseve the shotest path distances between vetices as much as possible. Fo a backbone B of a gaph G = (V, E), the stetch facto is defined as s(b) = max{ fb(u,v) f u, v V }, whee f G(u,v) B(u, v) and f G (u, v) ae the distances between vetices u, v in B and G, espectively. The stetch facto eflects the quality of the backbone. Thee have been esults showing that fo UDGs, bounded degee and plana spannes can be constucted when the distance function f(u, v) is defined as the minimum powe needed to send a message fom u to v [8][14]. In this pape, we pesent a distibuted algoithm that constucts a backbone B fo any quasi-udg G with a constant powe stetch facto. The node degees of the backbone B ae uppe bounded by a constant. In addition, although it is in geneal impossible to constuct plana backbones with constant stetch factos fo quasi-udg, we show that B is nealy plana, specifically, B has a constant uppe bound on the aveage numbe of edges cossing an edge. The latte popety is useful fo geogaphic outing algoithms based on coss link detection [1]. We evaluate the pefomance of the sepaatos, the outing potocol and the backbone constuction though extensive simulations. Thei pefomance is much bette compaed to the theoetical analysis of the wost cases. This shows that although the quasi-udg model is quite diffeent fom the UDG model, efficient algoithms can still be developed by exploiting the locality in the model. The est of the pape is oganized as follows. In section II, we pesent the gid gaph constuction and pove its sepaability esult. In section III, we pesent the backbone constuction though topology contol. In Section IV, we pesent the compact outing potocol based on the gid gaph and distance labelling, as well as the simulation esults. We conclude the pape in section V. II. GRID GRAPH OF QUASI-UDGS In this section, we pesent a distibuted algoithm fo constucting a gid gaph fo any quasi-udg, and pove that the gid gaph is well sepaable. The gid gaph, whose node density and edge density ae both uppe bounded by constants, is an abstaction of the quasi-udg. A quasi-udg may have highly vaiable node and edge densities, which pevent it fom having small sepaatos. The gid gaph is a spasified vesion of the quasi-udg, which etains the distance infomation fo vetices and well epesents the deployment egion of the quasi-udg. As a esult, the connectivity-elated esults fo the gid gaph can be easily mapped to esults fo the quasi-udg. An example of a quasi-udg and its coesponding gid gaph is shown in Fig. 1(a), (b). In the following, we pesent details on the gid gaph. (a) (c) Fig. 1. Gid Gaph Example. (a) A quasi-udg G with 1 vetices and R/ =.5; (b) The gid gaph coesponding to G; (c) The auxiliay gaph used to find the top level sepaato of G; (d) The backbone of G. A. Constuction of the gid gaph H To obtain a gid gaph H fo a quasi-udg G, we impose a gid on the plane and view each non-empty cell as a vetex. The constuction is shown in Fig.. Algoithm GidGaph INPUT: G = (V G, E G ): a quasi-udg with paametes R and OUTPUT: H = (V H, E H ): the gid gaph fo G 1. Impose a gid of cell size on the plane;. Fo each cell that has at least one vetex of G, H has a coesponding vetex, whose position is set at the cente of the cell;. Thee is an edge between two vetices of H if and only if thee is at least one edge connecting two vetices of G that ae, espectively, in the two coesponding cells. Fig.. (b) (d) Constucting gid gaph fo quasi-udg All the vetices of G in the same gid cell ae adjacent. The algoithm GidGaph can be easily implemented in a distibuted manne. The following theoem poves the constant uppe bounds fo the node density, edge density and the numbe of edges cossing any edge in the gid gaph H. Theoem 1: The algoithm GidGaph constucts a gid gaph H fo given quasi-udg G such that: (1) inside any disk of adius y, thee ae at most O( y ) vetices; () the degee of each vetex is uppe bounded by O( R );() the numbe of edges cossing any edge is uppe bounded by O( R4 ). 4 Poof: By the algoithm, the Euclidean distance between any two vetices of H is at least. Hence if we place an

3 open disk of adius centeed at evey vetex, no two disks will intesect. Theefoe given any disk of adius y, the numbe of such open disks intesecting it is uppe bounded by O( y ). So is the numbe of vetices of H inside the disk. Conside a vetex U of H, denote by v(u) the set of nodes of G inside the cell epesented by U. The numbe of vetices of H within distance R + to U is bounded by O( R ). No node of G in v(u) can be adjacent to w v(v ) whee V is moe than distance R + fom U. Hence the degee of U is uppe bounded by O( R ). Similaly, fo an edge {U, V } of H, the numbe of gid vetices within distance R + to any point in the line segment connecting U and V is also uppe bounded by O( R ). Theefoe, the total numbe of edges cossing {U, V } is uppe bounded by O( R4 4 ). If two vetices of H ae h hops away fom each othe, then two vetices of G in the two coesponding cells ae at most h + 1 hops away fom each othe. Note that the above method fo constucting gid gaphs, and the above esults, can be easily extended to thee and highe dimensional spaces. B. Sepaability of the gid gaph H Netwok sepaability is a fundamental popety that leads to efficient netwok algoithms (in paticula, those algoithms based on the divide and conque paadigms), fast paallel computation, and impovements in the study of computational complexity [11]. Many applications in wieless ad hoc netwoks (outing, infomation etieval, etc.), as well as quite a numbe of had theoetical poblems, have moe efficient solutions if the undelying gaph is well sepaable. Fo example, shotest path outing can be ealized with small outing tables when the gaph has small sepaatos, as in the case of plana gaphs o gaphs with bounded tee width [7]. Also, NP had poblems such as vetex cove and independent set ae solvable in polynomial time if the input gaph and all its subgaphs have bounded sepaatos. In this subsection, we study the sepaability of the gid gaph obtained above. We begin with a fomal definition of the sepaability of gaphs. Definition : Given a gaph G of n vetices, a b-sepaato of G is a set of vetices whose emoval splits G into two nonadjacent subgaphs, each of which contains at most bn vetices. We call a gaph G (f(n ), b)-sepaable if evey subgaph of G has a b-sepaato of at most O(f(n )) vetices, whee f(n ) is a function of the numbe of vetices n in that subgaph. In ode to compute a small sepaato fo the gid gaph H, we use the help of a plana auxiliay gaph T. Fist, we impose a lage gid on the plane and map the gid gaph H to an auxiliay gaph that is nealy plana. Then, we planaize it by adding a vitual vetex at the middle of each diagonal edge, eliminating all edge cossings. (Note that we see all the edges as being staight.) The detailed constuction of the auxiliay gaph T is pesented in Fig.. All the vitual vetices in T ae denoted by ed vetices and the othes which epesent cells ae denoted by black vetices. Each ed vetex has weight zeo, while each black vetex has a weight that equals the numbe of vetices of H in the coesponding cell. Algoithm AuxiliayGaph INPUT: H = (V H, E H ): a gid gaph with paametes R and OUTPUT: T = (V T, E T ): the auxiliay gaph fo H 1. Impose a gid of cell-size (R + ) (R + ) on the plane;. Fo each cell that has at least one vetex of H, T has a coesponding black vetex v, whose position is set at the cente of the cell; we assign to v a weight that equals the numbe of vetices of H in that cell;. Add an edge between two black vetices u, v of T if and only if thee is at least one edge connecting two vetices of H that ae, espectively, in the two coesponding cells; 4. Fo each pai of cossing edges {u, v}, {w, x}, add a ed vetex at the intesection of the two edges and eplace those two oiginal edges with fou new edges that connect the ed vetex, espectively, to the fou black vetices u, v, w and x; let the weight of the ed vetex to be ; 5. Fo each diagonal edge between two black vetices, we add a ed vetex of weight at the middle of the edge and eplace that oiginal diagonal edge with two new edges that connect the ed vetex, espectively, to those two black vetices. Fig.. AuxiliayGaph(H) Fig. 1(c) shows an example of the auxiliay gaph. The longest edge in the auxiliay gaph has length R +, and ed vetices ae eithe of degee o 4. Since the cell we apply in this algoithm is lage enough (of side length R + ) and all black vetices ae placed at the centes of thei coesponding cells, any black vetex may only connect to the eight black vetices aound it befoe the ed vetices wee added. Theefoe, aound each black vetex, thee can be at most fou ed vetices; and no two ed vetices ae adjacent to each othe. Fomally, we have the following lemma. Lemma 1: Let N T,b be the numbe of black vetices in the auxiliay gaph T. Then T is a plana gaph of at most N T,b vetices, and no two ed vetices ae adjacent. Lipton and Tajan poved in thei celebated Sepaation Theoem [11] that fo any vetex-weighted plana gaph of n vetices, thee exists a set of O( n) vetices that sepaates the gaph into two non-adjacent subgaphs, each of which weighs at most of the total weight of the gaph. The sepaato algoithm pesented in [11], howeve, is elatively complex. Fo the plana auxiliay gaph T, which has a constained stuctue, we pesent a simple and pactically moe efficient algoithm fo finding such a small sepaato. Based on that, the algoithm also finds a small sepaato fo the gid gaph H. The details of the algoithm ae pesented in Fig. 4. We now pove that the algoithm Sepaato constucts small balanced sepaatos fo H and T. We stat with a lemma. Lemma : Let ˆT be any subgaph of the auxiliay gaph T. If its oute face has k vetices, then the numbe of inne vetices (the vetices not on the oute face) is at most k π. Poof: The oute face of the plana gaph T is a closed cuve (o closed cuves, if ˆT is disconnected) on the plane. Let x = R + / be the side length of the cells in the constuction of the auxiliay gaph T. Fo each inne vetex of ˆT, we place a x x squae centeed at it, then otate

