Unison as a Self-Stabilizing Wave Stream Algorithm in Asynchronous Anonymous Networks

Unison as a Self-Stabilizing W a v e Stream Algorit hm in Async hronous Anon ymous Net w ork s Christian Boulinier LaRIA, CNRS FRE 2733 Univ ersit ´ e de Picardie Jules V erne, F rance Abstract How to pass from lo ca l to global scales in anonymous net works? In such netw or k s, how to organize a self-stabilizing propagatio n of information with feedback? F rom Angluin’s res ults, the deterministic leader election is imp os sible in general anonymous netw orks. Thus, it is imp ossible to build a r o oted spa nning tree. In this pap er we show how to use Unison to design a self- stabilizing b arrier synchr onization in an ano nymous netw ork. W e show that the comm unicatio n structure of this bar r ier sync hro nization designs a self-stabilizing wa ve stream, or pip elined wa ve, in anonymous netw orks. W e introduce t wo v aria nt s of w aves: Strong W av e and W av elet. Strong wa ves can b e used to s olve the idemp otent r -o p erator parametrized pr oblem, whic h implies well known problems like depth-fir st search tree constructio n – this instance requires identities for the pro cesso rs. W av elets deal with ρ -distance computation. W e show how to use Unison to design a self-stabilizing stro ng wa ve stre am, and wa velet strea m r esp ectively . Keyw ords : Anonymous Net work, Ba rrier Synchronization, Self-Stabilization, Unison, W av e. Corresp ondance : Christian BOULINIER Email: Ch r istian.Boulinier@u-picardie.fr LaRIA, Univ ersit ´ e de Picardie Jules V erne, Amiens, F rance T el. +33- 322-809-5 77 1 In tro du c tion Sev eral general message passing p r oblems are us eful to ac hieve man y tasks in d istr ibuted net works, lik e broadcasting information, global sync hronization, reset, termination detection, or calc u lation of a global fun ction whose the input d ep ends on several pro cesses or the totalit y o f the pro cesses in the n et w ork – see [RH90 , T el94, Lyn96]. In this pap er we consider the wave pr op agation problem in async hronous anonymous net w orks. 1.1 Related W orks In async hronous systems, there is no global signal. Sync hr onization is a crucial task. Inform ally , a sync hr on izer allo ws asynchronous sys tems to sim ulate synchronous ones. In async hr onous systems, one can at most ensure that no p ro cess starts to execute its ph ase i + 1 b efore all p r o cesses ha ve completed their phase i . This strongest sync hronization task, named Barrier Synchr onization , w as in tro d uced by Misra in [Mis91] in a complete graph. The researc h ab out sync hronization started with Aw erbuch [Awe 85 ]. Comm un ications wa ves are often used to ac hieve sync hronization. Designing efficien t fault-toleran t wa ve algorithms is an imp ortan t task. Self-stabilization [Dij74, Dol00] is a general tec hnique to design a system that tolerat es arbitrary transient faults, i.e. faults that m a y corrupt the state of pr o cesses or links. [KA98] prop oses a self-stabilizing solution for complet graphs . [HL01] designs a solution in u niform rings with an o dd size. A relaxed syn c hronization requirement is defined as follo ws: the clo c ks are in ph ase if th e v alues of t wo neigh b oring p ro cesses differ by no more than 1, and the clo c k v alue of eac h pro cess is incremente d by 1 infin itely often. The self-stabilizing asynchr onous unison [CF G92] deals with this criterium . A distributed p r oto col is uniform if ev ery pro cess with the same degree executes the s ame pro- gram. In particular, w e do not assume a uniqu e pro cess iden tifier – the n et w ork is anonymous – or some consistent orien tation of links in the net w ork suc h that an y dynamic election of a master clo c k can b e feasible. Numerous self-stabiliz ing wa v e algorithms use a ro oted spannin g tree or simply an only initiator, called the r o ot – s ee for in s tance [Kru79] [ABDT98].In these cases, p r oto cols are not uniform, they are only at most semi-uniform . S o, f or a u niform distribu ted proto col any p r o cessor ma y initiate a w a v e, and most generally a global computation. Any p ro cessor ma y b e an initiator. T o f ace this inheren t concurency , a solution is that ev ery pr o cessor main tains the identit y of the initiators – see for instance [CDPV02]. That is imp ossible in an anon ymous n etw ork. [KA98] designs a self-stabilizing Barrier S y n c honization algorithm in asynchronous an onymous complet n et w orks. F or the other top ologies the au th ors use the netw ork with a ro ot, the program is not un iform, but only semi-uniform. An interesting question is to giv e a solution to this problem in a general conn ected asyn chronous anon ymous net work. As far as w e kno w, the phase algorithm [T el91] is the only decen tralised uniform w a v e al gorithm for a general anonymous net w ork. This algorithm requires that the pro cessors know the d iameter, or most simply a common up p er b ound D ′ of the diameter. This algorithm is n ot self-stabilizing. 