Bounds for self-stabilization in unidirectional networks

apport d e r e c h e r c h e ISSN 0249-6399 ISRN INRIA/RR--6524--FR+ENG Thème NUM INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQ UE Bounds for self-stabilization in unidirectional netw orks Samuel Bernard ? — Stéphane De vismes ◦ — Maria Gradinariu Potop-Butucaru ?, † — Sébastien T ixeuil ?, ‡ ? Univ ersité Pierre et Marie Curie - Paris 6, LIP6, France ◦ CNRS, LRI, France † INRIA project-team Regal ‡ INRIA project-team Grand Large N° 6524 May 2008 Unité de recherche INRIA Futurs Parc Club Orsay Uni versité, ZA C des V ignes, 4, rue Jacques Monod, 91893 ORSA Y Cedex (France) Téléphone : +33 1 72 92 59 00 — Télécopie : +33 1 60 19 66 08 Bounds for self-stabilization in unidirectional netw orks Sam uel Bernard ? , St´ ephane Devismes ◦ , Maria Gradinariu P otop-Butucaru ?, † , S´ ebastien Tixeuil ?, ‡ ? Univ ersit´ e Pierre et Marie Curie - P aris 6, LIP6, F rance ◦ CNRS, LRI, F rance † INRIA pro ject-team Regal ‡ INRIA pro ject-team Grand Large Th ` eme NUM — Syst ` emes n um´ eriques Pro jet Grand Large Rapp ort de rec herc he n ° 6524 — May 2008 — 21 pages Abstract: A distributed algorithm is self-stabilizing if after faults and attac ks hit the system and place it in some arbitrary global state, the systems reco v ers from this catastrophic situation without external interv ention in finite time. Unidirec- tional net works preclude many common tec hniques in self-stabilization from b eing used, suc h as preserving lo cal predicates. In this pap er, w e in vestigate the intrinsic complexit y of achieving self-stabilization in unidirectional netw orks, and fo cus on the classical vertex coloring problem. When deterministic solutions are considered, w e pro ve a low er b ound of n states p er pro cess (where n is the netw ork size) and a reco very time of at least n ( n − 1) / 2 actions in total. W e present a deterministic algorithm with matching upp er bounds that p erforms in arbitrary graphs. When probabilistic solutions are considered, we observ e that at least ∆ + 1 states p er pro cess and a reco very time of Ω( n ) actions in total are required (where ∆ denotes the maximal degree of the underlying simple undirected graph). W e presen t a probabilistically self-stabilizing algorithm that uses k states p er process, where k is a parameter of the algorithm. When k = ∆ + 1, the algorithm recov ers in exp ected O (∆ n ) actions. When k may gro w arbitrarily , the algorithm recov ers in expected O ( n ) actions in total. Th us, our algorithm can be made optimal with resp ect to space or time complexity . Key-w ords: self-stabilization, lo wer b ounds, unidirectional netw orks, coloring Bornes p our l’auto-stabilisation dans les r´ eseaux unidirectionnels R ´ esum ´ e : Un algorithme r ´ eparti est auto-stabilisant si, apr` es que des fautes et des attaques aient frapp ´ e le syst ` eme et l’aient plac ´ e dans un ´ etat quelconque, le syst` eme corrige cette situation catastrophique sans in terv en tion ext´ erieure en temps fini. Les r ´ eseaux unidirectionels empˆ ec hent de nombreuses tec hniques habituelles dans le cadre de l’auto-stabilisation d’ˆ etre utilis ´ ees, comme la pr ´ eserv ation de pr ´ edicats lo caux. Dans cet article, nous ´ ev aluons la complexit ´ e in trins ` eque de la r ´ ealisation de l’auto-stabilisation dans les r ´ eseaux unidirectionels, et nous nous concen trons sur le probl ` eme du coloriage de nœuds. Quand les solutions d ´ eterministes son t envisag ´ ees, nous prouv ons qu’il existe une b orne inf´ erieure de n ´ etats par pro cessus (o` u n est la taille du r´ eseau) et un temps minimal a v an t retour ` a la normale de n ( n − 1) / 2 action au total. Nous pr ´ esen tons un algorithme d´ eterministe dans des r ´ eseaux de topologies quelconques qui est optimal en espace et en temps. Dans le cadre de solutions probabilistes, nous observons que ∆ + 1 ´ etats par pro cessus et Ω( n ) actions au total sont requises (o ` u ∆ repr´ esen te le degr ´ e du graphe non-orien t ´ e sous-jacen t). Nous pr´ esen tons un algorithme auto-stabilisant probabi- liste qui utilise k ´ etats par pro cessus, o ` u k est un param` etre de l’algorithme. Quand k = ∆ + 1, l’algorithme retrouve un comportement correct en O (∆ n ) actions en mo y enne. Mais si k augmente de mani` ere arbitraire, l’algorithme retrouv e un com- p ortemen t correct en O ( n ) actions en moy enne. Au final, notre algorithme p eut ˆ etre rendu optimal en espace ou en temps. Mots-cl ´ es : auto-stabilisation, b ornes inf´ erieures, r´ eseaux unidirectionels, colo- riage Bounds for self-stabilization in unidir e ctional networks 3 1 In tro duction One of the most v ersatile technique to ensure forw ard reco v ery of distributed systems is that of self-stabilization [9, 10]. A distributed algorithm is self-stabilizing if after faults and attacks hit the system and place it in some arbitrary