4 Algoithm Sepaato INPUT: H: a gid gaph with paametes R and OUTPUT: S H : a sepaato fo H. S T : a sepaato fo T. (T is the auxiliay gaph of H.) 1. Let T be the auxiliay gaph of H. Let T be a copy of T.. Build a beadth-fist seach (BFS) tee fo a dynamically changing gaph T (T changes because new edges ae added to it duing the BFS pocedue) in the following way: (1) pick a vetex v on the oute face of T to be the oot and stat the BFS; () duing the BFS pocess, when a vetex u is dicoveed (put into the BFS tee), fo evey face containing u, add edges fom u to as many othe vetices in the face as possible so long as T emains a simple plana gaph; if afte adding those edges, thee ae still faces containing u that ae not tiangulated, add edges to tiangulate them abitaily. Duing the BFS, a vetex s undiscoveed neighbos ae visited in the clockwise ode (stating with the vetex s paent in the BFS tee as the efeence point);. Check evey fundamental cycle (a cycle fomed by a non-tee edge and some tee edges) in the BFS tee. Let S T be a fundamental cycle that sepaates T (theefoe also T ) in the most balanced way, i.e. the diffeence between the summation of the weights of vetices in the two sepaated subgaphs A 1, B 1 is minimized. 4. Conside the gaph T. Let S T be a copy of S T. Fo each ed vetex u in S T with the set of neighboing vetices N(u), we distinguish two cases: Case (1) All vetices in N(u) belong to A 1 (espectively, B 1 ) except those in S T. Then, we move u fom S T to A 1(espectively, B 1 ); Case () Both A 1 and B 1 contain vetices of N(u). Then, we put all vetices in N(u) into S T and move u fom S T to A Let S H be the set of vetices of H in those cells coesponding to the black vetices of T in S T. Let A, B be the two sets of vetices of H in those cells coesponding to the black vetices of T in A 1 and B 1. Clealy, S H sepaates H into A and B. Fig. 4. Sepaato the squae by 45 degees. It is simple to see that now these (diamond shaped) squaes centeed at the inne vetices do not ovelap each othe. The aea of each squae is x. Fist conside the case when the oute face is connected, i.e. ˆT is connected. The oute face of ˆT consists of seveal (at least one) simple cycles. Suppose thee ae i such simple cycles of size k 1, k,..., k i in the oute face. i j=1 k j can be geate than k, the numbe of vetices in the oute face, because in the summation a vetex can be counted moe than once. The simple cycles fom the oute face of a plana gaph, so the numbe of times vetices ae ove-counted is exactly i 1. Thus i j=1 k j = k + i 1. [( i Fist we have k = j=1 k j ( i ) i j=1 k j l j k l i j=1 (i 1)k j+(i 1) i [ ( i i )] j=1 k j l j k l (i 1) ) ] i i + 1 = j=1 k j + j=1 k j + i j=1 k j. The last inequality holds because k j and l j k l contains exactly i 1 tems. The equality holds when i = 1. Each simple cycle of k j vetices has k j edges, thus the peimete of the cycle is at most k j x. Theefoe the aea of the egion inside the cycle k j is at most k j x 4π and the total aea of the egions inside the oute face is bounded by i j=1 k j x 4π k x 4π. Now if thee ae seveal disconnected cycles in the oute face, each connected pat say, of k vetices suounds a egion of aea no moe than k x 4π, since k ( k ) = k, the total aea of the egions suounded by the oute face is also bounded by k x 4π of inne vetices is bounded by k x 4π Thus, in all cases, the total numbe x / = k π. Define the depth of a tee to be the maximum numbe of edges in a path fom the oot to a leaf. We have: Lemma : Let N T be the numbe of vetices in the auxiliay gaph T. The BFS tee constucted in Step of the algoithm Sepaato is of depth at most N T. Poof: Let d be the depth of the BFS tee. Because of the tiangulation opeation enfoced on the gaph T duing the BFS pocess, fo i = 1,,, d 1, the vetices at level i (if i = 1, include the oot as well) of the BFS tee actually contain all the vetices on the oute face of the subgaph induced by the vetices at levels i, i + 1,, d. So it suffices to show that if we peel off one oute face fom T at each step, T becomes an empty gaph afte t N T steps. Let n x be the numbe of vetices emaining in the gaph T afte x steps. (By convention, define n = N T.) By Lemma, we know that in the x-th step we have peeled off at least πn x vetices. So n t 1 1, n i n i+1 + πn i+1 fo i = t, t,,. Now let us pove that n t j j by induction: when j = 1, we have n t 1 1 and when j =, we have n t 4; suppose ou claim is tue fo j i; conside the case j = i + 1, whee n t (i+1) n t i + πn t i i + π i i + i + 1 = (i + 1). We have N T = n = n t t t. So t N T. By Lemma in [11], if a vetex-weighted plana gaph has a spanning tee of depth h, then thee exists a fundamental cycle of size at most h + 1 that sepaates the gaph into two non-adjacent subgaphs each of which weighs no moe than / of the total weight of the gaph. As the BFS tee obtained in Step of Algoithm Sepaato is of depth at most N T, we have the following theoem immediately. Theoem : Let N T be the numbe of vetices in the auxiliay gaph T, and let N H be the numbe of vetices in H. Then, the total weight of the vetices of T is N H, and the set S T obtained in Algoithm Sepaato contains at most N T + 1 vetices and sepaates T into two non-adjacent subgaphs each of which weighs no moe than NH. We now pove that the algoithm Sepaato also finds a small balanced sepaato fo the gid gaph H. Theoem : Let N H be the numbe of vetices in the gid gaph H. Then, the algoithm Sepaato constucts a sepaato S H of size O( N H ) that sepaates H into two non-adjacent subgaphs each of which has no moe than NH vetices. Moeove, the gid gaph H is ( n, )-sepaable when the weights of all the vetices of H ae set to be 1. Poof: Let N be the numbe of black nodes in T. Clealy N N H ; and it is staightfowad that each cell coesponding to a black vetex of T contains at most (R+ /) vetices of H. Hence we have N = Θ(N H ). Fom lemma 1 we know that the numbe of ed vetices is no moe than N, and the total weight of vetices in T is N H. Hence the sepaato S T fo T contains no moe than N +1 vetices