1.2 Con tribution and pap er outline The main task of this pap er is to sh o w ho w Unison can b e view ed as a self- stabilizing wave str e am algorithm in async hronous anonymous n et w orks sc heduled b y an unfair daemon . Th e contribution is threefold: 1 Firstly , we in tro duce the ρ -d istance b arrier syn c hronization notion. It is a small exten tion of the barrier synchronizati on [Mis91] which ensures that no pro cess starts to execute its p hase i + 1 b efore all pr o cesses at distance less than or equal to ρ ha ve completed their phase i . W e sho w ho w to design a self-stabilizing b arrier synchr onization at distance ρ in an anon ymous net w ork. The s elf-stabilizing time complexity is in O ( n ) round s. It h as its space complexit y in O ( l og ( n ) + l og ( K )), where n is the num b er of p ro cesses in the net w ork and K t h e size of the clock. Seco nd ly , w e introd u ce tw o v ariants of W a v e: W a v elet and Strong W a ves. W e sho w that a str on g w a v e can b e used to solve the idemp oten t r -op erator parametrized problem, and a w a v elet deals with ρ -distance compu tation. Thirdly , we show that the comm unication structure of our ρ -distance b arr ier sync hr onization designs a self-stabilizi ng w a vele t stream, or p ip elined w a v elet, in any anonymous net w orks. W e s h o w that if ρ ≥ D the comm un ications design a self-stabilizing wa ve stream , and if ρ is greater than or equ al to the length of the longest simple path in the net w ork, then the p roto col designs a self-stabiliz ing strong-w a ve stream. The remainder of the p ap er is organized as follo ws. In th e next section (Section 2), w e describ e the underlying m o del for distributed system. W e also state wh at it means for a proto col to be self-stabilizing, we introd uce the notion of causal- D AG and we presen t the unison problem and its solutions. In Section 3 we define the ρ -distance b arrier sync hronization noti on and we in tro du ce a proto col wh ic h d esigns a self-stabilizing b arrier synchr onization at d istance ρ in an y anon ymous net wo rks. In Section 4 w e define t wo kinds of wa v es: wavelet and str ong waves , and we sho w the relationship b et wee n a strong w a ve and the idemp otent r -op erator parametrized computation problem . I n Section 5, we sho w ho w Unison can b e view as a wa ve stream, or a wa ve let stream, or a strong wa ve stream. I n Section 6, we giv e some concluding remarks . Because of the lac k of place, somme pro ofs are pu t b ack in an annexe. 2 Preliminaries In this section, firstly we d efine the m o del of distributed systems considered in this pap er, and state what it m eans for a p r oto col to b e self-stabilizing. Secondly , we presen t the notions of fin ite incre- men ting system and reset on it. Next, w e defin e what a self-stabilizing distributed Unison is. 2.1 The mo del Distributed Syst e m. A distribute d system is an un directed connected graph, G = ( V , E ), w h ere V is a set of no d es— | V | = n, n ≥ 2—and E is the s et of edges. No des represent pr o c esses , a nd edges repr esen t bidir e ctional c ommunic ation links . A comm unication link ( p, q ) exists iff p and q are neigh b ors. T he set of n eigh b ors of every p ro cess p is denoted as N p . The de gr e e of p is th e num b er of neigh b ors of p , i.e., equ al to |N p | . Th e distance b et w een t wo pro cesses p and q , d en oted by d ( p, q ) , is the length of the shortest path b et w een p and q . Let k b e a p ositiv e in teger. De fi n e V ( p, k ) as the set of pro cesses su ch th at d ( p, q ) ≤ k . D is the diameter of the netw ork. The program of a pro cess consists of a s et of registers (also referred to as v ariables) and a fi nite set of guard ed actions of the follo wing form : < label > :: < g uar d > − → < statement > . Eac h pro cess can only write to its o wn registers, and read its o wn registers and registers o wned by the neigh b oring pro cesses. The guard of an action in the program of p is a b o olean expression in v olving 2 the registers of p and its neigh b ors. The statemen t of an acti on of p u p d ates one or more registers of p . An act ion can b e executed only if its guard ev aluates to true. The actions are at omically executed, meaning the ev aluation of a guard and the execution of the corresp ond in g statement of an action, if executed, are d one in one atomic s tep. Th e state of a pro cess is d efined b y the v alues of its registers. The c onfigur ation of a s y s tem is the pro du ct of the s tates of all pr o cesses. Let a distribu ted proto col P b e a collectio n of binary tr an s ition relatio ns d en oted by 7→ , on C , the set of all p ossible configurations of the system. P describ es an oriented graph S = ( C , 7→ ), called the tr ansition gr aph of P . A sequence e = γ 0 , γ 1 , . . . , γ i , γ i +1 , . . . is called an exe cu tion of P iff ∀ i ≥ 0 , γ i 7→ γ i +1 ∈ S . A pro cess p is said to b e enable d in a configuration γ i ( γ i ∈ C ) if there exists an action A suc h that the guard of A is true in γ i . The v alue of a register r of a p ro cess p in th e state γ i , is denoted by p i .r . i is the momen t of the state γ i . When there is n o am biguit y , we will omit i . Similarly , an action A is said to b e enabled (in γ ) at p if the guard of A is true at p (in γ ). W e assu me that eac h tr an s ition from a configur ation to another is driv en b y a distribute d sche duler call ed daemon . In this pap er, w e consider only an Async hr on ou s distributed Daemo n . The Asynchr onous Daemon c ho oses an y nonempt y set of enabled pro cesses