global state, the systems reco vers from this catastrophic situation without external ( e.g. h uman) in terv ention in finite time. Self-stabilization makes no hypotheses ab out the extent or the nature of the faults and attacks that may harm the system, yet ma y induce some o verhead ( e.g. memory , time) when there are no faults, compared to a classical ( i.e. non-stabilizing) solution. Computing space and time b ounds for particular problems in a self-stabilizing setting is thus crucial to ev aluate the impact of adding forw ard reco v ery properties to the system. The v ast ma jority of self-stabilizing solutions in the literature [10] considers bidi- rectional comm unications capabilities, i.e. if a pro cess u is able to send information to another pro cess v , then v is alwa ys able to send information back to u . This assumption is v alid in many cases, but can not capture the fact that asymmetric situations ma y o ccur, e.g. in wireless netw orks, it is p ossible that u is able to send information to v y et v can not send an y information back to u ( u may hav e a wider range an tenna than v ). Asymmetric situations, that we denote in the follo wing under the term of unidir e ctional net w orks, preclude many common techniques in self-stabilization from b eing used, suc h as preserving local predicates (a pro cess u ma y tak e an action that violates a predicate in v olving its outgoing neighbors without u kno wing it, since u can not get an y input from them). Related w orks Self-stabilization in bidirectional net works mak es a distinction b et ween glob al tasks ( i.e. tasks whose sp ecification forbids particular state com bi- nations of pro cesses arbitrarily far from one another in the netw ork, such as leader election) and lo c al tasks (whose sp ecifications forbid particular state com binations only for pro cesses that are at distance at most d from one another, for some pa- rameter d ). Local tasks are often considered easier in bidirectional netw orks since detecting incorrect situations requires less memory and computing p o w er [3], recov- ering can b e done lo cally [2], and Byzantine con tainmen t can b e guaranteed [19, 21]. Since a self-stabilizing algorithm may start from an y arbitrary state, low er b ounds for non-stabilizing ( a.k.a. prop erly initialized) distributed algorithms still hold for self-stabilizing ones. As a result, relatively few w orks in vestigate low er b ounds that are sp ecific to self-stabilization [4, 11, 12, 14, 17, 22]. Results related to space low er b ounds deal with global tasks ( e.g. constructing a spanning tree [11], finding a cen- RR n ° 6524 4 S. Bernar d et al. ter [11], electing a leader [4, 11], passing a tok en [12, 14, 22], etc.). [17] pro vides a time lo w er b ound for self-stabilizing token passing, still a global task. Global tasks t ypically require Ω( n ) states p er pro cess ( i.e. Ω(log( n ) bits per pro cess) and Ω( n ) time complexit y to reco ver from faults. In v estigating the p ossibilit y of self-stabilization in unidirectional netw orks was recen tly emphasized in sev eral papers [1, 5, 6, 8, 13, 15, 16] 1 . In particular, [6] sho w that in the simple case of acyclic unidirectional netw orks, nearly any recursive function can b e computed anonymously in a self-stabilizing wa y . Computing global tasks in a general top ology requires either unique iden tifiers [1, 5, 13] or distinguished pro cesses [8, 15, 16]. Observ e that all aforemen tioned w orks consider global tasks, and provide constructiv e upper bound results ( i.e algorithms), lea ving the question of matc hing low er b ounds op en. Our contribution In this pap er, we inv estigate the in trinsic complexit y of ac hiev- ing self-stabilization in unidirectional netw orks, and fo cus on the classical v ertex coloring problem, a lo cal task with sev eral known efficient self-stabilizing solutions in bidirectional net w orks [18, 20]. Deterministic and probabilistic solutions require only a num b er of states that is prop ortional to the netw ork maximum degree ∆, and the n umber of actions p er pro cess in order to recov er is O (∆) (in the case of a deterministic algorithm) or exp ected O (1) (in the case of a probabilistic one). T o satisfy the vertex coloring sp ecification in unidirectional net w orks, an algorithm m ust ensure that no tw o neighboring no des ( i.e. tw o no des u and v suc h that either u can send information to v , or v can send information to u , but not necessarily b oth) ha v e iden tical colors. The main result of this pap er is to show that solving a lo c al task in unidirec- tional net works with a deterministic algorithm is as difficult as solving a glob al task in bidirectional netw orks, while nice complexit y guaran tees can b e preserv ed with probabilistic solutions: 1. When deterministic solutions are considered, w e prov e a lo wer b ound of n states per process (where n is the net work size) and a recov ery time of at least n ( n − 1) / 2 actions in total. W e presen t a deterministic algorithm with matc hing upper b ounds that p erforms in arbitrary graphs. 