Bounding the Locality of Distributed Routing Algorithms

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1 Nonme mncrip No. (will e inered y he edior) Bonding he Locliy of Diried Roing Algorihm Proenji Boe Pz Crmi Sephne Drocher Received: de / Acceped: de Arc We exmine ond on he locliy of roing. A locl roing lgorihm mke eqence of diried forwrding deciion, ech of which i mde ing only locl informion. Specificlly, in ddiion o knowing he node for which mege i deined, n inermedie node migh lo know 1) i locl neighorhood (he grph correponding o ll nework node wihin k hop of ielf, for ome fixed k), 2) he node from which he mege origined, nd 3) he incoming por (which of i neighor l forwrded he mege). Or ojecive i o deermine, k vrie, which of hee prmeer re necery nd/or fficien o permi locl roing on nework modelled y conneced ndireced grph. In priclr, we elih igh ond on k for he feiiliy of deerminiic k-locl roing for vrio cominion of hee prmeer, well correponding ond on dilion (he Some of hee rel ppered in preliminry form he 28h ACM SIGACT-SIGOPS Sympoim on Principle of Diried Comping (PODC 2009) [2]. Thi reerch w ppored in pr y he Nrl Science nd Engineering Reerch Concil of Cnd (NSERC). P. Boe School of Comper Science, Crleon Univeriy Ow, Cnd E-mil: ji@c.crleon.c P. Crmi Deprmen of Comper Science Ben-Grion Univeriy of he Negev Beer-Shev, Irel E-mil: crmip@c.g.c.il S. Drocher Deprmen of Comper Science, Univeriy of Mnio Winnipeg, Cnd E-mil: drocher@c.mnio.c wor-ce rio of cl roe lengh o hore ph lengh). Keyword diried lgorihm locl roing dilion 1 Inrodcion 1.1 Locl Roing Unic commnicion in nework i chieved y roing lgorihm h compe eqence of forwrding deciion h deermine he roe followed y mege (e.g., pcke) i rvel o i deinion. Trdiionlly, roing le re conrced fncion of he nework opology o provide efficien roing, where dilion decree memory nd le ize incree (e.g., ee [14 17, 24, 25]). In mny nework, implemening cenrlized roing lgorihm or, more generlly, roing lgorihm whoe iniilizion reqire knowledge of he enire nework opology i imprcicl; reon inclde h he nework i oo lrge, h he opology of he enire grph i nknown, or h he nework chnge dynmiclly [26], ch in n d hoc wirele nework, where ech node cn periodiclly cqire nd pde informion o i neighorhood, no necerily o din node in he nework. Alernively, locl roing lgorihm mke erie of diried forwrding deciion, comped ech of he inermedie node long he roe; when node receive mege, i elec por (i.e., one of i neighor) o which o forwrd he mege ing only locl informion. Specificlly, ech node i only wre of he e of he nework coniing of node wihin k hop from ielf, for ome k. Coneqenly, he roe cnno e precomped enirely in generl.

2 2 Frhermore, mege overhed nd locl memory re ofen limied [21]. In priclr, nework node cnno e expeced o minin hiory of mege h hve ped hrogh i (i.e., he nework i memoryle). Similrly, he mege overhed cnno ore he e of node viied y he mege (i.e., he roing lgorihm i ele). Alhogh righforwrd flooding lgorihm i poile, ch regy h ovio drwck, inclding high rffic lod [26], cyclic ehvior (if he nework i memoryle), nd reqiring knowledge of n pper ond on he dimeer of he nework o enre oh erminion nd ccefl delivery. In hi pper we conider ingle-ph deerminiic roing lgorihm. We repreen nework y conneced, nweighed, ndireced, imple grph G wih niqe verex lel. A nework node (grph verex) i idenified y i lel. In ome nework, node lel my provide informion o i neighorhood in he nework (e.g., grid grph node cn e lelled y i grid coordine). In generl, we ppoe h he verex lelling i independen of he grph; h i, node lel doe no encode ddiionl informion o he opology of he grph or he node neighorhood. Eqivlenly, we conider roing lgorihm h cceed on ny (poily dverril) permion of he verex lel of G. We me h every node know i own lel well he lel of i neighor. A mege lo reqire deinion node, idenified y h node lel. Some or ll of he following ddiionl informion my e ville o n inermedie node o compe he nex node o which mege hold e forwrded: 1. origin-wrene: knowledge of he node from which he mege origined, 2. predeceor-wrene: knowledge of he incoming edge (por) long which he mege w forwrded o (eqivlenly, he neighor of h l forwrded he mege), nd 3. k-locliy: knowledge of he k-neighorhood of (i.e., he grph of G coniing of ll ph rooed wih lengh mo k). Or ojecive i o deermine which of hee prmeer re necery nd/or fficien o permi locl roing k vrie. 1.2 Overview of Rel We idenify igh ond on he vle of he locliy prmeer k for he feiiliy of k-locl roing in ech of he for cominion of conrin: predeceor-wre T (n) origin-wre origin-olivio predeceor-wre n/4 n/3 predeceor-olivio n/2 n/2 Tle 1 Min rel: here exi k-locl roing lgorihm when k T (n), no k-locl roing lgorihm exi when k < T (n), where n denoe he nmer of nework node. Ronding operor re omied; ee Theorem 1 hrogh 3, Corollrie 2 nd 5, nd Theorem 5 hrogh 8 for exc vle of T (n). k n/4 n/3 n/2 lower ond pper ond Tle 2 Bond on he dilion inle y k-locl roing lgorihm. S(k) 2n/k 3 i lower ond on he wor-ce dilion of ny k-locl roing lgorihm. The ond on S(k) i igh when k {n/3, n/2} nd i onded from ove y 6 when k = n/4. See Theorem 4, 7, 8, nd 6. or predeceor-olivio, nd origin-wre or origin-olivio. In ech ce, le T (n) denoe he correponding hrehold. Th i, for every k < T (n), every k-locl roing lgorihm i defeed y ome conneced grph on n verice. Similrly, for every k T (n), here exi k-locl roing lgorihm h cceed on ll conneced grph on n verice. Or min rel i he idenificion of he vle of T (n); ee Tle 1. In ddiion, we elih lower ond of S(k) = 2n/k 3 on he wor-ce dilion of ny k-locl roing lgorihm nd how hi ond i igh for hree of he for cominion of conrin; ee Tle 2. Th, or ojecive i o idenify ond on he feiiliy of grneed delivery for memoryle, ele, deerminiic locl roing. Specificlly, we nwer he qeion of excly how mch of he nework ech node m e wre for locl roing o cceed. Locl roing cceed in priclr eing, e.g., on ome cle of geomeric grph, ch ni dic grph, plnr grph, nd ringlion (ee Secion 3). In hi pper, we eek o deermine wheher imilr locl roing lgorihm re poile eyond hee rericed cle of grph. A we how, grneeing delivery ing locl roing in rirry conneced grph reqire node o hve knowledge of neighorhood of ize Ω(n) node in ome ce, ggeing h locl informion (mll k) i infficien nd h he roing lgorihm m e modified or exended ing differen prdigm, ch cenrlized roing (roing le), rndomizion, increed memory (ped wih he mege or ored ech node), or leverging pecific ddiionl grph properie (e.g., geomeric emedding).