to execute an action in eac h computation step (Unfair Daemon). In ord er to compute the time complexit y , we use th e d efinition of r ound [DIM97]. This definition captures the execution rate of th e slow est pro cessor in an y computatio n . Gi ven an execution e , the first r ound of e (let us call it e ′ ) is the minimal p r efix of e con taining the execution of one acti on of the proto col or the neutralizatio n of every enabled pro cessor from the first configuration. Let e ′′ b e the suffix of e , i.e., e = e ′ e ′′ . Th en se c ond r ound of e is the first r ound of e ′′ , and so on. Self-Stabilization. Let X b e a set. A pr e dic ate P is a f u nction that h as a Bo olean v alue— true or false —for eac h elemen t x ∈ X . A predicate P is close d for a transition graph S iff ev ery state of an execution e that starts in a state satisfying P also satisfies P . A predicate Q is an attractor of the predicate P , denoted by P ⊲ Q , iff Q is closed for S and for ev ery execution e of S , b eginning b y a state satisfying P , there exists a configur ation of e for whic h Q is true. A tr an s ition graph S is self-stabilizing for a predicate P iff P is an attractor of the predicate true , i.e., true ⊲ P . 2.2 Causal DA Gs Definition 2.1 (Even t s and Causal D AGs) L et γ t 0 γ t 0 +1 .... b e a finite or infinite exe cu tion. ∀ p ∈ V , ( p, t 0 ) is an event. L et γ t → γ t +1 b e a tr ansition. If the pr o c ess p exe cutes a guar de d action during this tr ansition, we say that p exe cutes an action at time t + 1 , and we say that ( p, t + 1) is an event or a p -event. The causal D A G asso ciate d is the smal lest r elation ❀ on the set of events that satisfies: 1. L et ( p, t ) b e an event with t > t 0 . L et t ′ b e the gr e atest inte ger such that t 0 ≤ t ′ < t and ( p, t ′ ) is an ev e nt, then ( p, t ′ ) ❀ ( p, t ) 2. L et ( p, t ) b e an even t and let t > t 0 . L et q ∈ N p and let t ′ b e the gr e atest inte ger such that t 0 ≤ t ′ < t and suc h that ( q , t ′ ) is an event, then ( q , t ′ ) ❀ ( p, t ) . The causal order  on the set o f events is the r eflexive and tr ansitive closur e of the c ausal r elation ❀ . The past cone of an event ( p, t ) i s the c ausal- D AG induc e d by ev e ry event ( q , t ′ ) such that ( q , t ′ )  ( p, t ) . A past cone inv olves a pr o c ess q iff ther e is a q - event in the c one. The c over of an eve nt ( p, t ) is the set of pr o c e sses q c over e d by the p ast c one of ( p, t ) , this set is denote d by C ov er ( p, t ) . 3 Definition 2.2 (Cut ) A cut C on a c ausal D AG is a map fr om V t o N , which asso c i ates e ach pr o c ess p with a time t C p such that ( p, t C p ) is an event. We mix this map with its gr aph: C =  p, t C p  , p ∈ V  . The p ast of C i s the events ( p, t ) such that t ≤ t C p . It is denote d by ] ← , C ] . The futur e of C i s the events ( p, t ) such that t C p ≤ t . It is denot e d by [ C, → [ .A cut is coheren t if ( q , t ′ )  ( p, t ) and ( p, t )   p, t C p  then ( q , t ′ )   q , t C q  . A cut C 1 is less than or e qual to a cut C 2 , denote d by C 1  C 2 , if the p ast of C 1 is include d in the p ast of C 2 .If C 1 and C 2 ar e c oher ent and C 1  C 2 then [ C 1 , C 2 ] is the induced c ausal DAG define d by the events ( p, t ) such that  p, t C 1 p   ( p, t )   p, t C 2 p  .Any se gment [ C 1 , C 2 ] is a sequence of eve nts , e ach event of C 1 is c al le d an initial ev ent . 2.3 Distributed Unison Unison , or most pr ecisely Self-Stabilizing A synchr onous Unison , is a relaxed self-stabilizing Barrier Synchr onization in the follo wing meaning: the clocks are in p hase if the v alues of t w o neigh b orin g pro cesses differ by no m ore than 1, and the clock v alue of eac h pro cess is incremen ted b y 1 infinitly often. Self-stabilizing Unison w as introd uced b y [CF G92]. There is a p ossib ilit y of d eadlo c k if the size of the clo c k is to o short– see[BPV04]. A little alge b r aic framew ork, and some v o cabu lary are necessary . The vocabulary will b e used in the defin ition of the algorithm 1. Algebraic framew ork Let Z b e the set of in tegers and K b e a strictly p ositiv e in teger. Two in tegers a and b are said to b e c ongruent mo dulo K , denoted b y a ≡ b [ K ] if and only if ∃ λ ∈ Z , b = a + λK . W e d enote ¯ a the unique elemen t in [0 , K − 1] suc h that a ≡ ¯ a [ K ]. min( a − b, b − a ) is a distanc e on the torus [0 , K − 1] denoted by d K ( a, b ) . Two intege rs a and b are said to b e lo c al ly c omp ar able if and only if d K ( a, b ) ≤ 1. W e then define the lo c al or der r elationsh ip ≤ l as follo ws: a ≤ l b def ⇔ 0 ≤ b − a ≤ 1. If a and b are t wo lo cally comparable integ ers, w e define b ⊖ a as follo ws: b ⊖ a = def if a ≤ l b then b − a else − a − b . If a 0 , a 1 , a 2 , . . . a p − 1 , a p is a sequence of integ ers such that ∀ i ∈ { 0 , . . . , p − 1 } , a i is lo cally comparable to a i +1 , then S = p − 1 P i =0 ( a i +1 ⊖ a i ) is the lo c al variation of this sequence. Incremen ting system W e define X = {− α, . . . , 0 , . . . , K − 1 } , wher e α is a p ositiv e in teger. Let ϕ b e the fu nction from X to X defined b y: ϕ ( x ) = def if x ≥ 0 then x + 1 else x + 1. The pair ( X , ϕ ) is called a finite incr ementing system . K is called the p erio d of ( X , ϕ ). Let t ail ϕ = {− α, . . . , 0 } and stab ϕ = { 0 , . . . , K − 1 } b e the sets of “extra” v alues and “exp ected” v alues, resp ectiv ely . The set tail ∗ ϕ is equal to tail ϕ \ { 0 } . W e assume that eac h pr o cess p main tains a clo ck r egister r p with an incremen ting system ( X , ϕ ). Let γ the system confi gu r ation, w e defi ne the predicate W U : W U ( γ ) ≡ def ∀ p ∈ V , ∀ q ∈ N p : ( r p ∈ stab ϕ ) ∧ ( d ( r p , r q ) ≤ 1) in γ . In trinsic Path Delay [BPV04] Let γ a confi gurations in W U , the clo ck v alues of neigh b oring pro cesses are lo cally comparable. W e defin e the four notions: Dela y Th e dela y along a path µ = p 0 p 1 . . . p k , den oted b y ∆ µ , is the lo cal v ariation of the sequence r p 0 , r p 1 , . . . , r p k , i.e, ∆ µ = k − 1 P i =0  r p i +1 ⊖ l r p i  if k > 0, 0 otherwise ( k = 0). 