Bonding he Loliy of Diied Roing Algoihm [Exended A] Poenji Boe Shool of Compe Siene Cleon Univeiy Ow, Cnd ji@.leon. Pz Cmi Dep. of Compe Siene Ben-Gion Univ. of he Negev Bee-Shev, Iel mip@.g..il Sephne Dohe Dep. of Compe Siene Univeiy of Mnio Winnipeg, Cnd dohe@.mnio. ABSTRACT We exmine ond on he loliy of oing. A lol oing lgoihm mke eqene of diied fowding deiion, eh of whih i mde ing only lol infomion. Speifilly, in ddiion o knowing he node fo whih mege i deined, n inemedie node migh lo know ) he gph oeponding o ll newok node wihin k hop of ielf, fo ome vle of k, ) he node fom whih he mege oigined, nd ) whih of i neigho l fowded he mege. O ojeive i o deemine whih of hee pmee e neey nd/o ffiien o pemi lol oing k vie on newok modelled y onneed ndieed gph. In pil, we elih igh ond on k fo he feiiliy of deeminii k-lol oing fo vio ominion of hee pmee, well oeponding ond on dilion (he wo-e io of l oe lengh o hoe ph lengh). Cegoie nd Sje Deipo C.2.1 [Compe Syem Ognizion]: Compe- Commniion Newok Newok Ahiee nd Deign; F.2.2 [Theoy of Compion]: Anlyi of Algoihm nd Polem Complexiy Nonnmeil Algoihm nd Polem; G.2.2 [Mhemi of Comping]: Diee Mhemi Gph Theoy Genel Tem Algoihm, Theoy Keywod diied lgoihm, lol oing, dilion 1. INTRODUCTION Lol Roing. Uni ommniion in newok i hieved y oing lgoihm h ompe eqene Pemiion o mke digil o hd opie of ll o p of hi wok fo peonl o loom e i gned wiho fee povided h opie e no mde o diied fo pofi o ommeil dvnge nd h opie e hi noie nd he fll iion on he fi pge. To opy ohewie, o eplih, o po on eve o o ediie o li, eqie pio peifi pemiion nd/o fee. PODC 09, Ag 10 12, 2009, Clgy, Ale, Cnd. Copyigh 2009 ACM 978-1-60558-396-9/09/08...$5.00. of fowding deiion h deemine he oe followed y mege (e.g., pke) i vel o i deinion. In mny eing, enlized oing lgoihm o, moe genelly, oing lgoihm h eqie omplee knowledge of he newok opology e impil; eon inlde h he newok i oo lge, h he opology of he enie gph i nknown, o h he newok hnge dynmilly [13]. Alenively, lol oing lgoihm mke eie of diied fowding deiion, omped eh of he inemedie node long he oe. When node eeive mege, i ele po (i.e., one of i neigho) o whih o fowd he mege ing only lol infomion. In pil, eh node i only we of he e of he newok oniing of node wihin k hop fom ielf, fo ome k. Coneqenly, he oe nno e peomped eniely in genel. Fhemoe, mege ovehed nd lol memoy e ofen limied [11]. In pil, newok node nno e expeed o minin hioy of mege h hve ped hogh i (i.e., he newok i memoyle). Similly, he mege ovehed nno oe he e of node viied y he mege (i.e., he oing lgoihm i ele). Alhogh ighfowd flooding lgoihm i poile, h egy h ovio dwk, inlding high ffi lod [13], yli ehvio (if he newok i memoyle), nd eqiing knowledge of n ppe ond on he dimee of he newok o ene oh eminion nd efl delivey. In hi ppe we onide ingle-ph deeminii oing lgoihm. We epeen newok y onneed, nweighed, ndieed gph G = (V, E) wih niqe veex lel. A newok node (gph veex) i idenified y i lel. In ome newok, node lel my povide infomion o i neighohood in he newok (e.g., gid gph node n e lelled y i gid oodine). In genel, ppoe h he veex lelling i independen of he gph; h i, node lel doe no enode ddiionl infomion o he opology of he gph o he node neighohood. Eqivlenly, we onide oing lgoihm h eed on ny pemion of he veex lel of G. We me h evey node know i own lel well he lel of i neigho. A mege lo eqie deinion node, idenified y he node lel. Some o ll of he following ddiionl infomion my e ville o n inemedie node o ompe he nex node o whih mege hold e fowded: 1. oigin-wene: knowledge of he node fom whih he mege oigined,
T (n) oigin-we oigin-olivio pedeeo-we n/4 n/3 pedeeo-olivio n/2 n/2 Tle 1: Min el: hee exi k-lol oing lgoihm when k T (n), no k-lol oing lgoihm exi when k < T (n), whee n denoe he nme of newok node. B 1 k B 2 2. pedeeo-wene: knowledge of he inoming edge (po) long whih he mege w fowded o (eqivlenly, he neigho of h l fowded he mege), nd 3. k-loliy: knowledge of he k-neighohood of (i.e., he gph of G oniing of ll ph ooed wih lengh mo k). O ojeive i o deemine whih of hee pmee e neey nd/o ffiien o pemi lol oing k vie. Oveview of Rel. We idenify igh ond on he vle of he loliy pmee k fo he feiiliy of k- lol oing in eh of he fo ominion of onin: pedeeo-we o pedeeo-olivio, nd oigin-we o oigin-olivio. In eh e, le T (n) denoe he oeponding hehold. Th i, fo evey k < T (n), evey