Backward Consistency and Sense of Direction in Advanced Distributed Systems (Extended Abstract) Paola Flocchini Universite du Quebec a Hull

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1 Bckwrd Consisency nd Sense of Direcion in Adnced Disried Sysems (Exended Asrc) Pol Flocchini Uniersie d Qeec Hll (occhini@qh.qeec.c) Alessndro Ronco CDL Informic - Uniersi di Venezi (ronco@dsi.nie.i) Nicol Snoro Crleon Uniersiy, (snoro@scs.crleon.c) Asrc: The sdies on he relionship eween lel consisency, compiliy nd complexiy ssme he exisence of locl orienion; his ssmpion is in fc he sis of he poin-o-poin model nd is relisic for sysems where commnicion link cn connec only wo eniies. Howeer, in sysems which se more dnced commnicion nd inerconnecion echnology sch s ses, opicl neworks, wireless commnicion medi, ec., nd more impornly, heerogeneos sysems (sch s inerne) which inclde ny cominion of he oe, locl orienion cn no e ssmed. In his pper we consider new ype of consisency which we shll cll ckwrd consisency nd which, nlike sense of direcion, cn exis een wiho locl orienion. Ths, nlike ll preios forms of consisency, i cn e fond (or designed) in dnced disried sysems. We sdy ckwrd consisency oh in erms of is relionship wih he rdiionl properies of locl orienion nd (wek) sense of direcion, nd wih respec o symmeries of he edge lelings nd of he nming fncions. We proe h ckwrd consisency is compionlly eqilen o sense of direcion; in oher words, i is possile o ke dnge of he compionl power of sense of direcion een in sence of locl orienion. Inrodcion A disried sysem is collecion of compionl eniies commnicing y exchnging messges. Depending on how he commnicion is chieed, dieren models exis. In he poin-o-poin model, he commnicion opology of he sysem is iewed s n edge-leled ndireced grph (G = (V; E); ) where nodes correspond o he sysem eniies, edges represen pirs of neighoring eniies (i.e., eniies which cn commnice direcly), ech node x V hs locl lel (slly clled por nmer) x(x; y) ssocied Reserch sppored in pr y F.C.A.R nd N.S.E.R.C o ech of is inciden edges (x; y), nd = f x : x V g. In he sdies of comping in disried sysems, recrring chllnging qesions re on he relionships eween locl iews nd glol consisency, nd on heir impc on disried compiliy nd commnicion complexiy. In hese sdies, locliy is he eniy leel: he locl iew of n eniy x consiss of js he leling x of is commnicion pors (or, commnicion links, inciden edges) nd, someimes, of he nming x i ses o refer o he oher eniies. A mjor gol of hese inesigion is o derie nder wh properies of nd/or i is possile o infer glol properies o he enire sysem. A prcicl im is o ndersnd if nd how hese properies cn e seflly employed, e.g., o yield more ecien disried compions. In priclr, here is lrge ody of eidence on he posiie impc on complexiy of he se of glol consisency consrins sised y lelings wih sense of direcion (e.g., see [4, 5, 5, 7, 35]). Properies of Sense of Direcion he een sdied, for exmple, in [4, 5, 6, 8, 3, 9, 8, 0,, 36, 37]; for recen srey, see [7]. All sdies in he relen lierre, wih ery few excepions, mke he sic ( ofen nsed) fndmenl ssmpion h he eniies re le o disingish mong heir inciden links, i.e., he lelings x re injecie fncions. Een if he links migh no he lel ssigned ye, i is ssmed h n eniy is cple of sch n ssignmen (e.g., [0,,, 6]). This ssmpion, slly clled locl orienion, is in fc he sis of he poin-o-poin model nd is relisic for sysems where commnicion link cn connec only wo eniies. If we re o model sysems which se more dnced commnicion nd inerconnecion echnology sch s ses, opicl neworks, wireless commnicion medi, ec., nd more impornly, heerogeneos sysems (sch s inerne) which inclde ny cominion of he oe, locl orienion cn no e ssmed. This is ecse ny direc connecion eween k eniies will correspond, ech of hose eniies, o k? edges wih he sme lel; hence, if k > (e.g., s), is no injecie. Th is, nless he sysem is poin-o-poin (i.e., ech connecion is only eween wo eniies) here re nodes h cnno disingish eween some of heir inciden edges. This implies h ll he eslished ody of resls on he relionship eween consisency, compiliy nd complexiy does no hold nd, hs, cn no e pplied oside he poin-o-poin model.