4 In trinsic Dela y T he dela y b et w een tw o pro cesses p and q is intrinsic if it is indep endent on th e c hoice of th e p ath from p to q . T he dela y is intrinsic iff it is intrinsic for every p and q in V . In this case, and at time t , the intrinsic d ela y b et w een p and q is d enoted by ∆ ( p,q ) . WU 0 The pr edicate W U 0 is true f or a system configuration γ iff γ satisfies W U and th e dela y is in trinsic in γ . Precedence relationship When Dela y is in trinsic, it defines a total preordering on the pro cesses in V , named pr e c e denc e r elationship . This relationship dep end s on the state γ ∈ W U 0 . The ab s olute v alue of the dela y b et ween t wo p ro cesses p and q , is equal to or less than the distance d ( p, q ) in the net wo rk. This remark is imp ortan t for the follo wing. Cyclomatic Characteristic C G [BPV04] If G is an acyclic graph, then its cyclomat ic c harac- teristic C G is equal 2. Otherw ise G con tains cycles: Let Λ b e a cycle basis, th e length of the longest cycle in Λ is denoted λ (Λ). The cyclomatic c haracteristic of G , is equal to the lo w est λ (Λ) among cycle bases. It follo ws from the definition of C G that C G ≤ 2 D . Unison Definition W e assum e that eac h p ro cess p maint ains a r egister p .r ∈ χ . The self- stabilizing asynchr onous ( distribute d ) unison problem, or most shortly the unison pr ob lem, is to design a uniform proto col so that the follo wing prop er ties are true in ev ery execution [BPV05]: Safet y : W U is closed. Synchro nization : In W U , a pro cess can incr ement its clock r p only if the v alue of r p is lo wer than or equal to the clo c k v alue of all its neigh b ors. No Lo c k out (Liveness) : In W U , eve ry pro cess p incremen ts its clo ck r p infinitely often. Self-Stabilization : Γ ⊲ W U . The follo wing guard ed action solv es the synchr onization pr op erty and th e safety : ∀ q ∈ N p : ( r q = r p ) ∨ ( r q = ϕ ( r p )) − → r p := ϕ ( r p ); (1) The pr edicate W U 0 is closed for any execution of this guard ed action. Moreo v er, for an y execution starting from a configur ation in W U 0 , the no lo ckout pr op erty is guarant eed. Generally this p rop erty is not guarante ed in W U . A few general s chemes to self-stabilizing the n on-stabilizing proto cols ha v e b een prop osed. The fir st self-stabilizing async hron ou s u nison wa s introdu ced in [CF G92]. The deterministic proto col prop osed needs K ≥ n 2 . The stabilizatio n time complexit y is in O ( n D ). The second solution is prop osed in [BPV04]. Th e authors sho w that if K is greate r than C G then W U = W U 0 and the no lo ckout pr op erty is guarantee d in W U . (see Definition 2.3). The proto col is self-stabilizing if α ≥ T G − 2, where T G is the le ngth of the longest chordless cycle (2 in tree net wo rks). One can n otice that C G and T G are b oun ded by n . So, ev en if C G and T G are unkno wn , w e can c ho ose K ≥ n + 1 and α = n . Its self-stabilizing time complexit y is in O ( n ). In [BPV06], the authors presen t the Protocol W U M in , wh ic h is self-stabiliz ing to asyn chronous u nison in at most D rounds in trees. 3 Barrier Syn c hronization 3.1 Barrier synch ronization at distance ρ Barrier Synchr onization pr oblem has b een sp ecified in [KA98]. Let ρ b e an int eger greater than 0. The relaxation of this p roblem at d istance ρ is the follo wing. Let K b e an intege r greater than 1. W e assume that eac h p ro cess p maintai ns a K - or der clo ck r e gister p.R ∈ { 0 , 1 , ..., K − 1 } . Eac h pro cess executes a cyclique sequence of K terminating phases (the critical sectio n << cs >> ). The follo wing tw o prop erties are requir ed for eac h phase: 5 Global Unison (Sa f e ty) : for eac h phase x ∈ { 0 , ..., K − 1 } , no pro cess p can p ro ceed to phase x + 1 u n til all no des q , suc h that d ( p, q ) ≤ ρ , has executed its phase x . No lo ck out ( liveness) : ev ery p ro cess increments its clo c k infinitly often. F or ρ = 1, this sp ecification is the sp ecification of the standard stabilized unison. F or ρ ≥ D , this sp ecification is the sp ecificatio n of the global Barrier Synchronizatio n. 3.2 The general self-stabilizing Scheme The id ea is to stabilize an u nderlay er un ison in order to sync hr onize a δ K − clock, with δ large enough to guaran tee that the absolute v alue of the delay b et w een ev ery t wo pro cesses at distance less th an or equal to ρ is neve r larger than δ . It is sufficient that δ ≥ ρ holds. W e tak e χ = {− α, .., 0 , .., δ K − 1 } and α ≥ T G − 2 . W e u se the