Computing on rings y olivious roots: unifie pproh for ifferent tsks Ginlorenzo D Angelo, Griele Di Stefno, Alfreo Nvrr, Niols Nisse, Krol Suhn To ite this version: Ginlorenzo D Angelo, Griele Di Stefno, Alfreo Nvrr, Niols Nisse, Krol Suhn. Computing on rings y olivious roots: unifie pproh for ifferent tsks. Algorithmi, Springer Verlg, 2015, 72 (4), pp.1055-1096. <hl-01168428> HAL I: hl-01168428 https://hl.inri.fr/hl-01168428 Sumitte on 25 Jun 2015 HAL is multi-isiplinry open ess rhive for the eposit n issemintion of sientifi reserh ouments, whether they re pulishe or not. The ouments my ome from tehing n reserh institutions in Frne or ro, or from puli or privte reserh enters. L rhive ouverte pluriisiplinire HAL, est estinée u épôt et à l iffusion e ouments sientifiques e niveu reherhe, puliés ou non, émnnt es étlissements enseignement et e reherhe frnçis ou étrngers, es lortoires pulis ou privés.
Computing on rings y olivious roots: unifie pproh for ifferent tsks, Ginlorenzo D Angelo 1, Griele Di Stefno 2, Alfreo Nvrr 1, Niols Nisse 3, n Krol Suhn 4 1 Diprtimento i Mtemti e Informti, Università egli Stui i Perugi, Itly. ginlorenzo.ngelo@mi.unipg.it lfreo.nvrr@unipg.it 2 Diprtimento i Ingegneri e Sienze ell Informzione e Mtemti, Università egli Stui ell Aquil, Itly. griele.istefno@univq.it 3 COATI Projet, INRIA/I3S(CNRS/UNSA), Frne. niols.nisse@inri.fr 4 Fult e Ingenierì y Cienis, Universi Aolfo Iàñez, Chile krol.suhn@ui.l Astrt. A set of utonomous roots hve to ollorte in orer to omplish ommon tsk in ring-topology where neither noes nor eges re lele (tht is, the ring is nonymous). We present unifie pproh to solve three importnt prolems: the exlusive perpetul explortion, the exlusive perpetul lering, n the gthering prolems. In the first prolem, eh root ims t visiting eh noe infinitely often while voiing tht two roots oupy sme noe (exlusivity property); in exlusive perpetul lering (lso known s serhing), the tem of roots ims t lering the whole ring infinitely often (n ege is lere if it is trverse y root or if oth its enpoints re oupie); n in the gthering prolem, ll roots must eventully oupy the sme noe. We investigte these tsks in the Look-Compute-Move moel where the roots nnot ommunite ut n pereive the positions of other roots. Eh root is equippe with visiility sensors n motion tutors, n it opertes in synhronous yles. In eh yle, root tkes snpshot of the urrent glol onfigurtion (Look), then, se on the pereive onfigurtion, tkes eision to sty ile or to move to one of its jent noes (Compute), n in the ltter se it eventully moves to this neighor (Move). Moreover, roots re enowe with very wek pilities. Nmely, they re nonymous, synhronous, olivious, uniform (exeute the sme lgorithm) n hve no ommon sense of orienttion. In this setting, we evise lgorithms tht, strting from n exlusive n rigi (i.e. perioi n symmetri) onfigurtion, solve the three ove prolems in nonymous ring-topologies. 1 Introution In the fiel of root-se omputing systems, we onsier k 1 roots ple on the noes of n input grph. Roots re equippe with visiility sensors n motion tutors, n operte in Look-Compute-Move yles in orer to hieve ommon tsk (see [19]). The Look-Compute-Move moel onsiers tht in eh yle root tkes snpshot of the urrent glol onfigurtion (Look), then, se on the pereive onfigurtion, tkes eision to sty ile or to move to one of its jent noes (Compute), n in the ltter se it moves to this neighor (Move). Cyles re performe synhronously, i.e., the time etween Look, Compute, n Move opertions is finite ut unoune, n it is eie y the versry for eh root. Hene, roots tht nnot ommunite my move se on outte pereptions. From the prtil viewpoint, the Look-Compute-Move moel fithfully esries the ehvior of some rel roots. This work hs een prtilly supporte y the Reserh Grnt 2010N5K7EB PRIN 2010 ARS TehnoMei (Algoritmi per le Reti Soili Teno-meite) from the Itlin Ministry of University n Reserh, n y Projet ECOS-SUD Chile (Ation ECOS C12E03) n the Inri Assoite Tem AlDyNet. Preliminry results onerning this work hve een presente in the 15th IEEE IPDPS Workshop on Avnes in Prllel n Distriute Computing Moels (APDCM) [15].
In the ontinuous plne, this moel is referre in the literture lso s the CORDA moel [28]. The inury of the sensors use y roots to sn the surrouning environment motivtes its isretiztion. Moreover, roots n moel softwre gents moving on omputer network. Vrious prolems hve een stuie in this setting n severl lgorithms hve een propose for prtiulr topologies suh s lines, rings, trees n gris. Here, we propose unifie pproh to solve the exlusive perpetul explortion, the exlusive perpetul lering, n the gthering prolems on rings. The relevne of the ring topology is motivte y its ompletely symmetril struture. It mens tht lgorithms for rings re more iffiult to e evise s they nnot exploit ny topologil struture, s ll noes look the sme. In ft, our lgorithms re only se on roots isposl n not on topology. We onsier minimlist vrint of the Look-Compute-Move moel whih hs very wek hypothesis. Neither noes nor eges of the grph re lele n no lol memory is ville on noes. Roots re nonymous, synhronous, uniform (i.e. they ll exeute the sme lgorithm), olivious (memoryless) n hve no ommon sense of orienttion. Aprt for the gthering prolem, guie y physil onstrints, the roots my lso stisfy the exlusivity property, oring to whih t most noe n e oupie y t most one root [2]. In ontrst to the CORDA moel in the ontinuous plne, we ssume tht moves re instntneous, n hene ny root performing Look opertion sees ll other roots t noes n not on eges. Note tht, in isrete synhronous environment this oes not onstitute limittion to the moel. In ft, n lgorithm nnot tke vntges from seeing roots on the eges s the versry n eie to perform the Look opertions only when the roots re on the noes. On the other hn, if n lgorithm tkes vntge from the ssumption tht the roots lwys oupy noes, the sme lgorithm n e pplie y ing the rule tht if root sees nother root on n ege, it just on t move (i.e. it wits until ll the roots oupy only noes). In the following, we enote suh moel s the isrete CORDA moel. The isrete CORDA moel reeive lot of ttention in the reent yers. Most of the propose lgorithms onsier tht the strting onfigurtion is exlusive, i.e., ny noe is oupie y t most one root, n rigi, i.e., symmetri n perioi. An exlusive onfigurtion is lle symmetri if the ring mits geometril xis of symmetry, iviing the ring into two