2 In oher words, ery lile is known on hese sjec for sysems wih dnced commnicion echnology. (A s ody of knowledge does indeed exis on hese sysems, no on hese opics). There re few excepions: he erly work on comping in complee grphs wih \migos" lelings [34, ], he more recen sdies on compiliy in nonymos \wireless" neworks [3, 9], he resls on leder elecion nd fncion elion in nonymos neworks wih dieren ypes of \por wreness" [7, 38]. Noe h \migos", \wireless" nd \por wre" ll denoe he (possile) sence of locl orienion. In his pper we consider new ype of consisency which we shll cll ckwrd consisency. Bckwrd consisency is srongly reled o he clssicl \forwrd" consisency implied y sense of direcion; howeer, nlike sense of direcion, i cn exis een wiho locl orienion. Ths, nlike ll preios forms of consisency, ckwrd consisency cn e fond (or designed) in dnced disried sysems. In he following, we sdy ckwrd consisency oh in erms of is relionship wih he rdiionl properies of locl orienion nd (wek) sense of direcion, nd wih respec o symmeries of he edge lelings nd of he nming fncions. In priclr, we focs on he condiions nder which lelling cn he simlneosly oh forwrd nd ckwrd consisency. We hen consider he power of ckwrd consisency, nd proe h ckwrd consisency is compionlly eqilen o sense of direcion. In oher wrds, i is possile o ke dnge of he compionl power of sense of direcion een in sence of locl orienion. This coneriniie resl somehow exends nd generlizes he resls of [, 9] on he insensiiiy of ring neworks o locl orienion. In ddiion o he heoreicl resl, we lso proide simple eecie procedre o ecienly mp proocols exploiing sense of direcion ino ones sing \ckwrd sense of direcion". In he following we consider he ndireced cse (i.e., idirecionl commnicion cpiliies); his is done only for simpliciy of exposiion, s ll resls exend o nd hold lso in he direced cse. Bsic Deniions nd Properies. Lelings nd Sense of Direcion In his secion we rs recll he sic deniions of consisency, symmery nd sense of direcion in leled grphs; mos of he erminology follows [6]. Le G = (V; E) e simple ndireced grph, E(x) denoe he se of edges inciden o node x V. Gien G = (V; E) nd se of lels, locl leling fncion of x V is ny fncion x : E(x)! which ssocies lel l o ech edge e E(x). A se = f x : x V g of locl leling fncions will e clled leling of G, nd y (G; ) we shll denoe he corresponding (edge-)leled grph. If is injecie, we sy h i is locl orienion. Noe h locl orienion is ssmed in he poin-o-poin model nd in lmos ll lierre on informie lelings, inclding he one on nleled neworks (e.g., [0,, 6]). A leling is symmeric if here exiss ijecion :! sch h for ech hx; yi E, y(hy; xi) = ( x(hx; yi)); will e clled he edge-symmery fncion. A wlk in G is seqence of edges in which he endpoin of one edge is he sring poin of he nex edge. Le P [x] denoe he se of ll he wlks sring from x V, P [x; y] he se of wlks sring from x V nd ending in y V. Le x : P [x]! + nd = f x : x V g denoe he exension of x nd, respeciely, from edges o wlks; le [x] = f x() : P [x]g, nd [x; y] = f x() : P [x; y]g. Gien n edge symmery fncion, we shll denoe y : +! + is exension o srings; i.e., for = : : : p +, () = ( p) : : : ( ), where denoes he concenion operor. Informlly, leled grph (G; ), hs sense of direcion when i is possile o ndersnd, from he lels ssocied o he edges, wheher dieren wlks from ny gien node x end in he sme node or in dieren ones. A coding fncion of (G; ) is ny fncion c wih domin +. I is sid o e consisen if 8x; y; z V, 8 P [x; y], P [x; z] c( x( )) = c( x( )) i y = z. In oher words, wlks origining from he sme node re mpped o he sme le i hey end in he sme node. We shll denoe y N (c) he co-domin of c. Deniion WSD - Wek Sense of Direcion A leled grph (G; ) hs wek sense of direcion c i c is consisen coding fncion of (G; ). Alerniely, we shll sy h c is WSD in (G; ). I is immedie o see h consisency reqires locl orienion: Lemm Le (G; ) he WSD c. Then hs locl orienion. Gien coding fncion c, decoding fncion d for c is ny fncion d : N (c)! N (c) sch h 8hx; y) E(x), P [y; z] d( x(x; y); c( y())) = c( x(x; y) y()), where is he concenion operor. Deniion SD - Sense of Direcion A leled grph (G; ) hs sense of direcion (c; d) i c is WSD nd d is decoding fncion for c. Alerniely, we shll sy h (c; d) is SD in (G; ). Le L, W, nd D denoe he se of leld grphs (G; ) wih locl orienion, wek sense of direcion, nd sense of direcion, respeciely. The following relionship holds: Lemm [5; 6] D W L The proof follows from he fc h locl orienion is necessry (oiosly) no scien for consisency [6], nd he fc h here exis leled grphs wih WSD wiho SD [5].. Bckwrd Consisency nd Sense of Direcion In his secion we inrodce he noion of ckwrd consisency. A coding fncion c of (G; ) is ckwrd consisen if 8x; y; z V, 8 P [x; z], P [y; z] c( x( )) = c( y( )), x = y. In oher words, seqences of lels of wlks ermining in he sme node re mpped o he sme le i hey sr from he sme node. Noe h he dierence eween he consisency js de- ned nd he rdiionl one is of \iewpoin". In he ler, he focs is forwrds, on he seqence of lels on wlks leing from gien node; in he former he focs is ckwrds on he seqences of lels on wlks which ermine gien node.