3 2 Modelling Locl Roing In hi ecion we formlize or model for locl roing. 2.1 k-locl Roing Fncion Given (imple ndireced) grph G, we employ ndrd grph-heoreic noion, where V (G) denoe he verex e of G; E(G) denoe he edge e of G; for ech verex v V (G), Adj(v) = { {, v} E(G)} denoe he e of verice djcen o v; nd deg(v) = Adj(v) denoe i degree. Le di(, v) denoe he (nweighed) grph dince eween verice nd v, i.e., he nmer of edge in hore ph from o v. The girh of G i he lengh of i hore cycle or if G i cyclic. The k-neighorhood of verex V (G), denoed G k (), i he grph of G h conin ll ph rooed wih lengh mo k. A roing lgorihm i origin-wre, predeceor-wre, nd k-locl if i cn e defined fncion f(,,, v, G k ()), where V (G) i he origin node, V (G) i he deinion node, he mege i crrenly node V (G) (we y i he crren node), node received he mege from i neighor v Adj() (le v = efore forwrd he mege for he fir ime), G k () i he k-neighorhood of, nd f(,,, v, G k ()) rern he neighor of o which he mege hold e forwrded (i.e., he por o which node m forwrd he pcke). Every k-locl roing lgorihm A h correponding roing fncion f. A eqence of cll o fncion f rern eqence of forwrding deciion h correpond o wlk, i.e., he roe, hrogh G origining. We conider wo conrin on k-locl roing lgorihm: n origin-olivio k-locl roing lgorihm i no provided he prmeer, nd predeceorolivio k-locl roing lgorihm i no provided he prmeer v. To implify noion for predeceor-wre lgorihm, le f (v) denoe he locl roing fncion node for given,,, nd G k (), where f (v) = f(,,, v, G k ()). Th i, f (v) rern he neighor of o which he mege i forwrded fncion of he neighor v from which i i received. Oervion 1 Given ny k 1, ny predeceorwre k-locl roing lgorihm A, ny conneced grph G, nd ny {, } V (G), he direcion in which mege rvere given edge niqely deermine he nex forwrding deciion y A. In priclr, if A cceflly deliver mege o, hen he mege h rvered ech edge in E(G) mo once in ech direcion. Nrlly, no ll roing fncion cn e implemened efficienly locl roing lgorihm. The roing fncion model llow ronger negive rel o e elihed for more generl cl of roing lgorihm, regrdle of implemenion concern. Wih repec o poiive rel, he roing lgorihm we preen cn e implemened efficienly loclly; implemenion deil re no he foc of hi pper. Le C denoe conneced componen of G k () \ {}. We refer o C locl componen of. If v V (C) Adj(), hen we y C i rooed v (C cn hve mliple roo). If C conin verex z ch h di(, z) = k, hen (relive o ) C i n cive componen, edge {, v} i n cive edge (where v V (C)), v i n cive neighor of, nd hore ph from o z i n cive ph. In oher word, C exend o he limi of knowledge: node z my hve neighor oide C, hi informion i no known loclly. If C i n cive componen of nd every cive ph in C pe hrogh ome verex w, hen C i conrined cive componen nd w i conrin verex. If locl componen C i no n cive componen, hen we y C i pive componen. If i conneced o i locl componen C y ingle edge (i.e., C h niqe roo), hen we y C i n independen componen. Every independen cive componen i conrined cive componen. See Figre 1. A cycle C i locl cycle node if lie on C nd C h lengh mo 2k. Th i, C G k (), V (C) 2k, nd V (C). 2.2 Evling Roing Algorihm A roing lgorihm A defined y roing fncion f cceed (ynonymoly, grnee delivery) if for ll grph G nd ll origin-deinion pir (, ) in G, he eqence of vle rerned y f correpond o wlk from o in G. Oherwie, A i defeed y (or fil on) ome grph G nd ome pir (, ) in G. A roing lgorihm A h dilion onded y δ if for ll grph G nd ll origin-deinion pir (, ) in V (G) ch h, r A (, )/ di(, ) δ, where r A (, ) denoe he lengh of he roe from o rerned y A. A roing lgorihm h grnee dilion δ i omeime id o hve rech fcor [11] onded y δ or o e δ-compeiive [3]. 3

4 4 z B 4 B 1 v w k B 3 Fig. 1 In hi exmple, G 8 () coni of for locl componen, correponding o he for conneced componen of G 8 () \ {}. B 1, B 3, nd B 4 re cive componen. B 2 i pive componen. B 1 nd B 3 re conrined cive componen, B 2 nd B 4 re no. B 1 nd B 2 re independen componen, B 3 nd B 4 re no. Node v i n cive neighor of nd edge {, v} i n cive edge. Node w i conrin verex in he conrined cive componen B 3. All hree ph from o z of lengh eigh re cive ph. Oher mere of qliy in roing inclde congeion (rffic lod), rnning ime, nd cliliy. Thi pper conider qeion of exience of ccefl roing lgorihm. A ch, he mere of inere re grneed delivery nd ond on dilion. 3 Reled Work In poiion-ed roing, nework node re emedded in ome pce (ypiclly R 2 or R 3 ) nd ech node know i pil coordine (i.e., node re locion-wre). Poiion-ed roing i lo known geo-roing, geogrphic roing, or geomeric roing. Mny recen rel reled o locl roing re poiion ed (e.g., [1, 4,5,10,12,13,18,20 22,26]); we riefly decrie ome of hee nd dic he inerdependence eween poiioned nd poiion-olivio roing. Greedy roing [12] (forwrd he mege o he neighor cloe o he deinion), comp roing [21] (forwrd he mege long he edge h form he mlle ngle wih he line egmen o he deinion), nd greedy-comp roing [1] (pply greedy roing o he wo edge djcen o he line egmen o he deinion) re hree well-known poiion-ed roing lgorihm, ech of which cceed on pecific cle of grph i defeed y ome plnr grph [4]. All hree lgorihm re predeceor-olivio, originolivio, nd 1-locl. B 2 To how h roing lgorihm fil on ome cl of grph G, i ffice o idenify grph in G on which he lgorihm cycle infiniely wiho reching he deinion. Sronger negive rel re hoe h pply o ll roing lgorihm, howing h no roing lgorihm cceed on given cl of grph. Boe e l. [1] how h every poiion-ed, predeceor-olivio, origin-olivio, 1-locl roing lgorihm i defeed y ome convex diviion. Fce roing [21] w one of he fir poiion-ed 1-locl roing lgorihm o cceed on more generl cle of grph emedded in he plne. In rief, fce roing forwrd he mege in clockwie direcion long he edge of fce, nd long he eqence of fce h inerec he line egmen eween he origin nd deinion node. Forwrd progre i grneed y oring prmeer ch he frhe inerecion of he line egmen wih viied fce. A ch, fce roing i no ele ince i reqire Θ(log n) i o e ored wih he mege. Fce roing cceed on plnr grph [21], on ni dic grph [5], nd on d-qi ni dic grph for ny d [1/ 2, 1] [22]. See Boe e l. [5] nd Khn e l. [22] for definiion of ni dic grph nd qi ni dic grph, repecively. Frer conider generlizion of fce roing o grph emedded on ori [18]. See Gn [20] nd Sojmenović [26] for dicion of fce roing nd i vrin. Alhogh or dicion foce on deerminiic roing lgorihm, we riefly noe h rndomized olion permi k-locl roing on more generl cle of grph. Chen e l. [9] how h while rndomizion cn provide n (expeced) grnee of delivery, he expeced dilion remin high. Specificlly, hey how h for every rndomized poiion-ed, predeceor-olivio, origin-olivio, 1-locl roing lgorihm, here exi convex diviion in he plne on which he expeced roe lengh i Ω(n 2 ), mching he expeced lengh of rndom wlk from o. Flry nd Wenhofer conider he prolem of rndomized locl roing on ni ll grph [13] nd how h ny rndomized poiion-ed locl roing lgorihm h expeced roe lengh Ω(l 3 ), where l denoe he lengh of he hore ph. Drocher e l. [10] how h for every fixed k, every origin-wre, predeceor-wre, k-locl roing lgorihm fil on ome ni ll grph. The proof h wo pr. Fir, he correponding poiion-olivio rel i proven: for every fixed k, every origin-wre, predeceor-wre, k-locl roing lgorihm fil on ome grph. Nex, k-locl redcion from (nemedded) grph o ni ll grph i ed o how h if ome (poily poiion-ed) k-locl roing lgorihm cceed on ni ll grph, hen ome poiion-