2. When probabilistic solutions are considered, we observe that at least ∆ + 1 states p er pro cess and a reco v ery time of Ω( n ) actions in total are required. 1 W e do consider here the ov erwhelming n um b er of con tributions that assume a unidirectional ring shap ed net work, please refer to [10] for additional references INRIA Bounds for self-stabilization in unidir e ctional networks 5 W e present a probabilistically self-stabilizing algorithm that uses k states per pro cess, where k is a parameter of the algorithm. When k = ∆ + 1, the algorithm reco v ers in exp ected O (∆ n ) actions. When k may grow arbitrarily , the algorithm recov ers in exp ected O ( n ) actions in total. Th us, our algorithm can be made optimal with resp ect to space or time complexit y . Outline The remaining of the pap er is organized as follows: Section 2 presen ts the programming model and problem sp ecification, Section 3 provides imp ossibilit y results and low er b ounds for our problem, while Sections 4 and 5 present match- ing upp er b ounds (in the deterministic case) and asymptotically matc hing upp er b ounds (in the probabilistic case). Section 6 gives some concluding remarks and op en questions. 2 Mo del Program mo del A program consists of a set V of n pro cesses. A pro cess main- tains a set of v ariables that it can read or update, that define its state . Each v ariable ranges o ver a fixed domain of v alues. W e use small case letters to denote singleton v ariables, and capital ones to denote sets. A pro cess contains a set of c onstants that it can read but not up date. A binary relation E is defined ov er distinct pro cesses suc h that ( i, j ) ∈ E if and only if j can read the v ariables main tained by i ; i is a pr e de c essor of j , and j is a suc c essor of i . The set of predecessors (resp. successors) of i is denoted by P.i (resp. S.i ), and the union of predecessors and successors of i is denoted by N .i , the neighb ors of i . In some case, we are interested in the iterated notions of those sets, e.g. S.i 0 = i , S.i 1 = S.i , . . . , S.i k = ∪ j ∈ S.i S.j k − 1 . The v al- ues δ in .i , δ out .i , and δ .i denote resp ectiv ely | P .i | , | S.i | , and | N .i | ; ∆ in , ∆ out , and ∆ denote the maximum p ossible v alues of δ in .i , δ out .i , and δ.i o v er all pro cesses in V . An action has the form h name i : h g uar d i − → h command i . A guar d is a Bo olean predicate o v er the v ariables of the pro cess and its comm unication neighbors. A c om- mand is a sequence of statements assigning new v alues to the v ariables of the process. W e refer to a v ariable v and an action a of pro cess i as v .i and a.i resp ectively . A p ar ameter is used to define a set of actions as one parameterized action. A c onfigur ation of the program is the assignment of a v alue to every v ariable of each pro cess from its corresp onding domain. Eac h process con tains a set of actions. An action is enable d in some configuration if its guard is true in this configuration. A c omputation is a maximal sequence of configurations suc h that for eac h configuration γ i , the next configuration γ i +1 is obtained by executing the RR n ° 6524 6 S. Bernar d et al. command of at least one action that is enabled in γ i . Maximality of a computation means that the computation is infinite or it terminates in a configuration where none of the actions are enabled. A program that only has terminating computations is silent . A sche duler is a predicate on computations, that is, a sc heduler is a set of p ossible computations, suc h that every computation in this set satisfies the sc heduler predicate. W e distinguish t w o particular schedulers in the sequel of the pap er: the distribute d scheduler corresp onds to predicate true (that is, all computations are allo w ed); in con trast, the lo c al ly c entr al scheduler implies that in an y configuration b elonging to a computation satisfying the sc heduler, no t w o enabled actions are executed sim ultaneously on neigh b oring pro cesses. A configuration c onforms to a predicate if this predicate is true in this config- uration; otherwise the configuration violates the predicate. By this definition every configuration conforms to predicate true and none conforms to false . Let R and S b e predicates ov er the configurations of the program. Predicate R is close d with resp ect to the program actions if every configuration of the computation that starts in a configuration conforming to R also conforms to R . Predicate R c onver ges to S if R and S are closed and an y computation starting from a configuration conform- ing to R contains a configuration conforming to S . The program deterministic al ly stabilizes to R if and only if true conv erges to R . The program pr ob abilistic al ly stabilizes to R if and only if true conv erges to R with probability 1. Problem sp ecification Consider a set of colors ranging from 0 to k − 1, for some in teger k ≥ 1. Each pro cess i defines a function c olor .i that tak es as input the states of i and its predecessors, and outputs a v alue in { 0 , . . . , k − 1 } . The unidir e ctional vertex c oloring predicate is satisfied if and only if for every ( i, j ) ∈ E , c olor .i 6 = c olor .j . 