3 Deniion 3 WSD? Wek Bckwrd Sense of Direcion A sysem (G; ), hs wek ckwrd sense of direcion c i c is ckwrd consisen coding fncion. Alerniely, we shll sy h c is WSD? in (G; ). Gien coding fncion c, ckwrd decoding fncion for c is ny fncion : N (c)! N (c) sch h 8 P [x; y], (y; z) E(y), (c( x()); y(y; z)) = c( x() y(y; z)) where is he concenion operor. (Recll h N (c) denoes he codomin of c). We cn now dene ckwrd sense of direcion Deniion 4 SD? Bckwrd Sense of Direcion A sysem (G; ) hs ckwrd sense of direcion (c; d ) i c is WSD? nd is ckwrd decoding fncion for c. Alerniely, we shll sy h (c; d ) is SD? in (G; ). We shll denoe y W? nd D? he se of leled grphs (G; ) wih ckwrd wek sense of direcion nd ckwrd sense of direcion, respeciely. In he following, when no migiy rises, we shll omi he reference o (G; ). 3 Bckwrd Consisency 3. Asence of Locl Orienion The rs sic dierence eween forwrd nd ckwrd consisency is sriking: Locl Orienon is no implied y he deniion of SD?. Theorem (SD? ) 6 9L) (L ) 6 9SD? ) Locl orienion is neiher necessry nor scien for SD?. Proof Consider he leled grph (G; ) of Figre. I is esy o erify h here exiss oh consisen ckwrd coding nd decoding fncions in (G; ). Howeer here is no locl orienion. The fc h i is no scien is oios (see, for exmple, he grph of Figre 3 which hs locl orienion no sense of direcion ckwrd). s s Figre : SD? ) 6 9L Ths, in sysem wih SD?, nodes migh no e le o disingish mong some of heir inciden edges. As he exmple of Figre shows, in sysem wih ckwrd sense of direcion, his \lindness" cn e een complee (i.e., i exends o ll inciden edges of node) nd, in he exreme cse, ol (i.e., i occrs eery node). This phenomenon is no resriced o few specil grphs. On he conrry, eery grph G cn e leled so o he (complee nd) ol lindness, nd sill he ckwrd sense of direcion. Theorem For ny grph G here exiss leling sch h 8x V, 8y; z E(x), x((x; y)) = x((x; z)), (G; ) hs SD?. s s Proof Consider he leling dened s follows: 8(x; y), (w; z) E, x((x; y)) = w((w; z)) i x = w. Wih his leling, ll he links sring from he sme node x re leled wih he sme lel (hence, here is ol nd complee lindness). I is no dicl o show h he coding fncion c dened s follows: c( ) = for ll ; + is ckwrd consisen, nd h he fncion d(c(); ) = c() is ckwrd decoding for c; hence, (G; ) hs ckwrd sense of direcion. 3. Bckwrd Locl Orienion Sysems wih ckwrd consisency my e wiho locl orienion; howeer, hey do he propery which is he \ckwrd" nlog of locl orienion. A leling is ckwrd locl orienion if: 8x; y; z : y; z E(x); y((y; x)) 6= z((z; x)). Le L? denoe he se of leled grphs (G; ) wih ckwrd locl orienion. Firs osere h ckwrd locl orienion is (oiosly) no scien for ckwrd consisency. Theorem 3 L? ) 6 9WSD? Bckwrd locl orienion does no sce for ckwrd consisency. Proof Consider he leled grph of Figre. Sch grph hs clerly ckwrd locl orienion. By conrdicion, le s ssme h i hs ckwrd wek sense of direcion. By deniion of ckwrd coding fncion node y we wold he h c( ) = c(c ). Howeer he lels y deniion of ckwrd consisency node x, we ms he: c( ) 6= c(c ) " c """ " " " c@ Figre : L? ) 6 @ Noice h he leling in he proof of he preios heorem does no he locl orienion, hs i clly proes h (L?? W? )? L 6= ;. Howeer, in sysem wih WSD?, here is lwys ckwrd locl orienion. Theorem 4 WSD? ) L? If (G; ) hs WSD?, hen hs ckwrd locl orienion. Proof Le (G; ) e leled grph wiho ckwrd locl orienion nd wih ckwrd coding fncion c. Then here exis wo edges (y; x) nd (z; x) sch h y(y; x) = z(z; x). B his conrdics he exisence of ckwrd consisency, in fc y deniion of ckwrd consisency c( y(y; x)) 6= c( z(z; x)) Smmrizing, (ckwrd) locl orienion is necessry no scien for (ckwrd) consisency. Ths, W? L?. y

4 Two qesions nrlly rises: is he simlneos presence of oh locl orienion nd ckwrd locl orienion scien for consisency? Is i scien for ckwrd consisency? The nswer o hese qesions is negie s we proe he sronger resl h here exis leled grphs hing oh forms of locl orienion wiho eiher form of consisency. Theorem 5 (L \ L? )? (W [ W? ) 6= ; The simlneos presence of locl orienion nd ckwrd locl orienion does no imply neiher wek sense of direcion nor ckwrd wek sense of direcion. Proof Consider he leled grph (G; ) shown in Figre 3. The leling hs clerly oh locl orienion nd locl orienion ckwrd. By conrdicion, ssme (G; ) hs wek sense of direcion c. By deniion of consisen coding node x, i ms e: c( ) = c(c); howeer, node y i ms e c( ) 6= c(c) yielding conrdicion. Assme now, gin y conrdicion, h (G; ) hs ckwrd wek sense of direcion c. By deniion of ckwrd consisen coding node z, i ms e: c( ) = c(c); howeer node w,c( ) 6= c(c), which is conrdicion. x \ c \\ d c y \ e 3.3 Orhogonliy f f \ \\ g \ g c c w Figre 3: LO, LO? ) 6 9WSD In his secion we will show h he concep of ckwrd consisency is orhogonl o he one of sense of direcion; in priclr, een he presence of sense of direcion is no sf- cien o grnee he exisence of ckwrd consisency. Theorem 6 (D? L? 6= ;), (L?? D 6= ;) Sense of direcion is neiher necessry nor scien for he exisence of ckwrd locl orienion. Proof Consider ny grph wih more hn wo nodes leled wih neighoring leling (e.g., see Figre 4); ll sch leled grphs he sense of direcion [6]: he coding fncion c is c( ) = for ll ; +, nd he decoding fncion is d(; c()) = c(). On he oher hnd, (G; ) does no he locl orienion ckwrd. In oher words, SD is no scien for he exisence of LO?. The fc h i is lso no necessry follows from Theorem 5. In oher words, sense of direcion cnno grnee ckwrd consisency ecse i cnno een grnee ckwrd locl orienion. Wh if here is ckwrd locl orienion in ddiion o sense of direcion? The nswer is sill negie: Theorem 7 (D \ L? )? W? 6= ; Simlneos presence of sense of direcion nd ckwrd locl orienion is neiher necessry nor scien for he exisence of ckwrd consisen coding fncion. z S SSSS S 3 3 Figre 4: D? L? 6= ; Proof To proe h i is no necessry, consider he leled grph (G; ) of Figre 5. I is esy o erify h here exis oh consisen coding nd decoding fncion in (G; ). By conrdicion, ssme h here exiss ckwrd consisen coding fncion c in (G; ). Then, y deniion of ckwrd consisency node x, we he h c( ) = c( ). Howeer, y deniion of ckwrd consisency node y, c( ) 6= c( ); yielding he conrddicion. Bckwrd consisency clerly implies locl orienion ckwrd. Howeer, from Theorem 6 i follows h ckwrd consisency does no imply locl orienion nd, hs, does no imply sense of direcion. Which mens h he simlneos presence of sense of direcion nd ckwrd locl orienion is no scien for ckwrd consisency. x?@ P 9 Figre 5: (D \ L? )? W? 6= ; The resls of his secion indice h he presence of some ddiionl propery is necessry for one ype of consisency o imply he oher. We will show in he nex secion h edge-symmery is one sch propery. 4 Symmery nd Bckwrd Consisency The resls of he preios secion indice h he presence of some ddiionl propery is necessry for one ype of consisency o imply he oher. We will show h edgesymmery is one sch propery. Recll h leling is symmeric if here exiss ijecion :! sch h for ech (x; y) E, y((y; x)) = ( x((x; y))); will e clled he edge-symmery fncion, nd : +! + is is exension o srings. Noice h ll common lelings (e.g., \dimensionl" in hyperces, \compss" in meshes nd ori, \lef-righ" in rings, \disnce" in chordl rings, ec.) re symmeric. 3 z w

5 4. Edge-Symmery nd Bckwrd Consisency We will show h, while in generl no re (s shown y Theorem ), in sysems wih edge symmery locl orienion is necessry for ckwrd consisency. In fc, we shll show h, in sysems wih edge symmery, L = L?. Theorem 8 (ES: L, L? ) Le (G; ) e sysem wih edge symmery; hen, here is locl orienion i here is locl orienion ckwrd. Proof Le (G; ) e sysem wih edge symmery. )) By conrdicion, sppose here is no locl orienion ckwrd. Then, here ms exis wo edges (x; y), (x; z) sch h y((y; x)) = z((z; x)). Howeer, y locl orienion, we he h: x((x; y)) 6= x((x; z)) which conrdics he fc h here is edge symmery. () By conrdicion, sppose here is no locl orienion. Then, here ms exis wo edges (z; x), (z; y) sch h: z( (z; x)) = z((z; y)). Howeer, y locl orienion ckwrd we ms he x((x; z)) 6= x((y; z)) which conrdics he fc h here is edge symmery. Ths, sysem wih edge symmery eiher hs oh ypes of locl orienion or none; sill, presence of oh nd edge symmery re no scien for ckwrd consisency. Theorem 9 ES : L; L? ) 6 WSD? The simlneos presence of locl orienion nd edge symmery is no scien for he exisence of ckwrd consisency. Proof Consider he leled grph of Figre 6. The leling is coloring, i.e. he edge symmery fncion is he ideniy fncion; moreoer, i is locl orienion. By conrdicion, le s ssme h here exis ckwrd consisen coding fncion c. By deniion of ckwrd consisency node x, we he h c( ) = c(c d) nd node y, c(c d) = c(e f), which implies c( ) = c(e f). The seqence of lels corresponds o ph in P [w; z], nd f e corresponds o ph in P [; z]; hs we he h c( ) 6= c(e f) which is conrdicion. z c P PPPPP f x d g e e c f Figre 6: ES : L; L? ) 6 W? We will now show h edge symmery in sysem wih (wek) sense of direcion sces for he sysem o he (wek) ckwrd sense of direcion. w d y Theorem 0 (ES, 9(W)SD ) 9(W)SD? ) Le (G; ) e sysem wih edge symmery. If (G; ) hs WSD c, hen, here exiss WSD? c. Frhermore, if c hs consisen decoding, here exiss consisen ckwrd decoding of c. Conersely, edge symmery in sysem wih (wek) ckwrd sense of direcion sces for he sysem o he (wek) sense of direcion. Theorem (ES, 9(W)SD? ) 9(W)SD) Le (G; ) e sysem wih edge symmery. If (G; ) hs WSD? c, hen, here exiss WSD c. Frhermore, if c hs consisen ckwrd decoding, here exiss consisen decoding of c. From Theorems 0 nd, i follows h sysems wih edge symmery eiher he oh ypes of consisency or none. In oher words, in sysems wih edge symmery, W = W? nd D = D?. An immedie qesion is wheher sysems wih edgesymmery re he only ones wih sch propery. Theorem Edge symmery is no necessry for sysem o he oh forwrd nd ckwrd consisency. 4. Edge Symmery nd Biconsisency In he preios secion we he seen h ll sysems wih edge symmery eiher he oh ypes of consisency or neiher. For hose wo do he hem, he corresponding coding fncions re in generl dieren. The qesion we sk now is nder wh circmsnces single coding fncion sces; h is, when coding fncion is oh forwrd nd ckwrd consisen. We shll cll ny sch fncion iconsisen. The rs resl is h Theorem 3 (ES; WSD ) 6 WSD = WSD? ) Edge symmery is no scien for (ckwrd) consisen coding fncion o e iconsisen. Howeer, if here is nme symmery, hen ny consisen coding fncion is iconsisen. A wek sense of direcion c hs nme symmery i here exiss fncion : N (c)! N (c) sch h 8 P [x; y]: (c( x())) = c( y()): A sefl propery is he following [8]: Lemm 3 [8] Le e symmeric leling nd e he corresponding fncion. A consisen coding fncion c hs nme symmery i 8 P [s; ]; P [w; z] c( s( )) = c( w( )) ) c( ( s( ))) = c( ( w( ))): Theorem 4 (ES, NS ) WSD = WSD? ) In sysem (G; ) wih edge symmery, ny WSD wih nme symmery is lso WSD?. Proof Le c e WSD in (G; ) wih edge nd nme symmery nd le e he edge symmery fncion. By conrdicion, sppose h c is no consisen ckwrd coding fncion; h is, sppose 9x; y; z, P [x; z], P [y; z] wih x 6= y, sch h c( x( )) = c( y( )). Since here is nme symmery, y Lemm 3, we he h c( ( x( ))) = c( ( y( ))). B z( ) = ( x( )), z( ) = ( x( )) nd P [z; x], P [z; y] which conrdics he consisency of c in z.

