How To Calculate Stretch Factor Of Outig I Wireless Network

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1 Stretch Factor of urveball outig i Wireless Network: ost of Load Balacig Fa Li Yu Wag The Uiversity of North arolia at harlotte, USA Eail: {fli, yu.wag}@ucc.edu Abstract outig i wireless etworks has bee heavily studied i the last decade ad uerous routig protocols were proposed i literature. Most of the existig routig protocols are based o shortest path routig. Shortest path routig ejoys iiizig the total delay, but ay lead ueve distributio of traffic load i a etwork. For exaple, wireless odes i the ceter of a etwork usually have heavier traffic load sice ost of the shortest routes go through the ceter. To solve this proble, Popa et al. [] recetly proposed a ovel routig ethod, called curveball routig (B), which ca balace the traffic load ad vaish the crowded ceter effect. I B, odes are apped o a sphere ad packets are routed o those virtual coordiates o the sphere. While B achieves better load balacig for the etwork, it also uses loger routes tha the shortest paths. This ca be treated as the cost of load balacig. I this paper, we focus o studyig this cost of load balacig for curveball routig. Specifically, we theoretically prove that for ay etwork, the distace traveled by the packets usig B is o ore tha a sall costat factor of the iiu (the distace of the shortest path). The costat factor, we called stretch factor, is oly depeded o the ratio betwee the size of the etwork ad the radius of the sphere used i B. We the coduct extesive siulatios to evaluate the stretch factor ad load distributio of B ad copare the with the shortest path routig i both grid ad rado etworks. We also study the trade-off betwee stretch factor ad load balacig. I. INTOUTION outig is oe of the key topics i wireless etworks ad various routig protocols have bee proposed ad studied. Most of the routig protocols are based o shortest path algoriths where the packets are traveled via the shortest path betwee a source ad a destiatio. Takig the shortest path ca achieve saller delay ad shorter traveled distace, however it ay also lead ueve distributio of traffic load i a etwork. As observed i previous work [], [3], uder uifor couicatio sceario, the ceter of the etwork becoes crowded, sice ore shortest paths go through the ceter tha through the periphery of the etwork. This is also the observatio fro the etropolita trasportatio syste where the dowtow area is always the hot spot for traffic cogestio. Fig. shows a siple siulatio result of this sceario. The etwork is distributed o a grid as show i Figure (a). osider a uifor couicatio sceario where each ode seds oe packet to all other odes usig the shortest path routig (SP). Fig. (b) illustrates the cuulative This work was supported i part by the US Natioal Sciece Foudatio (NSF) uder Grat No. NS-7666 ad fuds provided by the Uiversity of North arolia at harlotte. ode traffic (i.e., uber of packets passig through) for each ode. It is clear that odes i the ceter area have the ost traffic load, thus they ay ru out of their batteries ore quickly tha odes i the periphery (a) topology (b) load of SP (c) load of B Fig.. rowded ceter effect: odes i the ceter area have uch heavier traffic load tha odes i the periphery. The curveball routig ca eliiate crowded ceter effect by spreadig the traffic o the sphere. To address the ueve load distributio proble, people ivestigate o load balacig routig for large wireless etworks. By spreadig the traffic across the wireless etwork via the elaborate desig of the routig algorith, load balacig routig averages the eergy cosuptio. This exteds the lifespa of the whole etwork by extedig the tie util the first ode is out of eergy. Load balacig is also