3 Imp ossibilit y results and lo w er b ounds General bounds W e first observ e tw o lo wer b ounds that hold for any kind of silen t program that is self-stabilizing or probabilistically self-stabilizing for the uni- directional coloring sp ecification: 1. The minimal numb er of states p er pr o c ess is ∆ + 1. Consider a bidirectional clique net work (that is (∆ + 1)-sized), and a terminal configuration of the program. Suppose that only ∆ states are used, then at least tw o pro cesses INRIA Bounds for self-stabilization in unidir e ctional networks 7 i and j hav e the same state, and ha ve the same view of their predecessors. As a result c olor .i = c olor .j , and i and j b eing neighbors, the unidirectional coloring predicate do es not hold in this terminal configuration. 2. The minimal numb er of moves over al l is Ω( n ). Consider a unidirectional c hain of processes whic h are all initially in the same state. F or ev ery process but one, the color is identical to that of its predecessor. Since a c hange of state ma y only resolv e t w o conflicts (that of the mo ving no de and that of its successor), a n um b er of ov erall mov es at least equal to b n/ 2 c is required, th us Ω( n ) mov es. Deterministic b ounds The rest of the section is dedicated to deterministic im- p ossibilit y results and low er b ounds. Theorem 1 (presented b elow) shows that if unconstrained sc hedules ( i.e. the sc heduler is distributed) are allow ed, some initial symmetric configurations can not b e broken afterwards, making the unidirectional coloring problem imp ossible to solve by a deterministic algorithm. This justifies the later assumption of a lo cally cen tral scheduler in Section 4, i.e. a scheduler that nev er sc hedules for execution t w o neigh b oring activ atable pro cesses simultane ously . Lemma 1 L et an unidir e ctional network { p 0 , p 1 , . . . , p n − 1 } of size n . Consider ev- ery no de exe cutes a uniform deterministic self-stabilizing uniform c oloring algorithm. Whenever ther e exists i such that s.p i = s.p ( i − 1) mod n , p i is activatable and if acti- vate d would change its state to s 0 .p i with s 0 .p i 6 = s.p i . Pro of: W e first show that p i is activ atable. Assume the contrary , and consider a uniform cycle of n no des that are all in the same state. If p i is not activ atable, none of the remaining pro cesses is activ atable either. Hence, the configuration is terminal. Now, a pro cess p i ma y only read its o wn state and that of its predecessor in the cycle, so the color c olor .p i is uniquely determined b y these t w o v alues only . Moreo v er, the c olor .p i is the same as c olor .p ( i − 1) mod n . So, the configuration is terminal and tw o neighboring pro cesses hav e the same color. This con tradicts the fact that the algorithm is a deterministic self-stabilizing unidirectional coloring one. Then we show that p i , if activ ated mov es to a different state s 0 .p i . Assume p i mo v es to the same state s.p i , then if the starting configuration is such that all no des ha v e the same state, then no no de is able to c hange its state, the algorithm b eing uniform. Since this configuration can not b e a coloring, the system never changes the global configuration and th us is not self-stabilizing. 2 Theorem 1 Ther e exists no uniform deterministic self-stabilizing c oloring algo- rithm that c an run on any unidir e ctional gr aph under a distribute d sche duler. RR n ° 6524 8 S. Bernar d et al. Pro of: Assume the con trary . Consider a unidirectional cycle { p 0 , p 1 , . . . , p n − 1 } of size n . Assume that in the initial state all no des are in the same state s (see Figure 1.(a)). By Lemma 1, all no des are activ atable, and if a no de is activ ated b y the sc heduler, it mo ves to a differen t state s 0 . Consider the sync hronous sc heduler that, at eac h step, activ ates all no des. Then, after one scheduler activ ation, all nodes ha v e state s 0 (see Figure 1.(b)). After another activ ation, all no des mov e to state s 00 , etc. In this infinite execution, every configuration has all nodes with the same state and th us, the coloration problem is not solved. As a result, the algorithm can not be self-stabilizing. 