unison of [BPV04] wh ic h stabilizes in O ( n ). T he pr oto col is d escrib e in Algorithm 1. T o ens u re self-stabilizatio n in W U 0 , we require δ K > C G . If we w ant to program a Barrier Sync hr on ization, we must tak e δ ≥ D , thus from C G ≤ 2 D , if K ≥ 3 then the inequalit y K δ > C G holds. In the remainder we supp ose that the inequalit y K δ > C G holds. Algorithm 1 ( S S − W S )Self-Stabilizing ρ -Barrier Sync hronization algorithm for the p ro cess p Constan t and v aria ble : N p : the set of neighbors of pro cess p ; p .r ∈ χ ; Bool ean F unctions : C onv er g enceS te p p ≡ p.r ∈ tail ∗ ϕ ∧ ( ∀ q ∈ N p : ( q.r ∈ tail ϕ ) ∧ ( p .r ≤ tail ϕ q .r )); Locally C or r ect p ≡ p.r ∈ stab ϕ ∧ ( ∀ q ∈ N p , q .r ∈ sta b ϕ ∧ ( ( p.r = q .r ) ∨ ( p.r = ϕ ( q .r )) ∨ ( ϕ ( p.r ) = q .r )) ); N ormal S tep p ≡ p.r ∈ stab ϕ ∧ ( ∀ q ∈ N p : ( p.r = q .r ) ∨ ( q .r = ϕ ( p.r ))); Reset I nit p ≡ ¬ Locally C or r ect p ∧ ( p.r 6∈ init ϕ ); Actions : N A : N ormal S tep p − → if p.r ≡ ρ − 1[ ρ ] then << CS 2 >> else << CS 1 >> ; p.r := ϕ ( p.r ); C A : C onve r g enceS tep p − → p.r := ϕ ( p.r ); RA : Reset I nit p − → p.r := α (reset); 3.3 Analysis Lifting construction In ord er to analyse the proto col 1 we in tro duce for eac h pro cess p , a global device, the register f p.r . Of course the v alue of this virtual register is in accessible to the pr o cess p . Informally f p.r is a wa y to unwind of the register p.r . Let γ t 0 γ t 0 +1 .... b e an in finite execution starting in W U 0 . Let p 0 b e a m aximal pro cess, according to the p r e c e denc e r elation – see Remark 2.3 – for the state γ 0 . Let ⊥ 0 = p 0 .r at time 0. F or eac h p ro cess p ∈ V , w e unwind the register p.r in the follo wing m anner. W e asso ciate a virtual register f p.r . F or the s tate γ 0 , w e initiate this virtual register b y the instruction f p.r := ⊥ 0 + ∆ 0 ( p 0 ,p ) . Durin g the execution, for eac h transition γ t → γ t +1 the in truction f p.r := f p.r + 1 holds if and only if p.r := p.r + 1 holds d uring the same transition. F or k ≥ ⊥ 0 w e define th e cut C k = { ( p, t p,k ) , p ∈ V } wher e t p,k is the smallest time suc h that f p.r := k . The first question is to pr o v e that th is cuts are coheren t. W e fir s t introd uce the easy lemma: Lemma 3.1 If ( p, t ) ❀ ( q , t ′ ) then: g q t ′ .r ∈ n g p t .r , g p t .r + 1 o . Ind uctiv ely, if ( q 0 , t 0 ) ❀ ( q 1 , t 1 ) ❀ ( q 2 , t 2 ) ... ❀ ( q i , t i ) then: g q t i i .r ∈ n g q t 0 0 .r , ..., g q t 0 0 .r + i o F rom th e Lemma 3.1, if ( q , t )  ( p, t p,k ) then ( q , t )  ( q , t q ,k ). It follo ws the prop osition: Prop osition 3.2 F or every k ≥ ⊥ 0 the cut C k is c oher ent. 6 Virtual register p.R a nd virtua l clo ck F or eac h pro cess p w e asso ciate the register p.R , whic h is virtu al. Its v alue is ev aluated b y the p r o cedure: if p.r ∈ st ab ϕ then p.R := p .r /δ else p.R := − 1, where the sym b ol / is the inte ger division op er ator. The virtual register p.R defines a clo c k on {− 1 , 0 , ..., K − 1 } . The algorithm 1 solves self-stabilizing Asynchronous Unison, so ev ery pro cess p incremen ts its cloc k p.r in finitly often. W e dedu ce that p.R incremen ts in finitly often, thus: Lemma 3.3 ( Liveness ) F or every pr o c ess p , the virtual r e g ister p.R is incr emente d infinitly often. Conse quently << CS 2 >> is exe cute d i nfinitly often. Theorem 3.4 If δ ≥ ρ , onc e the pr oto c ol is stabilize d, it solves the Barrier Synchr onization at distanc e ρ for the vi rtual c lo ck define d by the r e g ister p.R . Pro of. W e consider the ph ase U = [ C U δ , C U δ + δ − 1 ], for an y ev ent ( p, t ) in this sequence, th e register p.R is equal to U [ K ]. Let p and q b e tw o pro cesses, suc h that d ( p, q ) ≤ ρ . Let ( p, t p ) and ( q , t q ) b e in C U δ + δ . Sup p ose that t p ≤ t q , at time t p the r egister f q .r ∈ { U δ + δ − i, i ∈ { 0 , ..., ρ − 1 }} , th us at time t p , the critical section << CS 2 >> of the phase U is terminated for the pro cess q . ✷ Our proto col synchronizes p ro cessors at distance ρ in an y an onymous general net wo rk. On the general graph , this synchronizer do es n ot need any identi ty and do es n ot build any real or virtual spanning tree. Here, the broadcast run s in the b eginning of a phase from any d ecen tralised no de p . F or eac h n o de q ∈ V ( p, ρ ), at the end of the phase for q , t h e no de kno ws that in f ormation is gone to all th e others no des in V ( q , ρ ), the feedbac k is imp licit. The time complexit y of a ph ase [ C U δ , C U δ + δ − 1 ] is δ r ounds in worst case. Th e message complexit y is 2 δ | E | , wh ich is the price to pa y for uniformity . But is this message complexity usable? W e will giv e a p ositiv e answer. 4 W a v elet, W a v e and Strong W a v e During eac h ph ase of Algorithm 1, the structure of comm unications is a kind of w a ve d ep endin g of the v alue of δ . T hese comm unication structures are formally defin ed in this section. In the section 5 , follo wing the Theorem 5.2, w e will b e able to use these comm unications to compute some imp ortan t functions on the net w ork, for ins tance an infim u m if δ ≥ D , or most generally the idemp oten t r -op erator parametrized calculation problem when δ ≥ n , and so to solv e m an y silen t tasks [Duc98]. 