5 whose weights sum up to O( N H ), and sepaates T into two pats each of which weighs no moe than NH. Now we show that afte Step 4 of Algoithm Sepaato, S T is still a sepaato fo T of size O( N ), and A 1 and B 1 ae still of weight no moe than NH. Conside any ed vetex u S T in Step 4, in the case whee all of u s neighbos ae eithe in S T o A 1 (espectively, B 1 ), S T \{u} can sepaate T into A 1 {u} and B 1 (espectively, A 1 and B 1 {u}). Note that u has weight, so moving u fom S T to A 1 (o, B 1 ) does not change thei weights. In the complimentay case, the algoithm moves all u s neighbos into S T and moves u into A 1 ; clealy S T still sepaates A 1 and B 1. And by doing that, we decease the weights of both A 1 and B 1. The size of S T inceases by at most fo each ed vetex. Hence afte Step 4, we have eplaced all ed vetices in S T by black ones, inceasing the size of S T by at most thee times, not inceasing the weights of A 1 and B 1. Most impotantly, S T still sepaates A 1 and B 1. Theefoe S T is still of size O( N ) = O( N H ), and the weights of A 1 and B 1 ae no moe than NH. Each cell coesponding to a black vetex of T contains a bounded numbe of vetices of H, so S H is of size O( N H ). Also, the numbe of vetices in A (esp., B ) equals the weight of A 1 (esp., B 1 ) (at most NH ). By the constuction of the auxiliay gaph T, if no two black vetices ae joined by an edge o two edges with a ed vetex in the middle, thee is no edge connecting vetices of H in those two coesponding cells. A 1 and B 1 ae not adjacent in T, and S T has no ed vetex. So A and B obtained in Step 5 ae not adjacent in H, and S H sepaates A and B in H. It is simple to see that any subgaph of H can be used as the input of Algoithm Sepaato, and the above aguments still hold. Hence H is ( n, )-sepaable. Fo some applications, a pefectly balanced sepaato is desiable. By using the same technique descibed in [11], we can constuct a sepaato of size O( N H ) that sepaates H into two pats each of which has no moe than NH vetices. The idea is to sepaate the lage pat of the outcome of the algoithm ecusively. Hence we have Coollay 1: Let N H be the numbe of vetices in the gid gaph H. H is ( n,.5)-sepaable. Fo the gid gaph, we can develop a shotest path outing scheme based on its sepaatos, using the idea of distance labelling [7]. We can then tansfom it into a compact outing scheme fo the undelying quasi-udg G with a small stetch facto. The following theoem summaizes the esult. We leave the details of the outing algoithm, the poof of Theoem 4 and the extended esults to section IV. Theoem 4: Fo any quasi-udg G of N G vetices, let h(u, v) be the minimum hop distance between vetices u, v. Thee is a outing potocol that guaantees the outing path fom u to v to have at most h(u, v) + 1 hops, fo any two vetices u and v. The size of the outing table at each node and the message ovehead ae both O( N G log N G ). III. BACKBONE WITH CONSTANT STRETCH FACTOR We denote by backbone of a given gaph a subgaph that contains the same set of vetices but fewe edges. One example of backbones ae spanning tees. Backbones, paticulaly those with small stetch factos and degees, have vey impotant applications in wieless communication because they can help educe signal intefeences and simplify algoithms. In this section, we pesent a distibuted constuction of a backbone with constant stetch facto, constant node degee and a small numbe of edge cosses fo quasi-udgs. It is also an extension of the gid method descibed in Section II. We will show in Section IV that these backbones can also help educe the outing table size in ou outing scheme. A. Algoithm constucting the backbone Enegy is a majo limitation in wieless netwoks. Accodingly, the stetch facto of backbones is often defined based on enegy consumption. We stat with its fomal definition. Definition : Let u = u 1 u u k = v be a path fom u to v in the gaph G. Denote by ab G the Euclidian distance between any two vetices a and b. The communication cost between u, v following the given path is defined as: c G (u, v) = k 1 i=1 α u iu i+1 β G, whee β is the path loss exponent, β 5, and α is a scaling facto linea in the numbe of sent bits. If thee is no path fom u to v, c G (u, v) is defined as +. Definition 4: Given a gaph G = (V, E) and a backbone B of G, the stetch facto of B is defined as: { } cb,min (u, v) max, u,v V c G,min (u, v) whee c B,min (u, v) and c G,min (u, v) denote the minimum communication cost (ove all the paths) between u, v in gaph B and G, espectively. The stetch facto defined above is also called the powe stetch facto. We say that a backbone is enegy efficient if its powe stetch facto is bounded by a constant. We next pesent a distibuted localized algoithm that, when given a quasi-udg G, constucts a backbone whee the maximum degee of a node is bounded by O( R ), the aveage numbe of cossings of an edge is bounded by O( R4 ) and the 4 powe stetch facto is bounded by +ǫ, whee ǫ is a constant that can be made abitaily small. To un the algoithm, we classify the edges in the quasi-udg G into two types: shot edges whose lengths ae no geate than ; and long edges whose lengths ae stictly lage than. In ou algoithm, we fist educe the numbe of shot edges in the gaph by applying an opeation simila to Gabiel Planaization [5] to make the subgaph induced by all shot edges of G a plana gaph. In the second step, we apply an opeation descibed in [8] to bound the numbe of shot edges incident to any node. Finally, we apply a gid opeation to educe the numbe of long edges in the gaph. Figue 5 contains the details fo ou algoithm.