2 Notice that the result of Theorem 1 holds ev en if the program is not required to b e silen t or if participating pro cesses ha v e infinite n umber of states. F rom no w on, we assume the scheduler is lo cally central. W e demonstrate that a uniform silent deterministic self-stabilizing algorithm for the unidirectional coloring problem m ust use at least n states p er pro cess in general netw orks. The pro of is b y exhibiting a particular family of netw orks (namely , n -sized cycles) in which the b ound is reac hed by any such algorithm ev en assuming a lo cally central scheduler. Algorithm 1 A uniform deterministic coloring algorithm for unidirectional rings pro cess i const k : integer p.i : predecessor of i v ar c.i : color of node i action c.i = c.p.i → c.i := c.i + 1mo d k Lemma 2 Consider a unidir e ctional cycle { p 0 , p 1 , . . . , p n − 1 } of size n . Consider a sche duler that only activates a no de p i when s.p i = s.p ( i − 1) mod n . Assume every no de exe cutes a uniform deterministic self-stabilizing unidir e ctional c oloring algo- rithm that uses a finite numb er of states K . Ther e exists an initial c onfigur ation such that the state se quenc e starting fr om this c onfigur ation is isomorphic to that of A lgorithm 1, for some p ar ameter k ≤ K . INRIA Bounds for self-stabilization in unidir e ctional networks 9 Pro of: Consider a unidirectional cycle { p 0 , p 1 , . . . , p n − 1 } of size n . Assume a no de p i is activ ated only when its state s.p i is equal to s.p ( i − 1) mod n , the state of the predecessor of p i in the cycle. Then, the transition function of node p i is solely based on the state s.p i . Let s (0) , s (1) , . . . the sequence of states returned by the transition function of pro cess p i executing the self-stabilizing coloring algorithm started in an arbitrary state s (0). Note that (i) the nu mber of states is finite, (ii) a pro cess with the same state as its predecessor is alwa ys activ atable (Lemma 1), and (iii) the proto col is deterministic. Then, the shap e of the transition function of p i is as depicted in Figure 2.(a). That is, there exist i and l ( i < l ) such that s ( i ) = s ( l ) and l − i ≤ K . Since the proto col is self-stabilizing, it may b e started from an y arbitrary state, and in particular from state s ( i ) (in Figure 2.(a)). Let denote s ( j ) b y j − i , ∀ i ≤ j < l . The transition function of p i is isomorphic to that of the same pro cess executing Algorithm 1 (given in Figure 2(b)) and assuming k = l − i . 2 Theorem 2 A silent uniform deterministic self-stabilizing pr oto c ol for unidir e c- tional c oloring r e quir es at le ast n states p er pr o c ess in a n -size d network (with n ≥ 2 ). Pro of: Assume there exists a silen t uniform deterministic self-stabilizing proto col for coloring that requires less than n states for a particular no de. Since the proto col is uniform, every no de m ust use k < n states. Consider a unidirectional cycle { p 0 , p 1 , . . . , p n − 1 } of size n . In what follows, we consider executions of the protocol in whic h the scheduler only activ ates no des that ha v e the same state as their predecessor. By Lemma 2, the transition function of ev ery no de is isomorphic to that of Algorithm 1, so w e assume all no des execute Algorithm 1 with k < n . In the following, w e consider k = n − 1 but the pro of is easily exp endable to any k < n b y putting the n − k + 1 last pro cessors in the same state. Now consider the unidirectional ring presented in Figure 3.(a). The scheduler only activ ates the single no de with the same state 0 as its paren t, and reac h the configuration presented in Figure 3.(b). The scheduler may no w activ ate the single no de with the same state 1 as its parent and reac h the configuration presen ted in Figure 3.(c). W e rep eat the argumen t and reac h the configuration presented in Figure 3.(d). This configuration is symmetric to that of the configuration presented in Figure 3.(a), so the pro cess can repeat infinitely often. As a result, the proto col is not silen t. 2 W e now address the question of time low er b ounds for deterministic self-stabilizing programs for the unidirectional coloring problem. Theorem 3 A silent uniform deterministic self-stabilizing pr oto c ol for unidir e c- tional c oloring c onver ges in at le ast n ( n − 1) 2 steps in gener al gr aphs. RR n ° 6524 10 S. Bernar d et al. Pro of: Consider a chain top ology , and assume that pro cessors are ordered from the sink p 1 to the source p n . Assume all processors are initially in the same state (a self-stabilizing program ma y start from any arbitrary configuration). W e now consider a loc ally central sc heduler that activ ates no des according to the schedule presen ted in Sc hedule 1. Sc hedule 1 Our n ( n − 1) 2 -steps sc heduling in n -sized c hains v ar i , j : in teger sc heduler for j from n − 1 to 1 for i from 1 to j activ ate p i Sc hedule 1 selects a single pro cess at a time for execution, th us it satisfies the lo cally cen tral sc heduler prop erty . In addition, it only selects for execution a process that has the same state as its predecessor (and th us activ atable b y Lemma 1): if p 1 to p k ha v e the same state s then p 1 to p k − 1 are activ atable and if they are activ ated in ascending order, p 1 to p k − 1 will mov e to the same “next” state s 0 (see Figure 2.