5 5 olivio k-locl roing lgorihm cceed on ll grph. Thi inerdependence eween poiion-ed nd poiion-olivio roing lgorihm moive he qeion of exploring he ondry eween feiiliy nd impoiiliy of locl roing lgorihm fncion of he locl informion ville. In hi pper we conider he poiion-olivio ce. B 1 v k B 2 4 When Locl Roing i Impoile: Negive Rel In hi ecion we preen negive rel: every k- locl roing lgorihm fil on ome grph when he degree of locliy k i le hn he given ond. For ech cominion of origin-wrene/olivione nd predeceor-wrene/olivione, we demonre coner-exmple coniing of e of grph ch h ny k-locl roing lgorihm fil on le one of he grph in he e. 4.1 Properie of Locl Roing Fncion The proof of Theorem 1 hrogh 3 refer o Lemm 1 nd Corollry 1, which generlize n oervion of Drocher e l. [10] howing h if k-locl roing lgorihm grnee delivery, hen ech locl roing fncion correpond o circlr permion (nder cerin condiion). Recll h circlr permion of n diinc elemen i n ordering of hee elemen in cycle. Lemm 1 Given n rirry grph G nd ny node V (G) ch h 1. deg() 2, 2. every locl componen of i n independen cive componen, nd 3. neiher he origin node nor he deinion node i in G k (), if A i n origin-wre, predeceor-wre, k-locl roing lgorihm h grnee delivery, hen he locl roing fncion of A i circlr permion of Adj(). Proof Chooe ny k 1, ny node, nd ny k-neighorhood G k () ch h Properie 1 hrogh 3 hold. Sppoe A i ny k-locl roing lgorihm h grnee delivery for which he locl roing fncion f i no circlr permion. Ce 1. Sppoe f i no permion. Th i, f i no rjecive. Therefore, here exi ome v Adj() ch h for ll w Adj(), f (w) v. Le B 1 denoe Fig. 2 Thi exmple illre he grph conrced in Ce 1 of Lemm 1 for given G k () when k = 8. Noe h G k () coni of independen cive componen. he locl componen of h conin v nd le B 2 denoe ny oher locl componen of. By Propery 2, ech locl componen of i n cive componen. Le G denoe grph h conin G k () ch h node h degree one nd i he only node djcen o B 1 oide G k (). Similrly, le node hve degree one ch h i i he only node djcen o B 2 oide G k (). See Figre 2. Since for ll w Adj(), f (w) v, he mege will never ener B 1 nd, coneqenly, will never rech. Therefore, Algorihm A fil on grph G, deriving conrdicion. Ce 2. Sppoe f i permion no derngemen ( derngemen i complee permion). Therefore, f (v) = v for ome v Adj(). Le G e grph defined in Ce 1, wih he excepion h node nd re inerchnged. I follow h he mege will never ener ny locl componen oher hn B 1 nd, coneqenly, will never rech. Therefore, Algorihm A fil on grph G, deriving conrdicion. Ce 3. Sppoe f i derngemen no circlr permion. Therefore, f cnno e expreed ingle permion cycle. Le ( 1... k ) nd ( 1... j ) denoe ny wo diinc permion cycle of f. Oerve h { 1,... k } nd { 1,..., j } re dijoin e of Adj(). Le G e grph defined in Ce 1, wih he excepion h node i djcen o locl componen B 1 rooed node in { 1,... k } nd i djcen o locl componen B 2 rooed node in { 1,..., j }. I follow h he mege will never ener B 2 nd, coneqenly, will never rech. Therefore, Algorihm A fil on grph G, deriving conrdicion. All hree ce derive conrdicion nd or mpion m e fle. Therefore, he locl roing fncion f m e circlr permion.

6 6 In oher word, wiho ddiionl informion on which o e locl roing deciion, n inermedie node m ry ll poiiliie nd eqenilly forwrd he mege o ech of i neighor. When node h degree wo, niqe circlr permion i poile: mege received from one neighor of m e forwrded o he oppoie neighor. If node h degree j, hen (j 1)! circlr permion re poile. If roing Algorihm A i origin olivio, hen Lemm 1 give: Corollry 1 Given n rirry grph G nd ny node V (G) ch h 1. deg() 2, 2. every locl componen of i n independen cive componen, nd 3. he deinion node i no in G k (), if A i n origin-olivio, predeceor-wre, k-locl roing lgorihm h grnee delivery, hen he locl roing fncion of A i circlr permion on Adj(). Proof Given ny node, ny k 1, nd ny k-neighorhood G k (), if Properie 1 hrogh 3 hold ( defined in Lemm 1), hen he locl roing fncion f i circlr permion y Lemm 1. Since A i origin olivio, fncion f remin conn for ny given G k () nd, regrdle of. In priclr, f i circlr permion regrdle of wheher or no i conined in G k (). The rel follow. Theorem 1 hrogh 3 nd Corollry 2 elih lower ond correponding o ech of he for cominion of k-locl roing lgorihm: origin-wre/olivio nd predeceor-wre/olivio. 4.2 Predeceor Awre nd Origin Awre Theorem 1 For every k < (n + 1)/4, every originwre, predeceor-wre, k-locl roing lgorihm fil on ome conneced grph. Proof Chooe ny k < (n + 1)/4, k Z +. Therefore, k {1,..., r}, where r = (n 3)/4. Le G 1, G 2, nd G 3 denoe he grph illred in Figre 3, ch h ech ph P 1 hrogh P 4 coni of r verice h re lelled conienly relive o node in ll hree grph. In ech grph, G k () i ree coniing of for ph of lengh k rooed, none of which conin nor. In ddiion o he 4r node in ph P 1 hrogh P 4, ech grph inclde node,, nd. Depending roing regy circlr permion cceed fil 1 (P 1 P 2 P 3 P 4 ) G 1, G 3 G 2 2 (P 1 P 2 P 4 P 3 ) G 1, G 2 G 3 3 (P 1 P 3 P 2 P 4 ) G 2, G 3 G 1 4 (P 1 P 3 P 4 P 2 ) G 1, G 2 G 3 5 (P 1 P 4 P 2 P 3 ) G 2, G 3 G 1 6 (P 1 P 4 P 3 P 2 ) G 1, G 3 G 2 Tle 3 Ech roing regy correpond o circlr permion of he neighor of. on he vle of n mod 4, eween zero nd hree exr node remin; hee re dded eween nd P 1 o ring he ol nmer of node o n. Any ccefl roing lgorihm m p he mege cro P 1 o node. Since h degree for, i locl roing fncion i one of ix poile circlr permion y Lemm 1. The remining node hve degree mo wo. Therefore, when he mege i ped o node on ph h doe no conin or, y Lemm 1, he mege m conine forwrd nil i rern gin o. A hown in Tle 3, for ech of he ix poile roing regie, he mege never ener he ph conining in le one of he grph G 1, G 2, or G 3. Th i, for every roing regy A, here exi grph on which A fil. 4.3 Predeceor Awre nd Origin Olivio Uing n rgmen imilr o he proof of Theorem 1, we now how h he lower ond on he locliy prmeer k incree o (n + 1)/3 for origin-olivio k-locl roing lgorihm: Theorem 2 For every k < (n + 1)/3, every originolivio, predeceor-wre, k-locl roing lgorihm fil on ome conneced grph. Proof Chooe ny k < (n + 1)/3, k Z +. Therefore, k {1,..., r}, where r = (n 2)/3 }. Le G 1, G 2, nd G 3 denoe he grph illred in Figre 4, ch h ech ph P 1 hrogh P 3 coni of r verice h re lelled conienly relive o node in ll hree grph. In ech grph, G k () i ree coniing of hree ph of lengh k rooed, none of which conin. In ddiion o he 3r node in ph P 1 hrogh P 3, ech grph inclde node nd. Depending on he vle of n mod 3, eween zero nd wo exr node remin; hee re dded eween nd he correponding ph P i nere o o ring he ol nmer of node o n. Since node h degree hree, i locl roing fncion i one of wo poile circlr permion y Corollry 1. A roing regy m pecify he direcion in which mege iniilly leve node (hree direcion