speulr hlves, it is lle perioi if it is invrile uner non-trivil (i.e., non-omplete) rottions. In the following, we review the literture onerning the CORDA moel on vrious grph topologies. For the literture out the three prolems uner stuy in ifferent settings, the interest reer n refer to [1, 4, 9, 10, 20, 21, 27]. Relte work. In the prolem of grph explortion with stop [16 18], it is require tht eh noe (or eh ege) of the input grph is visite finite numer of times y t lest one root n, eventully, ll the roots hve to stop. Wheres, the exlusive perpetul grph explortion [2, 3, 7, 8] requires tht eh root visits eh noe of the grph infinitely mny times. Moreover, it s the exlusivity onstrint. In [7], first results on n-noe rings re given. In etil, the pper gives lgorithms for k = 3 n n 10, for k = n 5 (if n mo k 0), n shows tht the prolem is infesile for k = 3 n n 9, n for some symmetri onfigurtions where k n 4. Grph lering (lso lle grph serhing) hs een wiely stuie in entrlize [20] n istriute settings (e.g., [21]). The im is to mke the roots ler ll the eges of ontminte grph. An ege is lere if it is trverse y root or if oth its ens re oupie. However, lere ege is reontminte if there is pth without roots from ontminte ege to it. The stuy of grph lering in the isrete CORDA moel when the exlusivity property must e lwys 2
stisfie is introue in [6] where hrteriztion of the exlusive perpetul grph lering on tree topologies is given. As fr s we know, no results hve een propose in ring topologies for the exlusive perpetul grph lering prolem in the isrete CORDA moel. The gthering prolem onsists in moving ll the roots in the sme noe n remin there. In [11] n [14], full hrteriztion of the gthering on gri n tree topologies, respetively, without ny multipliity etetion is given. On rings, it hs een proven tht the gthering is unsolvle if the roots re not empowere y the so-lle multipliity etetion pility [26], either in its glol or lol version. In the former type, root is le to pereive whether ny noe of the grph is oupie y single root or more thn one (i.e., multipliity ours). In the ltter type, root is le to pereive the multipliity only if it is prt of it. Using the glol multipliity etetion pility, in [26], some impossiility results hve een proven. Then, severl lgorithms hve een propose for ifferent kins of exlusive initil onfigurtions in [12, 25, 26]. These ppers left open some ses whih hve een lose in [13] where unifie strtegy for ll the gtherle onfigurtions hs een provie. With lol multipliity etetion pility, n lgorithm strting from rigi onfigurtions where the numer of roots k is stritly smller thn n 2 hs een esigne in [22]. In [23], the se where k is o n stritly smller thn n 3 hs een solve. In [24], the uthors provie n lgorithm for the se where n is o, k is even, n 10 k n 5. Ppers [23] n [24] o not ssume tht the initil onfigurtion is rigi. The remining ses with lol multipliity etetion re left open n the esign of unifie lgorithm for ll the ses is still not known. Contriution. In this work, we provie unifie pproh for solving ifferent tsks in the isrete CORDA moel on ring topologies. Nmely, strting from ny rigi onfigurtion, we solve the exlusive perpetul explortion, the exlusive perpetul lering, n the gthering with lol multipliity etetion pility. Our lgorithms onsist of two phses. The first phse is ommon to ll prolems n llows k > 2 roots to hieve prtiulr rigi exlusive onfigurtion, enote elow y C, in n n-noe ring, k < n 2. The seon phse epens on the tsk. We present n lgorithm tht, strting from onfigurtion C, solves oth the exlusive perpetul explortion n the exlusive perpetul lering prolems, for ny tem of k roots in n-noe rings, n > 9, 5 k < n 3 (ut for k = 5 n n = 10). Moreover, we esign speifi lgorithm tht, strting from ny rigi onfigurtion, solves the exlusive perpetul lering prolem using n 3 roots in ny n-noe ring, n > 9. Finlly, we provie some impossiility results for the exlusive perpetul lering prolem, showing tht for 3 n 9 n k < n, or k {1,2,3,n 2,n 1} n n > 4, the prolem nnot e solve in n-noe ring with k roots. All together, we otin n lmost full hrteriztion of exlusive perpetul lering in rings, leving only open the ses (k = 4, n > 9) n (k = 5, n = 10). Conerning the gthering prolem, we esign n lgorithm tht strting from onfigurtion C solves the prolem with lol multipliity etetion for ny tem of k roots in n-noe rings, 2 < k < n 2 (note tht, if n = 2 or k n 2, no rigi onfigurtion exists). It is worth noting tht for the exlusive perpetul explortion n for the gthering prolems, esies proviing unifie pproh, we solve some open ses. Outline. In the next setion we efine the nottion use in the pper n esrie the isrete CORDA moel. In Setion 3, we propose n lgorithm to hieve the speil onfigurtion C. Exlusive perpetul lering is formlly efine n stuie in Setion 4. We note tht the lgorithms given in this setion lso solve the exlusive perpetul explortion prolem. The gthering prolem 3
x x y y z z Fig.1. A onfigurtion C in ring with 16 noes. The oupie noes re epite in grey. is onsiere in Setion 5. We then onlue the pper y Setion 6 with some possile future reserh iretions. 2 Moel n Nottions We onsier tem of k 1 roots lote in n n-noe ring, n 3. The ring is nonymous, tht is its noes n eges re unistinguishle, n no orienttion is provie. A onfigurtion onsists of the set of noes tht re oupie y root. Note tht, it oes not tke into ount the numer of roots in eh noe. A onfigurtion is si exlusive if eh noe is oupie y t most one root. For 2 k < n 2, we enote y C the onfigurtion tht onsists of k 1 onseutive oupie noes, one empty noe, one oupie noe, n t lest further two onseutive empty noes. An intervl in onfigurtion is n inlusion-mximl (possily empty) suset of onseutive empty noes, i.e., supth of empty noes tht stns etween two oupie noes. For instne, in C, there re k 2 intervls of length 0, one intervl of length 1, n one intervl of length n k 1 > 1. In onfigurtion C, view from some oupie noe r C is sequene of integers W(r) = (q 0,q 1,...,q j ),j < k, tht represents the sequene of the lengths of the intervls met when trversing the ring in one iretion (lokwise or nti-lokwise) strting from r. Ausing the nottion, for ny i j, we refer to q i s the orresponing intervl n its length. Note tht, if C is exlusive, then j = k 1 n 0 i<j q j = n k. Moreover, noe r my hve 2 istint views, epening on the iretion. Unless ifferently speifie, we refer to W(r) = (q 0,q 1,...,q j ) s