(a)). So every process activ ation leads to an effectiv e mov e and the total n um b er of activ ations is P n − 1 i =1 i = n ( n − 1) 2 . Finally , all executions of the proto col m ust be terminating (it is silen t), and from an initial configuration where all pro cesses ha ve the same state, a lo cally central sc hedule (like Schedule 1) may leads to n ( n − 1) / 2 steps at least b efore termination. Hence the result. 2 4 Self-stabilizing deterministic unidirectional coloring In this section we prop ose a time and space optimal silen t self-stabilizing determin- istic algorithm for unidirectional coloring. The algorithm is referred thereafter as Algorithm 2 and p erforms under the lo cally central sc heduler. The algorithm can b e informally describ ed as follo ws: eac h pro cess i has an integer v ariable c.i (that ranges from 0 to k − 1, where k is a parameter of the algorithm) that denotes its color; whenever a no de has the same color as one of its predecessors, it c hanges its color to the next av ailable color (using the classical total order on integers). Here, the c olor .i function simply returns the color v ariable c.i of i . INRIA Bounds for self-stabilization in unidir e ctional networks 11 Algorithm 2 A uniform deterministic coloring algorithm for general unidirectional net w orks pro cess i const k : integer P .i : set of predecessors of i parameter p : no de in P.i v ar c.i : color of node i action p ∈ P .i , c.i = c.p → do p ∈ P .i , c.i = c.p → c.i := c.i + 1mo d k o d A configuration is le gitimate if, for ev ery pro cess i , and for every predecessor p ∈ P .i , c.i 6 = c.p.i . Ob viously , a legitimate configuration satisfies the unidirec- tional coloring predicate (assuming c olor .i return c.i ) and is terminal (all guarded commands are disabled). There remains to sho w how fast the algorithm attains a legitimate configuration in the w orst case for every p ossible lo cally central schedule. Theorem 4 A lgorithm 2 is a (state-optimal) uniform silent deterministic self-stabilizing pr oto c ol for c oloring no des in unidir e ctional gener al networks of size n (when k = n ), assuming a lo c al ly c entr al sche duler and c onver ges in n ( n − 1) 2 steps to a le gitimate c onfigur ation. Pro of: Assume Algorithm 2 starts in an arbitrary initial configuration c . W e no w consider the table that lists, for ev ery possible color (in the set { 0 , . . . , n − 1 } since w e assume k = n ), the pro cesses that curren tly ha v e this color. An example of such a table is presen ted as T able 1, where pro cesses P 3 and P 2 ha v e color 0, process P n − 2 has color n − 1, etc. This table is denoted in the sequel as the c olor table . According to Algorithm 2 the ev olution of the color table follo ws tw o rules: 1. A cell con taining one pro cess can not b ecome empt y . That is, a pro cess having a color not used by an y other pro cess in the system can not b e activ ated. In RR n ° 6524 12 S. Bernar d et al. Pro cessors P 3 , P 2 P n − 1 , P 1 ... P n − 2 States 0 1 2 3 ... n − 1 T able 1: An example of color table our algorithm, this is due to the fact that pro cesses are activ atable only if they share their color with their predecessor. 2. A pro cess only mov es to the righ t (in a cyclic manner) and can not jump ov er an empty cell. Indeed, when activ ated, a pro cess chooses the first “next” (in the sense of the usual total order on in tegers) unconflictual color hence the pro cesses alwa ys mov e to the righ t. A process ma y mov e by sev eral positions, but nev er skips a free position (this w ould mean that a pro cess does not c ho ose the ”next” color although this color is not conflicting with any other pro cess and th us not with the process predecessors). Since there are n cells and n processes, ev ery pro cess could b e placed in a differen t cell if necessary . Since a pro cess can not jump ov er an empty cell, after n − 1 mov es, a pro cess is sure to find a free cell. In fact, the n umber of mov es a pro cess may ha v e to p erform to reach a free cell dep ends on the n um b er of free cells. With k free cells, there are at most n − k consecutiv e non-empty cells that could p otentially pro v oke further conflicts. A pro cess, in order to reach a free cell has to p erform at most n − k mov es. Once this pro cess o ccupies a free cell, the n um b er of free cells decreases to k − 1. Starting with n − 1 free cell (every pro cess has the same color), and finishing with 1, at most 1 + 2 + ... + ( n − 1) = n ( n − 1) 2 steps are needed to hav e ev ery pro cess in a free cell or with a non-conflicting color ( i.e. different from that of its predecessors) and th us reac h a legitimate configuration. 