6 L' & '$ ' $ & % % ' & & $ % % $ L? W W? D D? he fncion d? (c( ); (; )) = d(; c()) is ckwrd decoding of c if nd only if d? is ckwrd decoding of c. In oher words, (W)SD in (G; ) implies (W)SD in (G; ), nd (W)SD? in (G; ) implies (W)SD? in (G; ) Finlly osere h is symmeric; in fc, for eery (x; y) E, x(x; y) = ( y(y; x)) R. Ths, y Theorems 0 nd, (G; ) if i hs one ype of consisency, i hs oh. Figre 7: The consisency lndscpe In oher words, in sysems wih edge symmery, ny consisen coding fncion wih nme symmery is lso ckwrd consisen. We will now show h if ny sch coding fncion is decodle, hen i is lso ckwrd decodle; h is, if here is sense of direcion, here is lso ckwrd sense of direcion wih excly he sme coding fncion. Theorem 5 Le (c; d) e SD in sysem (G; ) wih edge nd nme symmery. Then, here exiss ckwrd decoding d of c; h is, (c; d ) is SD? in (G; ). 5 Consisency Lndscpe The resls of he preios secions cn e seen, nd he someimes een sed, in erms of he \consisency lndscpe" (see gre 7), i.e., of he relionship mong he ses L; W; D; L? ; W? ; D?. In his secion, we conine he nlysis of he \consisency lndscpe'. 5. Trnsformions nd Consrcions In his secion, we inrodce wo sefl operions on leled grphs. In he end, we shll derie generl resls on he srcre of he consisency lndscpe. The rs operion, clled \doling", llows o rnsform leling ino symmeric one. Hence, i llows o rnsform sysem wih only one ype of consisency ino one which hs oh. Gien (G; ), he doling of is he mpping x((x; y)) dened s follows: 8(x; y) E, x((x; y)) = ( x((x; y)), y((y; x))). Is exension from edges o wlks will e denoed y. The mos imporn propery of dole leling is h if (G; ) hs eiher form of consisency, hen (G; ) hs oh. Gien wo srings of eql lengh = ( 0; ; : : : ; k ) + nd = ( 0; ; : : : ; k ) +, le = (( 0; 0), ( ; ), : : :, ( k ; k )) ( ) + denoe heir prodc. Theorem 6 Dole Leling If (G; ) hs eiher (W)SD or (W)SD? hen (G; ) hs oh (W)SD nd (W)SD?. Proof Le c e coding fncion in (G; ). Consider he coding fncion c dened s follows: for ll ( ) + c ( ) = c(). Clerly, c is (resp. ckwrd) consisen in (G; ) if nd only if c is (resp. ckwrd) consisen in (G; ). Le c, nd hence c, e (resp. ckwrd) consisen. Gien fncion d : N (c)! N (c) dene he fncion d s follows: for ll (; ) nd ( ) +, d ((; ); c ( )) = d(; c()). Clerly, d is decoding of c if nd only if d is decoding of c. Similirly, gien d? : N (c)! N (c), The operion of doling is imporn in h i llows o rnsform leling ino symmeric one, nd hs o exend he exising consisency o inclde lso he oher ype. Anoher imporn spec of he dole leling is h i cn e consrced disriiely; i.e., sring from he locl leling x nd he gien form of coding nd (when pproprie) decoding, ech node x cn compe he leling x wih one rond of commnicion. The proof of Theorem 6 show lso how o employ he (resp. ckwrd) coding nd decoding of he originl sysem o consrc he nlogos ones of he new sysem. Howeer, i does no gie ny indicion on he nre of he coding nd decoding of he opposie ype. As we will see ler, sch nre is sefl lso for he oher rnsformion o e discssed in his secion. The prpose of he nex Lemms is o show how o consrc in he new sysems he coding nd decoding of he opposie ype. Gien sring = ( 0; ; : : : ; k ) +, le R = ( k ; k? ; : : : ; 0) denoe he reerse sring. Lemm 4 Le c e WSD in (G; ); hen c ( ) = c( R ) is WSD? in (G; ). Frhermore, if c hs decoding d, hen d (c ( ); (; )) = d(; c( R )) is ckwrd decoding of c. Proof Le x; y; z V, P [x; z], P [y; z]. Le x( ) = (( 0; 0 0), ( ; 0 ), : : :, ( h ; 0 h), nd y( ) = (( 0; 0 0); ( ; 0 ); : : :, ( k ; 0 k) Le x = y. By deniion of c we he h c ( x( )) = c( 0 h : : : 0 0) = c( z( ). Since x = y, y deniion of consisency of c we he h c( z( R )) = c( z( R ) = c( 0 k : : : 0 0). Since c( 0 k : : : 0 0) = c ( y( )), i follows h c ( x( )) = c ( y( )). Le x 6= y. By deniion of c we he h c ( x( )) = c( 0 h : : : 0 0). Since x 6= y, y deniion of consisency of c we he h c( z( R ) 6= c( z( R ). B z( R ) = ( 0 h : : : 0 0), nd z( R ) = c( 0 k : : : 0 0). Since c( 0 k : : : 0 0) = c ( y( )), i follows h c ( x( )) 6= c ( y( )). We now show h he corresponding ckwrd consisen decoding fncion is he following: 8 P [x; y], 8(y; z) E(y), le x() = ( )) nd y((y; z)) = (; ), hen: d (c ( ); (; )) = d(; c( R ) By deniion of consisen decoding fncion d(; c( R ) = c( R ). By deniion of c we he h c( R ) = c (( ) (; )) Ths, i follows h d is consisen, i.e,: d (c ( ); (; )) = c (( ) (; )). Similrily, Lemm 5 Le c e WSD? in (G; ). Then c f ( ) = c( R ) is WSD in (G; ). Frhermore, if c hs ckwrd decoding d, hen d f ((; ); c f ( )) = d(c( R ); ) is decoding of c f.

