A quadratic algorithm for road coloring

Transcription

1 A quadatic algoithm fo oad coloing Maie-Piee Béal and Dominique Pein Octobe 6, 0 axiv: v9 [cs.ds] 0 May 01 Abstact The Road Coloing Theoem states that evey apeiodic diected gaph with constant out-degee has a synchonized coloing. This theoem had been conjectued duing many yeas as the Road Coloing Poblem befoe being settled by A. Tahtman. Tahtman s poof leads to an algoithm that finds a synchonized labeling with a cubic wostcase time complexity. We show a vaiant of his constuction with a wost-case complexity which is quadatic in time and linea in space. We also extend the Road Coloing Theoem to the peiodic case. 1 Intoduction Imagine a map with oads which ae coloed in such a way that a fixed sequence of colos, called a homing sequence, leads the tavele to a fixed place whateve the stating point is. Such a coloing of the oads is called synchonized and finding a synchonized coloing is called the Road Coloing Poblem. In tems of gaphs, it consists in finding a synchonized labeling in a diected gaph. The Road Coloing Theoem states that evey apeiodic diected gaph with constant out-degee has a synchonized coloing (a gaph is apeiodic if it is stongly connected and the gcd of the length of the cycles is equal to 1). It has been conjectued unde the name of the Road Coloing Poblem by Adle, Goodwin, and Weiss [], and solved fo many paticula types of automata (see fo instance [], [], [9], [], [16], [5]). Tahtman settled the conjectue in [9]. In this pape, by Road Coloing Poblem we undestand the algoithmic poblem of finding a synchonized coloing on a given gaph (and not the existence of a polynomial algoithm which is solved by the Road Coloing Theoem). Univesité Pais-Est, Laboatoie d infomatique Gaspad-Monge CNRS UMR 8049, 5 boulevad Descates, Mane-la-Vallée, Fance, {beal,pein}@univ-mlv.f This wok is suppoted by Fench National Agency (ANR) though Pogamme d Investissements d Aveni (Poject ACRONYME n ANR-10-LABX-58). 1

2 Solving the Road Coloing poblem in each paticula case is not only a puzzle but has many applications in vaious aeas like coding o design of computational systems. These systems ae often modeled by finite-state automata (i.e. gaphs with labels). Due to some noise, the system may take a wong tansition. This noise may fo instance esult fom the physical popeties of sensos, fom uneliability of computational hadwae, o fom insufficient speed of the compute with espect to the aival ate of input symbols. It tuns out that the asymptotic behavio of synchonized automata is bette than the behavio of unsynchonized ones (see [1]). Synchonized automata ae thus less sensitive to the effect of noise. In the domain of coding, automata with outputs (i.e. tansduces) can be used eithe as encodes o as decodes. When they ae synchonized, the behavio of the code (o of the decode) is impoved in the pesence of noise o eos (see [4], []). Fo instance, the well-known Huffman compession scheme leads to a synchonized decode povided the lengths of the codewods of the Huffman code ae elatively pime. It is also a consequence of the Road Coloing Theoem that coding schemes fo constained channels can have sliding block decodes and synchonized encodes (see [1] and [1]). Tahtman s poof is constuctive and leads to an algoithm that finds a synchonized labeling with a cubic wost-case time complexity [9, 1]. The algoithm consists in a sequence of flips of edges going out of some state so that the esulting automaton is synchonized. One fist seaches a sequence of flips leading to an automaton which has a so-called stable pai of states (i.e. with good synchonizing popeties). One then computes the quotient of the automaton by the conguence geneated by the stable pais. The pocess is then iteated on this smalle automaton. Tahtman s method fo finding the sequence of flips leading to a stable pai has a wost-case quadatic time complexity, which makes his algoithm cubic. In this pape, we design a wost-case linea time algoithm fo finding a sequence of flips until the automaton has a stable pai. This makes the algoithm fo computing a synchonized coloing quadatic in time and linea in space. The sequence of flips is obtained by fixing a colo, say ed, and by consideing the ed cycles fomed with ed edges, taking into account the positions of the oots of ed tees attached to each cycle. The pize to pay fo deceasing the time complexity is some moe complication in the choice of the flips. We also extend the Road Coloing Theoem to peiodic gaphs by showing that Tahtman s algoithms povides a minimal-ank coloing. Anothe poof of this esult using semigoup tools, obtained independently, is given in [7]. Fo elated esults, see also [0] and []. The complexity of synchonization poblems on automata has been aleady studied (see [0] fo a suvey). It is well-known that thee is an O(n ) algoithm to test whethe an n-state automaton on a fixed-size alphabet is synchonized. The complexity of computing a specific synchonizing wod is O(n ) (see [14]). Howeve, the complexity of finding a synchonizing wod

3 of a given length is NP-complete [14] (see also [4], [7]). The complexity of poblems on automata has also been studied fo andom automata (see [8]). Seveal esults pove that, unde appopiate hypotheses, a andom ieducible automaton is synchonized [15], [8], and []. The aveage time complexity of these poblems does not seem to be known. In paticula, we do not know the aveage time complexity of the Road Coloing Poblem. The aticle is oganized as follows. In Section, we give some definitions to fomulate the poblem in tems of finite automata instead of gaphs. In Section we descibe Tahtman s algoithm and ou vaiant is detailed in Section 4. We give both an infomal desciption of the algoithm with pictues illustating the constuctions, and a pseudocode. The time and space complexity of the algoithm ae analyzed in Section 5. The peiodic case is teated in Section 6. A peliminay vesion of this pape was posted in []. The Road Coloing Theoem In ode to fomulate the Road Coloing Poblem we intoduce the notation concening automata. Let A be a finite alphabet and let Q be a finite set. We denote by A the set of wods ove A. A (finite) automaton A = (Q,E) ove the alphabet A with Q as set of states is a given by a set E of edges which ae tiples (p,a,q) whee p,q ae states and a is a symbol fom A called the label of the edge. Note that no initial o final states ae specified. Let F be the multiset fomed of the pais (p,q) obtained fom the set E by the map (p,a,q) (p,q). The multigaph having Q as set of vetices and F as set of edges is called the undelying gaph of A. A path in the automaton is sequence of consecutive edges. The label of the path ((p i,a i,p i+1 ) 0 i n is the wod a 0 a n. The state p 0 is its oigin and p n+1 is its end. The length of the path is n+1. The path is a cycle if p 0 = p n+1. An automaton is deteministic if, fo each state p and each lette a, thee is at most one edge stating at p and labeled with a. It is complete deteministic if, fo each state p and each lette a, thee is exactly one edge stating at p and labeled with a. This implies that fo each state p and each wod w thee is exactly one path stating at p and labeled with w. The end of this unique path is denoted by p w. An automaton is ieducible if its undelying gaph is stongly connected. The peiod of an automaton is the gcd of length of its cycles. An automaton is apeiodic if it is ieducible and of peiod Notethatthisnotion, whichisusualfo gaphs, isnotthenotionofapeiodic automata used elsewhee and which efes to the peiod of wods labeling the cycles (see e.g. [1]).