2 5 Probabilistic self-stabilizing unidirectional coloring In Section 3, we observed that there exist low ers b ounds ev en for probabilistic ap- proac hes to the unidirectional coloring problem. The space lo wer b ound is ∆ + 1 and the time lo w er b ound is n (where ∆ and n are the degree and the size of the underlying simple undirected graph, resp ectively). The algorithm presented as Algorithm 3 can be informally describ ed as follows. If a pro cess has the same color as one of its predecessors then it chooses a new color in the set of a v ailable colors ( i.e. the set of colors that are not already used b y any INRIA Bounds for self-stabilization in unidir e ctional networks 13 of its predecessors). The colors are c hosen in a set of size k , where k is a parameter of the algorithm. In the follo wing, w e show that Algorithm 3 is probabilistically self-stabilizing for the unidirectional coloring problem if k > ∆. T o reach that goal w e pro ceed in t w o steps: first we show that any terminal configuration satisfies the unidirectional coloring predicate (Lemma 3); secondly , we sho w that the exp ected n um b er of steps to reac h a terminal configuration starting from an arbitrary one is b ounded (Lemma 7). Algorithm 3 A uniform probabilistic coloring algorithm for general unidirectional net w orks pro cess i const k : integer P .i : set of predecessors of i C.i : set of colors of nodes in P .i parameter p : no de in P.i v ar c.i : color of node i action p ∈ P .i , c.i = c.p → c.i := random ( { 0 , . . . , k − 1 } \ C .i ) Lemma 3 A ny terminal c onfigur ation satisfies the unidir e ctional c oloring pr e dic ate. Pro of: In a terminal configuration, every pro cess i satisfies ∀ j ∈ P.i, c.i 6 = c.j and ∀ j ∈ S.i, c.i 6 = c.j . Hence, in a terminal configuration, every process i has a color that is different from those of its neighbors, which prov es the theorem. 2 Definition 1 (Conflict) L et p b e a pr o c ess and γ a c onfigur ation. The tuple ( p , γ ) is c al le d a conflict if and only if ther e exists q ∈ P .p (the pr e de c essors of p ) such that c.q = c.p in γ . RR n ° 6524 14 S. Bernar d et al. Lemma 4 Assume k > ∆ . L et ( p , γ ) b e a c onflict. The exp e cte d numb er of c onflicts cr e ate d by the exe cution of one step of p in or der to r esolve ( p , γ ) is: δ.p − δ in .p k − δ in .p (1) Pro of: When a pro cess p executes Action A from γ , it c ho oses a new color in a set of at least k − δ in .p colors. That is, there are k colors and it can not choose a color c hosen by one of its predecessors, therefore at most δ in .p colors are remov ed from the set of p ossible choices. F or each q ∈ S.p ∧ q / ∈ P.p , p and q are in conflict if and only if, p chooses the color of q . Notice that p has 1 ( k − δ in .p ) c hance to create a new conflict. Since the n um b er of successors of p not in the set of predecessors of p , ] { q ∈ S.p ∧ q / ∈ P .p } , is δ .p − δ in .p , the exp ected n umber of created conflicts is δ.p − δ in .p k − δ in .p . 2 Lemma 5 L et ( p, γ ) b e a c onflict. The exp e cte d numb er of c onflicts cr e ate d by the exe cution of one step of p in or der to r esolve ( p , γ ) is less than or e qual to: M = ∆ − 1 k − 1 , k > ∆ (2) Pro of: Observ e that ∀ p ∈ V , ∆ ≥ δ.p , therefore ∀ p ∈ V , δ.p − δ in .p k − δ in .p ≤ ∆ − δ in .p k − δ in .p . In order to find an upp er b ound for this v alue, let f : p ∈ V , ∆ < k , δ in .p ∈ [0 , ∆] , δ in .p 7→ ∆ − δ in .p k − δ in .p . Its deriv ative exists and is f 0 ( δ in .p ) = ∆ − k ( k − δ in .p ) 2 . By h yp othesis, k > ∆, so f 0 ( δ in .p ) < 0 and f is decreasing. Therefore, f ( δ in .p ) is maximum when δ in .p = 0 but for this v alue ( p , γ ) can not be a conflict. Therefore, δ in .p ≥ 1 and f is maximum for δ in .p = 1 whic h leads to δ.p − δ in .p k − δ in .p ≤ ∆ − δ in .p k − δ in .p ≤ ∆ − 1 k − 1 . 2 Lemma 6 L et ( p, γ ) b e a c onflict. The exp e cte d numb er of step cr e ate d in or der to r esolve this c onflict is less than: k − 1 k − ∆ , k > ∆ (3) Pro of: F rom Lemma 5, the exp ected num b er of conflicts created b y one step of p is less than M = ∆ − 1 k − 1 . Then, the pro cesses in S.p who received the created conflicts pro duce at most M new conflicts eac h since there are at most M exp ected such pro- cesses. They will p erform M steps and create at most M 2 =  ∆ − 1 k − 1  2 new conflicts. Then the M 2 pro cesses in S .p 2 (the successors at distance t w o from p ), after M 2 step INRIA Bounds for self-stabilization in unidir e ctional networks 15 (one for eac h) will pro duce M 3 new conflicts. By recurrence, at most M i exp ected steps will b e executed and M i +1 new conflicts will b e created by the pro cesses in S.p i (the successors at distance i from p ). Finally , the exp e cted n um b er of steps executed is P ∞ i =0 M i = P ∞ i =0  ∆ − 1 k − 1  i . Note that ∆ ≥ 1 and k > ∆, so 0 < ∆ − 1 k − 1 < 1. The exp ected n um b er of steps executed in order to solve the conflict ( p, γ ) is P ∞ i =0  ∆ − 1 k − 1  i = 1 1 − ∆ − 1 k − 1 = k − 1 k − ∆ 2 Lemma 7 Starting fr om an arbitr ary c onfigur ation, the exp e cte d numb er of steps to r e ach a c onfigur ation verifying the unidir e ctional c oloring pr e dic ate is less or e qual to: n ( k − 1) k − ∆ , k > ∆ (4) Pro of: In the w orst case the n um b er of initial conflicts is n . Then the pro of is a direct consequence of Lemma 6. 