the view t r tht is minimum in the lexiogrphil orer. For instne, given the onfigurtion epite in Figure 1, the possile views of the root t noe x re (2,1,3,1,2,1) n W(x) = (1,2,1,3,1,2). Let W(C) e the set of the t most 2k views (t most two views per oupie noe) in the onfigurtion C. The supermin onfigurtion view Wmin C of the onfigurtion C is the miniml view in W(C) in the lexiogrphil orer. Note tht, in Wmin C, no intervl hs length stritly smller thn q 0, n, moreover, if k < n, then q k 1 > 0. For instne, Wmin C = (q 0,...,q k 2,q k 1 ) with q 0 =... = q k 3 = 0, q k 2 = 1 n q k 1 = n k 1. For ny view W = (q 0,q 1,...,q j ) in onfigurtion C, we set W = (q 0,q j,q j 1,...,q 1 ), n W i = (q i,q (i+1) mo (j+1),...,q (i+j) mo (j+1) ) enotes the view otine y reing W strting from q i s first intervl. Note tht W(C) = {W i, W i, 0 i j}. Let I C e the set of intervls q i suh tht W i or W i re equl to Wmin C. The intervls in I C re the supermins of C. E.g., I C = 1. 4
An exlusive onfigurtion is lle symmetri if the ring mits geometril xis of symmetry, iviing the ring into two speulr hlves. An exlusive onfigurtion is lle perioi if it is invrile uner non-trivil (i.e., non-omplete) rottions. A onfigurtion whih is perioi n symmetri is lle rigi. In Figure 1, the intervls (onsisting of one noe) etween oupie noes y n z n etween y n z re the supermins of C n W C min = (1,2,1,2,1,3). I C = 2 n C is perioi n hs one unique xis of symmetry pssing through n. We now give some useful properties tht re prove in [13]. In prtiulr, Lemm 1 is use to etet possile symmetry or perioiity of onfigurtion. Property 1 ([13]). Given view W of onfigurtion C, there exists 0 < i j suh tht W = W i iff C is perioi; there exists 0 i j suh tht W = W i iff C is symmetri; C is perioi n symmetri iff there exists one unique xis of symmetry. It follows tht if onfigurtion is rigi, then eh oupie noe hs view whih is ifferent from ny other oupie noe. Lemm 1 ([13]). Given onfigurtion C, I C = 1 iff C is either rigi or it mits only one xis of symmetry pssing through the supermin; I C = 2 iff C is either perioi n symmetri with the xis not pssing through ny supermin or it is perioi with perio n 2 ; I C > 2 iff C is perioi, with perio t most n 3. We onsier isrete vrint of the CORDA moel introue in [28] where the roots hve no expliit wy of ommunite to eh other (e.g., they nnot exhnge messges). However, they re enowe with visiility sensors llowing eh root to pereive their own position in the grph n the positions of ll the other roots. However, when the exlusivity property oes not hol, n more thn one root resie t sme noe, root only pereives so lle multipliity, without the informtion of the ext numer of roots omposing it. The roots proee y yles of three phses Look-Compute-Move. In the Look-phse, root t some noe r esses snpshot of the network tht onsists of the view W(r). In the Computephse, the root eies its tion se on the informtion it reeive uring the Look-phse. Finlly, uring the Move-phse, the root exeutes its tion, i.e., it moves to neighoring noe or stys ile. The environment is fully synhronous whih, in prtiulr, mens tht the Computephse my e exeute se on n out-te view of the network. We onsier minimlist vrint of the moel, where the roots hve very wek ilities. Roots re nonymous, i.e., they o not hve ientifiers, uniform, i.e., they ll run the sme lgorithm, olivious, i.e., memoryless, n they hve no sense of iretion, i.e., they o not gree on ommon orienttion of the ring. Unless ifferently speifie, two or more roots nnot oupy the sme noe (exlusivity property). When the exlusivity property is not impose (e.g. for solving the gthering prolem), the roots hve the so lle lol multipliity etetion pility tht is, root is le to etet whether the noe where it resies is oupie y more thn one root or only y itself, ut it is not le to etet the ext numer of roots oupying the noe. Note tht this is the wekest ssumption tht hs to e me to solve the gthering sine it hs een shown tht the gthering is impossile if no multipliity etetion pility is llowe [26]. 5
In ontrst to the CORDA moel in the ontinuous plne, we ssume tht moves re instntneous, n hene ny root performing Look opertion sees ll other roots t noes n not on eges. We remrk tht, in isrete synhronous environment this oes not onstitute limittion to the moel. We ll suh moel the isrete CORDA moel. Our gol is to investigte the fesiility of severl ollortive tsks with these wek hypothesis. We ssume tht the strting onfigurtion is rigi n exlusive. 3 Rehing onfigurtion C In this setion, we propose n lgorithm, lle Align, in the isrete CORDA moel tht llows to reh onfigurtion C strting from ny exlusive rigi onfigurtion. Algorithm Align will e use in next setions to hieve the onfigurtions suitle for the exlusive perpetul explortion, lering, or gthering prolems. We first esrie the lgorithm tht llows to reh onfigurtion C. Then, we prove its orretness. 3.1 Algorithm Align The ie t the sis of Algorithm Align is to exploit the initil rigiity n exlusivity properties. In so oing, we n ensure tht one single root moves t time. The moves performe im to (lexiogrphilly) reue the unique supermin onfigurtion view of rigi onfigurtion in wy tht the otine onfigurtion is still rigi n exlusive, until onfigurtion C is hieve. By rigiity n exlusivity, the strting onfigurtion hs unique supermin intervl n eh noe hs unique supermin onfigurtion view (see Property 1 n Lemm 1). Therefore, the snpshots provie to the roots llow them to gree on ommon view (the unique minimum one) where eh root n ientify its position. This ensures tht single root will move n tht the next onfigurtion is still exlusive. Given onfigurtion C, four rules, lle reution i (C), i { 1,0,1,2}, re efine elow where, for eh rule, single root is ske to move to n empty noe. reution 0 (C) is exeute only if the supermin hs length t lest one. If the supermin hs null length, reution 1 (C) is exeute if the orresponing move oes not rete ny symmetry. Otherwise, reution 2 (C) is exeute if it oes not rete ny symmetry, n reution 1 (C) is exeute otherwise. We prove tht, strting from ny rigi onfigurtion, the move resulting from this lgorithm hieves new rigi onfigurtion. The only exeption is onfigurtion C s suh tht Wmin Cs = (0,1,1,2). In ft, from suh onfigurtion, ny single move woul generte either symmetri onfigurtion or onfigurtion C s itself. In this se, we first perform reution 1 (C s ), otining the symmetri onfigurtion