7 P 2 P 2 P 2 r P 1 P 1 P c c 1 P 3 P 3 P 3 c r r P 4 G P 1 4 G P 2 4 d r d d G 3 P 2 P 1 P 3 Fig. 3 The k-neighorhood G k () i idenicl in grph G 1, G 2, nd G 3. In hi exmple, ppoe n mod 4 = 0. Coneqenly, one exr node i dded eween nd P 1 ch h he ol nmer of node i n. r P 2 r P 1 P 3 r c G 1 P 2 P 1 P 3 c G 2 P 2 P 1 P 3 c G 3 P 2 d P 4 P 1 P 3 Fig. 4 The k-neighorhood G k () i idenicl in grph G 1, G 2, nd G 3. In hi exmple, ppoe n mod 3 = 0. Coneqenly, one exr node i dded nex o ch h he ol nmer of node i n. c 7 c re poile). The remining node hve degree mo wo. Therefore, when he mege i ped o node on ph h doe no conin, y Corollry 1, he mege m conine forwrd nil i rern gin o node. A hown in Tle 4, for ech of he ix poile roing regie, he mege never ener he ph conining in le one of he grph G 1, G 2, or G 3. Th i, for every roing regy A, here exi grph on which A fil. 4.4 Predeceor Olivio nd Origin Awre When knowledge of he predeceor node i wihheld, he lower ond on he locliy prmeer k incree o n/2 for k-locl roing lgorihm: Theorem 3 For every k < n/2, every origin-wre, predeceor-olivio, k-locl roing lgorihm fil on ome conneced grph. Proof Chooe ny k < n/2, k Z +. Therefore, k {1,..., r}, where r = n/2 1. Le G 1 denoe ph of n verice wih he origin node loced he (r+1) verex nd he deinion node loced he fr end. Le G 2 denoe he nlogo grph pon moving node o he oppoie end of he ph. Le he remining node e lelled conienly relive o node in r r G 1 G 2 r r Fig. 5 For ny k < n/2, he k-neighorhood of doe no conin. oh grph. See Figre 5. The k-neighorhood G k () i idenicl in G 1 nd G 2. If Algorihm A end he mege righ, hen A fil on grph G 2 ince i m evenlly end he mege lef, which poin i ehvior ecome cyclic. Similrly, if Algorihm A end he mege lef, hen i fil on grph G Predeceor Olivio nd Origin Olivio Finlly, if we frher conrin he knowledge ville o inermedie node (e.g, remove knowledge of he origin), hen he lower ond on he locliy prmeer k given in Theorem 3 pplie: Corollry 2 For every k < n/2, every origin-olivio, predeceor-olivio, k-locl roing lgorihm fil on ome conneced grph.

8 8 roing regy circlr permion iniil direcion cceed fil 1 (P 1 P 2 P 3 ) owrd G 1, G 3 G 2 2 (P 1 P 2 P 3 ) owrd G 1, G 2 G 3 3 (P 1 P 2 P 3 ) owrd c G 2, G 3 G 1 4 (P 1 P 3 P 2 ) owrd G 1, G 2 G 3 5 (P 1 P 3 P 2 ) owrd G 2, G 3 G 1 6 (P 1 P 3 P 2 ) owrd c G 1, G 3 G 2 Tle 4 Ech roing regy correpond o circlr permion of he neighor of pired wih n iniil direcion. Proof The rel follow y Theorem 3. k n 2k 1 k Dilion We now conider lower ond on dilion for k-locl roing lgorihm. Theorem 4 For ny k < n/2, no k-locl roing lgorihm cn grnee dilion le hn 2n 3k 1, (1) k + 1 regrdle of wheher he lgorihm i predeceor-wre/olivio or origin-wre/olivio. Proof Chooe ny n, ny k [1, n/2), nd ny k-locl roing lgorihm A. If A fil on ome grph, hen A h nonded dilion. In priclr, he dilion exceed (1). Therefore, ppoe h A cceed on ll grph. Given e of n diinc verex lel, le P denoe he correponding e of ll n!/2 diinc ph of lengh n. Sppoe he origin nd deinion node re lelled nd, repecively. For every ph P P, he locl neighorhood of every inernl node on P h wo independen componen; ince n 2k + 1, mo one of hee componen i pive. By Oervion 1, if he mege chnge direcion node h h wo cive componen, hen node on he ph eyond he correponding locl componen (where cold e loced) will never e viied. Coneqenly, for ny node, if G k () h wo cive componen, mege received from lef neighor m e forwrded o i righ neighor, nd vice-ver (i.e., he locl roing fncion i circlr permion). Conider ny locl neighorhood G k () h i ph of lengh 2k. Th, G k () h wo cive componen. There exi ph P nd P in P h conin G k () ch h lie o he lef of in oh P nd P, nd lie o he lef of G k () in P o he righ of G k () in P. Conider he fir ime node receive he mege. Algorihm A end he mege owrd he deinion in ph P or ph P, nd wy from in he oher. Coneqenly, here exi ph P in P ch h di(, ) = k + 1, A iniilly forwrd he Fig. 6 Node i he lefmo node righ of h cn confirm h doe no lie o he righ of c. mege wy from, nd A conine forwrding he mege wy from while oh locl componen of he crren node re cive. In priclr, Algorihm A cn end he mege wy from o n 2k 1 node efore he mege reche node h h pive componen ch h for ech node viied, di(, ) k + 1. See Figre 6. The mege m rern o efore proceeding in he oppoie direcion ck o. The correponding roe h lengh le 2(n 2k 1)+di(, ) = 2n 3k 1, while he hore ph h lengh di(, ) = k + 1. The ond on dilion (1) i perhp more clerly expreed in he limi he nmer of node pproche infiniy nd k = c n for ome conn c (0, 1). We denoe hi limi y S(k): 2n 3k 1 S(k) = lim = 2n n k + 1 k 3. (2) Of priclr inere re he vle of k {n/4, n/3, n/2}, for which he correponding ond on dilion re 5 (when k = n/4), 3 (when k = n/3), nd 1 (when k n/2). A hown in Theorem 7 nd 8 nd Corollry 5, hee ond re igh for k = n/3 nd k = n/2. 5 When Locl Roing i Poile: Roing Sregie In hi ecion we preen poiive rel: here exi ccefl k-locl roing lgorihm when he degree of locliy k exceed he given ond. We decrie k-locl roing lgorihm for ech cominion of origin-wrene/olivione nd predeceorwrene/olivione. c