4 A synchonizing wod of a complete deteministic automaton A = (Q, E) is a wod w A such that fo evey pai of states p,q Q, one has p w = q w. A synchonizing wod is also called a eset sequence [14], o a magic sequence [5, 6], o also a homing wod [6]. An automaton which has a synchonizing wod is called synchonized (see an example on the ight pat of Fig. 1). Two automata which have isomophic undelying gaphs ae called equivalent. Hence two equivalent automata diffe only by the labeling of thei edges. In the sequel, we shall conside only complete deteministic automata. Poposition 1. A synchonized complete deteministic automaton is apeiodic. Poof. We assume that the automaton has at least one edge. Let (p,a,q) be an edge of the automaton. Let w be a synchonizing wod focusing to a state. Since the gaph is stongly connected, thee is a wod v such that fom v = p. Thus p awvp = p wvp. The lengths of the cycles fom p to p labeled awv and wv diffe by 1. This implies that the peiod of automaton is 1. The Road Coloing Theoem can be stated as follows. Theoem (A. Tahtman [9]). Any apeiodic complete deteministic automaton is equivalent to a synchonized one Figue 1: Two complete apeiodic deteministic automata ove the alphabet A = {a,b}. A thick ed plain edge is an edge labeled by a while a thin blue dashed edge is an edge labeled by b. The automaton on the left is not synchonized. The one on the ight is synchonized. Fo instance, the wod aaa is a synchonizing wod. The two automata ae equivalent since thei undelying gaph ae isomophic. A tivial case fo solving the Road Coloing Theoem is the case whee the automaton has a loop edge aound some state []. Indeed, since the gaph of the automaton is stongly connected, thee is a spanning tee ooted at (with the edges of the tee oiented towads the oot). Let us label the edges of this tee and the loop by the lette a. This coloing is synchonized by the wod a n 1, whee n is the numbe of states. 4

5 An algoithm fo finding a synchonized coloing Tahtman s poof of Theoem is constuctive and gives an algoithm fo finding a labeling (also called a coloing) which makes the automaton synchonized povided it is apeiodic. In the sequel A denotes an n-state complete deteministic automaton ove an alphabet A. We fix a paticula lette a A. Edges labeled by a ae also called ed edges o a-edges. The othe ones ae called blue o b-edges. A pai (p,q) of states in an automaton is synchonizable if theeis a wod w with p w = q w. It is well-known that an automaton is synchonized if all its pais of states ae synchonizable (see fo instance Poposition.6.5 in [4]). Apai(p,q)ofstates inanautomaton isstable ifandonlyif, foanywod u, the pai (p u,q u) is synchonizable. This notion was intoduced in [10]. In a synchonized automaton, any pai of states is stable. Note that if (p,q) is a stable pai, then fo any wod u, (p u,q u) is also a stable pai, hence the teminology. Note also that if (p,q) and (q,) ae stable pais then (p,) is also a stable pai. It follows that the elation defined on the set of states by p q if (p,q) is a stable pai is an equivalence elation. As obseved in [, Lemma ], this elation is a conguence (i.e. p u q u wheneve p q) called the stable pai conguence. Moe geneally, a conguence is stable if any pai of states in the same class is stable. The conguence geneated by a stable pai (p,q) is the least conguence such that p and q belong to the same class. It is a stable conguence. Given a conguence on the states of an automaton, we denote by p the class of a state p. If A = (Q,E) is an automaton, the quotient of A by a stable pai conguence is the automaton B whose states ae the classes of Q unde the conguence. The edges of B ae the tiples ( p,c, q) whee (p,c,q) is an edge of A. The automaton B is complete deteministic when A is complete deteministic. The automaton B is ieducible (esp. apeiodic) when A is ieducible (esp. apeiodic). The following Lemma was obtained by Culik et al. [11]. We epoduce the poof since it helps undestanding Tahtman s algoithm (see the pocedue FindSynchonizedColoing below). Lemma (Culik et al. [11]). If the quotient of an automaton A by a stable pai conguence is equivalent to a synchonized automaton, then thee is a synchonized automaton equivalent to A. Poof. Let B be the quotient of A by a stable conguence and let B be a synchonized automaton equivalent to B. We define an automaton A equivalent to A as follows. The numbe of edges of A going out of p and ending in states belonging to a same class q is equal to the numbe of edges of B (and thus B ) going out of p and ending in q. We define A by labeling 5

6 these edges accoding to the labeling of coesponding edges in B. The automaton B is a quotient of A. Let us show that A is synchonized. Let w be a synchonizing wod of B and the state ending any path labeled by w in B. Let p,q be two states of A. Then p w and q w belong to the same conguence class. Hence (p w,q w) is a stable pai of A. Theefoe (p,q) is a synchonizable pai of A. Since all pais of A ae synchonizable, A is synchonized. Tahtman s algoithm fo finding a synchonized coloing of an apeiodic automaton A consists in finding an equivalent automaton A of A which has at least one stable pai (s,t), then in ecusively finding a synchonized coloing B fo the quotient automaton B by the conguence geneated by (s,t), andfinallyinliftingupthiscoloingtotheinitialautomatonasfollows. If thee is an edge (p,c,q) in A but no edge ( p,c, q) in B, then thee is an edge ( p,d, q) in B with c d. Then we flip the labels of the two edges labeled c and d going out of p in A. The algoithm fo finding a synchonized coloing is descibed in the following pseudocode. The pocedue FindStablePai, which finds an equivalent automaton which has a stable pai of states, is descibed in the next section. The pocedue Mege computes the quotient of an automaton by the stable conguence geneated by a stable pai of states. The pocedue Update updates some data needed fo the computation as descibed in Section 5.1. FindSynchonizedColoing(apeiodic automaton A, quotient automaton B) 1 B A while (size(b) > 1) do Update(B) 4 B,(s, t) FindStablePai(B) 5 lift the coloing up fom B to the automaton A 6 B Mege(B,(s,t)) 7 etun A Thetemination of thealgoithm is guaanteed by thefact that thenumbe of states of the quotient automaton of B is stictly less than the numbe of states of B. The computation of the quotient automaton (pefomed by the Pocedue Mege) is descibed in Section 7. 4 Finding a stable pai In this section, we conside an apeiodic complete deteministic automaton A ove the alphabet A. We design a linea-time algoithm fo finding an equivalent automaton which has a stable pai. In ode to descibe the algoithm, we give some definitions and notation. 6