C suh tht Wmin C = (0,0,2,2), then we perform reution 1 (C ) whih les to C. In ny se, in the entire lgorithm, only one root is llowe to move t one time. Moreover, we prove tht reution i (C), i {0,1,2} stritly ereses the supermin. Finlly, from some onfigurtion C, pplying reution 1 (C) my le to onfigurtion C with greter supermin onfigurtion view. However, we prove tht, in this se, the next move will reh new onfigurtion whose supermin onfigurtion view is stritly smller tht the one of C. Sine, lerly, C is the rigi onfigurtion with smllest supermin onfigurtion view, this will prove tht exeuting Algorithm Align eventully hieves C. We now formlly efine the four rules mentione ove. Let C e ny exlusive n rigi onfigurtion n let W C min = (q 0,q 1,...,q k 1 ) e its unique supermin onfigurtion view. Let l 1 e the smllest integer suh tht q l1 > 0 n let l 2 e the smllest integer suh tht q l2 > q l1. Tht is, if l 1 > 0 n l 2 > l 1 + 1, W C min = (0,...,0,q l 1,0,...,0,q l2,q l2 +1,...,q k 1 ). Let,, n 6
e the noes etween the intervls q 0 n q k 1, q l1 n q l1 +1, q l2 n q l2 +1, n q k 2 n q k 1, respetively. reution 0 (C): The root t moves to its neighor in the intervl q 0 > 0. Then, the new onfigurtion is (q 0 1,q 1,...,q k 2,q k 1 +1); reution 1 (C): The root t moves to its neighor in the intervl q l1 > 0. Then, the new onfigurtion is (q 0,q 1,...,q l1 1,q l1 1,q l1 +1 +1,...,q k 1 ); reution 2 (C): The root t moves to its neighor in the intervl q l2 > 0. Then, the new onfigurtion is (q 0,q 1,...,q l2 1,q l2 1,q l2 +1 +1,...,q k 1 ); reution 1 (C): The root t moves to its neighor in the intervl q k 1 > 0. Then, the new onfigurtion is (q 0,q 1,...,q k 2 +1,q k 1 1). The pseuo-oe of lgorithm Align is formlly given in Figure 2 n esrie lter. It is ler from the efinition of the rules tht, from n exlusive rigi onfigurtion, only one root n exeute move n tht the rehe onfigurtion is still exlusive. Note tht, in the se tht the onfigurtion is C s (i.e. Wmin Cs = (0,1,1,2)), ny reution move retes symmetri onfigurtion. In this se, we perform reution 1 whih proues the symmetri onfigurtion C suh tht Wmin C = (0,0,2,2). After this, reution 1 is gin performe n it les to C (i.e Wmin C = (0,0,1,3)). As C is symmetri, the supermin onfigurtion view n e otine y reing the ring in oth possile iretions (i.e. Wmin C = (WC min )). However root is unequivolly ientifie s the single root on the xis of symmetry n reution 1 orrespons to moving in n ritrry iretion. In ny se C is hieve. In next susetion, we formlly prove tht C is eventully hieve n tht, exept for the se of C s, the otine intermeite onfigurtions re lwys rigi. Pseuo-oe of Align. The pseuo-oe of Align is given in Figure 2 n it is performe y generi root r. It mkes use of proeure reution i whose pseuo-oe is given in Figure 3 n esrie lter. Let q min e the first intervl of Wmin C. If q min > 0, the lgorithm performs reution 0 (lines 2 3). Otherwise, it first tries to perform reution 1 y omputing the onfigurtion C tht woul e otine (line 5) n y heking whetherc is symmetri (line 6). In the negtive se, reution 1 is performe (line 7), Otherwise, the lgorithm tries to perform reution 2 (lines 9 11) n then reution 1 (lines 13 15). If the onfigurtion otine is still symmetri, then it must e C s suh tht Wmin Cs = (0,1,1,2). In this se, reution 1 is performe t line 17. The onfigurtion otine is C suh tht Wmin C = (0,0,2,2). At the next step, reution 1 is gin performe t line 7. We now esrie the pseuo-oe of reution i whih is given in Figure 3 s performe y eh root. Let W = (q 0,q 1,...,q k 1 ) e the view of C re y the root r whih performs the proeure n let Wmin C = ( q 0, q 1,..., q k 1 ). At lines 1 6, the lgorithm moves the lst root in supermin onfigurtion view, tht is it performs reution 1. If (Wmin C ) k 2 (Wmin C ) k 1, then the root hs to move if n only if W 1 = Wmin C (line 2), tht is, q 0 = q k 1, q 1 = q 0,..., q k 1 = q k 2, n it hs to move towrs q 0 (line 3) orer to reue q k 1 y enlrging q k 2. If (Wmin C ) k 2 (Wmin C ) k 1, then the root hs to move if n only if W k 2 = Wmin C (line 5) n it hs to move towrs q k 1 (line 6). Lines 7 9 implement reution 0 whih onsists in reuing the supermin intervl y moving the root on the lrgest sie of suh intervl, tht is the root whose view is the supermin one. At lines 10 15 the lgorithm performs reution i for i {1,2}. 7
Algorithm: Align Input: Rigi n exlusive onfigurtion C with view W = (q 0,q 1,...,q k 1 ) s seen from root r 1 Let q min e the first intervl of W C min; 2 if q min > 0 then 3 Apply reution 0(C,W); 4 else 5 Let C e the onfigurtion otine fter reution 1(C,W); 6 if not symmetri(c ) then 7 Apply reution 1(C,W); 8 else 9 Let C e the onfigurtion otine fter reution 2(C,W); 10 if not symmetri(c ) then 11 Apply reution 2(C,W); 12 else 13 Let C e the onfigurtion otine fter reution 1(C,W); 14 if not symmetri(c ) then 15 Apply reution 1(C,W); 16 else 17 Apply reution 1(C,W); Fig. 2. Algorithm Align. If (W C min ) l i (W C min ) l i +1, then root hs to move if n only if there exists n integer m suh tht q 0 = q m, q 1 = q m 1,..., q li = q 0, tht is if n only if W m = W C min n m = l i (line 11). In this se, suh root hs to move towrs q 0 (line 12). If (W C min ) l i (W C min ) l i +1, then root hs to move if n only if there exists n integer m suh tht q 0 = q m, q 1 = q m+1,..., q li = q k 1, tht is if n only if W m = W C min n k 1 m = l i (line 14). In this se, suh root hs to move towrs q k 1 (line 15). 3.2 Corretness We onsier rigi exlusive onfigurtion C with unique (y Lemm 1) supermin onfigurtion view W C min = (q 0,q 1,...,q k 1 ). We prove tht, when one of the four rules is pplie y Algorithm Align, the resulting onfigurtion C is still rigi. Moreover, in the se of the first three rules, the supermin onfigurtion view of C is stritly smller thn W C min. In the se of reution 1, we must onsier the next move to stritly reue the supermin onfigurtion view. Sine W C min = (q 0,q 1,...,q k 1 ) is the supermin onfigurtion view, no intervl hs length smller thn q 0 n q 1 q k 1. Therefore, if q 0 > 0 n reution 0 is pplie, the view (q 0 1,q 1,...,q k 2,q k 1 +1) is lerly the unique supermin onfigurtion view of the resulting onfigurtion C. By Lemm 1, we otin: Property 2. The onfigurtion C otine y pplying reution 0 in the rigi exlusive onfigurtion C with q 0 > 0 is rigi. Moreover, Wmin C > WC min (in lexiogrphil orer). Algorithm Align performs reution 0 until it rehes rigi exlusive onfigurtion C with supermin onfigurtion view W C min = (0,q 1,...,q k 1 ) (i.e., q 0 = 0). In this se, reution 0 nnot e pplie s otherwise there woul e ollision, tht is, multipliity is rete ut t this stge we wnt to voi it. Therefore reution 1, reution 2 or reution 1 re pplie 8