9 9 k A Fig. 7 The red rrow denoe he neighor o which he mege i iniilly forwrded from he origin. Ble rrow denoe eqen forwrding deciion fncion of he neighor from which he mege w received. (A) The righ-hnd rle grnee delivery on ny ree. (B) The righ-hnd rle cn fil if ome cycle h lengh greer hn 2k. Frhermore, i cn occr h he mege i never forwrded o node whoe k-neighorhood conin he deinion. In hi exmple, if k 4 hen he righ-hnd rle forwrd he mege from coner-clockwie rond he cycle wih eing exclded from every viied node k-neighorhood. B 5.1 Predeceor Awre nd Origin Awre Given ny k n/4, we decrie predeceor-wre, origin-wre, k-locl roing lgorihm h cceed on ll conneced grph on n verice. Moivion: Generlizing he Righ-Hnd Rle Roing on ree i eily ccomplihed ing righhnd rle. Th i, he mege i ped long he eqence of edge on he fce deermined y ny noncroing emedding of he ree in he plne. Specificlly, if every locl roing fncion i circlr permion, hen every mege i grneed o rech i deinion (i.e., knowledge of he emedding i no reqired). A righ-hnd rle cn e implemened on ny grph G y elecing e of i edge h form pnning ree of G. If G conin only locl cycle, hen ny cycle on which node lie i enirely viile in G k (). Node cn lel one edge on every locl cycle dormn. If hi lelling rle were pplied conienly ll node nd he mege were forwrded only cro roing (non-dormn) edge, hen i wold ffice o define ech locl roing fncion o e cyclic permion of i roing edge. If G my conin cycle of rirry lengh, hen he righ-hnd rle cnno e pplied direcly ince node h no knowledge of cycle no enirely conined in G k (). In priclr, cyclic ehvior cn occr ch h he mege never come wihin dince k of he deinion. See Figre 7. Since k n/4, however, he nmer of cycle of lengh 2k + 1 or greer i limied. Uing imple e of k-locl rle which we now define, we how how o grnee delivery in ny grph G. Preproceing: Idenifying Roing Edge in G k () When mege rrive node, he lgorihm egin wih k-locl preproceing ep o idenify he edge of G k () on which roing ke plce. We cll hee edge roing edge nd denoe he correponding edge-indced grph of G k () y G k (). Specificlly, ome edge of G k () my e idenified dormn edge loclly. Once dormn edge re removed from G k (), he remining edge h lie on ph rooed wih lengh mo k re idenified roing edge. Grph G k () i no lwy pnning grph of G k (); ny remining edge of G k () (hoe h re neiher roing nor dormn edge) re edge whoe dince from long roing edge exceed k nd, coneqenly, re no inclded in G k (). Every edge djcen o, however, i idenified eiher roing or dormn edge; hi clificion form he i of or roing lgorihm. The niqe lelling of node deermine ric ol order on he edge of G (e.g., lel ech edge y concening he lel of i endpoin nd order edge lel lexicogrphiclly). We refer o he lel of n edge e i rnk, denoed rnk(e), where edge precede edge in he order if nd only if rnk() < rnk(). In priclr, ny e of edge h n edge of minimm rnk. Uing echniqe imilr o hoe pplied y Li e l. [23] nd Chávez e l. [7] (ed o rek cycle in G k () o conrc k-locl minimm pnning ree on ni dic grph) grph G k () i conrced loclly node y clifying he edge of minimm rnk on every locl cycle of dormn edge. See Figre 8. A locl cycle my conin mliple dormn edge if wo or more locl cycle hre common edge. See Figre 9. If edge e i no clified dormn edge in ny locl neighorhood, hen we y e i conien. Oherwie, we y e i inconien. A conien ph or cycle i one whoe edge re ll conien. Similrly,

10 10 v 3 B 1 v 1 v 2 B 3 B 2 A B B 1 v v 1 v B 2 B 3 B 1 v 1 v 3 v 2 B 2 B C D Fig. 8 k-locl preproceing. Sppoe node h hree cive neighor, v 1 hrogh v 3, nd G k () conin locl cycle h inclde verice v 1,, nd v 2 (A). The preproceing ep clifie one of he edge on he cycle dormn edge (mgen). The eleced edge my e din from (B) or djcen o (C nd D). The choice of dormn edge doe no ffec node no on he cycle (e.g., v 3 ) ince ll edge of he locl cycle lie in he me locl componen of he correponding verex (E); in priclr, none of he cycle edge re djcen o v 3. B 3 v 3 B 1 v 1 v 2 B 2 v 3 v 1 v 2 E C 1 e 1 e 2 C 2 A B C Fig. 9 (A) Sppoe k = 5 nd edge e 1 nd e 2 hve he lowe nd econd-lowe rnk, repecively, mong ll edge in G k (). G k () conin wo locl cycle: C 1 (ligh red) nd C 2 (ligh le). G k () lo conin hird cycle (ligh green), i lengh exceed 2k. Edge e 1 nd e 2 re clified dormn for cycle C 1 nd C 2, repecively. (B) The reling grph G k () i illred. Edge nd verice whoe dince from long roing edge i greer hn k re no inclded in G k (), even if hee re roing edge h pper in G k (). (C) Thi implificion of G k () illre h h wo independen cive componen (ligh le) nd one independen pive componen (ligh red). ome edge in n inconien ph or cycle i inconien. We elih ome properie of he e of conien edge in Lemm 2 nd 3, nd Propoiion 1. Lemm 2 Every edge djcen o in G k () i conien. Proof Sppoe here exi n inconien edge e = {, v} ch h e i roing edge in G k (). Therefore, edge e i dormn in G k (w) for ome node w. Frhermore, here exi locl cycle C in G k (w) h conin node, v, nd w on which edge e h minimm rnk. Since he crdinliy of C i mo 2k, cycle C m e conined in G k () nd, coneqenly, edge e i clified dormn in G k (), deriving conrdicion. Lemm 3 i imilr o h of Li e l. [23, pge 5, Lemm 2]. Lemm 3 i inclded here for compleene ince he definiion of edge coniency ed y Li e l. differ lighly from he one ed in hi pper. Lemm 3 Given ny wo node nd v in G, here exi conien ph from o v. Proof Le D denoe he e of inconien edge h lie on ph from o v in G nd le e = {, } denoe he edge of mximm rnk in D. Since edge e i clified dormn in G k (w) for ome node w, i m lie on locl cycle C coniing of edge whoe rnk re greer hn rnk(e). Since e h mximm rnk mong ll edge in D, herefore, he ph C \ {e} joining nd i conien. Conider he e D = D \ {e} nd le e = {, } denoe he edge of mximm rnk in D. Uing n nlogo rgmen i follow h here exi conien ph from o. In priclr, if ome ph from o inclde edge e, here exi correponding conien ph from o h void e y following he ph C \ {e}. Thi rgmen cn e repeed recrively nil D =. Oerve h ll componen in G k () re independen componen. The nmer of cive neighor of