7 The second rnsformion is \reersl". Gien leling ) of grph G, he reerse leling ~ of G is oined in he following wy: 8(x; y) E, ~ x((x; y)) = y((y; x)). The reersl operion is reled o doling s follows, where c, c f, d nd d f re s dened in Lemms 4 nd 5: Lemm 6 Le c e WSD in (G; ) hen c is WSD? in (G; ). ~ Frhermore, if c hs decoding d, hen (c ; d ) is SD? in (G; ). ~ Lemm 7 Le c e WSD? in (G; ) hen c f is WSD in (G; ~ ). Frhermore, if c hs ckwrd decoding d, hen (c f ; d f ) is SD in (G; ~ ). Proof I direcly follows from Theorem 6 h he following coding fncion is consisen: 8 P [x 0], = [(x 0; x ), : : :, (x m?; x m)]: c (~ x()) = c( ~ x m?((x m?; x m)), : : :, ~ x0 ((x 0; x ))). The corresponding ckwrd decoding fncion is he following: 8 P [x 0], = ((x 0; x ), : : :, (x m?, x m)), 8(x m; y) E(x m): d (c (~ x0 ()), ~ xm((x m; y))) = d(~ xm ((x m; y)), c(~ x m? ((x m?; x m)), : : :, x0 ~ ((x 0; x ))). The oe Lemms he ery imporn conseqence: Theorem 7 (G; ) hs (W)SD? if nd only if (G; ~ ) hs (W)SD This resl implies h he srcre of he \forwrd" consisency lndscpe nd h of he ckwrd consisenc re speclr. This fc will e insrmenl o simplify he proof of some of he resls o e eslished in he following of his secion. 5. Core We now focs on he srcre of he \core" of he consisency lndscpe; i.e he se W [ W?. The rs resl we eslish is h ckwrd sense of direcion is sronger form of consisency hn wek sense of direcion. From Lemms nd 7, we he: Theorem 8 D? W? There re sysems wih ckwrd consisency where no coding fncion is ckwrd decodle. Noe h his is he speclr of Lemm. The nex qesion we consider is wheher he simlneos presence of oh consisencies sces for he exisence of eiher form of decoding. The nswer is negie, s here exis sysems wih oh WSD nd W? where no (ckwrd) coding fncion is (ckwrd) decodle. Le G w e he leled grph shown in Figre 8. Lemm 8 [5] G w W? D Theorem 9 (W \ W? )? (D [ D? ) 6= ; Exisence of oh ypes of consisency is no scien o decodiliy of eiher ype. Proof The grph G w hs wek sense of direcion nd ny coding fncion is no decodle. Sch leled grph hs edge symmery since he leling is coloring. Ths, y Theorem 0 we he h i hs lso ckwrd sense of direcion. Howeer, i is esy o erify h no ckwrd consisen coding fncion cn e ckwrd decodle. c e d m!!!!!!! d n f p p?? e! m f l q h??? l ll XXXXXXXX??? q Figre 8: G w Theorem 0 (D \ W? )? D? 6= ; Simlneos presence of sense of direcion nd ckwrd wek sense of direcion is neiher necessry nor scien for he exisence of ckwrd decoding fncion. From Theorems 7 nd 0, we he. Theorem (D? \ W)? D 6= ; 5.3 Oer Srcre In his secion we focs on he oer srcre of he consisency lndscpe: (L [ L? )? (W \ W? ). We rs inrodce wo sefl properies. Gien wo leled grphs (G ; ) nd (G ; ), he melding of (G ; ) nd (G ; ) y x V ; x V, denoed y G [x ; x ]G, is he nion of he wo grphs resriced y imposing x = x. Lemm 9 Le G nd G e erex- nd lel-disjoin leled grphs wih WSD. Then 8x V ; x V, lso G = G [x ; x ]G hs WSD. Frhermore, if G nd G he SD lso (G; ) hs SD. Theorem (W? D)? L? 6= ; Proof Consider he leled grph of Figre 9. Noice h sch grph is he melding in x of he grph of Figre 8 nd line wih wo edges (x; y) nd (y; z). Since he leled line hs riilly WSD nd since, y Lemm 8, he he grph of Figre 8 hs WSD, i follows from lemm 9 h he grph of Figre 9 hs wek sense of direcion s well. On he oher hnd, y Lemm 8, he grph of Figre 8 does no he SD, s conseqence, lso he grph of Figre 9 does no he SD. Moreoer his leled grph does no he locl orienion ckwrd, in fc x(x; y) = z(z; y). G w r s z r x y Figre 9: (W? D)? L? 6= ; c