Proeure: reution i Input: Rigi n exlusive onfigurtion C with view W = (q 0,q 1,...,q k 1 ) s seen from root r 1 if i = 1 then 2 if W 1 = W C min then 3 move towrs q 0; 4 else 5 if W k 2 = (W C min) then 6 move towrs q k 1 ; 7 if i = 0 then 8 if W = W C min then 9 move towrs q 0; 10 if i {1,2} then 11 if for some m, W m = Wmin C n m = l i then 12 move towrs q 0; 13 else 14 if for some m, W m = Wmin C n k 1 m = l i then 15 move towrs q k 1 ; Fig. 3. Proeure reution. epening on the onfigurtion C. In prtiulr, reution 1 is pplie if it oes not rete ny symmetry. If q 0 = 0, y performing reution 1 we nnot otin symmetry exept for some prtiulr onfigurtions given in the next lemm. Lemm 2. Let C e rigi exlusive onfigurtion with supermin onfigurtion view W C min = (q 0,q 1,..., q k 1 ), with 2 < k < n 2 n q 0 = 0. Then, the onfigurtion C resulting from the pplition of reution 1 is perioi. Moreover, C is symmetri if n only if the following onitions hol: q i = 0, for eh i = 0,1,...,l 1 1; (1) q l1 = 1; (2) q l1 +1 +1 = q k 1 ; (3) the sequene q l1 +2,q l1 +3,...q k 2 is symmetri. (4) Proof. By rigiity of C, only one root n perform reution 1 n then C is well efine n mits view W = (q 0,q 1,...,q k 1 ) = (q 0,q 1,...,q l1 1,q l1 +1 +1,...,q k 1 ). IfC is perioi, y Property 1, there must existj > 0 suh tht(q j mo k,q (j+1) mo k,...,q (j+l 1 ) mo k ) = (q 0,q 1,...,q l1 1) = (0,...,0,q l1 1). Note tht, s q l 1 +1 = q l 1 +1+1 > 0, then j > l 1 +1. Hene, in tht se, the view (q j,...,q k 1,q 0,...,q j 1 ) is view of C stritly smller thn W C min, ontrition. Therefore, C is perioi. If equtions 1 4 hol, then C hs view W = (0,...,0,q l1 +1 +1,q l1 +2,q l1 +3,...q k 2,q l1 +1 + 1) whih is symmetri with the xis of symmetry pssing through the mile of the sequenes q 0,q 1,...,q l1 1 n q l1 +2,q l1 +3,...q k 2. We now show the only if sttement. Note tht Conition 1 is lwys stisfie y the hypothesis tht q 0 = 0 n the efinition of l 1. Let us ssume tht C is symmetri n let W = (q 0,q 1,...,q l1 1,q l1 1,q l1 +1 +1,...,q k 1 ) = (0,0,...,0,q l1 1,q l1 +1 +1,...,q k 1 ). 9
For the ske of ontrition, let us ssume tht q l1 > 1. Then, sine q l1 q k 1 n q l1 1 > 0, it is esy to hek tht W is the supermin onfigurtion view of C, n W < Wmin C. Hene, q 0 must e the unique supermin of C sine otherwise, supermin intervl ifferent from q 0 woul hve een supermin intervl in C, ontriting the ft tht Wmin C is the supermin minimum view of C. By Lemm 1, sine I C = 1 n C is symmetri, the (unique) xis of symmetry of W psses through the ege orresponing to q 0. However, sine q l1 1 < q k 1, C is not symmetri, ontrition. It follows tht q l1 = 1. In this se, the first l 1 elements of W re 0 n, s efore, this sequene is unique n the possile xis of symmetry of C psses through the mile of suh unique sequene. This implies tht C is symmetri only if q l1 +1+1 = q k 1 n tht the sequene q l1 +2,q l1 +3,...,q k 2 is symmetri. It follows tht if W C min oes not stisfy Conitions 1 4, the pplition of reution 1 results in rigi onfigurtion. Otherwise, reution 2 is pplie unless it retes symmetries. The following Lemm 3 shows tht tully, when Conitions 1 4 hol, reution 2 n rete symmetries only for some speifi onfigurtions. For the next lemmt, we nee further nottion. A pttern is the set of possile onfigurtions mitting view tht fulfills some rules efine y string of integer numers n the following symols. Let x e n integer numer: x enotes the repetition of x zero or more times; x + enotes the repetition of x one or more times; x {n} enotes the repetition of x extly n times. Given onfigurtion C we sy tht C elongs to pttern P if it hs view W tht mthes the rules of the pttern. We enote it y W P. As n exmple, the onfigurtion C with view (0,0,0,1,...,1,2,2,...,2) elongs to (0 {3},1,2 + ). Lemm 3. Let C e rigi exlusive onfigurtion with supermin onfigurtion view W C min = (q 0,q 1,..., q k 1 ), suh tht 2 < k < n 2, q 0 = 0, n Conitions 1 4 hol. Then, the onfigurtion C resulting from the pplition of reution 2 is perioi. Moreover, C is symmetri if n only if one of the following onitions hol: W C min (0,1,1+,2); (5) W C min (0{l 1},1,{0 {l 1 1},1} +,0 {l 1 2},1). (6) Proof. By rigiity of C, only one root n perform reution 2 n then C is well efine n mits view W = (q 0,...,q k 1 ) = (q 0,q 1,...,q l2 1,q l2 +1 +1,...,q k 1 ). Beuse C stisfies Conitions 1 4, it is strightforwr to see tht C is perioi. If W C min (0,1,1+,2), y performing reution 2 we otin either W = (0,1,0,3) or W = (0,1,0,2,1,2). In the first se, C is symmetri with the xis of symmetry pssing through the intervls of size 1 n 3. In the seon se, C is symmetri with the xis of symmetry pssing through the single noe of intervl q 1 n either in the mile of the sequene 1 or in the oupie noe whih seprtes the two intervls of size 2. If W C min (0{l 1},1,{0 {l 1 1},1} +,0 {l 1 2},1), y performing reution 2 we otin either W (0 {l 1},1,0 {l 1},1,0 {l 1 2},1,{0 {l 1 1},1},0 {l 1 2},1) orw (0 {l 1},1,0 {l 1},1,0 {l 1 3},1). In oth ses C is symmetri with the xis of symmetry pssing through the single noe of intervl q 1 n either in the mile of the sequene 1,{0 {l 1 1},1} in the first se, or in the mile of the sequene 0 {l 1 3} in the seon se. Let us ssume tht C is symmetri. We prove the only if sttement y se nlysis on q l1 +1. q l1 +1 > 0. Let us first ssume tht l 1 + 2 < k 1. The hypothesis q l1 +1 > 0, implies tht l 2 = l 1 + 1 n hene, W (0 {l 1},1,q l1 +1 1,q l1 +2 + 1,S,q l1 +1 +1), for some sequene S. 10