k- lol oing lgoihm i defeed y ome onneed gph on n veie. Similly, fo evey k T (n), hee exi k-lol oing lgoihm h eed on ll onneed gph on n veie. O min el i he idenifiion of he vle of T (n); ee Tle 1. In ddiion, we elih lowe ond of S(k) = 2 3k/n on he wo-e dilion of ny k-lol oing lgoihm nd how hi ond i igh fo hee of he fo ominion of onin. 2. MODELLING LOCAL ROUTING In hi eion we fomlize o model fo lol oing. k-lol Roing Fnion. Given gph G = (V, E), we employ ndd gph-heoei noion, whee fo eh veex v V, Adj(v) = { {, v} E} denoe he e of veie djen o v nd deg(v) = Adj(v) denoe i degee. Le di(, v) denoe he (nweighed) gph dine eween veie nd v. The k-neighohood of veex v V, denoed G k (v), i he gph of G h onin ll ph ooed v wih lengh mo k. A oing lgoihm i oigin-we, pedeeo-we, nd k-lol if i n e defined fnion f(,,, v, G k ()), whee V i he oigin node, V i he deinion node, he mege i enly node V, node eeived he mege fom i neigho v Adj(), G k () i he k-neighohood of node, nd f(,,, v, G k ()) en he neigho of o whih he mege hold e fowded (i.e., he po o whih node m fowd he pke). B 4 B 3 Fige 1: In hi exmple, G 8() oni of fo lol omponen, oeponding o he fo onneed omponen of G 8()\{}. B 1, B 3, nd B 4 e fonie omponen B 2 i no. B 1 nd B 3 e onined fonie omponen B 2 nd B 4 e no. Sy v = if he mege h no ye een fowded (i.e, he mege leve node fo he fi ime). Evey k- lol oing lgoihm A h oeponding oing fnion f. A eqene of ll o fnion f en eqene of fowding deiion h oepond o wlk hogh G oigining (i.e., he oe). We onide wo onin on k-lol oing lgoihm: n oiginolivio k-lol oing lgoihm i no povided he pmee, nd pedeeo-olivio k-lol oing lgoihm i no povided he pmee v. To implify noion fo pedeeo-we lgoihm, le f (v) denoe he lol oing fnion node fo given,,, nd G k (), whee f (v) = f(,,, v, G k ()). Th i, f (v) en he neigho of o whih he mege i fowded fnion of he neigho v fom whih i i eeived. Nlly, no ll oing fnion n e implemened effiienly lol oing lgoihm. The oing fnion model llow onge negive el o e elihed fo moe genel l of oing lgoihm, egdle of implemenion onen. Wih epe o poiive el, he oing lgoihm we peen n e implemened effiienly lolly; implemenion deil e no he fo of hi ppe. Le C denoe onneed omponen of G k () \ {}. We efe o C lol omponen of. If v C Adj(), hen we y C i ooed v (C n hve mliple oo). If C onin veex v h h di(, v) = k, hen (elive o ) C i fonie omponen, v i fonie veex, nd hoe ph fom o v i fonie ph. In ohe wod, C exend o he limi of knowledge: node v my hve neigho oide C, hi infomion i no known lolly. If C i fonie omponen of nd evey fonie ph in C pe hogh ome veex w, hen C i onined fonie omponen nd w i onin veex. See Fige 1. Evling Roing Algoihm. A oing lgoihm A defined y oing fnion f eed (ynonymoly,
gnee delivey) if fo ll gph G nd ll oigin-deinion pi (, ) in G, he eqene of vle ened y f oepond o wlk fom o in G. Ohewie, A i defeed y ome gph G nd ome pi (, ) in G. A oing lgoihm A h dilion onded y δ if fo ll gph G nd ll oigin-deinion pi (, ) in G, A(, )/ di(, ) δ, whee A(, ) denoe he lengh of he oe fom o ened y A. 3. RELATED WORK In poiion-ed oing, newok node e emedded in ome pe (ypilly R 2 o R 3 ) nd eh node know i pil oodine (i.e., node e loion-we). Poiioned oing i lo known geo-oing, geogphi oing, o geomei oing. Mny een el eled o lol oing e poiion ed. We iefly deie ome of hee eled el nd di he inedependene eween poiion-ed nd poiion-olivio oing. Geedy oing [7] (fowd he mege o he neigho loe o he deinion), omp oing [11] (fowd he mege long he edge h fom he mlle ngle wih he line egmen o he deinion), nd geedy-omp oing [1] (pply geedy oing o he wo edge djen o he line egmen o he deinion) e hee well-known poiion-ed oing lgoihm, eh of whih eed on peifi le of gph i defeed y ome pln gph [2]. All hee lgoihm e pedeeo-olivio, oigin-olivio, nd 1-lol. To how h oing lgoihm fil on ome l of gph G, i ffie o idenify gph in G on whih he lgoihm yle infiniely wiho ehing he deinion. Songe negive el e hoe h pply o ll oing lgoihm, howing h no oing lgoihm eed on given l of gph. Boe e l. [1] how h evey poiion-ed, pedeeo-olivio, oigin-olivio, 1-lol oing lgoihm i defeed y ome onvex diviion. Fe oing [11] w one of he fi poiion-ed 1- lol oing lgoihm dioveed o eed on moe genel le of gph emedded in he plne. In ief, fe oing fowd he mege in lokwie dieion long he edge of fe, nd long he eqene of fe h inee he line egmen eween he oigin nd deinion