8 Speclrly, y Theorems 7 nd, we he h: Theorem 3 (W?? D? )? L 6= ; Theorem 4 ((W? D) \ L? )? W? 6= ; Proof Consider he leled grph (G; ) of Figre 0. Noice h i he melding in x of G w (Figre 8 nd leled grph (G 0 ; 0 ). I is esy o erify h (G 0 ; 0 ) hs WSD which implies h (G; ) hs WSD. Howeer, since (G ; ) does no he SD, i follows h lso (G; ) does no he SD. Clerly here is no locl orienion ckwrd. By conrdicion, le s ssme h here is WSD? in (G; ). By deniion of ckwrd consisency node x we wold he h c() = c(); howeer y deniion of ckwrd consisency node y i ms e c( ) 6= c( ). G w Speclrly, we he x 0 7???@ y P 3 PPPPP 3 P Figre 0: ((W? D) \ L? )? W? 6= ; Theorem 5 ((W?? D? ) \ L)? W 6= ; 6 Compionl Eqilence 9 From compionl iewpoin, nonymos sysems re he les powerfl disried sysems, nd hence he idel seing o sdy he cpiliies of specic properies (e.g., [,, 8, 3, 8, 9, 30, 3, 33, 39, 40]). The compionl power of sense of direcion in nonymos sysems hs een inesiged for specic leled grphs (e.g., meshes nd ori [3], hyperces [3, 4], Cyley grphs [], ec.). The generl chrcerizion hs een gien in [8]; s n indicion of is cpiliies: mny nsolle prolems in nonymos neworks (e.g., comping he XOR in reglr nework wiho knowledge of he nework size) cn e soled if he sysem hs sense of direcion (nd wiho reking nonymiy). To exis, nd hs o e exploied, sense of direcion reqires he exisence of consrins mch sronger hn locl orienion. Wh hppens, from compionl poin of iew, if he sysem does no een grnee locl orienion? Wh cn e comped if nodes cnno een disingsh mong heir links? There re seerl sdies on compiliy in sence of locl orienion, inesiging hese qesions for specic opologies [9], for specic prolems [38], or clsses of fncions [7, 3, 38]. In ccording o expecion, hese resls indice h sence of locl orienion drmmiclly redce he compionl power in lmos ll grphs; ( noicele excepion is he ring [9]). In his secion we sdy he compionl cpiliy of he newly inrodced ckwrd consisency, which cn exis een in presence of complee nd ol lindness. We proe he nexpeced resl h ckwrd consisency hs he sme compionl cpiliies of sense of direcion. In 5 z w oher words, no only i oercomes he hndicp of no hing locl orienion, i clly empowers een olly lind sysem wih he ddiionl cpiliies of sense of direcion. 6. Compiliy A sic concep when comping on nonymos neworks is he one of iew, inrodced in [40]. The iew T (G;) () of node in leled grph (G; ) is n innie, leled, rooed ree \ downwrd loclly isomorphic" o G; i.e., sch h here exiss mp from he erices of he ree o he erices of G which mps he roo of he ree o, he children of he roo o he neighors of nd, recrsiely, he children of node o he neighors of h node. The leling of he rcs re lso downwrd presered. When no migiy rises, we shll denoe iew T (G;) () simply y T (). Le G denoe he se of leled grphs. Gien C G nd grph G, le C=G denoe he resricion of C o G; e.g., L? =G is he se of leled grphs oined y leling G wih ckwrd locl orienions. A prolem P is solle in C=G if i is solle in ll leled grphs (G; ) C=G. Gien opologicl knowledge K, prolem P is K-solle in C=G if i is solle in ll (G; ) C=G where ll he nodes re empowered wih - priori knowledge K. We rs repor wo resls on he compionl cpiliies of sense of direcion. The rs resl is h, from compionl poin of iew, here is no dierence eween wek sense of direcion nd sense of direcion. Theorem 6 [8] 8G; 8P, P is solle in W=G i P is solle in D=G The second resl shows h, wih sense of direcion, no oher knowledge is necessry. Theorem 7 [8] 8G; 8P; 8K, P is K-solle in L=G ) P is solle in W=G We will now focs on ckwrd consisency. Theorem 8 8G; 8P, P is solle in D=G i P is solle in D? =G Proof To proe his heorem we will need seerl seps. Gien wo leled grphs (G = (V; E); ) nd (G 0 = (V 0 ; E 0 ), 0 ), leled grph isomorphism is ijecion : V! V 0 which preseres edges nd edge lels; h is, h; i E, h(), ()i E 0 nd (h; i) = 0 ((), ()i). The complee opologicl knowledge (denoed y T K) is knowledge of n isomorphic imge (G; ) of (G; ) nd of he isomorphism. The impornce of T K follows from he following fc: Lemm 0 [8] 8G; 8P, P is T K-solle in L=G i P is solle in D=G In oher words, T K represens he mximm informion oinle wih SD. Hence, o proe he min heorem, we need only o show h eery node cn consrc T K wih WSD?. To do his, rs osere h: Lemm Le e n isomorphism of (G; ). If (G; ) L, hen gien (G; ) nd (x), x V cn reconsrc. Hence, if is locl orienion, o conrc he isomorphism i is scien for node o know he imge of he leled grph nd is own imge.