Note tht S my e empty if l 1 +2 = k 2, otherwise we set S = (S,q k 2 ) (where S my e n empty sequene). The possile xis of symmetry nnot pss through the mile of the initil sequene of 0s euse, uner the hypothesis tht q l1 +1 > 0, we hve tht q k 1 = q l1 +1 + 1 > 1 = q l1 n hene q l1 q k 1. Therefore, W ontins susequene (q j,...,q j+l 1 ) = (1,0 {l1} ) where the sequene of l 1 0s is isjoint from the initil sequene of 0s, i.e., l 1 < j n j + l 1 < k 1. If l 1 + 1 < j, then the view Wmin C hs to ontin (q j,...,q j+l 1 ) or (q j 1,...,q j+l 1 ) (the seon se ours only if j = l 1 + 2) s susequene isjoint from (q 0,...,q l1 1). This woul onstitute nother supermin, smller thn or equl to the originl one, ontriting the rigiity of C. Therefore, j = l 1 +1 n, thus, q l1 +1 = 1. By similr rguments, we show tht l 1 must e equl to 1. Therefore, W = (0,1,0,q l1 +2 + 1,S,q l1 +1 + 1), n the xis in C psses through the single noe of q 1 n the mile of sequene S whih thus is symmetri. Hene, q l1 +2 + 1 = q k 1 n, s q l1 +1 = 1 n q k 1 = q l1 +1 + 1, then q k 1 = 2 n q l1 +2 = 1. Sine sequene S is symmetri, we hve tht q l1 +2+m = q k 1 m, for ll m = 1,2,..., k 1 l 1 4 2. Moreover, y Conition 4, q l1 +1+m = q k 1 m, for ll m = 1,2,..., k 1 l 1 3 2. As q l1 +2 = 1, this implies tht (q l1 +2,q l1 +3,...q k 2 ) (1 + ). In onlusion, Wmin C (0,1,1,1+,2). If q l1 +1 > 0 n l 1 + 2 = k 1, we hve tht Wmin C (0{l1},1,q l1 +1,q l1 +1 + 1) n W (0 {l1},1,q l1 +1 1,q l1 +1+2). By similr rguments s ove, C is symmetri only if l 1 = 1 n q l1 +1 1 = 0 whih implies tht Wmin C = (0,1,1,2). Summrizing if q l1 +1 > 0 n C is symmetri, then Wmin C (0,1,1+,2). q l1 +1 = 0. In this se q k 1 = q l1 +1 + 1 = 1 n then Wmin C (0{l1},1,0,S,1), where, y Conition 4, S is symmetri sequene. We first show tht the possile xis of symmetry nnot pss through the sequene 0 {l1}. Let W = (q 0,q 1,...,q k 1 ) (0{l1},1,0,S,1), for some sequene S, n note tht q i = q i for ll i {0,1,...,k 1}\{l 2,l 2 +1}. If the xis psses through the sequene 0 {l1}, then the sequene (q l 1 +1,q l 1 +2,...,q k 2 ) = (0,S ) is symmetri. Therefore, sine q l 1 +1 = 0, then q k 2 = 0. Sine q l 2 +1 1, it follows tht q k 2 q l 2 +1, tht is j 1 l 2 +1 n then q k 2 = q k 2 = 0. Moreover, sine lso S is symmetri, q l 1 +2 = 0 n l 1 +2 l 2, whih implies tht q l 1 +2 = q l 1 +2 = 0. By iterting these rguments, we hve tht q i = q i = 0 for ll i {l 1 +1,...,k 2} whih implies tht k = n 2, ontrition. Let us ssume tht there is n xis not pssing through the sequene of 0 {l1}. This implies tht W ontins susequene (q j,...,q j+l 1 ) = (1,0 {l1} ) where the sequene of l 1 zeros is isjoint from the initil sequene of zeros, i.e., l 1 < j n j +l 1 < k 1. Three ses my rise: reution 2 retes sequene 0 {l 1+1} (i.e., there ws in W C min sequene 0{l 1} istint from the initil one). In this se, the xis of symmetry of C hs to pss through the mile of the unique sequene 0 {l 1+1}. This implies tht W (0 {l 1},1,0 {l 1+1},1,0 {l 1},S ), where S is symmetri sequene. Therefore, W C min = (0{l 1},1,0 {l 1},1,0 {l 1+1},S ) whih is ontrition to the ft tht W C min is the supermin onfigurtion view s there is sequene of l 1 +1 of zeros. reution 2 retes sequene 0 {l 1} (isjoint from the initil one). Uner this hypothesis, q l2 = 1, either W C min = (0{l 1},1,0 {l 1 1},1,1) or W C min (0{l 1},1,0 l 1 1,1,q l2 +1,S,1) where S is sequene tht my e empty. Beuse W C min stisfies Conitions 1 4, the first se my our only for l 1 = 2, n in tht se, W C min (0{l 1},1,{0 {l 1 1},1} +,0 {l 1 2},1). 11
Assume tht Wmin C (0{l1},1,0 l1 1,1,q l2 +1,S,1). In tht se, W (0 {l1},1,0 {l1},q l2 +1+ 1,S,1). We first show tht the possile xis of symmetry psses through the mile of the initil susequene (0 {l1},1,0 {l1} ). By ontrition, let us ssume tht the xis of symmetry psses through nother intervl whih implies tht there exists n inex m l 2 +1 suh tht W = W m (see Property 1). However, W < Wmin C while W m > (Wmin C ) m (euse q l2 +1 inrese) n (Wmin C ) m > Wmin C (euse WC min is the unique supermin). Therefore W < W m, ontrition. It follows tht the xis of symmetry psses through the mile of the initil susequene (0 {l1},1,0 {l1} ) n therefore, q l2 +1 = 0 n S is symmetri sequene. Summrizing, W (0 {l1},1,0 {l1},1,s,1) nwmin C (0{l1},1,0 {l1 1},1,0,S,1) = (0 {l1},1,0,0 {l1 2},1,0,S,1), where S is symmetri n, y Conition 4, (0 {l1 2},1,0,S ) is lso symmetri. By the ltter symmetry, we hve tht S ens with (0,1,0 {l1 2} ) n y the former one it follows tht S strts with (0 {l1 2},1,0). By iterting these rguments, we otin S (0 {l1 2},1,0,0 {l1 2},1,0,...,0,1,0 {l1 2},0, 1,0 {l1 2} ) = (0 {l1 2},1,{0 {l1 1},1},0 {l1 2} ) n hene, y plugging S into Wmin C, WC min (0 {l1},1,{0 {l1 1},1} +,0 {l1 2},1). reution 2 oes not rete sequene 0 {x}, for ny x l 1. In this se, the sequene (1,0 {l1} ) is ontine lso in Wmin C. Let m e the position of the first 0 of this sequene in W. Note tht, suh sequene oes not ontin neitherq l2 norq l2 +1. HeneW (0 {l1},1,0,...,q l2 1,q l2 +1+1,...,1,0 {l1},...,1). Moreover W < Wmin C while W m > (Wmin C ) m. Hene W m nnot e equl to W. It follows tht no suh xis of symmetry n exist. In onlusion, if q l1 +1 = 0 n C is symmetri, then W C min (0{l 1},1,{0 {l 1 1},1} +,0 {l 1 2},1). It follows tht we n use reution 2 in ll the onfigurtions whih stisfy Conitions 1 4 ut not Conitions 5 6. The next lemm shows tht in the remining ses we n use reution 1, still ensuring tht the resulting onfigurtion is rigi. Lemm 4. Let C e rigi exlusive onfigurtion with supermin onfigurtion view Wmin C. If either Wmin C (0,1,1,1+,2) or Wmin C (0{l1},1,{0 {l1 1},1} +,0 {l1 2},1), then, the onfigurtion C resulting from the pplition of reution 1 is rigi. Proof. By rigiity of C, only one root n perform reution 1 n then C is well efine. If W C min (0,1,1,1+,2), then C mits view W (0,1,1,1,2,1) whih is lwys rigi. Inee, there is only one intervl of size 0 n only one intervl of size 2 whih implies tht possile xis n pss only through these two intervls. However, the numer of noes etween theses two intervls on one sie is ifferent from tht on the other sie. IfW C min (0{l 1},1,{0 {l 1 1},1} +,0 {l 1 2},1), thenc mits vieww (0 {l 1+1},1,{0 {l 1 1},1} +,0 {l 1 3},1) whih is rigi. Inee, the xis of symmetry n pss only through the mile of the initil sequene 0 {l 1+1} ut the two sies of suh sequene re ifferent. By the ove lemm, it follows tht if we pply reution 1 to supermin onfigurtion view Wmin C fulfilling Conition 5 or 6, the only se in whih the otine onfigurtion n e symmetri is when Wmin C = (0,1,1,2). The orretness of Align then follows from next theorem. 12