node. Fowd poge i gneed y oing pmee h he fhe ineeion of he line egmen wih viied fe. A h, fe oing i no ele ine i eqie Θ(log n) i o e oed wih he mege. Fe oing eed on pln gph [11], on ni di gph [3], nd on d-qi ni di gph fo ny d [1/ 2, 1] [12]. See [3] nd [12] fo definiion of ni di gph nd qi ni di gph, epeively. Fe onide genelizion of fe oing o gph emedded on oi [9]. Alhogh o diion foe on deeminii oing lgoihm, we iefly noe h ndomized olion pemi k-lol oing on moe genel le of gph. Fly nd Wehofe onide he polem of ndomized lol oing on ni ll gph [8] nd how h ny ndomized poiion-ed lol oing lgoihm h expeed oe lengh Ω(l 3 ), whee l denoe he lengh of he hoe ph. Dohe e l. [6] how h fo evey fixed k, evey oiginwe, pedeeo-we, k-lol oing lgoihm fil on ome ni ll gph. The poof h wo p. Fi, he oeponding poiion-olivio el i poven: fo evey fixed k, evey oigin-we, pedeeo-we, k-lol oing lgoihm fil on ome gph. Nex, k-lol edion fom (nemedded) gph o ni ll gph i ed o how h if ome (poily poiion-ed) k-lol oing lgoihm eed on ni ll gph, hen ome (poiionolivio) k-lol oing lgoihm eed on ll gph. Thi inedependene eween poiion-ed nd poiionolivio oing lgoihm moive he qeion of exploing he ondy eween feiiliy nd impoiiliy of lol oing lgoihm fnion of he lol infomion ville. In hi ppe we onide he poiionolivio e. 4. WHEN LOCAL ROUTING IS IMPOSSI- BLE: NEGATIVE RESULTS In hi eion we peen negive el: evey k-lol oing lgoihm fil on ome gph when he degee of loliy k i le hn he given ond. Fo eh ominion of oigin-wene/olivione nd pedeeowene/olivione, we demone one-exmple oniing of e of gph h h ny k-lol oing lgoihm fil on le one of he gph in he e. 4.1 Popeie of Lol Roing Fnion The poof of Theoem 3 hogh 6 efe o Lemm 1 nd Coolly 2, whih genelize n oevion of Dohe e l. [6] howing h if k-lol oing lgoihm gnee delivey, hen eh lol oing fnion oepond o il pemion (nde ein ondiion). Rell h il pemion of n diin elemen i n odeing of hee elemen in yle. Lemm 1. Le denoe node h h 1. deg() 2, 2. fo ll {, } Adj(), nd elong o diffeen fonie omponen of, nd 3. neihe he oigin node no he deinion node i in G k (). If A i n oigin-we, pedeeo-we, k-lol oing lgoihm h gnee delivey, hen he lol oing fnion of A i il pemion of Adj(). Poof. Chooe ny k 1, ny node, nd ny k- neighohood G k () h h Popeie 1 hogh 3 hold. Sppoe A i ny k-lol oing lgoihm h gnee delivey nd h he lol oing fnion f i no il pemion. Ce 1. Sppoe f i no pemion. Th i, f i no jeive. Theefoe, hee exi ome v Adj() h h fo ll w Adj(), f (w) v. Le B 1 denoe he lol omponen of h onin v nd le B 2 denoe ny ohe lol omponen of. By Popey 2, eh lol omponen of exend o he fonie of G k (). Le G denoe gph h onin G k () h h node h degee one nd i he only node djen o B 1 oide G k (). Similly, le node hve degee one h h i i he only node djen o B 2 oide G k (). See Fige 2. Sine fo ll w Adj(), f (w) v, he mege will neve ene B 1 nd, oneqenly, will neve eh. Theefoe, lgoihm A fil on gph G, deiving ondiion.
oing egy il pemion eed fil B 1 v k B 2 1 (P 1P 3P 4) G 1, G 3 G 2 2 (P 1P 4P 3) G 1, G 2 G 3 3 (P 1P 3P 4) G 2, G 3 G 1 4 (P 1P 3P 4) G 1, G 2 G 3 5 (P 1P 4P 3) G 2, G 3 G 1 6 (P 1P 4P 3) G 1, G 3 G 2 Tle 2: Eh oing egy oepond o il pemion of he neigho of. Fige 2: G k () oni of fonie omponen, eh of whih i ooed niqe neigho of. Thi exmple ille he gph oned in Ce 1 fo given G k () when k = 8. Ce 2. Sppoe f i pemion no dengemen ( dengemen i omplee pemion). Theefoe, f (v) = v fo ome v Adj(). Le G e gph defined in Ce 1, wih he exepion h node nd e inehnged. I follow h he mege will neve ene ny lol omponen ohe hn B 1 nd, oneqenly, will neve eh. Theefoe, lgoihm A fil on gph G, deiving ondiion. Ce 3. Sppoe f i dengemen no il pemion. Theefoe, f nno e expeed ingle pemion yle. Le ( 1... k ) nd ( 1... j) denoe ny wo pemion yle of f. Oeve h { 1,... k } nd { 1,..., j} e dijoin e of Adj(). Le G e gph defined in Ce 1, wih he exepion h node i djen o lol omponen B 1 ooed node in { 1,... k } nd i djen o lol omponen B 2 ooed node in { 1,..., j}. I follow h he mege will neve ene B 2 nd, oneqenly, will neve eh. Theefoe, lgoihm A fil on gph G, deiving ondiion. All hee e deive ondiion nd o mpion m e fle. Theefoe, he lol oing fnion f m e il pemion. In ohe wod, wiho ddiionl infomion on whih o e lol oing deiion, n inemedie node m y ll poiiliie nd eqenilly fowd he mege o eh of i neigho. When node h degee wo, only one il pemion i poile: mege