4.1 W alk and W a v e Definition 4.1 Walk. A W alk is a finite non empty wor d m = q 0 q 1 .....q r on the alphab et V , such that for al l i ∈ { 0 , r − 1 } , q i = q i +1 or q i +1 ∈ N q i . A walk i s cir cu lar i f r > 1 and q 0 = q r . The walk m is b e g i nning in q 0 denote d head ( m ) , and is ending in q r . Its length is r . L et m b e a walk , if ther e exists two wor ds m 1 and m 2 , and a c i r cular walk u such that m = m 1 um 2 , ( u is a factor of m ), then m 1 head ( u ) m 2 is a walk and we write: m → m 1 head ( u ) m 2 The tr ansitive closur e of the r elationship → defines a strict p artial or dering ∗ → i n the set of w alks . A simple walk is a minimal walk ac c or ding to the ∗ → p artial or dering. Most simply, a simp le wa lk is a w alk without any rep etition . An elemen tary w alk is a w alk suc h that if for i < j , q i = q j then for al l k ∈ { i, ..., j } , q k = q i . A redu cing of a wa lk m is a simple w alk m ′ such that m ∗ → m ′ . Walk c over of an event in a se quenc e. L et S = [ C 1 , C 2 ] b e a sequence of ev en ts . If in S , ( q , t ′ )  ( p, t ) then ther e exists a causalit y c hain fr om ( q , t ′ ) to ( p, t ) : ( q , t ′ ) = ( q 0 , t 0 ) ❀ ( q 1 , t 1 ) ❀ 7 ( q 2 , t 2 ) ... ❀ ( q r , t r ) = ( p, t ) , its asso ciate d walk is the w alk q 0 q 1 ... q r . The w alk co ver of an event ( p, t ) ∈ S is th e set of wa lks asso ciate d to the c ausality chains of S ending to ( p, t ) . This set is denote d by W al k C ov er ( p, t ) . Of c ourse, this set c ontains the w alk of length 0 denote d by p . Lemma 4.2 If m ∈ W alkCo v er(p,t) then ther e exists an elementary walk m ′ in W alkCo v er(p,t) such that m ∗ → m ′ Pro of. Let m = q 0 q 1 .....q r the a sso ciated walk of the causa lity chain ( q 0 , t 0 ) ❀ ( q 1 , t 1 ) ❀ ( q 2 , t 2 ) ... ❀ ( q r , t r ). Suppose that m = m 1 um 2 where u is a cir cular walk q i q .....q j . F r om the definition of ❀ re la tionship, there ex ists a chain: ( q i , t i ) ❀ ( q i , t i 1 ) ❀ ( q i , t i 2 ) ... ❀ ( q i , t j ). Let l the length of this chain. If v = l Q k =1 q i then ¯ m = m 1 v m 2 is an elemen t of WalkCover(p,t) . Such a r ewriting op er ation is po ssible only a finite n umber of times, at the end, the word is elementary . ✷ Definition 4.3 (W av e le t , W av e, and Strong W a v e) F ol lowing [T el94], we assume that ther e ar e sp e cial events c al le d decide ev ents , the natur e of these events dep e nds of the algorithm. L et k an inte ger. A k -wa velet is a sequence of ev ent s [ C 1 , C 2 ] that satisfies the fol lowing two r e qui r ements: The causal D A G induc e d by [ C 1 , C 2 ] c ontains at le ast one de cide event. F or e ach decide eve nt ( p, t ) , the p ast of ( p, t ) in [ C 1 , C 2 ] c overs V ( p, k ) . We simply c al l it a w av e when k ≥ D , wher e D is the diameter of the network. A strong w a ve is a wa ve [ C 1 , C 2 ] that satisfies the fol lowing adde d r e quir ement: F or e ach decide ev ent( p, t ) in [ C 1 , C 2 ] , and for e ach simple walk m 0 = q 0 q 1 ... q n − 1 p ending in p , ther e exists a ca us ality c hain ( q 0 , t 0 ) ❀ ( q ′ 1 , t 1 ) ... ❀  q ′ r − 1 , t r − 1  ❀ ( p, t ) in [ C 1 , C 2 ] , such that its asso ciate d walk m is elementary, and m ∗ → m 0 . 4.2 Infima and r − op erators T el, in his work ab out w a ve algorithms [T el94], introd uces th e infim um op er ators. An in fimum ⊕ o v er a set S , is an asso ciativ e, c ommutati ve and idemp oten t (i.e. x ⊕ x = x ) binary op er ator. I f P = { a 1 , a 2 , ..., a r } is a finite part of ( S ) then, from th e asso ciativit y , ⊕ P mak es sens as a 1 ⊕ a 2 ⊕ ... ⊕ a r . And if a ∈ S , th en a ⊕ P mak es sens as a ⊕ a 1 ⊕ a 2 ⊕ ... ⊕ a r . Such an op erator defin es a partial ord er relation ≤ ⊕ o v er S , by x ≤ ⊕ y if an d only if x ⊕ y = x . W e supp ose that S has a greater element e ⊕ , suc h that x ≤ ⊕ e ⊕ for every x ∈ S . Hence ( S , ⊕ ) is an Ab elian idemp oten t semi-group with e ⊕ as iden tit y elemen t for ⊕ . Ducourthial in tro d uces in [Duc98] the notion of r -op erator wh ic h generalizes the infimum op erators. Definition 4.4 The b inary op er ator ⊳ on S is a r − op er ator if ther e exits a ( S , ⊕ ) - endomorph ism r , c al le d r -fu nction , such that: ∀ x, y ∈ S , x ⊳ y = x ⊕ r ( y ) . L et ⊳ b e a r − op er ator on S , and let r b e its asso ciate d r − function , ⊳ is idemp otent if and only if: ∀ x ∈ S , x ≤ ⊕ r ( x ) . A mapping ⊳ fr om ( S ) n to S is an n − ary r − op er ator if ther e exists n − 1 ( S , ⊕ ) -endomorph isms r 1 , r 2 , ..., r n − 1 such that for al l ( x 0 , x 1 , ..., x n − 1 ) ∈ ( S ) n : ⊳ ( x 0 , x 1 , ..., x n − 1 ) = x 0 ⊕ r 1 ( x 1 ) ⊕ ... ⊕ r n − 1 ( x n − 1 ) Remark 4.5 r is an e ndomorphism, which me ans tha t for al l x, y in S , r ( x ⊕ y ) = r ( x ) ⊕ r ( y ) . F r om the definition of ≤ ⊕ , we de duc e that r is c omp atible with ≤ ⊕ , formal ly: ∀ x, y ∈ S , x ≤ ⊕ y ⇒ r ( x ) ≤ ⊕ r ( y ) 8 4.3 Infim um and r -op erator parametrized computation problem Let [ C 1 , C 2 ] b e a wa ve . W e denote b y N t p the set of pro cesses such that there exists a time t q suc h that ( q , t q ) ❀ ( p, t ) . Note that p m a y b e in N t p . Because of the lac k of place, th e pro of of Theorem 4.10 is in the annexe . Infim um computation Giv e eac h pro cess p , an extra v ariable p.r es : S . Eac h register p.r es is initialised du ring the initial