useful for reducig cogestio hot spots thus reducig wireless collisios. There are already several load balacig routig protocols [4] [6] i literature. However, ost of the try to dyaically adjust the routes to balace the real tie traffic load based o the kowledge of curret load distributio (or curret reaiig eergy distributio), which is ot very scalable for large wireless etworks. Multi-path routig [] was also used for load balacig. However, [7] showed uless usig a very large uber of paths the load distributio is alost the sae as sigle path routig. Hyytia ad Virtao [3] also studied how to avoid the crowded ceter proble by aalyzig the load probability i a dese etwork. They proposed a radoized choice betwee shortest path ad routig o ier/outer radii to level the load. ecetly, Popa et al. [] proposed a ovel load balacig routig ethod, called curveball routig (B). I curveball routig, odes are apped o a sphere ad packets are routed o those virtual coordiates. Istead of usig the Euclidea distace as the routig etric, B uses the spherical distace betwee the virtual coordiates. Fig. (c) shows the load distributio of B i the sae grid etwork uder uifor couicatio sceario. It is clear that B ca vaish the

2 crowded ceter effect. I [], B is evaluated i both s siulator ad a large testbed i Berkeley. All siulatios cofir that B ca achieve better load balacig. However, there s o such thig as a free luch. While B achieves better load balacig, it also uses loger routes tha the shortest paths. I geeral, this eas B ay eed ore relayig odes to deliver the packets thus leads to large eergy cosuptio. We treat the icrease of path legth as the cost of load balacig for B. I [], the authors did ot provide ay foral study o this cost, except claied that I the preseted siulatio, curveball routig icreases the average path legth by less tha 7.% copared to the greedy paths. Siilarly, the logest path icreases by 9%. I this paper, we focus o theoretically studyig the cost of load balacig for B. We forally defie the copetitiveess ad stretch factor of ay routig ethod copared to SP. Give a routig ethod A, let P A (s, t) be the path foud by A to coect the source ode s ad the target ode t. A routig ethod A is called α-copetitive if for every pair of odes s ad t, the total legth of path P A (s, t) is withi a costat factor α of the legth of the shortest path coectig s ad t i the etwork. The costat factor α is called stretch factor of A. The, we theoretically prove that for ay etworks, the stretch factor of B is bouded by ax( ( + ɛ),), where ɛ is a costat paraeter oly depeds o the ratio betwee the size of the etwork ad the radius of the sphere used i B. I other words, B ca guaratee the total distace traveled by packets is costat copetitive eve i the worst case. II. UVEBALL OUTING The basic idea of curveball routig [] is appig all odes i a etwork oto a 3 sphere. Sice the surface of the sphere is syetric, if odes oly couicate o the surface ad couicatio is uifor, there will be o crowded ceter effect. Oe-to-oe appig betwee odes i a plae ad odes o a sphere has bee well-studied i projective geoetry. A siple stereographic projectio [8] ca ap a ifiite plae oto a sphere ad vice versa. Fig. (a) illustrates the appig ethod used by B, which is a stereographic projectio. For a wireless etwork, the area i which the wireless odes lie correspods to a fiite regio i the plae. Let this regio be P. With the iforatio of the etwork regio, we ca place a sphere S cetered at the ceter O(,, ) of the etwork. The radius of S is a adjustable paraeter for B. Ay poit (x, y, ) i P aps to its projectio (x,y,z ) o the sphere S, which is the itersectio of S ad the lie through ad the orth pole (N(,,)). As show i Fig. (a), the projectio of a ode iside the equator is o the