7 Let R be the subgaph of the gaph of A made of the ed edges. The gaph R is a disjoint union of connected components called clustes. Since each state has exactly one outgoing edge in R, each cluste contains a unique (ed) cycle with tees attached to the cycle at thei oots. If is the oot of such a tee, its childen ae the states p such that p is not on the a ed cycle and (p,a,) is an edge. If p,q belong to the same tee, p is an ancesto of q (o q is a descendant of p) in the tee if thee is a ed path fom q to p. Note that in these tees, the edges ae oiented fom the child to the paents and the paths fom the descendant to the ancestos. If q belongs to some ed cycle of length geate than 1, its pedecesso is the unique state p belonging to the same cycle such that (p,a,q) is an edge. In the case the length of the cycle is 1, we set that the pedecesso is q itself. Fo each state p belonging to some cluste, we define the level of p as the distance between p and the oot of the tee containing p. If p belongs to the cycle of the cluste, its level is thus null. The level of an automaton is the maximal level of its states. A maximal state is a state of maximal level. A maximal tee is a tee containing at least one maximal state and ooted at a state of level 0. A maximal oot is the oot of a maximal tee and a maximal child of a maximal oot is a child of having at least one maximal state as descendant. The algoithm fo finding a coloing which has a stable pai elies on the following key lemma due to Tahtman [9]. It uses the notion of minimal images in an automaton. An image in an automaton A = (Q,E) is a set of states I = Q w, whee w is a wod and Q w = {q w q Q}. A minimal image in an automaton is an image which does not popely contain anothe image. In an ieducible automaton two minimal images have the same cadinality which is called the minimal ank of A. Also, if I is a minimal image and u is a wod, then I u is again a minimal image and the map p p u is one-to-one fom I onto I u. Note that the hypotheses in the statement below depend on the choice of the lette a defining the ed edges. Lemma 4 (Tahtman [9]). Let A be an ieducible complete deteministic automaton with a positive level. If all maximal states in A belong to the same tee, then A has a stable pai. Poof. Since A is ieducible, thee is a minimal image I containing a maximal state p. Let l > 0 the level of p (i.e. the distance between p and the oot of the unique maximal tee). Let us assume that thee is a state q p in I of level l. Then the cadinal of I a l is stictly less than the cadinal of I, which contadicts the minimality of I. Thus all states but p in I have level stictly less than l. Let m be a common multiple of the lengths of all ed cycles. Let C be the ed cycle containing. Let s 0 be the pedecesso of in C and s 1 the child of containing p in its subtee. Since l > 0, we have s 0 s 1. Let 7

8 J = I a l 1 and K = J a m. Since the level of all states of I but p is less than o equal to l 1, the set J is equal to {s 1 } R, whee R is a set of states belonging to the ed cycles. Since fo any state q in a ed cycle, q a m = q, we get K = {s 0 } R. Let w be a wod such that Q w is a minimal image. Fo any wod v, the minimal images J vw and K vw have the same cadinal equal to the cadinal of I. We claim that the set (J K) vw is a minimal image. Indeed, J vw (J K) vw Q vw, hence all thee ae equal. But (J K) vw = R vw s 0 vw s 1 vw. This foces s 0 vw = s 1 vw since the cadinality of R vw cannot be less than the cadinality of R. As a consequence (s 0 v,s 1 v) is synchonizable and thus (s 0,s 1 ) is a stable pai. In the sequel, we call Condition C the assumption of Lemma 4: all maximal states belong to the same tee. In the subsections below, we descibe sequences of flips of edges that make the esulting equivalent automaton satisfy Condition C and hence have a stable pai. We conside seveal cases coesponding to the geomety of the automaton. 4.1 The case of null maximal level In this section, we assume that the level of the automaton is l = 0. The subgaph R of ed edges is a disjoint union of cycles. A set of edges going out of a state p is called a bunch if these edges all end in a same state q. Note that if a state q has two incoming bunches fom two states p,p, then (p,p ) is a stable pai. If the set of outgoing edges of each state is a bunch, then thee is only one ed cycle, and the automaton is not apeiodic unless the tivial case whee the length of this cycle is 1. We can thus assume that thee is a state p whose set of outgoing edges is not a bunch. Thee exists b a and q such that (p,a,q) and (p,b,) ae edges. We flip these two edges. This gives an automaton A which satisfies Condition C. Let s be the state which is the pedecesso of in its ed cycle. It follows fom the poof of Lemma 4 that the pai (p,s) is a stable pai. This case is descibed in the pseudocode LevelZeoFlipEdges whee GetPedecesso() etuns the pedecesso of on its ed cycle. The function LevelZeoFlipEdges(A) etuns an automaton equivalent to A togethe with a stable pai. 8