eeived fom one neigho of m e fowded o he oppoie neigho. If node h degee j, hen (j 1)! il pemion e poile. If oing lgoihm A i oigin olivio, hen Lemm 1 give: Coolly 2. Le denoe node h h 1. deg() 2, 2. fo ll {, } Adj(), nd elong o diffeen fonie omponen of, nd 3. he deinion node i no in G k (). If A i n oigin-olivio, pedeeo-we, k-lol oing lgoihm h gnee delivey, hen he lol oing fnion of A i il pemion on Adj(). Poof. Given ny node, ny k 1, nd ny k-neighohood G k (), if Popeie 1 hogh 3 hold ( defined in Lemm 1), hen he lol oing fnion f i il pemion y Lemm 1. Sine A i oigin olivio, fnion f emin onn fo ny given G k () nd, egdle of. In pil, f i il pemion egdle of whehe o no i onined in G k (). The el follow. Theoem 3 hogh 6 elih lowe ond oeponding o eh of he fo ominion of k-lol oing lgoihm: oigin-we/olivio nd pedeeo-we/olivio. 4.2 Pedeeo Awe nd Oigin Awe Theoem 3. Fo evey k < (n + 1)/4, evey oiginwe, pedeeo-we, k-lol oing lgoihm fil on ome onneed gph. Poof. Chooe ny k < (n + 1)/4, k Z +. Theefoe, k {1,..., }, whee = (n 3)/4. Le G 1, G 2, nd G 3 denoe he gph illed in Fige 3, h h eh ph P 1 hogh P 4 oni of veie h e lelled onienly elive o node in ll hee gph. In eh gph, G k () i ee oniing of fo ph of lengh k ooed, none of whih onin no. In ddiion o he 4 node in ph P 1 hogh P 4, eh gph inlde node,, nd. Depending on he vle of n mod 4, eween zeo nd hee ex node emin; hee e dded eween nd P 1 o ing he ol nme of node o n. Any efl oing lgoihm m p he mege o P 1 o node. Sine h degee fo, i lol oing fnion i one of ix poile il pemion y Lemm 1. The emining node hve degee mo wo. Theefoe, when he mege i ped o node on ph h doe no onin no, y Lemm 1, he mege m onine fowd nil i en gin o. A hown in Tle 2, fo eh of he ix poile oing egie, he mege neve ene he ph onining in le one of he gph G 1, G 2, o G 3. Th i, fo evey oing egy A, hee exi gph on whih A fil. 4.3 Pedeeo Awe nd Oigin Olivio Uing n gmen imil o he poof of Theoem 3, we now how h he lowe ond on he loliy pmee k inee o (n + 1)/3 fo oigin-olivio k-lol oing lgoihm: Theoem 4. Fo evey k < (n + 1)/3, evey oiginolivio, pedeeo-we, k-lol oing lgoihm fil on ome onneed gph.
P 1 P 1 P 1 P 3 P 3 P 3 P 4 G P 1 4 G 4 d d d P 1 P 3 G 3 d P 4 Fige 3: The k-neighohood G k () i idenil in gph G 1, G 2, nd G 3. In hi exmple, ppoe n mod 4 = 0. Coneqenly, one ex node i dded eween nd P 1 h h he ol nme of node i n. G 1 G 2 G 3 P 3 P 1 P 3 P 1 P 3 P 1 P 3 P 1 Fige 4: The k-neighohood G k () i idenil in gph G 1, G 2, nd G 3. In hi exmple, ppoe n mod 3 = 0. Coneqenly, one ex node i dded nex o h h he ol nme of node i n. Poof. Chooe ny k < (n + 1)/3, k Z +. Theefoe, k {1,..., }, whee = (n 2)/3 }. Le G 1, G 2, nd G 3 denoe he gph illed in Fige 4, h h eh ph P 1 hogh P 3 oni of veie h e lelled onienly elive o node in ll hee gph. In eh gph, G k () i ee oniing of hee ph of lengh k ooed, none of whih onin. In ddiion o he 3 node in ph P 1 hogh P 3, eh gph inlde node nd. Depending on he vle of n mod 3, eween zeo nd wo ex node emin; hee e dded eween nd he oeponding ph P i nee o o ing he ol nme of node o n. Sine node h degee hee, i lol oing fnion i one of wo poile il pemion y Coolly 2. A oing egy m peify he dieion in whih mege iniilly leve node (hee dieion e poile). The emining node hve degee mo wo. Theefoe, when he mege i ped o node on ph h doe no onin, y Coolly 2, he mege m onine fowd nil i en gin o node. A hown in Tle 3, fo eh of he ix poile oing egie, he mege neve ene he ph onining in le one of he gph G 1, G 2, o G 3. Th i, fo evey oing egy A, hee exi gph on whih A fil. 4.4 Pedeeo Olivio nd Oigin Awe When knowledge of he pedeeo node i wihheld, he lowe ond on he loliy pmee k inee o n/2 fo pedeeo-olivio k-lol oing lgoihm: Theoem 5. Fo evey k < n/2, evey oigin-we, G 1 G 2 Fige 5: illion in ppo of Theoem 5 pedeeo-olivio, k-lol oing lgoihm fil on ome onneed gph. Poof. Chooe ny k < n/2, k Z +. Theefoe, k {1,..., }, whee = n/2 1. Le G 1 denoe ph of n veie wih he oigin node loed he ( + 1) veex nd he deinion node loed he f end. Le G 2 denoe he nlogo gph pon moving node o he oppoie end of he ph. Le he emining node e lelled onienly elive o node in oh gph. See Fige 5. The k-neighohood G k () i idenil in G 1 nd G 2. If lgoihm A end he mege igh, hen A fil on gph G 2 ine i m evenlly end he mege lef, whih poin i ehvio eome yli. Similly, if lgoihm A end he mege lef, hen i fil on gph G 1.