6 Algoithm QuasiUDG-Backbone INPUT: G: a quasi-udg with paametes R and OUTPUT: B: a backbone of G 1. Planaize the subgaph induced by shot edges of G The subgaph B will contain the same vetex set as G. Initially, the edge set of B is set to empty. Fo each edge e{u, v} in G, if thee is no common neighbo of u and v in G esiding in the disk whose diamete is the edge e{u, v}, we add e{u, v} into B. Simila to the Algoithm 1 descibed in [14], this pocess can be done in a distibuted manne by exchanging no moe than O(m) messages whee m is the numbe of edges in G.. Reduce the numbe of shot edges incident to each vetex Let G be the subgaph of B that includes all the vetices and shot edges of B. Note that hee G is in fact the Gabiel gaph constucted fom a UDG (with communication ange ); so G is plana. We apply the algoithm descibed in [8] on G. Hee is a bief desciption of the algoithm that is pefomed by each vetex: Diect the edges in G (using the classical acyclic oientation of a plana gaph) so that evey vetex in G has at most 5 incoming edges; Pefom a standad Yao step [8] on the set of outgoing edges; Select cetain edges that fom lage angles with consecutive edges (see [8] fo details); Finally, communicate with all the neighbos of the vetex and keep edges that have been selected by least one of thei ends. When the above algoithm ends, we emove fom B those edges that have been emoved by the algoithm fom G. This step will educe the numbe of shot edges incident to evey vetex to a constant k + 5, whee k is a selectable paamete, and it can be done locally. Compaed to the subgaph of G that contains all the shot edges of G, B inceases the minimum communication cost between any two vetices by a facto of at most 1+( sin(π/k)) β, whee k is a paamete, and β is the path loss exponent.. Reduce the numbe of long edges incident to each vetex Add all the long edges of G to B. We impose a gid of cell-size on the plane. Clealy, any long edge must be connecting vetices in two diffeent cells. Fo each pai of cells, we emove fom B all the long edges between them except fo the shotest one. Fig. 5. Constuct a backbone fo a given quasi-udg Theoem 5: The algoithm QuasiUDG-Backbone constucts a backbone of the given quasi-udg G such that its maximum degee is O( R ), the aveage numbe of edges cossing an edge is O( R4 ), and the powe stetch facto is 4 +ǫ (whee ǫ is a constant that can be made abitaily small). Poof: Let G be the subgaph of G that includes all the vetices and shot edges of G. It is easy to see that G is a UDG. Theefoe afte Step 1 and Step of the algoithm, we have emoved the cossings between shotest edges, and educed the numbe of shot edges incident to any vetex to no moe than k + 5, whee k > 8 is the paamete to the algoithm [8]. Note that in Step, we keep at most one edge between any two cells, and the numbe of cells eachable fom any vetex is bounded by O( R ). The total numbe of long edges incident to any vetex is then bounded by the same constant. Thus in the final backbone, the degee of a node is bounded by O( R ). On the othe hand, any edge cossing in the final backbone has to involve a long edge since the subgaph induced by shot edges is plana. Fo an abitay edge e, we will bound the numbe of long edges that can coss it. Any long edge that cosses e must connect one cell at one side of e to anothe cell on the othe side. We can veify that the numbe of cells on one side of e that can connect to a cell on the othe side is ω = O( R ). Theefoe, the numbe of long edges that can coss e is at most ω = O( R4 ). Suppose that the total numbe 4 of edges in the final backbone is m. Then the total numbe of edge cossings is bounded by O( R4 )m. Theefoe the aveage 4 numbe of edges cossing an edge is bounded by O( R4 ). 4 Afte Step 1 and, we have constucted a plana powe spanne fo G of stetch facto bounded by 1+ β sin β (π/k) [8]. In Step, by emoving all the edges between any two cells C 1 and C except the shotest among them, the stetch facto is inceased but still bounded by + β+1 sin β (π/k). To pove this bound, we only need to pove that fo any edge {x, y} of G that is emoved, thee is a path fom u to v in the final backbone such that the atio of the communication cost of the path and that of the edge {u, v} is at most + β+1 sin β (π/k). If the edge {u, v} is emoved in the Step 1, we know that the communication cost between u, v did not change (because β ). Othewise we distinguish two cases: Case 1, the edge {u, v} is emoved in step. In this case, [8] guaantees that by the end of step, thee is a path fom u to v consisting of edges of length at most and the stetch facto of the path is bounded by 1+ β sin β (π/k). Since step only emoves edges of length geate than, the above path fom u to v is peseved in the backbone and the stetch of the path is bounded by 1 + β sin β (π/k) < + β+1 sin β (π/k). Case, the edge {u, v} is emoved in Step. In this case, the length of {u, v} is geate than and thee is anothe edge {u, v } in the final backbone such that u and u belong to the same cell, v and v belong to the same cell, and d(u, v ) d(u, v). By an agument simila to that in case 1, thee must exit a path between u and u in the final backbone whose communication cost is at most (1 + β sin β (π/k))d(u, u ) β (1 + β sin β (π/k)) β < (1+ β sin β (π/k))d(u, v) β. Similaly, thee is a path between v and v in the final backbone whose communication cost is at most (1 + β sin β (π/k))d(u, v) β. Since d(u, v ) d(u, v), the stetch facto of the path (u, u, v, v) is at most (1 + β sin β (π/k)) + 1 = + β+1 sin β (π/k). Note that β+1 sin β (π/k) can be made abitaily small by choosing a sufficiently lage paamete k. This completes the poof. IV. APPLICATIONS AND PERFORMANCE EVALUATION In this section, we fist pesent out outing algoithm based on the sepaatos, then pove the bound fo the path stetch facto of ou outing potocol. As the second pat of the section, we show the simulation esults of the backbone constuctions and the outing pefomance of ou outing algoithms to veify the theoetical bounds we pove. A. A outing scheme based on the sepaatos As one of the applications of the small sepaatos of the gid gaphs, we pesent a outing scheme fo quasi-udg based on the gid gaph and analyze its pefomance. Ou outing scheme is suitable fo systems in which the size of the messages itself is elatively lage. We will give the simulation esults late in this section.