11 11 in G k () i i cive degree. Since n cive edge join o componen conining le k n/4 node nd 4k + 1 > n, we ge he following propoiion: c Propoiion 1 Every node h cive degree mo 3. Roing Algorihm Rle S1 Rle S2 Rle S3 Fig. 10 Algorihm 1, Ce 2: The mege i he origin node. Sppoe node re lelled ch h rnk() < rnk() < rnk(c). Ligh le region denoe cive componen of. If h i cive componen, hen rle Si i pplied. Pive componen re no illred. The red rrow denoe he neighor o which he mege i iniilly forwrded from. Ble rrow denoe eqen forwrding deciion fncion of he neighor from which he mege w received.. We now decrie he locl roing lgorihm pplied ech node pon receiving mege. If he deinion i in G k (), hen he mege i forwrded long hore ph o (ee Algorihm 1, Ce 1). If i no in G k () (nd herefore no in G k ()), hen he mege i forwrded long roing edge ino n cive componen of (ince pive componen re ded end wih repec o conien edge). The lgorihm m enre h previo forwrding deciion re no repeed (wiho explicily recording hee). In rief, he lgorihm pplie regy inpired y he righ-hnd rle o eqenilly explore ll cive componen, excep when doing o cold led o ded end or o cyclic ehvior. Poenil fre cycling i idenified nd vered y ing he origin node poin of reference; if he mege i en owrd, i direcion i evenlly lered o void repeing previo roing deciion (ee Algorihm 1, Ce 2 nd 4). The preproceing ep need no e repeed nle he nework opology chnge. Once node h idenified i roing edge, imple e of rle deermine forwrding deciion, defined fncion of five locl nvigionl ce : 1. wheher he deinion node lie in G k (), 2. wheher node i he origin node (i.e., = ), 3. wheher he origin node lie in pive componen of G k (), 4. he nmer of cive componen in G k (), nd 5. he neighor of from which he mege w received. The roing lgorihm coni of for ce olined elow, ech of which pplie imple deerminiic rle o mke forwrding deciion. Le denoe he crren node. Rle U1 Rle U2 c Rle U3 Fig. 11 Algorihm 1, Ce 3: The mege i node ch h eiher i in n cive componen of G k () or V (G k ()). Ligh le region denoe cive componen of. If h i cive componen, hen rle Ui i pplied. Pive componen re no illred. Ble rrow denoe forwrding deciion fncion of he neighor from which he mege w received.. Algorihm 1: (n/4)-locl, origin-wre, predeceor-wre roing lgorihm Ce 1. Sppoe di(, ) k. Th i, V (G k ()). The lgorihm forwrd he mege o ny neighor of on hore ph from o nil he mege rrive. Ce 2. Sppoe di(, ) > k nd =. Forwrding deciion re illred in Figre 10. If v = (i.e., he mege i eing en from he origin for he fir ime) hen forwrd he mege o i cive neighor of lowe rnk (node ). Ce 3. Sppoe di(, ) > k,, nd eiher V (G k ()) or i in n cive componen of. Forwrding deciion re illred in Figre 11. Ce 4. Sppoe di(, ) > k,, nd i in pive componen of G k (). Forwrding deciion re illred in Figre 12. If he mege i received from he pive componen conining he origin node, hen forwrd he mege o i cive neighor of lowe rnk (node ).

12 12 c C, i follow h h pive componen conining le k 1 node. The rel follow y pplying n nlogo rgmen o Rle S1 nd US1. Rle US1 Rle US2 Rle US3 Fig. 12 Algorihm 1, Ce 4: The mege i node ch h i in pive componen of G k (). Sppoe node re lelled ch h rnk() < rnk() < rnk(c). Ligh le region denoe cive componen of. If h i cive componen, hen rle USi i pplied. The ligh red region denoe he pive componen of h conin. Oher pive componen re no illred. The red rrow denoe he neighor o which he mege i iniilly forwrded from he pive componen conining. Ble rrow denoe eqen forwrding deciion fncion of he neighor from which he mege w received.. The cce of Algorihm 1 relie on he propery h ech node h cive degree mo 3. Noice h he lower ond rgmen of he proof of Theorem 1 coni of grph h hve one node wih cive degree 4; y Propoiion 1, hi cnno occr when k n/4. Properie of Algorihm 1 We egin y elihing he following properie which re ed in Lemm 7 o how he correcne of Algorihm 1. Corollrie 3 nd 4 follow from he definiion of Algorihm 1 nd y Lemm 2. Corollry 3 Algorihm 1 forwrd mege only long conien edge. Corollry 4 Any mege h ener pive componen m p hrogh i roo. Frhermore, Algorihm 1 forwrd mege ino pive componen if nd only if h componen conin he deinion node. Lemm 4 If Rle S1, U1, or US1 re pplied node o revere he direcion of mege, hen G k () h pive componen conining le k 1 node. Proof Sppoe Rle U1 i pplied node ch h receive he mege from i neighor nd rern he mege immediely ck o. By he definiion of Rle U1, node i he niqe cive neighor of. Frhermore, y he definiion of Algorihm 1, Rle U1 i pplied only if di(, ) > k. I follow h di(, ) > k, oherwie, (y Algorihm 1) he mege wold no hve een forwrded o. Coneqenly, node forwrded he mege o n independen cive componen C of G k (). Every cive componen conin le k node. Since i conrin verex in Lemm 5 Every conien cycle in G h lengh le 2k + 1. Proof Some edge on every cycle C of lengh mo 2k i clified dormn in G k () for every node in C. The rel follow. Expreed in grph-heoreic erm, he grph indced y he conien edge of G h girh le 2k + 1. Lemm 6 Any grph of girh le g h conin wo or more cycle h le 3g/2 1 verice. Proof Chooe ny g nd ny grph G wih girh le G h conin le wo cycle. Ce 1. Sppoe G conin wo verex-dijoin cycle. Ech cycle h le g verice. Therefore, V (G) 2g. Ce 2. Sppoe G conin wo cycle h hve ingle verex in common. Similrly, V (G) 2g 1. Ce 3. Sppoe ll pir of cycle in G hve le wo verice in common. Any wo inerecing cycle define le one ddiionl cycle. I ffice o how h grph of girh g wih excly hree cycle h le 3g/2 1 verice. Sch grph coni of hree ph joined wo verice of degree hree. Le,, nd c denoe he nmer of verice on ech ph, repecively, no inclding he wo verice of degree hree. Therefore, min{ +, + c, + c} + 2 g + + c + 2 3g 2 1. The rel follow ince V (G) + + c + 2. Correcne of Algorihm 1 We now prove he correcne of Algorihm 1 nd derive igh ond on he correponding dilion. Lemm 7 Given ny conneced grph G on n node, ny k n/4, nd ny {, } V (G), Algorihm 1 cceflly deliver mege from node o node. Proof By Propoiion 1, every node h mo hree cive componen. Coneqenly, one of Ce 1 hrogh 4 of Algorihm 1 i pplicle ech ep.