ev ent of p by the v alue p.v 0 . let ( p, t ) b e any even t in [ C 1 , C 2 ]. Whenev er ( p, t ) holds, p.r es is set to the v alue p.v 0 L  q t q .r es, q ∈ N t p  . T el sho ws the follo win g theorem: Theorem 4.6 [T el94] A wave c an b e use d to c ompute an infimum. Idemp otent r -op e ra tor parametrized computation problem Let [ C 1 , C 2 ] b e a strong wa ve . W e associate to eac h orien ted link ( p i , p j ) of G = ( V , E ) a idemp oten t r − f unction : r p i ,p j . By exten tion, for the sequence ( p i , p i ) w e asso ciate the id en tit y: r ii = id . Lik e ab o v e, giv e eac h pro cess p , an extra constan t p.v 0 : S an d a register p.r es . Eac h register p.r es is initialised d uring the initial ev en t of p by the v alue p.v 0 . let ( p, t ) b e an y eve nt in [ C 1 , C 2 ]. Whenever ( p, t ) h olds, p.r es is set to the v alue p .v 0 ⊕  r q ,p ( q .r es ) , q ∈ N t p  . Eac h n o de p can b e seen as a ( d + 1)-a ry r -op erator if d is the degree of the no d e. Definition 4.7 F or any walk µ = p 0 p 1 ....p n , we define ev al ( µ ) = r µ ( p 0 .v 0 ) , with r µ = r p n − 1 ,p n or p n − 2 ,p n − 1 o...or p 0 ,p 1 , wher e o is the c omp osition of functions. F or any p ∈ V , the sets Λ ′ p and Λ p ar e define d by: Λ ′ p =  ev al ( µ ) , µ ∈ Σ ′ p  and Λ p = { ev al ( µ ) , µ ∈ Σ p } , wher e Σ ′ p is the set of the w alks ending to p , and Σ q is the set of the simple wa lks ending to p . F rom th e definitions and the idemp otence of th e r -op erators, the follo wing lemma holds: Lemma 4.8 A ssume that m and m ′ ar e two walks with p = head ( m ) . We supp ose that m ∗ → m ′ . Then r m ( p.v 0 ) ≥ ⊕ r m ′ ( p.v 0 ) , and if m is elementary then r m ( p.v 0 ) = r m ′ ( p.v 0 ) Definition 4.9 (Legitimate output ) W e define the le gitimate output of a pr o c ess p as the quan- tity: ⊕ Λ p Theorem 4.10 A strong w a v e c an b e use d to solve the idemp otent r -op er ator p ar ametize d pr oblem. 5 Unison as a self-stabilizing w a v e stream algorithm, applications 5.1 Analysis of the Unison Beha vior start ing in W U 0 Lemma 5.1 L et k ≥ ⊥ 0 . If ( p, t ) is an e v ent in the interval [ C k , → [ , then: V ( p, g p t .r − k ) ⊂ C ov er ( p, t ) and Σ g p t .r − k p ⊂ W al k C ov er ( p, t ) . Wher e Σ ρ p is the set of simple walks of length less than or e qual to ρ , ending to p . Pro of. The lemma is true for the initial even ts of [ C k , → [. Let A b e the set of even ts ( p, t ) in [ C k , → [ such that the se ntence: V ( p, g p t .r − k ) ⊂ C over ( p, t ) ∧ Σ g p t .r − k p ⊂ W alk C over ( p, t ) 9 do es not hold. W e as sume that A is not empt y , let ( q , τ ) a minimal even t in A according to  . Let δ = g q τ .r − k , and let p 1 ∈ V ( q , δ ). If p 1 = q then p 1 ∈ C over ( q , τ ), e ls e there exists q 1 ∈ N q such tha t p 1 ∈ V ( q 1 , δ − 1 ) . ( q , τ )is not a initial even t, so q 1 ∈ N τ q and there exists τ q 1 such that ( q 1 , τ q 1 ) ❀ ( q , τ ), and b y the minimality of ( q , τ ) the inclusio n V ( q 1 , δ − 1) ⊂ C ov e r ( q 1 , τ q 1 ) holds and thus V ( q 1 , δ − 1) ⊂ C ov er ( q , τ ) and p 1 ∈ C ov er ( q , τ ). F ollowing the same way , let m b e a walk in Σ δ p , if m = q then m ∈ W alk C ov er ( q , τ ), else if m = p 1 p 2 ....p r q then p r ∈ N τ q bec ause( q , τ ) is not a initial e ven t, so there exists τ p r such that ( p r , τ p r ) ❀ ( q , τ ), and by the minimalit y of ( q , τ ) the inclusion Σ δ − 1 p r ⊂ W alk C ov er ( p r , τ p r ) holds , and thus p 1 p 2 ....p r q ∈ W alk C ov er ( q , τ ) . So ( q , τ ) is not in A . Thus A = ∅ , and the lemma is prov ed. ✷ As corollary , w e deduce the imp ortan t follo wing theorem: Theorem 5.2 L et k ≥ ⊥ 0 and δ b e a p ositive inte ger, then [ C k , C k + δ ] , with C k + δ as the set of de cide events, is a δ -wa vel et , and a wa v e if δ ≥ D . If δ is gr e ater than or e qual to the length of a longest simple walk in G , then [ C k , C k + δ ] is a str ong wave. 5.2 Self-stabilizing c omputation of an infim um at distance ρ If ρ ≥ D F or eac h pro cess p , the r egisters p.v 0 and p.r es are intiali zedby the same v alue. W e need one step for the initilisat ion, and D steps for the wa v e of calculation. So we tak e δ ≥ D + 1. F or an y in teger U , [ C U δ , C U δ + δ − 1 ] is a w av e. So w e defin e the cr itical secti ons as follo ws: << C S 2 >> ≡ initializ ation of p.v 0 and p.re s << C S 1 > > ≡ p.res := p.v 0 M { q .re s, q ∈ N p } F rom T heorem 4.6, at the cut C U δ + δ − 1 the register p.r es con tains the righ t v alue L { q .v 0 , q ∈ V } . If ρ < D W e tak e δ = ρ + 1. W e supp ose that the register q .v 0 is in itialised dur ing the critical section << C S 2 >> at the b eginning of the phase, precisely when the register p.r tak es the v alue U δ . T o reac h the ob jectiv e, w e defin e for eac h pr o cess p tw o added registers p.v 1 and p.v 2 . These t w o r egisters are initialized at the date C U δ during th e critical section << C S 2 >> , by the v alue p.v 0 . F or α ∈ { 1 , 2 , ..., ρ } , at the date C U δ + α , the action << C S 1 >> is the f ollo wing: p.v 1 := p.v 2 ; p.v 2 := p.v 0 M  q .v ϕ ( q ) , q ∈ N p  with, if q .r = p.r th en ϕ ( q ) = 2, and if q .r = p.r + 1 then ϕ ( q ) = 1. Theorem 5.3 At the c u t C U δ + δ − 1 the r e gi ster p.r es c ontains the right value: L { q .v 0 , q ∈ V ( p, ρ ) } . The pro of is in