oing egy il pemion iniil dieion eed fil 1 (P 1P 3) owd G 1, G 3 G 2 2 (P 1P 3) owd G 1, G 2 G 3 3 (P 1P 3) owd G 2, G 3 G 1 4 (P 1P 3) owd G 1, G 2 G 3 5 (P 1P 3) owd G 2, G 3 G 1 6 (P 1P 3) owd G 1, G 3 G 2 Tle 3: Eh oing egy oepond o il pemion of he neigho of pied wih n iniil dieion. 4.5 Pedeeo Olivio nd Oigin Olivio Finlly, if we fhe onin he knowledge ville o inemedie node, hen he lowe ond on he loliy pmee k doe no inee: Theoem 6. Fo evey k < n/2, evey oigin-olivio, pedeeo-olivio, k-lol oing lgoihm fil on ome onneed gph. Poof. The lowe ond of n/2 follow y Theoem 5. 4.6 Dilion We now onide lowe ond on dilion fo k-lol oing lgoihm. Theoem 7. Fo ny k < n/2, no k-lol oing lgoihm n gnee dilion le hn 2n 3k 1. (1) k + 1 Poof. Chooe ny n nd ny k < n/2. Le P denoe he e of ph of lengh n wih veex e {,, v 1,..., v n 2}. Chooe ny k-lol oing A h eed on ll ph in P. Sppoe he oigin nd deinion node e lelled nd, epeively. Le d P (i) = mx{di(, v) v V i}, whee V i denoe he e of veie of P o whih lgoihm A h ped he mege ding he fi i ep. Th, d P (i) i. Chooe ny i {0,..., n 2k 2}. Oeve h hee exi ph P 1 nd in P h h fo ll v V i, ) G k (v) i idenil in P 1 nd, ) G k (v) h wo fonie omponen, neihe of whih onin, nd ) lie oppoie endpoin of ph P 1 nd elive o. The node of V i oepond o ph onining mo i + 1 node; eh endpoin of he ph h n nexploed fonie omponen (k node eh) nd node emin o of igh. Smming hee give i + 1 + 2k + 1 n node. When he mege ehe he oeponding node dine d P1 (i) = d P2 (i), A end he mege in he me dieion in P 1 nd, i.e., owd in one ph nd wy fom in he ohe. In he le e, A m evenlly p he mege k o efoe i n eh. Rening he mege o eqie le i + 2 ep nd ehing eqie le k + 1 ddiionl ep. Theefoe, he ol nme of ep i le 2i + k + 3. Thi give A(, ) 2n 3k 1 when i = n 2k 2. Sine di(, ) n e lile k + 1, he el follow. The ond on dilion (1) i pehp moe lely expeed y king he limi he nme of node ppohe infiniy (nd k = n fo ome onn ). We denoe hi limi y S(k): 2n 3k 1 S(k) = lim = 2n n k + 1 k 3. Of pil inee e he vle of k {n/4, n/3, n/2}, fo whih he oeponding ond on dilion e 5 (when k = n/4), 3 (when k = n/3), nd 1 (when k n/2). Thee ond e igh fo k = n/3 nd k = n/2 hown in Theoem 14, 16, nd 17. 5. WHEN LOCAL ROUTING IS POSSIBLE: ROUTING STRATEGIES In hi eion we peen poiive el: hee exi efl k-lol oing lgoihm when he degee of loliy k exeed he given ond. We deie k-lol oing lgoihm fo eh ominion of oigin-wene/olivione nd pedeeo-wene/olivione. 5.1 Pedeeo Awe nd Oigin Awe We deie n (n/4)-lol pedeeo-we nd oiginwe oing lgoihm. Given ny k n/4, he lgoihm egin wih k-lol pepoeing ep o idenify he edge on whih oing ke ple. Evey node G ild gph of G k () y deleing n edge fom evey yle h onin in G k (). The ode in whih he yle e oken i impon. By o mpion h node hve niqe lel, we oin n odeing on he edge of G. In pil, ny e of edge h minimm edge. The node onide evey yle h onin i in G k () nd delee he minimm edge e mong ll yle. See Fige 6. The node hen ompe ll emining yle in G k () \ {e} h onin nd delee he minimm edge mong hee yle. Thi poe i epeed nil no yle onining emin. The finl gph denoed G k() i pnning gph of G k () in whih i veex. Conide he lol omponen in G k(). The edge in G k() joining o lol omponen e lled oing edge. If lol omponen in G k() i fonie omponen, he omponen i lled ive. Ohewie, oh he omponen nd oing edge e lled pive. The nme of ive edge djen o veex i i ive degee. Sine n ive edge join o omponen wih le n/4 veie, node n hve ive degee mo 3. An edge {, } i lled onien povided h oh nd onide he edge eween hem o e ive. A node i lled onien povided h ll of i ive edge e onien. Ohewie, he node i lled inonien Oevion 8. An inonien node h ive degee 1.