13 Sppoe Algorihm 1 i defeed y ome grph G, for ome k n/4 nd ome origin-deinion pir (, ) V (G) V (G). Since n i finie nd Algorihm 1 i deerminiic, he mege m vii repeing eqence of verice nd edge in G. Le R denoe he correponding repeing grph of G. Le S V (G) denoe he e of verice of G no in R viied prior o enering he repeing eqence R. Finlly, le T = V (G) \ (V (R) S) denoe he e of verice of G h re never viied. The e {V (R), S, T } priion V (G). Since he mege doe no rech i deinion, Ce 1 of Algorihm 1 never occr. Coneqenly, v V (R) S, di(v, ) k + 1 Frhermore, T k + 1. (3) V (R) = n T S, y definiion of R, S, nd T, n T 4k T, ince k n/4, 3k 1, y (3). (4) By Lemm 5 nd 6, nd Ineqliy (4), R conin mo one conien cycle. By Lemm 3, here exi conien ph in G from o ome verex in V (R). Le {q R, q T } denoe he l edge on ch ph ch h q R V (R) nd q T T. I follow h q T i n cive neighor of q R h i never viied. Coneqenly, neiher Rle S1, U1, nor US1 i pplied q R. Similrly, ince q R i viied mliple ime (i.e., q R V (R)), neiher Rle S2, U2, nor US2 i pplied q R, ince ech of hee rle implie q T hving received he mege. Th, only Rle S3, U3, or US3 my e pplied q R, implying he following oervion: Oervion 2 Node q R h cive degree hree. Frhermore, wo of i cive neighor re in V (R) while he hird, q T, i in T. Ce 1. Sppoe grph R conin conien cycle. By Lemm 5, hi cycle m conin le 2k + 1 node. Coneqenly, V (R) 2k + 1. (5) Since k n/4, y (3) nd (5), we hve, S k 2. (6) ined of Rle US2 node p =. The following oervion follow from Corollry 4, Lemm 4, nd (6): Oervion 3 Neiher Rle S1, U1, nor US1 cn e pplied in he repeing eqence. 13 Therefore, he mege cn revere i direcion node in R if nd only if Rle S2 or US2 i pplied. Coneqenly, every verex in R h cive degree wo or greer in G. Rle S2 pplie if nd only if h wo cive componen. Converely, Rle US2 cn e pplied only if h only one cive componen ince i in pive componen of. Therefore mo one of Rle S2 or US2 i pplicle, implying here i mo one node in R which he mege cn revere i direcion. Boh Rle S2 nd US2 iniilly forwrd he mege in he oppoie direcion from h in which he reverl occr. Coneqenly, he cive componen in which he mege i originlly forwrded cnno e in R if he mege revere i direcion y Rle S2 or US2 in R. Therefore, eiher he iniil cive componen i in S, conrdicing (6), or he mege revere i direcion in h componen, reqiring h one of Rle S1, U1, or US1 e pplied, conrdicing Oervion 3. Th, none of Rle S1, U1, US1, U2, or US2 cn e pplied o revere he mege direcion in R. We conclde h R coni of ingle conneced cycle (wiho ny dngling rnche). Ce 1. Sppoe he origin node i in pive componen of ome node p V (R) h h cive degree wo. Conider he fir ime node p receive he mege. Rle US2 i pplied fir ime, forwrding he mege in one direcion rond cycle R. Since V (R) i follow h Rle S2 i no pplied nd he mege conine rond cycle R nil i rern o node p. Rle US2 i pplied once gin, forwrding he mege in he oppoie direcion rond cycle R. Upon rerning o node p, he mege i gin forwrded in he me direcion, nd conine cycling in hi direcion infiniely. The mege h viied every node in R le once from ech direcion. By Oervion 2, Rle U3 i pplied q R. In priclr, Rle U3 i pplied from wo direcion. In one of hee direcion he mege will e forwrded long he edge {q R, q T }. Since q T V (R), we derive conrdicion. Ce 1. Sppoe he origin node i in pive componen of ome node p V (R) h h cive degree hree. We derive conrcion y n rgmen nlogo o Ce 1, y pplying Rle US3 ined of Rle US2 node p. Ce 1c. Sppoe he origin node i in V (R) nd h cive degree wo. We derive conrcion y n rgmen nlogo o Ce 1, y pplying Rle S2 Ce 1d. Sppoe he origin node i in V (R) nd h cive degree hree. We derive conrcion y n rgmen nlogo o Ce 1, y pplying Rle S3 ined of Rle US3 node p =.

14 14 Ce 1e. Sppoe he origin node i in n cive componen of ome node p V (R) nd S. Therefore, p h cive degree hree. Since he mege cnno revere i direcion in R nd R i cycle, he enire cive componen conining m e in S, implying S k. We derive conrdicion y (6). Ce 2. Sppoe grph R i cyclic. Since R m e conneced, R i ree. Coneqenly, he mege croe every edge in E(R) from oh direcion. By Oervion 2, Rle U3 i pplied q R. Agin, Rle U3 i pplied from wo direcion. In one of hee direcion he mege will e forwrded long he edge {q R, q T }. Since q T V (R), we derive conrdicion. Ech ce derive conrdicion, implying or mpion m e fle. Therefore, he mege doe no vii ny repeing eqence of verice. Since he grph i finie, he mege m evenlly rech node. n k 6 Fig. 13 The red rrow denoe he neighor o which he mege i iniilly forwrded from. Ble rrow denoe eqen forwrding deciion fncion of he neighor from which he mege w received. Algorihm 1 forwrd he mege from he origin node clockwie rond he cycle ck o node. The mege i hen forwrded coner-clockwie rond he cycle ck o node c efore eing forwrded long he ph from c o he deinion node. Thi roe h lengh 2n k 3, where he hore ph h lengh k + 3. c k d e Lemm 8 Algorihm 1 h dilion mo 7. Frhermore, for ny ɛ > 0, here exi nework on which Algorihm 1 h dilion le 7 ɛ. Proof Lemm 7 how h every mege evenlly reche i deinion. By Oervion 1, mege cn rvere ech edge mo wice, once in ech direcion. Node which Ce 1 of Algorihm 1 pplie re viied mo once. By Corollry 3, he mege rvel only long conien edge of G. We priion he conien edge ino hoe h forwrd he mege excly once nd hoe h forwrd he mege mo wice, nd ond he crdinliie of hee wo e. Ce 1. Sppoe neiher Ce 2, 3, nor 4 of Algorihm 1 i pplied. Therefore, only Ce 1 i pplied, he mege follow hore ph o i deinion, nd he dilion i 1. Ce 2. Sppoe Ce 2, 3, or 4 of Algorihm 1 re pplied. Therefore, di(, ) k + 1. (7) In priclr, Ce 1 of Algorihm 1 i pplied he l k + 1 node, ech of which i viied excly once. Thee node form ph T of lengh k. Th i, V (T ) = k + 1 nd E(T ) = k. (8) Le Q denoe he conien grph of G indced y he remining node which Ce 2, 3, or 4 my e pplied. Since k n/4 nd y (8), V (Q) = n V (T ) 4k (k + 1) = 3k 1. (9) By Lemm 5 nd 6, nd y (9), i follow h Q conin mo one cycle. Therefore, E(Q) V (Q) 3k 1, (10) where ech edge in E(Q) i viied mo wice. Ech edge in E(T ) i viied excly once. One ddiionl edge of G i followed once from Q o T. Therefore, he dilion, S(k), i onded y S(k) E(T ) + 2 E(Q) + 1 di(, ) k + 2(3k 1) + 1 k + 1 < 7, (11) y (7), (8), nd (10). We now how h (11) i igh. Chooe ny ɛ > 0 nd le n 96/ɛ. Le G denoe he grph on n verice illred in Figre 13 long wih he correponding locl roing fncion for node nd c (Rle S2 nd U3, repecively). Oerve h Rle U2 pplie ll remining node on he cycle nd Ce 1 of Algorihm 1 pplie ll node on he ph from node d o he deinion node. The roe followed y Algorihm 1 h lengh 2n k 3, where he hore ph h lengh k +3. Therefore, when k = n/4, Algorihm 1 h dilion 2n n/4 3 n/4 + 3 = 7 96 n + 12 > 7 ɛ, howing h (11) i igh. Theorem 5 follow from Lemm 7 nd 8:

15 A Fig. 14 (A) In he originl definiion of Algorihm 1, he repeed pplicion of Rle U2 forwrd he mege righ from nil i reche. Here, Rle S2 i pplied, ending he mege ck o. (B) Algorihm 1B revere he direcion of he mege pre-empively node (Rle U2c). Theorem 5 For every k n/4, here exi n originwre, predeceor-wre, k-locl roing lgorihm h cceed on ll conneced grph while grneeing dilion mo 7. Redcing Dilion Lemm 8 how h Algorihm 1 cnno grnee dilion le hn 7 ɛ for ny ɛ > 0. A hown in Theorem 4, no (n/4)-locl roing lgorihm cn grnee dilion le hn 5. In Appendix A we decrie refinemen of Rle U2 h redce he dilion of Algorihm 1 o mo 6. We refer o hi new roing regy Algorihm 1B. Informlly, he modificion o Rle U2 pplie Rle S2 or US2 pre-empively if he crren node h fficien informion o deermine h one of hee wo rle i o o e pplied node in i locl neighorhood. Th i, ined of forwrding he mege in given direcion only o hve he mege revere i direcion few hop wy, he direcion of he mege i revered immediely. See Figre 14. We prove he following heorem in Appendix A: Theorem 6 For every k n/4, here exi n originwre, predeceor-wre, k-locl roing lgorihm h cceed on ll conneced grph while grneeing dilion mo Predeceor Awre nd Origin Olivio Given ny k n/3, we decrie predeceor-wre, origin-olivio, k-locl roing lgorihm h cceed on ll conneced grph on n verice. Roing Algorihm The k-locl preproceing ep of Algorihm 1 i pplied o idenify e of roing edge in G k () which we denoe y G k (). Since 3k + 1 > n nd n cive componen conin le k node, we ge he following propoiion, nlogo o Propoiion 1: 1 2 B Propoiion 2 Every node h cive degree mo 2. Forwrding deciion re deermined y he following e of rle, imilr o hoe defined in Algorihm 1. Le denoe he crren node. Algorihm 2: (n/3)-locl, origin-olivio, predeceor-wre roing lgorihm Ce 1. Sppoe di(, ) k. Th i, V (G k ()). The lgorihm forwrd he mege o ny neighor of on hore ph from o nil he mege rrive. Ce 2. Sppoe di(, ) > k, =, nd v =. Th i, he mege i eing en from he origin for he fir ime. The lgorihm forwrd he mege long ny cive edge of. Ce 3. Sppoe di(, ) > k nd v. Forwrding deciion re illred y Rle U1 nd U2 in Figre 11. If he mege rrived vi pive componen of, hen he lgorihm forwrd he mege long ny cive edge of. Correcne of Algorihm 2 We now prove he correcne of Algorihm 2 nd derive igh ond on he correponding dilion. Oerve h Lemm 2, 3, 5, nd 6, nd Corollrie 3 nd 4 pply direcly o Algorihm 2. Frhermore, he following lemm follow y n rgmen nlogo o he proof of Lemm 4: Lemm 9 If Rle U1 i pplied node o revere he direcion of mege, hen G k () h pive componen conining le k 1 node. Uing rgmen imilr o he proof of Lemm 7 nd 8, we now prove he correcne of Algorihm 2 nd derive igh ond on he correponding dilion. Lemm 10 Given ny conneced grph G on n node, ny k n/3, nd ny {, } V (G), Algorihm 2 cceflly deliver mege from node o node. Proof By Propoiion 2, h mo wo cive componen. Coneqenly, one of Ce 1 hrogh 3 of Algorihm 2 m pply every node. Sppoe Algorihm 2 i defeed y ome grph G, for ome k n/4 nd ome origin-deinion pir (, ) V (G) V (G). Since n i finie nd Algorihm 2 i deerminiic, he mege m vii repeing eqence of verice nd edge in G. Le R denoe he correponding repeing grph of G. Le T = V (G) \ V (R) denoe he e of remining verice of G, inclding ll hoe h re never viied. By Lemm 3, here

16 16 exi conien ph in G from o ome verex in V (R). Le {q R, q T } denoe he l edge on ch ph ch h q R V (R) nd q T T. Since he mege doe no rech i deinion, Ce 1 of Algorihm 2 never occr. Coneqenly, v V (R), di(v, ) k + 1, implying h q T i n cive neighor of q R. By Propoiion 2, node q R h mo wo cive neighor, one of which i node q T. Since he mege i in repeing eqence, Ce 3 of Algorihm 2 pplie. In priclr, Rle U2 pplie, nd he mege i forwrded o node q T. Since q T V (R), we derive conrdicion. Therefore, he mege doe no vii ny repeing eqence of verice. Since he grph i finie, he mege m evenlly rech node. Lemm 11 Algorihm 2 h dilion mo 3. Proof Lemm 10 how h every mege evenlly reche i deinion. By Oervion 1, mege cn rvere ech edge mo wice, once in ech direcion. Node which Ce 1 of Algorihm 2 pplie re viied mo once. By Corollry 3, he mege rvel only long conien edge of G. We priion he conien edge ino hoe h forwrd he mege excly once nd hoe h forwrd he mege mo wice, nd ond he crdinliie of hee wo e. Ce 1. Sppoe neiher Ce 2 nor 3 of Algorihm 2 i pplied. Therefore, only Ce 1 i pplied, he mege follow hore ph o i deinion, nd he dilion i 1. Ce 2. Sppoe Ce 2 or 3 of Algorihm 2 re pplied. Therefore, di(, ) k + 1. (12) In priclr, Ce 1 of Algorihm 2 i pplied he l k + 1 node, ech of which i viied excly once. Thee node form ph T of lengh k. Th i, V (T ) = k + 1 nd E(T ) = k. (13) Le Q denoe he conien grph of G indced y he remining node which Ce 2 or 3 my e pplied. Since k n/3 nd y (13), V (Q) = n V (T ) 3k (k + 1) = 2k 1. (14) By Lemm 5 nd 6, nd y (14), i follow h Q i cyclic. Therefore, E(Q) < V (Q) 2k 1. (15) Ce 2. Sppoe Rle U1 i never pplied. Th, he mege never chnge direcion. Since Q i cyclic, he mege never vii ny edge in E(Q) more hn once. One ddiionl edge of G i followed once from Q o T. Ech edge in E(T ) i viied excly once. Therefore, he dilion, S(k), i onded y S(k) E(T ) + E(Q) + 1 di(, ) y (12), (13), nd (15). 3k k + 1 < 3, Ce 2. Sppoe Rle U1 i pplied ome node. By Lemm 9 nd (15), le k 1 edge in E(Q) re no viied. Coneqenly, mo k edge of E(Q) re viied, ech of which i viied mo wice. One ddiionl edge of G i followed once from Q o T. Ech edge in E(T ) i viied excly once. Therefore, he dilion, S(k), i onded y S(k) 3k + 1 k + 1 < 3. In ll ce, we ge h S(k) < 3. Theorem 7 follow from Lemm 10 nd 11: Theorem 7 For every k n/3, here exi n originolivio, predeceor-wre, k-locl roing lgorihm h cceed on ll conneced grph while grneeing dilion mo 3. A hown in Theorem 4, no (n/3)-locl roing lgorihm cn grnee dilion le hn 3. Therefore, Algorihm 2 i opiml wih repec o wor-ce roe lengh. 5.3 Predeceor Olivio nd Origin Olivio Given ny k n/2, we decrie predeceor-olivio origin-olivio k-locl roing lgorihm h cceed on ll conneced grph on n verice. Since 2k + 1 n nd n cive componen conin le k node, we ge he following propoiion, nlogo o Propoiion 1 nd 2: Propoiion 3 Every node h cive degree mo 2. We egin y elihing he following propery which i ed o how he correcne of Algorihm 3 in Lemm 13: Lemm 12 Given ny conneced grph G on n node, ny k n/2, nd ny {, } V (G), eiher di(, ) k or G k () h one conrined cive componen.