9 LevelZeoFlipEdges (automaton A of level l = 0) 1 fo each state p on a ed cycle C do if the set of outgoing edges of p is not a bunch then let e = (p,a,q) and f = (p,b,) be edges with b a and q 4 Flip(e,f) 5 s GetPedecesso() 6 etun A, (p,s) 7 etun Eo(A is not apeiodic) The pocedue Flip(e,f) exchanges the labels of two edges e,f. It also pefoms the coesponding update of data as explained in Section The case of non-null maximal level In this section, we assume that the level of the automaton is l > Main teatment We descibe a sequence of flips of edges such that the automaton obtained afte this sequence of flips has a unique maximal tee. Note that the levels and othe useful data will not be ecomputed afte each flip (which would incease the time complexity too much). Let C be a ed cycle containing a maximal tee T ooted at. We denote by 1 =,,... k the maximal oots of C in the ode given by the oientation of the ed edges of the cycle. Fo k > 1 and 1 i k we denote by I( i ) the set of states contained in the ed simple path fom the oot j with j = (i 1 mod k) + 1 to i with j included and i excluded. Fo k = 1 we define I() as the set of all states of C. Similaly, fo k > 1 and 1 i k we denote by J( i ) the set of states contained in the ed simple path fom the oot j with j = (i 1 mod k)+1 to i with j excluded and i included. Fo k = 1 we define J() as the set of all states of C. We denote by s 0 the pedecesso of in C. If the length of C is 1, s 0 =. We denote by S() the set of maximal childen of (i.e. which ae ancestos of some maximal state). Let ρ be the cadinality of S(). Fo each s in S(), we choose a maximal state p in the subtee ooted at s (see Fig. ). Thee may be seveal possible choices fo the state p and we select one of them abitaily. We denote by P() the set of these maximal states. This set has cadinality ρ. The key idea, in ode to guaantee the global linea complexity, is to pefom opeations fo each maximal oot, whose time complexity is linea in the numbe of nodes belonging to tees attached to the states contained in J(). Since the automaton is ieducible, fo each p P() thee is at least one blue edge ending in p. Each blue edge (t,b,p) ending in a state p P() 9

10 p 1 p p 1 p s 1 s s 1 s s 0 s 0 k k Figue : On the left pat the figue, the dashed (blue) edge ending in p 1 has type 1 while the one ending in p has type. On the ight pat, the set L s1 of dashed (blue) edges cove all maximal states of the subtee ooted at the child s 1. can be of one of the following type depending on the position of t in the gaph: type 0: t is not in the same cluste as, o t has a positive level and t is not an ancesto of p in T. type 1: t is in the same cluste as, has a null level, and t is outside the inteval I(). type : t is in the same cluste as, has a null level, and t is contained in the inteval I(). This includes the paticula case whee k = 1 and t =. type : t is an ancesto of p in T and t. Note that it is possible that t = p. In this case the edge (t,b,p) has type 0 since t has a positive level. A pocedue FindEdges(), that will be descibed late in detail (see Section 4..), fist flips some edges and etuns a value of one of the following foms. A pai (0,e), whee e is an edge of type 0. A tiple (1,e,f), whee e,f ae two edges of type 1 o ending in distinct states of P(). A pai (,e), whee e is an edge of type 1 o. Moeove, in this case, the pocedue modifies the tee T in such a way that has a unique maximal child. A pai (,e), whee e is an edge of type stating at a state which is an ancesto of all maximal nodes of T. 10

11 Fo each maximal oot, the pocedue FlipEdges(A, ) etuns eithe an automaton equivalent to A togethe with a stable pai, o an automaton equivalent to A togethe with one edge (t,b,p ). Its execution depends on the value etuned by FindEdges() accoding to the following fou cases descibed below. Afte unning FlipEdges(A, ) on each maximal oot, we obtain eithe an automaton satisfying Condition C (i.e. which has a stable pai) o an automaton whee each maximal oot has a unique maximal child and such that the potential flip of (t,b,p ) with the ed edge stating at t makes the oot not maximal anymoe. In the fist case, ou goal is achieved. In the latte case, we flip the blue edge (t,b,p ) and the ed one stating at t fo all maximal oots but one. We get an equivalent automaton which has unique maximal tee and thus has a stable pai by Lemma 4. The combination of all these tansfomations is ealized by the pocedue FindStablePai given at the end of this section. The possible values etuned by the pocedue FlipEdges(A, ) ae the following. Case 0. The value etuned by FindEdges() is (0,e) with e = (,b 1,p 1 ) of type 0. The pocedue FlipEdges(A,) etuns the automaton obtained by flipping the edge (,b 1,p 1 ) and the ed edge going out of. This automaton is equivalent to A and satisfies Condition C. Indeed, one may easily check that, afte the flip, all states of maximal level belong to the same tee as p 1. p 1 p p 1 p s 1 s s 1 s s 0 8 s Figue : The pictue on the left illustates Case 1.1. The edge (,b 1,p 1 ) if of type 1. Afte flipping the edge (,b 1,p 1 ) and the ed edge going out of, we get the automaton on the ight. It satisfies the Condition C, i.e. it has a unique maximal tee (hee ooted at ). Maximal states ae coloed and the (dashed) b-edges of the automaton ae not all epesented. Case 1. The value etuned by FindEdges() is (1,e 1,e ), with e 1 = (,b 1,p 1 ), e = (t,b,p ) of type 1 o. Recall that p 1 p and that 11

12 b 1,b a. Case 1.1. If e 1 (o e ) has type 1, the same conclusion as in Case 0 holds by flipping the edge (,b 1,p 1 ) and the ed edge going out of, as is shown in Fig.. Case 1.. In the case both edges e 1,e have type and t, without loss of geneality, we may assume that < t in the inteval I() (see Fig. 4). We flip the edge (,b 1,p 1 ) and the ed edge going out of. We denote by T the tee ooted at afte this flip. Case If the height of T is geate than l, the automaton satisfies Condition C (see the ight pat of Fig. 4). Case1... OthewisetheheightofT isatmostl(seetheleft pat of Fig. 5). In that case, we also flip the edge (t,b,p ) and the ed edge going out of t. The new equivalent automaton satisfies Condition C (see the ight pat of Fig. 5). The computation of the size of T is detailed in Section 5. p 1 p p 1 p 0 s 1 s 1 t 8 0 s 1 s 1 t Figue 4: The pictue on the left illustates Case 1..1 of the main teatment. Thee ae two edges (,b 1,p 1 ), (t,b,p ) of type. The height of the tee T obtained afte flipping the edge (,b 1,p 1 ) and the ed edge going out of, is, which is geate than the maximal level. We get a unique maximal tee ooted at in the same cluste. The pictue on the ight illustates the esult. Case 1.. In this case both edges e 1,e have type and = t. We denote by s 1 (esp. s ) the child of ancesto of p 1 (esp. p ). We denote by T 0 the tee ooted at obtained by the potential flip of (,b 1,p 1 ) and the ed edge going out of, keeping only and the subtee ooted at the child s 0. The nodes of the tee T 0 ooted at ae epesented in salmon in the left pat of Fig. 6. This step again needs a computation of the height of T 0 explained 1