7 Ou outing scheme is based on the distance labelling scheme descibed in [7]. The basic idea of distance labelling is to give each vetex a label such that the distance between two vetices can be computed using only thei labels. A staightfowad labelling scheme is to stoe in each node a full table of the distances to all the othe vetices. The goal of the distance labelling scheme in [7] is to find the labels of minimum length. The sepaability of the undelying gaph is a key facto of how good a distance labelling scheme is available fo the netwok. In [7] the authos poved that fo a gaph which has a sepaato of size k, thee is a distance labelling scheme of label size O(k log n + log n), and the distance between two nodes can be computed in time O(log n), whee n is the numbe of nodes in the netwok. Although a quasi-udg G may not possess a small sepaato, we have poved that the gid gaph H with n vetices constucted fo G does have a balanced sepaato of size O( n). Conceptually, ou outing potocol utilizes two-level outing: vitually, the message is sent in the gid gaph fom the cell containing the souce to the cell containing the destination, via the shotest path in the gid gaph; in eality, the outing is implemented in the undelying quasi-udg to oute fom cell to cell. (Note that in each cell, the quasi-udg vetices ae fully connected, so outing fom one cell to the next takes at most two hops.) The basic idea to achieve shotest path outing in the gid gaph is to split H into two non-adjacent pats using the small sepaato. Each vetex of H emembes the distance to all sepaato vetices. Thus, two vetices in the two pats (o the sepaato) can compute thei shotest path distance using that infomation, because thei shotest path must go though a sepaato vetex. We ecusively apply the same pocess to patition each pat into small pats, to enable any two vetices to compute thei shotest path distance using thei stoed infomation (thei labels). We stop patitioning a pat when its size is below a cetain constant. (We call such a pat a basic block.) Since we use balanced sepaatos, the pocess ends afte O(log n) levels of patitioning. Fo a vetex W of H, let v(w) be the set of quasi-udg vetices of G that eside in the cell coesponding to W. The following list contains the infomation that each vetex u v(w) in G stoes in ou potocol. the minimum distances (in H) to all the sepaato vetices that ae on the boundaies of all the patitions W is in; the neighboing quasi-udg vetex though which it can get to othe cells adjacent to W in H; a shotest path outing table fo the vetices of H in the basic block whee u esides. The outing potocol assumes that the souce knows the label of the destination. This piece of infomation can be obtained fom location sevice. Since location sevice is not diectly elated to ou topic, we skip the details hee. If the destination is not in the same cell as the souce, the message will follow a shotest path in H fom the souce cell to the destination cell. By utilizing the second pat of the list (label), a vetex can send a message to any of its neighboing cell in two hops. Within a basic block, the thid pat of the outing table points out the shotest path between cells diectly. Ou outing potocol compaes favoably with shotest path outing algoithms and compact outing algoithms fo geneal netwoks fo its significantly smalle outing table size and maintained constant stetch facto. Poof of Theoem 4 Poof: In the outing potocol descibed above, the fist pat of a node s outing table is of size O( N log N). The second and thid pats of the outing table both consist of a constant numbe of enties because the numbe of neighboing cells and the numbe of cells in each basic block ae both constants. The size of the outing table is then O( N log N). Inside each message we need only to cay the label of the destination vetex, so the ovehead in the message size is also bounded by O( N log N). Given a path p fom u to v, let d(p) denote its numbe of hops, and let c(p) denote the numbe of times the path p tavels fom one cell to anothe. Let p opt be the shotest path fom u to v, and let p be the outing path of ou potocol. Clealy, c(p opt ) d(p opt ), and c(p ) c(p opt ) because ou potocol uses shotest path outing in the gid gaph. p tavels fom one cell to the next in at most two hops, so d(p ) c(p ) + 1. So d(p ) d(p opt ) + 1. Sometimes we ae moe concened about the enegy consumption than the hop distance if the wieless nodes ae able to adjust thei communication ange to save powe. Let the communication cost be as defined in Section III. In eality, it is infeasible fo a node to educe its communication ange to infinitely small. Thee is always a constant ange δ below which the wieless node cannot educe its communication ange. With this assumption, we pove the following theoem. Theoem 6: Let the communication cost be as defined in Section III, and assume that the minimum communication ange is δ. (Theefoe, the communication cost of an edge of Euclidean length d is α (max{d, δ}) β.) Then, the