2 Notice that with a minimal num b er of colors ( i.e. , k = ∆ + 1), the exp ected n um b er of steps to reach a terminal configuration starting from an arbitrary con- figuration is less than n ∆. Moreov er, when the num b er of colors increases ( i.e. , k → ∞ ), the exp ected n um b er of steps to reach a terminal configuration starting from an arbitrary configuration con v erges to n . Theorem 5 A lgorithm 3 is a pr ob abilistic self-stabilizing solution for the unidir e c- tional c oloring when k > ∆ . Pro of: The pro of is a direct consequence of Lemma 3 and Lemma 7. 2 6 Conclusion W e inv estigated the in trinsic complexity of performing lo cal tasks in unidirectional net w orks in a self-stabilizing setting. Con trary to “classical” bidirectional net works, lo cal vertex coloring now induces global complexity ( n states p er process at least, n mo v es p er pro cess at least) for deterministic solutions. W e presented state and time optimal solutions for the deterministic case, and asymptotically optimal solutions for the probabilistic case. This work raises sev eral imp ortan t open questions: 1. Our probabilistic solution can b e tuned to b e optimal in space (and is then with a ∆ multiplicativ e penalty in time), or optimal in time, but not both. Ho wev er, RR n ° 6524 16 S. Bernar d et al. our lo wer b ounds do not preclude the existence of probabilistic solutions that are optimal for b oth complexity measures. 2. Sev eral of the lo w er b ounds w e provide in the deterministic case rely on the silence prop ert y of the expected solution. W e question the p ossibility of de- signing deterministic algorithms that are not silent y et pro vide coloring with less than n colors in general graphs. References [1] Y ehuda Afek and Anat Bremler-Barr. Self-stabilizing unidirectional netw ork algorithms b y p ow er supply . Chic ago J. The or. Comput. Sci. , 1998, 1998. [2] Y ehuda Afek and Shlomi Dolev. Lo cal stabilizer. J. Par al lel Distrib. Comput. , 62(5):745–765, 2002. [3] Joffro y Beauquier, Sylvie Dela¨ et, Shlomi Dolev, and S ´ ebastien Tixeuil. T ran- sien t fault detectors. Distribute d Computing , 20(1):39–51, 2007. [4] Joffro y Beauquier, Maria Gradinariu, and Colette Johnen. 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International Journal of Principles and Applic ations of Information Scienc e and T e chnolo gy (P AIST) , 1(1):1–13, Decem b er 2007. [20] Nathalie Mitton, Eric Fleury , Isabelle Gu´ erin-Lassous, Bruno S´ ericola, and S ´ ebastien Tixeuil. On fast randomized colorings in sensor net w orks. In Pr o- c e e dings of ICP ADS 2006 , pages 31–38. IEEE Press, July 2006. RR n ° 6524 18 S. Bernar d et al. [21] Mikhail Nesterenk o and Anish Arora. T olerance to unbounded b yzantine faults. In 21st Symp osium on R eliable Distribute d Systems (SRDS 2002) , pages 22–. IEEE Computer So ciet y , 2002. [22] S ´ ebastien Tixeuil. On a space-optimal distributed tra v ersal algorithm. In Datta and Herman [7], pages 216–228. INRIA Bounds for self-stabilization in unidir e ctional networks 19 s s s s s s s s (a) uniform configuration in state s s' s' s' s' s' s' s' s' (b) uniform configuration in state s 0 Figure 1: A p ossible execution with synchronous scheduling RR n ° 6524 20 S. Bernar d et al. s(i) s(i+1) s(l-2) s(l-1) s(0) s(2) s(1) (a) state transition function of a uniform de- terministic self-stabilizing algorithm 0 1 2 3 4 k-2 k-1 (b) state transition function of Algorithm 1 Figure 2: Isomorphism of the state transition function when s.p i = s.p i − 1 mod n INRIA Bounds for self-stabilization in unidir e ctional networks 21 0 1 2 3 4 n-2 0 activation (a) starting configuration 1 1 2 3 4 n-2 0 activation (b) after 1 step 1 2 2 3 4 n-2 0 activation (c) after 2 steps 1 2 3 4 n-2 0 0 activation (d) after n − 1 steps Figure 3: A p ossible execution of Algorithm 1 RR n ° 6524 Unité de recherche INRIA Futurs Parc Club Orsay Uni versité - ZA C des V ignes 4, rue Jacques Monod - 91893 ORSA Y Cedex (France) Unité de recherche INRIA Lorraine : LORIA, T echnopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 V illers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cede x (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de V oluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Éditeur INRIA - Domaine de V oluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France) http://www.inria.fr ISSN 0249-6399