B 1 B 1 B 1 B 1 v 1 v 1 v 1 v 1 v 1 v 3 v 2 B 3 B 2 v 3 v 2 B 3 B 2 v 3 v 2 B 3 B 2 v 3 v 2 B 3 B 2 v 3 v 2 A B C D E Fige 6: k-lol pepoeing. Sppoe he en node h hee neigho, v 1 hogh v 3, nd G k () onin yle h inlde veie v 1,, nd v 2 (A). The pepoeing ep lifie one of he edge on he yle non-oing edge (mgen). The eleed edge my e din fom (B) o djen o (C nd D). The hoie of oing edge doe no ffe node no on he yle (e.g., v 3) ine he edge of he yle ll lie in he me lol omponen fom he poin of view of he oeponding veex (E); in pil, none of he yle edge e djen o v 3. Rle S3 Rle S2 Rle U2 Rle S1 Rle U3 Fige 7: Algoihm 1, Ce 2: Mege eeived/fowded y. Rle U1 One eh node h idenified i oing edge, imple e of le deemine he fowding deiion. The e of he oing lgoihm elie on he popey h eh node h ive degee mo 3. Noie h he lowe ond gmen of he poof of Theoem 3 oni of gph h hve one node wih ive degee 4; hi nno o when k > n/4. The oing lgoihm oni of fo e olined elow. Fige 8: Algoihm 1, Ce 3: Mege eeived/fowded y node h nno ee nd n poily ee in n ive omponen. Algoihm 1: (n/4)-lol, oigin-we, pedeeowe oing lgoihm. Ce 1. Sppoe di(, ) k. Th i, G k (). The lgoihm fowd he mege o ny neigho of on hoe ph fom o nil he mege ive. Ce 2. Sppoe di(, ) > k nd =. Fowding deiion e illed in Fige 7. Ce 3. Sppoe di(, ) > k,, nd eihe di(, ) > k o i in n ive omponen of. Fowding deiion e illed in Fige 8. Ce 4. Sppoe di(, ) > k,, nd i in pive omponen of. Fowding deiion e illed in Fige 9. Rle US3 Rle US2 Lemm 9. Algoihm 1 eflly delive he mege fom he oigin node o he deinion node. Poof. Sppoe, fo ke of ondiion, h mege fom doe no eh. Sine he nme of node in G i finie, he mege m vii epeing eqene of node of G. Denoe y X = x 1,..., x m he eqene of Fige 9: Algoihm 1, Ce 4: Mege eeived/fowded y node h nno ee nd ee in pive omponen.
node viied, whee R = x i,..., x m i epeing eqene of node fo ome 1 i < m. Clim 1: All node in eqene R e onien. Thi follow fom he popey h n inonien node n only eeive nd fowd mege mo one. Thee i e I of node in G h e inviile o ll node in X. By inviile, we men h x X, G k (x) I =. In pil, I. Conide hoe ph fom o node v V \I. By onion, v m e on he fonie of ome veex x j X, i.e., di(x j, v) = k n/4. Ohewie, ll neigho of v wold e viile o x j, ondiing he f h v i he loe viile veex o. Fhemoe, no node on hoe ph eween x j nd v n ppe in X. Sine v i fonie veex of x j, he fonie omponen of x j onining v m e ive. Howeve, x j neve fowd he mege o hi omponen. Coneqenly, x j h ive degee le 2, whih y Oevion 8 implie h x j i onien node. Le e = {x j, y} denoe he oeponding ive edge (long whih he mege i no fowded). To oin ondiion, we how h in ll e, x j limely fowd he mege long edge e. Clim 2: x j 1 x j+1. Only fowding le S1, S2, U1, nd US2 wold hve x j 1 = x j+1. In ll fo e, he mege i fowded o ll ive edge djen o x j, ondiing he f h v i he loe viile veex o. Clim 1 nd 2 imply h x j h ive degee 3. Conide he ndieed gph S of G h oni only of he edge followed y he lgoihm in he epeing eqene R. Oeve h y onion, S h mximm degee 3. Ce 1. Sppoe S onin no yle. Th i, S i ee. Roo hi ee x j. When x j fowd he mege o x j+1, he mege i ped ino he ee of S ooed x j+1. By Clim 2, ine S i ee, ll ph fom node in he ee ooed x j+1 o node in he ee ooed x j 1 m go hogh x j. Thi implie h ome poin in he eqene R, he node x j+1 fowd he mege o x j, whih will hen fowd i o y, deiving ondiion. Ce 2. Sppoe S onin imple yle. By Clim 1, ll edge in he yle e onien. Evey yle of lengh mo 2k onin le one pive edge. Theefoe, he yle m onin gee hn 2k n/2 veie. The node x j m lie on h yle, ohewie, in Ce 1, fowding he mege long x j+1 will e i o en o x j vi he edge {x j+1, x j}, deiving ondiion. We now how h he mege nno e ped ond hi yle. The key o eking he yle i he loion of he oigin node. Conide wo e: eihe node i inviile o evey node in he yle o i i viile o le one node in he yle. Ce 2. Sppoe i no viile o ny node in he yle. Theefoe, when iniilly fowd he mege, he mege follow ph fom o ome node on he yle. Thi ph onin le k n/4 diin node h e no p of he yle ine no node of he yle n ee. The yle onin moe hn n/2 diin node nd he ph fom x j o v onin le n/4 node, none of whih e viied y he eqene X. Theefoe, we oin he deied ondiion ine hee e exly n node in G. Ce 2. Sppoe i viile o