souther heisphere, while the projectio of a ode outside is o the orther heisphere. Note that stereographic projectio preserves circles perfectly. That is, a circle o the sphere aps to a circle i the plae ad vice versa. After appig the odes o the sphere, B routes packets o the spherical shortest paths o the sphere istead of the Eu- (a) Fig.. N(,,) O(,,) S(,, ) (b) * z Projectig the etwork o a sphere. N(,,) O(,,) S(,, ) clidea shortest paths i the plae. Note that the appig preserves topological eighborhoods, it oly chages the cost of the liks. For ay existig lik betwee two odes ad i the etwork, B uses the shortest distace o the sphere betwee their projected odes ad (deoted by d( )) as the cost of lik. The, routig decisio ca be ade either by shortest path algorith or greedy algorith o the ew costs. Hereafter, we assue the shortest path algorith is used, i.e., B chooses the route with sallest total spherical distace. B is easy to be ipleeted by the siple odificatio of shortest path routig or greedy routig ad has egligible additioal coputatioal overhead. The oly itroduced overhead is that each ode eeds to copute its eighbors spherical coordiates. III. STETH FATO OF UVEBALL OUTING I this sectio, we study the stretch factor of B, i.e., the ratio betwee the total legth of the path take by B ad the legth of the shortest path. Before givig the proof, we eed to preset soe preliiaries for stereographic projectio. Lea : Assue that the furthest wireless ode is of distace fro the ceter, the the z value of the highest projectio o the sphere (deoted this value as k) is ɛ ɛ+, where ɛ = /. Proof: First, we assue. Let be the furthest ode, be its projectio, ad be the projectio of o lie seget NS, as show i Fig. (b). The k = z ax = si O. Note that O = ON = ( NO) = NO. Thus, k = si( NO )=(si NO cos NO) =( + + )= +. Whe we choose = ɛ (i.e., ɛ = / ), thus k = ɛ ɛ+.if <, we ca draw the sae coclusio through a siilar proof. ecall that circles o the sphere ap to circles i the plae i stereographic projectio, thus the projectio of a great circle o the sphere S is also a circle i the plae. Let d( ) be the legth of a arc fro a projectio to a projectio alog a great circle o the surface of S, d() be the legth of the arc betwee ad alog the projectio of the great circle i the plae, ad be the Euclidea distace betwee ad i the plae. The ext two leas show that d( ) is bouded fro above ad below by the Euclidea distace i the plae. Lea : osider ay two odes ad i the plae with their projectios ad o the sphere S, wehave β d( ),

3 where β =whe both two odes are iside the equator ad β = +ɛ otherwise. Proof: First, sice the Euclidea distace of two poits is always saller tha the distace alog ay arc passig the, d(). Thus, we oly eed to prove d() β d( ). Let us cosider the followig three cases. ase : both ad are outside. See Fig. 3(a) for a illustratio. It is a oe-to-oe appig betwee oe poit o arc ad oe poit o arc. dx = d( ), where is a iiature seget o. Siilarly, dx = d(), where dx is the projectio of i the plae. p q is a tiy seget o with its legth, ad = p q. The projectio of p q is pq with the legth dx = pq. eote p as the projectio of p o lie seget NS. The z value of p (or p ) is deoted by z p. The Np NO = zp. Whe,dx, i.e., pq, p q, pq ad p q are i the sae plae (the plae defied by odes N,p ad q). The, due to the siilarity of triagles Np p ad NpO, dx = p q = Np pq Np = Np NO = zp. Because the highest value of z p is k fro Lea, we have dx k Thus, d() = dx = ɛ ɛ+ = +ɛ. +ɛ dx = +ɛ d( ). ase : both ad are iside. See Fig. 3(b) for a illustratio. Siilar to ase, we ca defie dx ad. Now, sice ode p is always below the plae, dx = p q = Np pq Np = Np NO. Thus, d() = dx = d( ). ase 3: oe of ad is iside ad the other is outside. Without loss of geerality, we