communication cost of a outing path fom u to v geneated by ou outing potocol is uppe bounded by a constant times the minimum communication cost ove all the paths fom u to v. Poof: Let p opt be the optimal path fom u to v with the minimum communication cost C opt, and let p be the outing path of ou algoithm with cost C. If u, v ae in the same cell of the gid gaph H, then C opt αδ β, and C α β since vetices in the same cell fom a clique. So C ( β /δ β )αδ β ( β /δ β )C opt = C opt O(1). Now assume that u, v ae in diffeent cells of H. Let l opt and l denote, espectively, the numbe of hops in p opt and p. By Theoem 4, l l opt +1. So C l αr β (l opt +1)αR β l opt+1 l opt Rβ δ β B. Simulations l opt αδ β Rβ δ β C opt = C opt O(1). We conducted extensive simulations to evaluate the pefomance of ou backbone constuction algoithm and outing potocol. The pefomance has been stable and consistent. In the following expeiment, we andomly deploy N quasi-udg nodes in a -D space of size We incease the numbe of nodes, N, in the system fom 1, 15 to

8 to veify the effects of density change on the pefomance. We also incease the value R/ fom 1, 1.5,, to 1 to see the pefomance of ou algoithms fo diffeent wieless connectivity models. To mimic nontivial netwok topologies, we andomly geneate a big hole of adius andomly picked in the ange [R, R] and five small andom holes of adius picked in the ange [, R]. The centes of the holes ae unifomly andomly chosen in the plane. If the distance between two nodes is in the ange (, R], we put a diect link between them pobabilistically. Fo each configuation, we un the simulation times and take the aveage of the pefomance metics. We would like to point out that the pefomances of ou outing algoithm and backbone constuction method ae elatively independent of the size of the netwok. Ou theoetical bounds and the simulation esults both show that the quality of the backbone constucted and the stetch of the outing paths ae closely elated to the atio of to R. 1) Backbone constuction: In the backbone constuction simulations, we measue the powe stetch facto, maximum degee, the aveage degee and the aveage numbe of edge cossings on an edge in the backbone constucted and compae them to the oiginal gaph. Fo node pais whose distances ae between and R, we adjust the pobability of thei being connected to ensue an expected aveage degee of 1 in ode to compae the esults between diffeent densities and values of R/. The esults shown in Table I and Fig. 6 ae fo backbones constucted by only pefoming the fist step and the last step in Algoithm Backbone. We eliminated the esults fo the case when R = since thee the backbones ae known to be plana with powe stetch facto being 1 because of the Gabiel opeation in the fist step of the algoithm. Ou esults showed that fo all configuations the backbones have vey small powe stetch factos, much smalle maximum degee and in most cases, we can bing the aveage degee to below 6 (which is the uppe bound of the aveage degee fo plana gaphs). Even when R/ = 1, the aveage degee of ou backbones is no moe than 8. As fo the numbe of cossings, ou algoithm educed it by at least 6% in all cases. TABLE I POWER STRETCH FACTOR FOR THE BACKBONES(β = ) Stetch Facto N R/=1 R/= R/= R/= Fom the thee ba gaphs in Fig. 6, the eduction in the metics is quite unifom. It implies that the pefomance of ou algoithm is stable fo diffeent sizes of the netwok. ) Routing pefomance: We apply ou outing potocol not only to the oiginal quasi-udgs but also to the backbones we obtain. To study the pefomance, we measue the maximum label size, aveage label size and the stetch facto of outing path that is defined as the distance in the actual outing path ove the shotest path between the souce and the destination. 4 1/1 1/ 1/ / /R 6 4 N=1 1/1 1/ 1/ / /R 4 N=15 1/1 1/ 1/ / /R N= Fig. 6. The Maximum degee, the aveage degee, and the aveage numbe of edges cossing an edge fo quasi-udgs and thei backbones. The 6 bas in each goup ae, fom left to ight (1) maximum degee in quasi-udg; () maximum degee in backbone; () aveage degee in quasi-udg; (4) aveage degee in backbone; (5) aveage numbe of cossings pe edge in quasi-udg; (6) aveage numbe of cossings pe edge in backbone. Note that in some goups, the last ba is not shown, because the aveage numbe of cossings pe edge in backbones equals thee. The length of the path fo outing in the oiginal gaphs is defined as the hop distance between two nodes, while in the backbones, we use the communication cost with β = as the length of the path. In both cases we andomly pick 1 souce-destination pais in the gaph, simulate the outing pocess and compae the length of the path with the shotest. Due to the page limit we only pesent the esults on the quasi- UDG with expected degee 1 and emak that the esults ae consistent fo gaphs with othe edge densities. Table II shows the aveage values of the maximum label size and the aveage label size (with a node ID as a unit) ove the expeiments fo two cases. We obseve that the label sizes with the algoithm applied to the backbones ae smalle than those to the oiginal gaphs. This is mainly because the backbones ae spase than the oiginal quasi-udgs, hence the gid gaphs we get ae also spase and have smalle sepaatos. We will see late that this advantage comes at a cost of slightly lage stetch factos. Fig. 7(a) shows the aveage hop distance stetch factos of the outing path fo the outing algoithm applied to the oiginal gaphs diectly. In all cases we have the path stetch