9 Lemm [8] Le c e consisen coding fncion of (G; ). Then 8 V, T (G; =c = (G; ) Th is, consisency wold llow ech node o consrc n isomorphic imge of (G; ) from ech iew. Since in is iew node knows is locion (i is he roo), i knows is locion in he imge. Hence, y Lemm (since consisency implies LO) consisency wold llow node o consrc he enire isomorphism s well. Therefore, o complee he proof, we he o show how o empower ech node wih locl orienion nd cosisen coding. If here is ckwrd consisency in (G; ) locl orienion wih forwrd consisency cn e esily consrced y employing he resl of Lemm 7: if (G; ) hs ckwrd consisency, hen (G; ) ~ hs locl orienion nd consisency. Since ~ is disriely consrcle (i.e., ech node x cn consrc ~ x) y simple rond of commnicion), lso T (G; ~) () is consrcle eery node. Ths, we ms only show how o consrc ech node consisen coding fncion c for (G; ), ~ gien, x, nd x. ~ This is done y osering h, y Theorem 6, if (G; ) hs ckwrd consisen coding, hen we cn consrc consisen coding c of (G; ), nd y Lemm 7, c is consisen coding for (G; ). ~ Smmrizing, if (G; ) hs ckwrd consisency, ech node cn consrc he iew T (G; ~) () nd consisen coding of (G; ) ~ which will empower i o consrc oh n isomorphisc imge of (G; ) nd of he isomorphism. This knowledge is eqilen o complee opologicl knowledge; since he clss of prolems solle wih T K is excly he sme solle wih SD lone, i follows h SD? is compionlly eqilen o SD. 6. Complexiy The resl of Theorem 8 opens he prolem of how o eeciely se he compionl power of ckwrd consisency. The pproch sggesed y he proof of Theorem 8, is o se ckwrd consisency o \simle" sense of direcion. While heoreiclly lid, his pproch sideseps he prolem since i does no exploi ckwrd consisency direcly. Frhermore, he echniqe employed in he proof is no lgorihmiclly fesile since i reqires he consrcion of he iews T (G; ~) (), sk wih formidle commnicion complexiy. While no soling he generl prolem, we will proide some complexiy relief y poining o how he simlion pproch cn e crried o simply nd ecienly, wiho consrcion of he iews. Le A e n lgorihm which soles prolem P in ny sysem wih SD. The simlion S(A) of A in sysem (G; ) in which here is SD? is performed in wo sges: ) Preprocessing: Ech processor x compes he se x() = f : x(x; y) = nd y(y; x) = g for ech of is lels x. Noice h [() = x(e(x)). ~ ) Simlion: wheneer, in lgorihm A, node x send messge m on edge leled l, in he simlion S(A) i will send messge (m; l) o he se of edges leled y he (niqe) lel p x sch h l x(p); wheneer, in lgorihm A, node performs n cion O pon recepion of messge m from link leld l, in he simlion of A, i will perform he sme cion O pon recepion of messge (m; l) from ny of he edges leled p, where p is he (niqe) lel sch h l x(p). I is esy o see h lgorihm S(A) ehes on (G; ) in exc he sme wy s he lgorihm A wold ehe on (G; ~ ) (he only dierence is h S(A) sends pir (m; lel) eery ime h A sends messge m. In fc, Theorem 9 Algorihm S(A) soles P in ny sysem wih SD? i A soles P in ny sysem wih SD. Proof Le A soles P in ny sysem wih SD. Since (G; ) hs SD?, from Theorem 7, we he h (G; ~ ) is sysem wih SD nd, hs, A soles P on (G; ~ ). B, y consrcion, lgorihm S ehes on (G; ) in exc he sme wy s he lgorihm A wold ehe on (G; ~ ) nd he proof follows. The ice-ers is nlogos. Moreoer, he nmer of messge rnsmissions of S(A) i is he sme s in he originl lgorihm A. The nmer of messge recepions oiosly depend on he crdinliy of he ses x(p). Gien (G; ), le h(g) = Mx xv;f x()g; clerly, h(g) d(g). Theorem 30 M T (S(A); G; ) = M T (A; G; ~ ) M R(S(A); G; ) h(g)m R(A; G; ~ ) In oher words, i is possile o se simlion wih some leel of eciency. Howeer, he rel sk is o deelop proocols nd echniqes which exploi ckwrd consisency direcly (no js o simle forwrd consisency). References [] D. Anglin. Locl nd glol properies in neworks of processors. In Proc. of h A.C.M. Symposim on Theory of Comping, 8{93, 98. [] H. Aiy, M. Snir, nd M.K. Wrmh. Comping on n nonymos ring. Jornl of he A.C.M., 35(4):845{ 875, 988. [3] P.W. Beme nd H.L. Bodlender. Disried comping on rnsiie neworks: he ors. In Proc. of 6h Symposim on Theoreicl Aspecs of Comper Science, 94{303, 989. [4] G.. Bochmnn, P. Flocchini, nd D. Rmzni. Disried ojecs wih sense of direcion. In s Workshop on Disried D nd Srcres, {5, Orlndo, 998. [5] P. Boldi nd S. Vign. On he complexiy of deciding sense of direcion. In Proc. of nd Colloqim on Srcrl Informion nd Commnicion Complexiy, 39{5, Olympi, 995. [6] P. Boldi nd S. Vign. On some consrcions which presere sense of direcion. In 3rd Inernionl Colloqim on Srcrl Informion nd Commnicion Complexiy, nmer 6, 47{57, Sien, 996. [7] P. Boldi nd S. Vign. Comping ecor fncions on nonymos neworks. In Proc. of 4h Colloqim on Srcrl Informion nd Commnicion Complexiy, 0{4, Ascon, 997. [8] P. Boldi nd S. Vign. Miniml sense of direcion nd decision prolems for Cyley grphs. Informion Processing Leers, 64:99{303, 997.

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