assue is the oe outside. The arc itersects arc at ode l(l ) which is also o the equator. By dividig the arc ito two parts l ad l, we ca apply results fro ase ad ase to the. Thus, we have d() = d(l)+d(l) +ɛ d( l )+d(l ) ax( +ɛ, )(d( l )+d(l )) = ax( +ɛ, )d( ). Note that sice there is at least oe ode outside, >, i.e., ɛ>. Thus d() +ɛ d( ). (a) p q p q d() d( ) N p* O S (b) N O d() q p p* q p S d( ) Fig. 3. The legth of the projectio d() is bouded by the legth of the seget of great circle d( ) o the sphere. d() β d( ). Lea 3: osider ay two odes ad i the plae with their projectios ad o the sphere S, wehave d( ) β, where β = whe both two odes are outside the equator ad β = otherwise. Proof: Sice arc is a seget betwee ad o a circle (which ay ot be cetered at O), we have d() (Note that whe is a half circle, the ratio of d() to reaches its axiu of ). Thus, we oly eed to prove that d( ) is less or equal to a costat tie of d(). Siilar to the proof of Lea, assue that is a iiature seget o ad dx is the projectio of i the plae. ase : both ad are outside. Fro the proof of ase i Lea, we kow dx dx = z p. Thus, dx, ad d( )= dx = d(). ase : both ad are iside. Fro the proof of ase i Lea, we kow dx dx = Np NO (whe p = S, this ratio reaches axiu). Thus, dx, ad d( )= dx =d(). ase 3: oe of ad is iside ad the other is outside. Agai let l(l ) be the itersectio poit o ad assue is the oe outside, we have d( ) = d( l )+d(l ) d(l)+d(l) (d(l)+d(l)) = d(). This cocludes the proof. Now we are ready to prove the ai theore of this paper about the stretch factor of curveball routig. ecall that the stretch factor of a routig ethod A is a costat α, ifad oly if for every pair of odes s ad t, the total legth of path P A (s, t) foud by A is withi α ties of the legth of the shortest path coectig s ad t i the etwork. I other words, here we wat to prove that B ca fid a path whose legth is withi a sall costat factor of the iiu eve i the worst case sceario. There are four paths we will use i the proof: () P SP (s, t) is the shortest path betwee the source s ad the destiatio t i the plae; () P SP(s, t) is the surface path coectig all the projectios o the sphere

4 of each ode alog P SP (s, t) usig the spherical distace; (3) P B (s, t) is the path foud by B protocol i the plae; ad (4) P B(s, t) is the surface path o the sphere coectig all the projectios of each ode alog P B (s, t). Note that i ay two poits alog a path i the plae, the shortest distace is the legth of the straight lie coectig the, eawhile the shortest spherical distace of its projectio o the sphere is the legth of a seget (a arc) of a great circle. For a path P A i the plae, we defie P A as the suatio of the Euclidia distace of each lik i P A. For a path P A o the sphere, we defie P A as the suatio of the legth of each arc i P A. Theore 4: The stretch factor of B routig is bouded by β 3, i.e., P B (s, t) β 3 P SP (s, t), where β 3 = ( + ɛ) whe ɛ ad β 3 = otherwise. I.e., β 3 = ax( ( + ɛ),). Proof: Let P B (s, t) =v,v,v,,v, where v = s ad v = t. Let the projectio of P B (s, t) o the sphere P B(s, t) =v,v,v,,v. Siilarly, let P SP (s, t) = u,u,u,,u, where u = s = v ad u = t = v. Let the projectio of P SP (s, t) o the sphere P SP(s, t) = u,u,u,,u, where u = s = v ad u = t = v. Note that ay ot equal to. ase : ɛ <. I.e., < ad all odes are iside the equator. Fro Lea, we kow v i v i d(v i v i ), therefore, P B(s, t) = i= v i v i i= d(v i v i ) = P B(s, t). Accordig to the B protocol, P B(s, t) P SP(s, t) sice P B(s, t) has the shortest total spherical distace aog all routes o the sphere surface coectig s ad t. Fro Lea 3, we have d(u i u i ) u i u i. Thus, P SP(s, t) = i= d(u i u i ) i= u i u i = P SP (s, t). osequetly, P B (s, t) P B(s, t) P SP(s, t) P SP (s, t). ase : ɛ. I.e., ad soe odes are outside the equator. Fro Lea, we kow v i v i ax( +ɛ, )d(v i v i ) = +ɛ d(v i v (+ɛ) i ), sice >. Thus, P B (s, t) +ɛ P B(s, t). Accordig to the B protocol, P B(s, t) P SP(s, t). Fro Lea 3, we have d(u i u i ) ax(,) u i u i = u i u i. Thus, P SP(s, t) P SP (s, t). osequetly, P B (s, t) +ɛ P B(s, t) +ɛ P SP(s, t) ( + ɛ) P SP(s, t). Theore 4 gives a theoretical boud of the stretch factor of B protocol. It shows that the path legth i B protocol is ot too uch differet fro the shortest path routig. The stretch factor is oly related to ɛ =. ecall that is the distace of the furthest ode to the ceter of the etwork ad is the radius of the projectio sphere. Give a etwork with fixed, we ca cotrol to adjust the stretch factor. IV. SIMULATION We ow evaluate the perforace of B via extesive siulatios for both grid etworks ad rado etworks by our ow developed siulator. I both cases, wireless odes are distributed i a square area ad a siple uit disk graph propagatio odel is used. I B, the ceter of the sphere S is located at the ceter of the deployet area. The virtual coordiates of all projectio odes o the sphere are the geerated. The perforace of B is easured i ters of traffic load ad stretch factor. It is clear that the size of the sphere affects the distributio of the apped odes o the sphere, thus affects the perforace. I our siulatios, we try differet radii of the sphere (i.e., various ratio of.) ad plot the perforace figures with differet ratio of. Grid Networks: We first deploy the odes o a grid iside the square area, the set the trasissio rage of all odes to 3. The resulted topology is show i Fig. (a). We copare the perforace of the shortest path routig (SP) ad the curveball routig (B) uder the uifor couicatio sceario where every pair of odes has uit essage to couicate. Fig. 4(a-c) show the average, axiu, ad stadard deviatio (ST) of traffic load for all odes i the etwork for SP ad B with differet radii. It is clear that the axiu load (or ST of load) of B is saller tha SPT whe <.9. This shows the B ideed balaces load aog all odes. The iiu values occur whe equals to.4 or. (For exaple, the axiu load ca be reduced about 4% by B copared to SPT at =.4). As the cost of load balacig, we also study the stretch factor (SF) of B. Fro Theore 4, the SF of S is bouded by ( + ) whe ad whe <. We easure the SF for each route geerated by B i our siulatio. Fig. 4(d) shows the average ad axiu stretch factor of B o all routes. The siulatio results of SFs cofir our theoretical bouds. Actually the practical SFs are uch saller tha the bouds. The average SFs are all very close to, i.e., the distace traveled by the packets i B is alost the sae as the iiu (the distace of the shortest path). The axiu SFs of B icrease with the value of icreases which cofirs our theoretical proof. Therefore, give a etwork, the cost of load balacig icreases with the decrease of the radius of the sphere. There is a clear tradeoff betwee load balacig ad stretch factor. We also studied the ipact of the uber of odes. We deploy the 4 odes o a grid iside the square area ad set the trasissio rage of all odes to.. Fig. 4(e-h) show the siulatio results which basically have the sae patter as of the results with -odes etwork. The iiu values of axiu load (Fig. 4(f)) or ST of load (Fig. 4(g)) occur whe equals to.3 or.4. ado Networks: We test the perforace of S with rado etworks with both odes ad 4 odes. Trasissio rage is set to 4 (for -odes etwork) or (for 4-odes etwork). I each case, we geerate rado etworks ad take the average for all results. The results are also plotted i Fig. 4(a-d) ad Fig. 4(e-h). All the patters of perforaces are still alost the sae as of grid etworks. But the iiu values of axiu load (Fig. 4(b)(f)) or ST