ome x k on he yle. Conide wo fhe e. Ce 21. Sppoe i onined in he yle. Node m hve ive degee 2 o 3. If i h ive degee 2, hen Rle S2 implie h he mege i en long he yle in wo diffeen dieion, ondiing he f h he mege i no fowded long edge {x j, y}. If h ive degee 3, hen only one of hee pi of node djen o n e involved in he yle. Le exmine eh pi in n. Fo wh follow, onide he lelling given in Fige 7. The node nd nno e involved in he yle ee when he mege i eeived y fom, i i fowded o nd when i i eeived fom y, i i lo fowded o. Node nd nno e involved in he yle ee iniilly fowd he mege o, whih implie h o omplee he yle, i m eeive he mege fom. Howeve, when eeive mege fom, i i fowded o. Theefoe, oding o Rle S3, if p of he yle, hen on he yle, i eeive he mege fom nd fowd i o o vie ve. Howeve, iniilly fowd he mege o nd he mege en o. Thi implie h none of he node in he ive omponen of ooed e p of he yle. Thi omponen h ize le n/4. Moeove, none of hee node n e p of he ph fom x j o v. Theefoe, we gin ondi he f h G h n node. Ce 22. Sppoe i no onined in he yle. The gmen i nlogo o Ce 21 pon iing le US2 nd US3 fo le S2 nd S3, epeively. Lemm 10. Algoihm 1 h dilion mo 8. Poof. If mege fom i en o nd i viile o, hen he ph followed y Algoihm 1 h dilion 1. When i no viile o, hen he lengh of he hoe ph fom o i le n/4 + 1. By looking he poof of Lemm 9, we ee h if mege follow yle, hen i vii evey edge of he yle one. Ohewie, i vii evey edge mo wie. Thi implie h he lengh of he ph followed y Algoihm 1 i mo 2n. Sine he lengh of he hoe ph i le n/4 + 1, we ge he deied el. Theoem 11. Fo evey k n/4, hee exi n oiginwe, pedeeo-we, k-lol oing lgoihm h eed on ll onneed gph while gneeing dilion mo 8. 5.2 Pedeeo Awe nd Oigin Olivio We deie (n/3)-lol pedeeo-we nd oiginolivio oing lgoihm. Chooe ny inege k n/3. Theefoe, 3k + 1 > n. Sine fonie omponen onin le k node, evey node h mo wo fonie omponen. In pil, i h mo wo onined fonie omponen. Le denoe he en node. The lgoihm eqie k-lol pepoeing ep imil o h deied in Algoihm 1 o idenify e of ive edge in G k () whih we denoe y G k(). Anlogoly, fe pplying he pepoeing ep, evey ive neigho of i he oo of niqe onined omponen of in G k() (hee n e mo wo). The following lgoihm i hen pplied o G k() o mke he fowding deiion. Algoihm 2: (n/3)-lol, oigin-olivio, pedeeo-we oing lgoihm.
Ce 1. Sppoe di(, ) k. Th i, G k (). The lgoihm fowd he mege o ny neigho of on hoe ph fom o nil he mege ive. Ce 2. Sppoe di(, ) > k. Fowding deiion e illed y Rle U1 nd U2 in Fige 8. If he mege oigined o ived vi n inonien edge, hen he lgoihm fowd he mege long ny ive edge of. Lemm 12. Algoihm 2 eflly delive he mege o he deinion node. Poof. (Skeh) Sine 3k + 1 > n, node nno hve hee o moe ive omponen. Theefoe, eh node h ive degee mo 2. Sppoe Algoihm 2 doe no delive he mege o node. The mege m vii epeing eqene of node of G. Uing n gmen imil o ( imple hn) he poof of Lemm 9, we deive ondiion y howing h hi epeing eqene m inlde node h h n ive omponen h i no viied (nd evenlly led o node ). Sine eh node h ive degee mo 2, hi omponen m e viied oding o le U2. Lemm 13. Algoihm 2 h dilion mo 3. We omi he poof of Lemm 13 de o pe onin. Theoem 14 follow fom Lemm 12 nd 13: Theoem 14. Fo evey k n/3, hee exi n oiginolivio, pedeeo-we, k-lol oing lgoihm h eed on ll onneed gph while gneeing dilion mo 3. 5.3 Pedeeo Olivio nd Oigin Olivio We deie (n/2)-lol pedeeo-olivio nd oiginolivio oing lgoihm. Chooe ny k n/2. Theefoe, 2k + 1 n. Sine fonie omponen onin le k node, evey node h mo wo fonie omponen. Le denoe he en node. Algoihm 3: (n/2)-lol, oigin-olivio, pedeeo-olivio oing lgoihm. Ce 1. Sppoe di(, ) k. Th i, G k (). The lgoihm fowd he mege o ny neigho of on hoe ph fom o nil he mege ive. Ce 2. Sppoe h zeo o wo fonie omponen. Sine 2k + 1 n, he enie newok i onined in G k () nd, heefoe, d(, ) k. Roing poeed in Ce 1. Ce 3. Sppoe h one nonined fonie omponen. Sine evey nonined fonie omponen onin le 2k veie, he enie newok i onined in G k () nd, heefoe, d(, ) k. Roing poeed in Ce 1. Ce 4. Sppoe di(, ) > k nd h one onined fonie omponen. Le v denoe he onin veex in G k () h i fhe fom. The lgoihm fowd he mege o ny neigho of h ede he dine o v. Thi poede onine nil he lgoihm ene Ce 1, 2, o 3. Lemm 15. Algoihm 3 eflly delive he mege o he deinion