13 p 1 p p 1 p 0 s 1 s 1 t 8 0 s 1 s 1 t Figue 5: The pictue on the left illustates Case 1... The two edges (,b 1,p 1 ), (t,b,p ) ae of type. The height of the tee T obtained afte flipping the edge (,b 1,p 1 ) and the ed edge going out of, is equal to l =. In this case, we also flip the edge (t,b,p ) and the ed edge going out of t. We get a unique maximal tee ooted at in the same cluste. The pictue on the ight gives the esulting cluste. in the complexity issue. Case 1. occus when ρ > 1, k = 1 and =. In the paticula case whee the length of C is 1, the tee T 0 is educed to the node (it coesponds to the Case 1.. below). Case If the height of T 0 is geate than the height of T, we flip (,b 1,p 1 ) and the ed edge going out of. The equivalent automaton satisfies Condition C. Case 1... If the height of T 0 is less than the height of T, we flip (,b 1,p 1 ) and the ed edge going out of. We then call again the pocedue FlipEdges(A, ) with this new ed cycle. This time the (new) tee T 0 has the same height as T. Hence this call is done at most one time fo a given maximal oot. Case 1... Finally, we conside the case whee the heights of T and T 0 ae equal (see the left pat of Fig. 6). Case If the set of outgoing edges of s 0 is a bunch and thee is a state s i S() whose set of outgoing edges is also a bunch, we get a tivial stable pai (s 0,s i ). Case 1... If the set of outgoing edges of s 0 is a bunch and, fo any state s S(), the set of outgoing edges of s is not a bunch (as in the left pat of Fig. 6), we flip (,b 1,p 1 ) and the ed edge going out of. The (new) tee T 0 (obtained by the potential flip of (t,b,p ) and the ed edge going out of, keeping only and the 1

14 subtee ooted at the child s 1 ) has the same height as T. We then call again the pocedue FlipEdges(A, ) with this new ed cycle. This time the height of the new tee T 0 is still equal to the height of T and the set of outgoing edges of the pedecesso of on the cycle is not a bunch. This call is thus pefomed at most one time. Case 1... If the set of outgoing edges of s 0 is not a bunch, let (s 0,b 0,q 0 ) be a b-edge going out of s 0 with q 0. If q 0 does not belong to T, we get an equivalent automaton satisfying Condition C by flipping (s 0,b 0,q 0 ) and the ed edge going out of s 0. If q 0 belongs to T, we flip (s 0,b 0,q 0 ) and the ed edge going out of s 0. We also flip (,b 1,p 1 ) and the ed edge going out of if q 0 is not a descendant of s 1, o (,b,p ) and the ed edge going out of, in the opposite case. Note that s 0 since the height of T 0 is equal to the non-null height of T. We get an equivalent automaton satisfying Condition C (see the ight pat of Fig. 7). p 1 p p 1 p 0 s 1 s 1 s s 1 s 1 s Figue 6: The pictue on the left illustates Case 1... of the main teatment. The two edges (,b 1,p 1 ) and (,b,p ) ae of type. Let T 0 be the tee ooted at obtained by the potential flip of (,b 1,p 1 ) and the ed edge going out of, keeping only and the subtee ooted at the child s 0. The nodes of the tee T 0 ooted at ae epesented in salmon in the left pat of the figue. The state s 0 is a bunch. Afte flipping the edge (,b 1,p 1 ) and the ed edge going out of, we get the automaton pictued in the ight pat of the figue. The tee T 0 is now tee ooted at obtained by the potential flip of (,b,p ) and the ed edge going out of, keeping only and the subtee ooted at the child s 1. Its states ae coloed in salmon. The height of T 0 is. Case. We now come to the case whee the value etuned by Find- 14

15 p p 1 p p 1 0 s s 1 1 s s s 1 1 s Figue 7: The pictue on the left illustates Case 1... The two edges (,b 1,p 1 ) and (,b,p ) ae of type. Let T 0 be the tee ooted at obtained by the potential flip of (,b 1,p 1 ) and the ed edge going out of, keeping only and the subtee ooted at the child s 0. The nodes of the tee T 0 ooted at ae epesented in salmon in the left pat of the figue. The state s 0 is not a bunch: it has a b-edge (s 0,b 0,q 0 ) with q 0 = s. Afte flipping the edge (,b 1,p 1 ) and the ed edge going out of, and flipping (s 0,b 0,q 0 ) and the ed edge going out of s 0, we get a unique maximal tee ooted at in the same cluste (see the ight pat of the figue). Edges() is a pai (,e) with e = (,b 1,p 1 ) of type 1 o, and T is modified in such a way that has a unique maximal child, i.e. ρ = 1. Case.1. If (,b 1,p 1 ) has type 1, we flip the edge (,b 1,p 1 ) and the ed edge going out of. We get an equivalent automaton satisfying Condition C. Case.. If (,b 1,p 1 ) has type, we denote by T 0 thetee ooted at obtained by the potential flip of (,b 1,p 1 ) and the ed edge going out of, keeping only and the subtee ooted at the child s 0. Case. occus when ρ = 1, k = 1 and =. In the paticula case whee the length of C is 1, T 0 is educed to the node which coesponds to the Case.. below. Case..1. If the height of T 0 is geate than the height of T, we do the flip and the equivalent automaton satisfies Condition C. Case... If the height of T 0 is less than the height of T, we do not do the flip, and etun the automaton togethe with the edge (,b 1,p 1 ). Note that a possible futue flip of (,b 1,p 1 ) and the ed edge stating at makes the oot not maximal anymoe. Case... We now come to the case whee the height of T 0 is equal to the height of T. 15