5 SP_Grid B_Grid SP_ado B_ado Maxiu Traffic Load SP_Grid B_Grid SP_ado B_ado Stadard eviatio of Traffic Load SP_Grid B_Grid SP_ado B_ado Stretch Factor of B Grid_Avg Grid_Max ado_avg ado_max (a) (b) Maxiu Traffic Load (c) ST of Load (d) Stretch Factor of B SP_Grid B_Grid SP_ado B_ado Maxiu Traffic Load 3 x 4... SP_Grid B_Grid SP_ado B_ado Stadard eviatio of Traffic Load SP_Grid B_Grid SP_ado B_ado Stretch Factor of B Grid_Avg Grid_Max ado_avg ado_max (e) (f) Maxiu Traffic Load (g) ST of Load (h) Stretch Factor of B SP_Square B_Square SP_oud B_oud Maxiu Traffic Load 3 x 4... SP_Square B_Square SP_oud B_oud Stadard eviatio of Traffic Load SP_Square B_Square SP_oud B_oud Stretch Factor of B Square_Avg Square_Max oud_avg oud_max (i) (j) Maxiu Traffic Load (k) ST of Load (l) Stretch Factor of B Fig. 4. Average, axiu, ST of load, ad stretch factor of B copared with SP. (a-d) are for -odes grid ad rado etworks i a square area, (e-h) are for 4-odes grid ad rado etworks i a square area, ad (i-l) are for 4-odes rado etworks i either a square area or a disk area. of load (Fig. 4(c)(g)) occur whe equals to. or.. It is iterestig that rado etworks have saller average load tha grid etworks. eeber that the trasissio rage i rado etworks is larger tha the oe i grid etworks (i order to guaratee the etwork coectivity). Thus, average hop cout of routes i rado etworks is less tha the oe i grid etworks which leads to lower total load i rado etworks. O the other had, rado etworks have larger axiu load tha grid etworks. This is due to ore ueve distributio of odes i rado etworks tha perfect grid etworks. This also causes that rado etworks have larger stretch factors tha grid etworks. We also studied the ipact of the shape of the etwork. Istead of a square area, we deploy the odes radoly i a disk area with radius equal to. Fig. 4(i-l) deostrates the results i both square ad roud etworks with 4 odes. The siilar coclusios ca be draw fro these figures. V. ONLUSION Popa et al. [] recetly proposed curveball routig (B), which ca balace the traffic load ad vaish the crowded ceter effect uder uifor couicatio sceario. I this paper, we studied the stretch factor of B. We theoretically proved that for ay etwork the distace traveled by the packets usig B is o ore tha a sall costat factor ax( (+ɛ),) of the iiu (the distace of the shortest path). The stretch factor is oly depeded o ɛ =, where is the ratio of the size of the etwork to the radius of the sphere used i B. Via siulatio, we showed that there is a trade-off betwee balacig the traffic load ad reducig the stretch factor, i.e., appropriate radius of the sphere should be selected to optiize the load distributio without icreasig stretch factor too uch. We leave further theoretical study of load distributio of B as our future work. EFEENES [] L. Popa, A. ostai,. Karp,. Papadiitriou, ad I. Stoica, Balacig the traffic load i wireless etworks with curveball routig, i Proc. of AM Mobihoc, 7. [] P.P. Pha ad S. Perreau, Perforace aalysis of reactive shortest path ad ultipath routig echais with load balace, i Proc. of IEEE INFOOM, 3. [3] E. Hyytia ad J. Virtao, O traffic load distributio ad load balacig i dese wireless ultihop etworks, EUASIP Joural o Wireless ouicatios ad Networkig, Article I 693, 7. [4] S.-J. Lee ad M. Gerla, yaic load-aware routig i ad hoc etworks, i Proc. of IEEE I,. [] H. Hassaei ad A. Zhou, outig with load balacig i wireless ad hoc etworks, i Proc. of AM MSWIM,. [6] Y. Yoo ad S. Ah, A siple load-balacig approach i cheat-proof ad hoc etworks, i Proc. of IEEE Globeco, 4. [7] Y. Gajali ad A. Keshavarzia, Load balacig i ad hoc etworks: sigle-path routig vs. ulti-path routig, i IEEE INFOOM, 4. [8] H.S.M. oxeter, Itroductio to Geoetry, Joh Wiley & Sos, New York, d editio, 969.