node long hoe ph fom o. Poof. In Ce 1 hogh 3, he dine fom he en node o he deinion, di(, ), deee y one ni eh ime he mege i fowded. In Ce 4, oeve h di(, ) = di(, v) + di(v, ). Theefoe deee in di(, v) implie n eql deee in di(, ). I follow h he oe fom o h lengh di(, ). Theefoe, Algoihm 3 find hoe ph fom o. Theoem 16 follow fom Lemm 15: Theoem 16. Fo evey k n/2, hee exi n oiginolivio, pedeeo-olivio, k-lol oing lgoihm h eed on ll onneed gph nd find hoe ph fom he oigin o he deinion. Algoihm 3 n e defined nlogoly o Algoihm 1 nd 2. Th i, when node, he me pepoeing ep old e inlded o lify edge ive o pive. Upon doing o, node h ive degee mo 3 in Algoihm 1 (k n/4), ive degee mo 2 in Algoihm 2 (k n/3), nd ive degee mo 1 in Algoihm 3 (k n/2). Ignoing pedeeo- nd oiginwene/olivione, hi end gge h he loliy pmee k i inveely popoionl o he nme of poile fowding deiion h k-lol oing lgoihm m onide eh node. 5.4 Pedeeo Olivio nd Oigin Awe A we now how, n oigin-we k-lol oing lgoihm doe no eqie gee loliy pmee k hn doe n oigin-olivio k-lol oing lgoihm o gnee delivey. Theoem 17. Fo evey k n/2, hee exi n oiginwe, pedeeo-olivio, k-lol oing lgoihm h eed on ll onneed gph nd find hoe ph fom he oigin o he deinion. Poof. Fo evey k-lol oigin-olivio oing lgoihm A hee i oeponding k-lol oigin-we lgoihm A whoe oing fnion mhe h of A. In ohe wod, poviding knowledge of he oigin nno hinde n oigin-olivio oing lgoihm. The el follow y Theoem 16. 6. DIRECTIONS FOR FUTURE RESEARCH Dieed Gph. The el of hi ppe onen k- lol oing on ndieed gph. Of oe, he nlogo qeion n e poed in he eing of dieed gph, mny of whih emin open. Peliminy inveigion of lol oing on dieed gph hve een mde y Chávez e l. [5] who deie 1-lol oing lgoihm fo Elein gph nd oepln gph nd Fe e l. [10] who how h evey 1-lol oing lgoihm eqie Ω(n) i of memoy on dieed gph (i.e., no ele 1-lol oing lgoihm exi). Uing Addiionl Memoy. Relxing onin nd lloing ddiionl memoy he mege ovehed o oe e infomion llow moe genel olion o he lol oing polem. Bvemn [4] how h hee exi poiion-olivio 1-lol oing lgoihm ing Θ(log n) e i h eed on ll gph. An ineeing open
qeion i o deemine whehe hee i oeponding lowe ond. Alenively, doe hee exi poiionolivio, oigin-we, pedeeo-we, k-lol oing lgoihm wih o(log n) e i fo k O(1)? In genel, n we idenify igh ond on memoy eqiemen fo deeminii k-lol oing nde vio model? Aknowledgemen The ho wih o hnk Theee Biedl who oeved h o oigin-olivio nd pedeeo-olivio (n/2)-lol oing lgoihm (Algoihm 3) idenifie hoe ph. 7. REFERENCES [1] P. Boe, A. Bodnik, S. Clon, E. D. Demine, R. Fleihe, A. López-Oiz, P. Moin, nd I. Mno. Online oing in onvex diviion. Inenionl Jonl of Compionl Geomey nd Appliion, 12(4):283 295, 2002. [2] P. Boe nd P. Moin. Online oing in inglion. SIAM Jonl on Comping, 33(4):937 951, 2004. [3] P. Boe, P. Moin, I. Sojmenović, nd J. Ui. Roing wih gneed delivey in d ho wiele newok. Wiele Newok, 7(6):609 616, 2001. [4] M. Bvemn. On d ho oing wih gneed delivey. In Poeeding of he ACM SIGACT-SIGOPS Sympoim on Piniple of Diied Comping (PODC), volme 27, pge 418, 2008. [5] E. Chávez, S. Doev, E. Knki, J. Opny, L. Sho, nd J. Ui. Roe diovey wih onn memoy in oiened pln geomei newok. Newok, 48(1):7 15, 2006. [6] S. Dohe, D. Kikpik, nd L. Nynn. On oing wih gneed delivey in hee-dimenionl d ho wiele newok. Wiele Newok, 2008. To ppe. [7] G. G. Finn. Roing nd ddeing polem in lge meopolin-le inenewok. Tehnil Repo ISI/RR-87-180, Infomion Siene Inie, 1987. [8] R. Fly nd R. Wenhofe. Rndomized 3D geogphi oing. In Poeeding of he IEEE Confeene on Compe Commniion (INFOCOM), pge 834 842, 2008. [9] M. Fe. Lol oing on oi. In Poeeding of he Confeene on Ad-Ho, Moile, nd Wiele Newok, volme 4686 of Lee Noe in Compe Siene, pge 1 14. Spinge, 2007. [10] M. Fe, E. Knki, nd J. Ui. Memoy eqiemen fo lol geomei oing nd vel in digph. In Poeeding of he Cndin Confeene on Compionl Geomey (CCCG), volme 20, 2008. [11] E. Knki, H. Singh, nd J. Ui. Comp oing on geomei newok. In Poeeding of he Cndin Confeene on Compionl Geomey (CCCG), volme 11, pge 51 54, 1999. [12] F. Khn, R. Wenhofe, nd A. Zollinge. Ad-ho newok eyond ni dik gph. In Join Wokhop on Fondion of Moile Comping, pge 69 78, 2003. [13] I. Sojmenović. Poiion ed oing in d ho newok. IEEE Commniion Mgzine, 40(7):128 134, 2002.