16 Case...1. If the set of outgoing edges of s 0 and s 1 ae bunches, thee is a tivial stable pai (s 0,s 1 ). Case... If the set of outgoing edges of s 0 is a bunch and the set of outgoing edges of s 1 is not a bunch (see the left pat of Fig. 8), we flip the edge (,b 1,p 1 ) and the ed edge going out of. We then call the pocedue FlipEdges(A,)with this new ed cycle. Theoot has now a unique child (s 1 ) ancesto of maximal state whose set of outgoing edges is a bunch (see the ight pat of Fig. 8). This call is thus pefomed at most one time. Case... Finally, if s 0 is a not a bunch, let (s 0,b 0,q 0 ) be a b-edge with q 0. If q 0 does not belong to T we flip the edge (s 0,b 0,q 0 ) and the ed edge going out of s 0. The equivalent automaton satisfies Condition C. It q 0 belongs to T and is not a descendant of s 1, we flip the edge (,b 1,p 1 ) and the ed edge going out of, and we also flip the edge (s 0,b 0,q 0 ) and the ed edge going out of s 0. The equivalent automaton satisfies Condition C. If q 0 belongs to T and is a descendant of s 1, we etun the automaton togethe with the edge (s 0,b 0,q 0 ). p p 0 s 0 s s 1 s Figue 8: The pictue on the left illustates Case... of the main teatment. The edge (,b 1,p 1 ) has type. Afte flipping the edge (,b 1,p 1 ) and the ed edge going out of, we get the automaton on the ight pat of the figue. The oot has a new single child s 1 ancesto of a maximal state, whose set of outgoing edges is a bunch. The new tee ooted at has hee the same level l = as befoe and FlipEdges(A,) is called a second and last time. Case. If the value etuned by FindEdges() is an edge (,b 1,p 1 ) of type and is an ancesto of all maximal nodes of T the pocedue FlipEdges(A, ) etuns this edge. 16

17 Afte unning FlipEdges(A, ) on all maximal oots, we get eithe an automaton with a stable pai, o an automaton whee each cluste fulfills the following conditions. the oot of each maximal tee has a unique maximal child; fo each maximal oot, thee is an edge (t,b,p ) such that the potential flip of (t,b,p ) and the ed edge stating at t makes the oot not maximal anymoe. If the latte case, we flip the blue edge (t,b,p ) and the ed one stating at t fo all maximal oots but one. We get an equivalent automaton which satisfies Condition C as is shown in Fig. 9. The pseudocode fo this final teatment is given in pocedue FindStablePai. FindStablePai (automaton A) 1 if the maximal level l = 0 then etun LevelZeoFlipEdges(A) else fo each maximal oot 4 do A,S FlipEdges(A,) 5 if S is a (stable) pai of states (s,t) 6 then etun A, (s,t) 7 else (S is a b-edge (t,b,p )) set e() = S 8 fo each maximal oot 0 9 do flip the edge e() and the ed edge stating at t 10 s GetPedecesso( 0 ) 11 t the child of 0 ancesto of p 0 1 etun A, (s,t)

18 p p s 1 s 0 0 s s Figue 9: The pictue on the left illustates the case whee FlipEdges(A, ) has etuned a b-edge e() fo all maximal oots. We flip e() and the ed edge stating at the same state fo all but one maximal oot. The new cluste is pictued on the ight pat of the figue. It has a unique maximal tee. By Lemma 4 the pai (6,15) is stable. 4.. The auxiliay pocedue FindEdges In this section, we descibe the pocedue FindEdges() which is a peliminay step of the pocedue FlipEdges(). Let be a maximal oot, S() be the set of maximal childen of. Fo each s in S(), we choose a maximal state p in the subtee ooted at s and we denote by P() the set of these maximal states (see Fig. ). Recall that the pocedue FindEdges() flips some edges and etuns an equivalent automaton togethe with one o two edges of the following foms. One edge e of type 0. Two edges e,f of type 1 o ending in distinct states of P(). One edge e of type 1 o. Moeove, in this case, the pocedue modifies the tee T in such a way that has a unique maximal child. One edge e of type stating at a state which is an ancesto of all maximal nodes of T. Fo each maximal child s, we denote by T s the subtee of T ooted at s. The pocedue FindEdges(,s) computes a list L s of b-edges (q,b,p), whee p is a maximal node of T s and q is an ancesto of p in T distinct fom. The stating states q of edges of this list cove the maximal nodes of T s in the following sense: fo each maximal node p in T s, thee is a unique edge (t,b,p) L s such that t is an ancesto of p (see fo instance the ight pat of Fig. ). The list L s is computed by scanning at most one time each node of the tee T s. Fo each maximal leaf p, we follow the ed edges up

19 to s and eithe find s o an aleady scanned node, o find a node with an outgoing b-edge ending in p. In the latte case, this edge is added to L s and wecontinue with anothe maximal leaf. Inthe case thelist L s does not cove all maximal nodes of T s, and since the gaph of the automaton is stongly connected, the pocess finds an edge (t s,b s,p s ) whee p s is a maximal node of T s, of type 0, 1 o. If thee is a maximal child s such that an edge (t s,b s,p s ) of type 0 is found, then FindEdges() etuns this edge. Othewise, if thee ae two maximal childen s 1 s such that two edges (t s1,b s1,p s1 ), (t s,b s,p s ) of type 1 o ae found, then FindEdges() etuns these two edges. If thee is a maximal child s 1 such an edge e = (t s1,b s1,p s1 ) of type 1 o and coveing lists L s fo the othe maximal childen s s 1 ae found, then we pefom the following flips. Fo any maximal child s s 1 and any edge (t,b,p) L s, we flip the edge (t,b,p) and the ed edge going out of t. We update the data of the tees attached to the nodes fom p to t in the new ed cycle ceated by the flip. Afte this tansfomation the node has s 1 as unique maximal child. The pocedue FindEdges() etuns the edge e of type 1 o and has a unique maximal child. Finally, if one obtains coveing lists fo all maximal childen, then, fo all these childen s but one, say s 1, we flip each edge (t,b,p) L s and the ed edge going out of t. We also flip all edges (t,b,p) L s1 but one, (,b 1,p 1 ). We update the data of the tees attached to the nodes fom p to t in the new ed cycle ceated by each flip. The pocedue FindEdges() etuns the edge (,b 1,p 1 ) of type. Its stating state is disctinct fom ans is an ancesto of all maximal states of T. 5 The complexity issue In this section, we establish the time and space complexity of ou algoithm. We denote by k the size of the alphabet A and by n the numbe of states of A. Since A is complete deteministic, it has exactly kn edges. 5.1 Data stuctues and thei updating Some data attached to the states is useful to obtain the claimed complexity. This data is updated afte the computation of each quotient automaton with the pocedue Update with a time complexity which is linea in the size of the quotient automaton. The edges of the automaton can be stoed in tables indexed by the states and labels. The updating pocedue computes the level of each state, the oot of its tee in its cluste. It also computes a list of maximal oots and the pedecesso of a state on the cycle. The function GetPedecesso(q) etuns the pedecesso of state q on its ed cycle in constant time.