9 TABLE II LABEL SIZES OF ROUTING SCHEME BASED ON SEPARATORS On oiginal On backbone N R/ Max Size Avg Size Max Size Avg Size Stetch Facto Stetch Facto Path Hop distance Stetch Factos.5 N=1 N=15 N= /1 1/ 1/ / 1 /R (a) based on G Powe Stetch Factos 1/1 1/ 1/ / 1 /R (c) based on B N=1 N=15 N= Stetch Facto Stetch Facto Path Hop distance Stetch Factos.5 N=1 N=15.5 N= /1 1/ 1/ / 1 /R (b) based on B Geedy+flooding Stetch Factos N=1 N=15 N= 1/1 1/ 1/ / 1 /R (d) Geedy+flooding Fig. 7. Stetch factos fo outing algoithms. G is the oiginal gaph, and B is the backbone. factos no lage than 1.. Fig. 7(c) shows the powe stetch factos and Fig. 7(b) shows the hop distance stetch factos of the outing paths when the algoithm is applied to the backbones. The hop stetch factos shown in Fig. 7(b) ae modeately lage than the ones shown in Fig. 7(a). It is the pice we paid fo the eduction in the size of the outing tables. It looks inteesting fom the figues that when R/ is lage(1), the algoithm geneally pefoms bette than the othe cases. This is because to maintain the same aveage node degee of the gaphs we have to decease the value of. In that case a gid gaph actually descibes the oiginal gaph moe accuately and with moe details. Hence the sizes of the labels ae lage(see Table II), but the paths we discoveed ae close to the shotest ones. We have also implemented the well known geedyfowading plus local-flooding (expanding ing seach with doubling adius) outing algoithm, and pefomed the same numbe of expeiments on the same set of gaphs. The aveage stetch factos ae shown in Figue 7(d). Ou esults indicate that compaed to that algoithm, the outing potocol based on sepaatos has a much bette stetch facto because of its obustness to the existence of holes. V. CONCLUSION In this pape, we have studied two stuctual popeties of quasi-udgs: sepaability and the existence of powe efficient spannes. Such esults lead to a deepe undestanding of the locality popeties of quasi-udg netwoks and an impovement in the development of netwoking potocols. As the futue wok, we will exploe the sepaability of quasi-udgs deployed in D space, othe popeties of quasi-udgs, and thei netwok applications. REFERENCES [1] K. ALZOUBI, X. LI, Y. WANG, P. WAN AND O. FRIEDER, Geometic spannes fo wieless ad hoc netwoks, IEEE Tans. Paallel and Distibuted Systems, vol. 14, no. 4, pp ,. [] L. BARRIÈRE, P. FRAIGNIAUD AND L. NARAYANAN, Robust positionbased outing in wieless ad hoc netwoks with unstable tansmission anges, Poc. of the 5th intenational wokshop on Discete algoithms and methods fo mobile computing and commnunications(dialm 1), pp. 19-7, (1). [] P. BOSE, P. MORIN, I. STOJMENOVIC, AND J. URRUTIA, Routing with guaanteed delivey in ad hoc wieless netwoks, Poc. of the d intenational wokshop on Discete algoithms and methods fo mobile computing and commnunications(dialm 99), pp , [4] Q. FANG, J. GAO AND L. J. GUIBAS, Landmak-based infomation stoage and etieval in senso netwoks, Poc. of INFOCOM 6, 6. [5] K. GABRIEL, AND R. SOKAL, A new statistical appoach to geogaphic vaiation analysis. Systematic Zoology, 18:59-78, [6] D. GANESAN, B. KRISHNAMACHARI, A. WOO, D. CULLER, D. ES- TRIN, AND S. WICKER, Complex behavio at scale: an expeimental study of low-powe wieless senso netwoks. Technical Repot UCLA/CSD-TR -1, UCLA,. [7] C. GAVOILLE, D. PELEG, S. PÈRENNES, AND R. RAZ, Distance labeling in gaphs, Jounal of Algoithms 5(1), pp.85-11, 4. [8] I. A. KANJ AND L. PERKOVIC, Impoved stetch facto fo boundeddegee plana powe spannes of wieless ad-hoc netwoks. To appea in the poceedings of ALGOSENSOR 6. [9] B. KARP AND H. T. KUNG, GPSR: Geedy peimete stateless outing fo wieless netwoks, Poc. of the 6th annual intenational confeence on Mobile computing and netwoking, (), pp [1] Y. KIM, R. GOVINDAN, B. KARP AND S. SHENKER, Geogaphic outing made pactical, Poc. of the nd USENIX/ACM Symposium on Netwoked System Design and Implementation (NSDI 5), Boston, MA, (May, 5) [11] R. J. LIPTON AND R. E. TARJAN, A sepaato theoem fo plana gaphs, SIAM Jounal on Applied Mathematics, Vol. 6, No., (1979), pp [1] F. KUHN AND A. ZOLLINGER, Ad-hoc netwoks beyond unit disk gaphs. Poc. joint wokshop on Foundations of mobile computing, pages 69 78,. [1] K. SOHRABI, B. MANRIQUEZ AND G. POTTIE, Nea gound wideband channel measuement. IEEE Vehicula Technology Confeence, vol. 1, pp , [14] W.-Z. SONG, X.-Y. LI, Y. WANG, AND O. FRIEDER, Localized algoithms fo enegy efficient topology in wieless ad hoc netwoks, Mobile Netwoks and Applications, 1(6):911-9,5.