Theorem 1. Let 2 < k < n 2 roots stning in n n-noe ring n forming rigi exlusive onfigurtion, Algorithm Align eventully termintes hieving onfigurtion C n ll intermeite onfigurtions otine re exlusive n either rigi or suh tht the supermin view is (0,0,2,2). Proof. As Align strts from rigi exlusive onfigurtion, y Lemm 1, there exists unique supermin in the initil onfigurtion. Hene extly one root moves t one time. Moreover, ll the performe movements reue n intervl whih is stritly greter thn 0 n hene the otine onfigurtion is exlusive. First, we ssume tht the initil onfigurtion is not C s. In urrent rigi exlusive onfigurtion C with unique supermin onfigurtion view Wmin C = (q 0,q 1,...,q k 1 ), we prove tht the next move is unique n result in rigi exlusive onfigurtion. If q 0 > 0, the lgorithm performs reution 0. This involves unique root n the resulting onfigurtion is rigi y Property 2. If q 0 = 0, unique root exeutes reution 1 if the resulting onfigurtion is rigi. Otherwise, y Lemm 2, Wmin C stisfies Conitions 1 4. In tht se, unique root exeutes reution 2 if the resulting onfigurtion is rigi. Otherwise, y Lemm 3, Wmin C (0,1,1+,2) or Wmin C (0 {l1},1,{0 {l1 1},1} +,0 {l1 2},1). In this se, unique root exeutes reution 1. By Lemm 4, s the initil onfigurtion is ifferent from C s, this results in rigi onfigurtion. Sine onfigurtion C is the onfigurtion with the smllest supermin onfigurtion view, it only remins to show tht eh movement reues the supermin. Hene, in the following, we show tht eh movement (or eh two movements) of Align reues the supermin. Let us enote y W = (q 0,q 1,...,q k 1 ) the view of the onfigurtion C otine fter the movement. W is the view of C t the sme noe n in the sme iretion s Wmin C. Let WC min e the supermin onfigurtion view of C. If the movement is reution 0, then q 0 = q 0 1 n hene Wmin C W < WC min. If the movement is reution i, i {1,2} then W = (q 0,q 1,...,q li 1,q li +1+1,...,q k 1 ) < Wmin C n therefore WC min W < WC min. If the movement is reution 1 it follows tht Wmin C (0,1,1,1+,2) or Wmin C (0{l1},1,{0 {l1 1},1} +,0 {l1 2},1). In the ltter se, W (0 {l1+1},1,{0 {l1 1},1} +,0 {l1 3},1) n hene Wmin C W < WC min. In the former se, W (0,1,1,1,2,1) n hene W > Wmin C. However, C is rigi n oes not stisfy Conitions 1 4 n hene the movement performe in C is reution 1. Therefore, the onfigurtion C otine fter performing reution 1 on C is W (0,0,2,1,2,1). Therefore, W < Wmin C. Let us now ssume tht the initil onfigurtion is C s. Note tht, this is the only initil onfigurtion with k = 4 n n = 8 whih is rigi n ifferent from C. From C s, reution 1 is performe n the symmetri onfigurtion C suh tht Wmin C = (0,0,2,2) is hieve. The next movement performe is gin reution 1 whih les to C (i.e. Wmin C = (0,0,1,3)) inepenently from the supermin view. In ft, even if onfigurtion C is symmetri, root is unequivolly ientifie s the single root on the xis of symmetry n reution 1 orrespons to moving in n ritrry iretion. In ny se C is hieve. 4 Clering n Exploring ring In this setion, we stuy the exlusive perpetul lering n explortion prolems in the isrete CORDA moel. In etil, we stuy the exlusive perpetul lering n note tht ny lgorithm provie to this respet lso solves the exlusive perpetul explortion n hene in the following we only refer to the lering. 13
Let us onsier n n-noe ring (n 3) n tem of 1 k n roots forming rigi n exlusive onfigurtion. In the se, 4 < k < n 3 n n > 9 (or n > 10 if k = 5), we propose n lgorithm tht mkes use of Algorithm Align presente in the previous setion. We then propose speifi lgorithm for the se k = n 3 n n > 9. On the impossiility sie, we show tht for k {1,2,3,n 2,n 1} n n > 3, or for 3 n 9 n k < n, there is no lgorithm tht solves the prolem, even if the initil onfigurtion is rigi. The ses k = 4 n (k = 5,n = 10) re left s open prolems. 4.1 Exlusive perpetul lering Given grph G where ll eges re ontminte, the grph lering prolem onsists in oorinting tem of roots to eventully ler ll eges. The roots oupy the noes of G n root n move long n ege from its urrent position to neighoring noe. An ege is lere when it is trverse y root or if oth its enpoints re simultneously oupie y some roots. However, lere ege is instntneously reontminte if there is pth from one of its enpoint to the enpoint of ontminte ege n no noe of this pth is oupie y some root. This vrint of grph lering is lssilly referre s mixe grph serhing [5]. Motivte y physil onstrints n following [6], we moreover impose the exlusivity onstrint, i.e., noe n e oupie y t most one root. A lering strtegy using 1 k n roots onsists of hoosing set of k noes, the initil positions, n sequene of moves of the roots, sliing the roots long the eges to empty neighors, tht eventully ler ll eges. For instne, there is no lering strtegy tht lers n- noe ring using one root. On the other hn, possile strtegy using two roots is the following: first ple two roots t jent noes u n v, then slie the root t u long the empty noes of the ring until it rehes the other neighor w of v. In this setion, we onsier the grph lering prolem in n-noe rings in the isrete CORDA moel. More preisely, we im t esigning lgorithms tht llow roots to ler n-noe ring strting from ny rigi exlusive onfigurtion. As our lgorithms ensure tht ll met onfigurtions re rigi n exlusive, n s the roots re olivious of the lere eges, the resulting strtegies ler the ring perpetully, i.e., the whole ring is lere infinitely often. Moreover, we stuy the exlusive perpetul explortion. Exlusive perpetul lering n exlusive perpetul explortion re not equivlent. For instne, one root lwys moving lokwise will perpetully explore ring without lering it. On the other hn, the ove lering strtegy using two roots perpetully lers ring (one root is t v n the other one lternte its move from u to w n then from w to u) ut oes not perpetully explore it sine the root t v never moves. The lgorithms we propose in the sequel oth perpetully explore n ler the rings, exlusively. 4.2 Impossiility results In this setion, we show tht for k {1, 2, 3, n 2, n 1} or for n 9, no lgorithm in the isrete CORDA moel llows to ler n n-noe ring using k roots. For these results we o not ssume tht the initil onfigurtions re rigi, tht is the impossiility results hol on stronger moel. We strt with simple result. Lemm 5. For ny n > 2 n for ny exlusive onfigurtion C, there is no lgorithm tht solves the exlusive perpetul lering prolem in n-noe ring using n 1 roots strting from C. 14