20 One computes fo each oot of a tee T, the height of T, fo each maximal oot, the list of its maximal childen, fo each maximal child, the list of the maximal nodes belonging to the subtee ooted at this child. This data can be moeove updated in time linea in the size of the tee. We also maintain an invese stuctue of the quotient automaton. Giving a label c and a state q, it gives, fo each lette c, an unodeed list of states p such that thee is an edge (p,c,q) in the quotient automaton. Thepocedue Flip(e,f) exchanges the labels of the two edges e = (p,b,q),f = (p,a,q ). It also updates in the invese stuctue the lists of edges coming in p and p. Its time complexity is thus uppe bounded by the numbe of edges going out of p,p o coming in p,p. 5. Complexity of the algoithm Poposition 5. The wost-case complexity of FindSynchonizedColoing applied to an n-state apeiodic automaton is O(kn ) in time, and O(kn) in space. Poof. The complexity of FindSynchonizedColoing is at most n times the complexity of the pocedues Update and FindStablePai. Indeed, each call in the pocedue Mege educes the numbe of states of the automaton so that it is called at most n 1 times. Since each of its steps without the ecusive calls takes a time at most kn, the contibution of Mege in FindSynchonizedColoing is at most kn. As the pocedue Update has a time complexity O(kn), we just have to show that the time complexity of FindStablePai is O(kn). Since LevelZeoFlipEdges contains only one Flip call, we show that the calls to FlipEdges(A,) fo all maximal oots can be pefomed in time O(kn). We fist examine the complexity of the auxiliay step FindEdges() fo a given maximal oot. This pocedue equies a scan of the nodes of tees T s ooted at the maximal childen s of togethe with thei outgoing edges. Since the edges contained in the lists L s have distinct taget states in T, the flips of edges in L s can be pefomed with a time complexity at most E(), whee E() is the numbe of edges going out of o coming in a node of the tee T ooted by. Indeed, the update of the invese stuctue fo nodes in T can be pefomed one time fo all the flips of edges in L s. Note that the updating of the data afte the flips is at most the size of T. Indeed, afte a flip of (t,b,p) and (t,a,p ) only the nodes belonging to tees ooted at nodes along the ed path fom p to t ae updated. As a 0

21 consequence, the contibution of the auxiliay step in FindStablePai is O( E()) = O(kn). We now come to the complexity induced by the main teatment. We denote by Sect() the numbe of edges coming in o going out of a node belonging to the secto J(), i.e. the nodes contained in a tee attached to a node of the cycle between and ( included and ), whee is the maximal oot peceding on C. Let us compute fo instance the complexity of the pocedue UniqueChildFlipEdges(A,,e = (,b 1,p 1 )) (see Section 7). It contains at most two flips of edges ending in T. The height of the tee T 0 is easily computed by scanning all nodes attached to some node of C between and ( and both excluded). In the case whee this height is equal to l and the set of outgoing edges of s 0 is a bunch, we flip the edge e. We pefom the pocedue UpDateSecto(, e) fo updating the data of the nodes contained in the tees whose oots belong to J(). Then we call a second (and last) time FlipEdges(A,). Since the time complexity of UpDateSecto(, e) is at most Sect(), we get that the time complexity of UniqueChildFlipEdges(A,, e) is also Sect(). Similaly, the time complexity of the pocedues ChildenFlipEdgesEqual and ChildenFlipEdgesUnEqual is also Sect(). Hence the oveall time spent fo computing FlipEdges(A,) fo all maximalootsiso( Sect()) = O(kn). Thespacecomplexity iso(kn). Indeed, only linea additional space is needed to pefom all opeations. 6 The case of peiodic gaphs Recall that the peiod of an automaton is the gcd of the lengths of its cycles. If the automaton A is an n-state complete deteministic ieducible automaton which is not apeiodic, it is not equivalent to a synchonized automaton. Nevetheless, the pevious algoithm can be modified as follows fo finding an equivalent automaton with the minimal possible ank. It has a quadatic-time complexity. PeiodicFindColoing (automaton A) 1 B A while (size(b) > 1) do Update(B) 4 B,(s, t) FindStablePai(B) 5 lift the coloing up fom B to the automaton A 6 if thee is a stable pai (s,t) 7 then B Mege(B,(s,t)) 8 else etun A 9 etun A It may happen that FindStablePai etuns an automaton B which has no stable pai (it is made of a cycle whee the set of outgoing edges of any 1

22 state is a bunch). Lifting up this coloing to the initial automaton A leads to a coloing of the initial automaton whose minimal ank is equal to its peiod. This esult can be stated as the following theoem, which extends the Road Coloing Theoem to the case of peiodic gaphs. Theoem 6. Any ieducible automaton A is equivalent to a an automaton whose minimal ank is the peiod of A. Poof. Let us assume that A is equivalent to an automaton A which has a stable pai (s,t). Let B be the quotient of A by the conguence geneated by (s,t). Let d be the peiod of A (equal to the peiod of A) and d the peiod of B. Let us show that d = d. It is clea that d divides d (which we denote d /d). Let l be the length of a path fom s to s in A, whee s is equivalent to s. Since (s,s ) is stable, it is synchonizable. Thus thee is a wod w such that s w = s w. Since the automaton A is ieducible, thee is a path labeled by some wod u fom s w to s. Hence d/(l+ w + u ) and d/ (w + u ), implying d/l. Let s be the class of s and z be the label of a cycle aound s in B. Then thee is a path in A labeled by z fom s to x, whee x is equivalent to x. Thus d/ z. It follows that d/d and d = d. Suppose that B has ank. Let us show that A also has ank. Let I be a minimal image of A and J be the set of classes of the states of I in B. Two states of I cannot belong to the same class since I would not be minimal othewise. As a consequence I has the same cadinal as J. The set J is a minimal image of B. Indeed, fo any wod v, the set J v is the set of classes of I v which is a minimal image of A. Hence J v = J. As a consequence, B has ank. Let us now assume that A has no equivalent automaton which has a stable pai. In this case, we know that A is made of one ed cycle whee the set of edges going out of any state is a bunch. The ank of this automaton is equal to its peiod which is the length of the cycle. Hence the pocedue PeiodicFindColoing etuns an automaton equivalent to A whose minimal ank is equal to its peiod. Since the modification of FindSynchonizedColoing into PeiodicFindColoing does not change its complexity, we obtain the following coollay. Coollay 7. Pocedue PeiodicFindColoing finds a coloing of minimal ank fo an n-state ieducible automaton in time O(kn ). 7 Pseudocode This section contains the pseudocode of some main pocedues.