the ann exe. 6 Concluding r emarks W e show ed ho w the stucture of the comm unications b et we en pro cesses of Unison can b e viewed as a w a v e stream. T hanks to this stru cture, w e ha v e b een able to build a self-stabiliz ing w a ve stream algorithm in async hr onous anonymous netw orks scheduled by an u nfair daemon. Precisely , we sho wed that the b eha vior of Unison can b e viewed as a self-stabilizing wa ve, k -wa velet or bidirected link flo o d streams. F rom these remarks, in an y async hronous anon ymous netw ork sc heduled by an unfair daemon, we deduced self-stabilizing solutions to the barrier sync hr onization p r oblem, the infi mum calculatio n problem, and the idemp oten t r -op erator p arametrized c alculation problem. No w, an imp ortant qu estion w ould b e to r educe the self-stabilizing time complexit y of un ison f rom O ( n ) to O ( D ) in a general graph . 10 References [ABDT98] L. O. A lima, J. Beauqu ier, A. K. Datta, and S. Tixeuil. Self-stabilization with global ro oted synchronizers. In IEEE 18th International Confer enc e on Distribute d Computing Syst ems (ICDCS 98) , pages 102–109, 1998. 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Intr o duction to Di stribute d A lgorithms . Ca mbridge Universit y Press, 1994. 7 Annexe 7.1 Pro of of Theorem 4.10 Prop osition 7.1 F or any p ∈ V , ⊕ Λ ′ p exists, and the e qu ality ⊕ Λ ′ p = ⊕ Λ p holds. Pro of. Λ p is fin ite, s o ⊕ Λ p exists, furtherm ore Λ p ⊂ Λ ′ p . If m = p 0 p 1 ....p n b e a not elemen tary wal k , there exists a simp le w alk m ′ suc h that m ∗ → m ′ . F rom the lemma 4.8, r m ( p.v 0 ) ≥ ⊕ r m ′ ( p.v 0 ) and r m ′ ( p.v 0 ) ≥ ⊕ ⊕ Λ p hold. The pr op osition follo ws. 11 ✷ Lemma 7.2 L et ( p, t ) b e an ev e nt in [ C 1 , C 2 ] , then at time t , p t .r es = M { ev al ( µ ) , µ ∈ W al k C ov er ( p, t ) } Pro of. Let A b e the set of even ts ( p, t ) in [ C 1 , C 2 ] su c h that the equ ality is not true. Note that th e minimal ev ents in [ C 1 , C 2 ] are n ot in A . If A is empty , the pro of is finished. Supp ose that A is n ot empt y . Let ( p, t ) a m inimal ev ent of A acco rd ing to the relation  : p t .r es := p.v 0 M  r q ,p ( q t q .r es ) , q ∈ N t p  But, by defin ition: W al k C ov er ( p, t ) = S q ∈N t p { µp, µ ∈ W alk C ov er ( q , t q ) } ∪ { p } . F rom th e minimalit y of ( p, t ) in A , the ev en ts ( q , t q ) are not in A , so: p t .r es = p.v 0 M q ∈N t p r q p  M { ev al ( µ ) , µ ∈ W al k C ov er ( q , t q ) }  But r pq is compatible with ≤ L (remark 4.5), thus: p t .r es = p.v 0 M q ∈N t p { r q p ( ev al ( µ )) , µ ∈ W al k C ov er ( q , t q ) } ( p, t ) is not an initial eve nt, so p ∈ N t p and: p.v 0 ≥ ⊕ M { r pp or µ ( head ( µ ) .v 0 ) , µ ∈ W alk C ov er ( p, t p ) } W e deduce, from associativit y of ⊕ and from r q p or µ = r µp that : p t .r es = M q ∈N t p { ev al ( µp ) , µ ∈ W alk C ov er ( q , t q ) } but W al k C ov er ( p, t ) = S q ∈N t p { µp, µ ∈ W al k C ov er ( q , t q ) } , so: p t .r es = { ev al ( µ ) , µ ∈ W al k C ov er ( p , t ) } W e deduce that ( p, t ) is not in A , whic h is a con tradiction. W e ded uce that A = ∅ and the lemma. ✷ Theorem 7.3 (4.10) A strong wa ve c an b e use d to solve the idemp otent r - op er ator p ar ametize d pr oblem. Pro of. If ( p, t ) is a decide even t then, from the Lemma 7.2, w e eta blis h that p t .res = { ev al ( µ ) , µ ∈ W al k C over ( p, t ) } holds, and that W al k C over ( p, t ) satisfies the Definition 4.3. Recall that Λ p = { ev al ( µ ) , µ ∈ Σ p } . F or an y m ∈ W al k C over ( p, t ), there exists m 0 ∈ Σ p such that m ∗ → m 0 and from the Lemma 4.8 the inequal- it y ev al ( m ) ≥ ev al ( m 0 ) holds. W e deduce that p t .res ≥ ⊕ Λ p . Conversely , if m 0 ∈ Λ p , there exists m ∈ W alk C ov er ( p, t ) such that m ∗ → m 0 , but from the Lemma 4.2, there ex is ts also a walk m 1 ∈ W al k C ov er ( p, t ) such that m ∗ → m 1 and m 1 ∗ → m 0 , and fr o m Lemma 4.2 ev al ( m 1 ) = ev a l ( m 0 ). W e deduce that p t .res ≤ ⊕ Λ p . F rom thes e tw o inequalities, we deduce p t .res = ⊕ Λ p and the theo r em is prov ed. ✷ 12 7.2 Pro of of Theorem 5.3 Prop osition 7.4 F or p ∈ V and α ∈ { 1 , ..., ρ } , at the date C U δ + α , hold the e qualities: p.v 1 = M { q .v 0 , q ∈ V ( p, α − 1) } and p.v 2 = M { q .v 0 , q ∈ V ( p, α ) } Pro of. A t the date C U δ , an y pro cess p s atisfies p.v 1 = p.v 0 and p.v 2 = p.v 0 , it is the initializi n g step. Let A the set of eve nts in [ C U δ +1 , C U δ + δ − 1 ], for whic h the prop osition is not true. W e assume that A is not empt y . Let ( p, t ) a minimal eve nt in A . let α ∈ { 1 , 2 , ..., ρ } such that ( p, t ) ∈ C U δ + α . There exists t 0 suc h that ( p, t 0 ) ( p, t ). W e hav e p t .v 1 = p t 0 .v 2 = and p t 0 .v 2 = L { q .v 0 , q ∈ V ( p, α − 1) } . This equalit y is true even if α = 1. No w, p t .v 2 = p .v 0 L  q t q .v ϕ ( q ) , q ∈ N p  . F rom the minimalit y of the e ven t ( p, t ), the ev ents ( q , t q ), where t q < t , are not in A and are in [ C U δ , C U δ + δ − 1 ]. So, p.v 0 L  q t q .v ϕ ( q ) , q ∈ N p  = L { q .v 0 , q ∈ V ( p, α ) } . W e obtain a contradict ion. W e ded u ce that A is empt y , and the prop osition follo ws ✷ As corollary , w e obtain the theorem: Theorem 7.5 (5.3) At the cut C U δ + δ − 1 the r e gister p.r es c ontains the right value L { q .v 0 , q ∈ V ( p, ρ ) } . 13