Proof. In ny onfigurtion with n 1 oupie noes, only two roots my move without violting the exlusivity property: the two roots jent to the empty noe. Sine these two roots hve the sme view of the network, whtever e the lgorithm in the isrete CORDA moel, they re fore y the versry to tke the sme eision. Either they never move n the ring nnot e lere, or oth eie to move to their empty neighor. In the ltter se, their moves n e sheule (ue to the synhroniity) suh tht they ollie, hene violting the exlusivity property. Let us onsier the se of two roots in ring with t lest three noes. Two noes u n v of n n-noe ring re lle imetrl if either n is even n there re two shortest pths etween u n v; or n is o n the length of the two pths from u to v iffer y one. We sy tht two roots oupy imetrl onfigurtion if they oupy two imetrl noes. We show tht ny lgorithm for exlusive perpetul lering on rings with two roots nees to reh onfigurtion where the two roots oupy two imetrl noes. Then, we show tht one the two roots oupy two imetrl noes they nnot rek the symmetry n hene they nnot ler the ring. The next theorem follows. Theorem 2. For ny n > 2 n for ny initil onfigurtion C, there is no lgorithm tht solves the exlusive perpetul grph lering prolem in n-noe ring using k 2 roots strting from C. Proof. Sine there is no strtegy to exlusively n perpetully ler ring using one root, it follows tht t lest two roots re neessry. We first give generl remrks on the lering of ring with two roots, inepenently of the istriute moel of omputtion. Let us ssume only two roots re oupying the noes of n-noe ring, n > 2, ll eges of whih re initilly ontminte. If the two roots never oupy jent noes, then the ring will never e lere. Therefore, onsier the first time tht suh sitution ours. Let u n v e the two neighors oupie y the roots t this step. Then, ll eges ut {u,v} re ontminte t this step. Moreover, for the ring to e eventully lere, there must e lter step suh tht, up to symmetry, the root tht ws oupying v is now t w v n the other root rehes w the neighor of w on the pth etween u n w not ontining v. In prtiulr, this proves tht, t some step of ny lering strtegy of the ring, the two roots re oupying imetrl noes. In wht follows, we show tht no lgorithm in the isrete CORDA moel n ensure the ove properties euse the symmetry nnot e roken when the roots pss through imetrl noes. This will prove the theorem. It is worth noting tht in symmetri onfigurtions, ny root llowe to move either resies on the xis of symmetry or it mits symmetri root lso llowe to move. If n lgorithm llows to move two symmetri roots, it might hppen tht while one root moves, its symmetri one hs only performe its Look phse. This results in possile pening move, tht is, the symmetri root will perform the sheule move, eventully. We onsier n versril sheuler tht lwys lterntes the moves of the two roots until it rehes imetrl onfigurtion for the first time. Tht is, it first mkes one root o its Look-Compute-Move yle, n then o the sme with the seon root, n so on. By the ove remrks, if imetrl onfigurtion is never rehe, then the ring nnot e lere. Moreover, when imetrl onfigurtion is rehe, then there re no pening moves. Now there re two ses epening on the prity of n. In wht follows, the roots lwys re in imetrl onfigurtion. Therefore, whtever e the lgorithm use, oth roots re fore to move when they look suh onfigurtion. 15
Assume first tht n is even. Then, sine there re no pening moves, the versril sheuler n synhronize the two roots suh tht fter their respetive moves the onfigurtion hs not hnge n there still re no pening moves. Inee, the two roots look n eie efore ny move n then oth of them move efore the next look. Going on this wy, the roots remin in imetrl onfigurtion n the ring nnot e ler. Now onsier the se when n is o. Consier the pth etween the two roots with n o numer of noes n let v e the noe on this pth t sme istne from oth roots. Then, the versril sheuler mkes the two roots o their Look-Compute tions n then their Move tion. Sine they re in symmetril onfigurtion (with xis pssing through v), they move symmetrilly. Doing so, fter eh move of oth roots, i.e., eh time they re looking, the noe v remins t equl istne from oth roots. Therefore, it nnot e rehe unless oth roots ollie in it. Hene, the ring nnot e lere. Let us now onsier the se of three roots in ring with t lest four noes. For ese of presenttion, we give ientifiers to the roots. Of ourse, the roots re nonymous in the sense tht they re not wre of these ientifiers n tht no lgorithm for lering the ring n mke use of them. However, the versril sheuler will use them. Hene, let us ll the three roots y r,r n r. At ny step s, we enote y ist s (x,y) the istne (i.e., the numer of onseutive eges) etween the noes oupie y roots x n y t this step (if there is no miguity, the susript will e omitte). Let C e the onfigurtion where the three roots oupy three onseutive noes. Given ny lgorithm Alg for exlusively n perpetully lering ring with 3 roots, we sy tht onfigurtion C is if, in this onfigurtion, ist(r,r ) ist(r,r ) n there exists root suh tht, if this root exeutes Alg in onfigurtion C, then the onfigurtion rehe fter its move is suh tht ist(r,r ) > ist(r,r ). In wht follows, we show tht ny lgorithm for exlusively n perpetully lering ring with three roots must lwys voi the onfigurtion C. Then, we show tht suh n lgorithm nnot voi to reh onfigurtion. Finlly, we show tht from ny onfigurtion, it is possile to sheule the three roots suh tht either they reh the onfigurtion C, or (1) eh root is sheule t lest one; (2) this rehes onfigurtion suh tht ist(r,r ) ist(r,r ) n r hs een jent to r in the mentime; n (3) if the new onfigurtion is not C, then from this new onfigurtion, Alg will reh nother onfigurtion efore r is jent to r. Sine ny lgorithm for exlusively n perpetully lering the ring must ensure tht r is infinitely mny times jent to r, this proves tht suh n lgorithm nnot exist. Theorem 3. For ny n > 3 n for ny initil onfigurtion C, there is no lgorithm tht solves the exlusive perpetul grph lering prolem in n-noe ring using 3 roots strting from C. Proof. First, if n = 4, the single noe tht is not oupie nnot e rehe without ollision. Therefore, let us ssume tht n > 4. For purpose of ontrition, let us onsier ny lgorithm Alg tht exlusively n perpetully lers the ring with 3 roots. Let us onsier the perioi infinite sequene S of moves of the roots following Alg, sujet to sheuler tht lternte the roots, i.e., first r mkes its Look-Compute- Move tions, then r, then r n so on. The gol of onsiering suh sheuler is to e le to nlyze the ehvior of Alg in some prtiulr onfigurtions (without pening moves). Then, tking use of more lever sheuler, Alg n fil. There re two ses to e onsiere. Cse 1. Let us first ssume tht S ontins the onfigurtion C where the three roots re oupying three onseutive noes. Consiering the moves just efore n just fter this onfigurtion, we 16