23 7.1 Pocedue Mege The computation of the conguence geneated by (s, t) can be pefomed by using usual Union/Find functions computing espectively the union of two classes andthe leade of the class of astate. Afte megingtwo classes whose leades ae p and q espectively, we need to mege the classes of p l and q l fo any l A. A pseudocode fo meging classes is given in Pocedue Mege below. Mege (automaton A, stable pai (s, t)) 1 x Find(s) y Find(t) if x y 4 then Union(x, y) 5 fo l A 6 do Mege(A,(x l,y l)) 7 etun A 7. Pocedue FlipEdges We give below a pseudocode of the pocedue FlipEdges(A, ). Fo each maximal oot, it etuns eithe an automaton equivalent to A togethe with a stable pai, o an automaton equivalent to A togethe with one edge. It pefoms some flips depending on the type of the edges etuned by FindEdges(). It calls UniqueChildFlipEdges(, e) in the case has a unique maximal child and e is an edge of type etuned by FindEdges(). It calls ChildenFlipEdgesUnequal(A, ) in the case has at least two maximal childen and FindEdges() etun a pai of edges with distinct stating states. It calls ChildenFlipEdgesUnequal(A, ) in the case has at least two maximal childen and FindEdges() etuns a pai of edges which have the same stating state. Recall that GetPedecesso() etuns the pedecesso of state on its ed cycle.

24 FlipEdges( automaton A, maximal oot ) 1 esult FindEdges() if ( a unique maximal child s 1 ) and (esult (,e)) then if (esult = (0,e) o (esult = (,e) whee e has type 1) 4 then Flip(e) 5 etun A and the stable pai (s 1,GetPedecesso()) 6 else (esult = (,e) whee e has type ) 7 etun UniqueChildFlipEdges(, e) 8 if ( at least two maximal childen) and (esult = (1,e 1,e ) whee e 1 = (,b 1,p 1 ),e = (t,b,p ) have type 1 o ) 9 then if t 10 then etun ChildenFlipEdgesUnequal(,e 1,e ) 11 else etun ChildenFlipEdgesEqual(,e 1,e ) 1 if esult = (,e) whee e is an edge of type 1 then etun A,e UniqueChildFlipEdges (automaton A, maximal oot, edge e = (,b 1,p 1 ) of type ) 1 let s 1 be the unique child of s 0 GetPedecesso()) let T 0 be the tee ooted at obtained by the potential flip of e and the ed edge going out of, keeping only and the subtee ooted at the child s 0 4 if height(t 0 ) > height(t) 5 then Flip(,b 1,p 1 ) 6 etun A and the stable pai (s 1,s 0 ) 7 if height(t 0 ) < height(t) 8 then etun A and the edge e 9 if height(t 0 ) = height(t) 10 then if the set of outgoing edges of s 0 and s 1 ae bunches 11 then etun A and the stable pai (s 0,s 1 ) 1 if the set of outgoing edges of s 0 is a bunch and the set of outgoing edges of s 1 is not a bunch 1 then Flip(,b 1,p 1 ) 14 UpDateSecto(,e) (we still have height(t 0 ) = height(t)) 15 etun FlipEdges(A, ) 16 if the set of outgoing edges of s 0 is not a bunch then let (s 0,b,q 0 ) a b-edge going out of s 0 with q 0 if q 0 / T then Flip(s 0,b 0,q 0 ) 0 if the level of q 0 is positive 1 then 0 the oot of the tee containing q 0 s GetPedecesso( 0 ) t the child of 0 ancesto of q 0 4 etun A and the stable pai (s,t) 5 else 0 the oot of the tee containing q 0 6 s GetPedecesso( 0 ) 7 etun A and the stable pai (s,s 0 ) 8 else (q 0 T and q 0 ) 9 etun A and the edge (s 0,b,q 0 ) 4

25 ChildenFlipEdgesEqual (automaton A, maximal oot, edges e 1,e ) of type 1 set e 1 = (,b 1,p 1 ) and e = (,b,p ) s 0 GetPedecesso() let T 0 be the tee ooted at obtained obtained by the potential flip of (,b 1,p 1 ) and the ed edge going out of, keeping only and the subtee ooted at s 0 4 if height(t 0 ) > height(t) 5 then Flip(,b 1,p 1 ) 6 etun A and the stable pai (s 1,s 0 ) 7 if height(t 0 ) < height(t) 8 then Flip(,b 1,p 1 ) 9 UpDateSecto(,e 1 ) 10 etun FlipEdges(A, ) 11 if height(t 0 ) = height(t) 1 then if the set of outgoing edges of s 0 is a bunch and thee is an intege i 1 such that the set of outgoing edges of s i is a bunch 1 then etun A and the stable pai (s 0,s i ) 14 if the set of outgoing edges of s 0 is a bunch and the sets of outgoing edges of s i fo i 1 ae not bunches 15 then Flip(,b 1,p 1 ) 16 UpDateSecto(,e 1 ) (we still have height(t 0 ) = height(t)) etun FlipEdges(A, ) if the set of outgoing edges of s 0 is not a bunch then let (s 0,b,q 0 ) a b-edge going out of s 0 with q 0 0 if q 0 / T 1 then Flip(s 0,b 0,q 0 ) if the level of q 0 is positive then 0 the oot of the tee containing q 0 4 s GetPedecesso( 0 ) 5 t the child of 0 ancesto of q 0 6 etun A and the stable pai (s,t) 7 else 0 the oot of the tee containing q 0 8 s GetPedecesso( 0 ) 9 etun A and the stable pai (s,s 0 ) 0 else (q 0 T) 1 if q 0 is not a descendant of s 1 then Flip(,b 1,q 1 ) Flip(s 0,b 0,q 0 ) 4 t the child of ancesto of q 0 5 etun A and the stable pai (s 1, ) 6 else (q 0 is a descendant of s 1 ) 7 Flip(t,b,q ) 8 Flip(s 0,b 0,q 0 ) 9 etun A and the stable pai (s 1,s ) The pocedue UpDateSecto(,e = (,b 1,p 1 )) is called afte a flip of the edge e and the ed edge going out of. It updates the data of the nodes (and thei tees attached to) along the ed path going fom p 1 to s 1, whee s 1 is the unique maximal child of. 5