Parys has recently proposed a quasi-polynomial version of Zielonka's recursive algorithm for solving parity games. In this brief note we suggest a variation of his algorithm that improves the complexity to meet the state-of-the-art complexity of broadly $2^{O((\log n)(\log c))}$, while providing polynomial bounds when the number of colours is logarithmic

Lehtinen, Karoliina

Schewe, Sven

Wojtczak, Dominik

English

arXiv

Parys has recently proposed a quasi-polynomial version of Zielonka's
recursive algorithm for solving parity games. In this brief note we suggest a
variation of his algorithm that improves the complexity to meet the
state-of-the-art complexity of broadly $2^{O((\log n)(\log c))}$, while
providing polynomial bounds when the number of colours is logarithmic

University of Liverpool Repository

arXiv:1904.11810v4  [cs.GT]  5 Jun 2019Improving the complexity of Parys’ recursivealgorithm⋆K. Lehtinen, S. Schewe, and D. WojtczakUniversity of Liverpool, Liverpool, UK[k.lehtinen,sven.schewe,d.wojtczak]@liverpool.ac.ukAbstract. Parys has recently proposed a quasi-polynomial version ofZielonka’s recursive algorithm for solving parity games. In this brief notewe suggest a variation of his algorithm that improves the complexityto meet the state-of-the-art complexity of broadly 2O((logn)(log c)), whileproviding polynomial bounds when the number of colours is logarithmic.1 IntroductionIn 2017 Calude et al. published the ﬁrst quasi-polynomial algorithm for solv-ing parity games [CJK+17]. Since then, several alternative algorithms have ap-peared [Leh18,JL17], the most recent of which is Parys’s quasi-polynomial ver-sion of the Zielonka’s recursive algorithm [Par19].Parys’s algorithm, although enjoying much of the conceptual simplicity ofZielonka’s algorithm [Zie98], has a complexity that is a quasi-polynomial factorlarger than [CJK+17], [JL17], and [FJS+17]. More precisely, their complexity is,modulo a small polynomial factor,(c′+ll), with c′ being c or c/2 and l ∈ O(log n),for games with n positions and c colours. This also provides ﬁxed-parametertractability and a polynomial bound for the common case where the number ofcolours is logarithmic in the number of states. We propose a simpliﬁcation thatbrings the complexity of Pary’s algorithm down to match this. Note, however,that in a ﬁne grained comparison the recursive algorithm still operates symmet-rically, going through every colour, rather than just half of them, and O(log n)hides a factor of 2. Thus, a very careful analysis still reveals a small gap.We also brieﬂy comment on the relationship between this recursive algorithmand universal trees.2 PreliminariesA parity game G = (V, VE , E,Ω : V → [0..c]) is a two-player game betweenplayers Even and Odd, on a ﬁnite graph (V,E), of which positions are partitionedbetween those belonging to Even, VE and those belonging of Odd VO = V \ VE ,and labelled by pi with integer colour from a ﬁnite co-domain [0..c] by pi. Weassume that every position has a successor.⋆ Supported by EPSRC project Solving Parity Games in Theory and Practice.2 K. Lehtinen, S. Schewe, and D. WojtczakA play pi is an inﬁnite path through the game graph. It is winning for Evenif the highest colour occurring inﬁnitely often on it is even; else it is winning forOdd. We write pi[i] for the ith position in pi and pi[0, j] for its preﬁx of lengthj + 1.A strategy for a player maps every preﬁx of a play ending in a position thatbelongs to this player to one of its successors. A play pi agrees with a strategyσ for Even (Odd) if whenever pi[i] ∈ VE (VO), then σ(pi[0, i]) = pi[i + 1]. Astrategy for a player is winning from a position v if all plays beginning at v thatit agrees with are winning for that player. Parity games are determined: fromevery position, one of the two players has a winning strategy [Mar75].Even’s (Odd’s) winning region in a parity game is the set of nodes fromwhich Even (Odd) has a winning strategy. We are interested in the problem ofcomputing, given a parity game G, the winning regions of each player.Given a set S ⊆ V , the E-attractor of S in G, written AttrE(S,G), is theset of nodes from which Even has a strategy which only agrees with plays thatreach S. O-attractors, written AttrO(S,G) are deﬁned similarly for Odd.An even dominion is a set of nodes P ⊆ V such that nodes in P ∩ VE haveat least one successor in P and nodes in P ∩ VO have all of their successors inP , and Even has a winning strategy within the game induces by P . An odddominion is deﬁned similarly.We will use the following simple lemmas to prove the correctness of ouralgorithm.Lemma 1. If a dominion D for player P in a game G does not intersect withX, then it does not intersect with AttrP¯ (X,G) either, where P¯ is the opponentof P .Proof. From the deﬁnition of a dominion, player P has a strategy that fromwithin D only agrees with plays staying within D, contradicting any node in Dbeing within the attractor of X .Lemma 2. Let D be a dominion for player P in a game G. Then for all setsX, D \AttrP (X,G) is a dominion for P in G \AttrP (X,G).Proof. The same strategy that witnessesD being a dominion for P inGwitnessesD \AttrP (X,G) being a dominion for P in G \AttrP (X,G).Lemma 3. If the highest priority h in a dominion D for a player P in a playG is not of P ’s parity, then D contains a non-empty sub-dominion without h.Proof. Otherwise, every position in D would be in the attractor of the nodes ofpriority h, and the opponent would have a strategy to see h inﬁnitely often.3 The AlgorithmWe ﬁrst recall Parys’ quasi-polynomial version of Zielonka’s algorithm inAlgorithm 1. In brief, the diﬀerence between this algorithm and Zielonka’s is thatImproving the complexity of Parys’ recursive algorithm 3Algorithm 1 SolveE(G, h, pE , pO)1: if G = ∅ ∨ pE ≤ 1 then2: return ∅;3: end if4: while WO 6= 0 do5: Nh := {v ∈ G|pi(v) = h};6: H := G \AttrE(Nh, G)7: WO := SolveO(H,h− 1, ⌊pO/2⌋, pE);8: G := G \AttrO(WO, G);9: end while10: Nh := {v ∈ G|pi(v) = h};11: H := G \AttrE(Nh, G, )12: WO := SolveO(H,h− 1, pO, pE);13: G := G \AttrO(WO, G, );14: while WO 6= 0 do15: Nh := {v ∈ G|pi(v) = h};16: H := G \AttrE(Nh, G)17: WO := SolveO(H,h− 1, ⌊pO/2⌋, pE);18: G := G \AttrO(WO, G);19: end while20: return Gthis procedure takes a pair of parameters that bound the size of the dominions,for Eve and Odd respectively, that the procedure looks for; it ﬁrst removes oneplayer’s dominions (and their attractors) of size up to half the parameter untilthis does not yield anything anymore, then searches for a single dominion of thesize up to the input parameter, then again carries on with searching for smalldominions. In each of the recursive calls, the algorithm solves a parity game withone colour less, and either half the input parameter (most of the time) or thefull input parameter (once). The correctness hinges on the observation that onlyone dominion can be larger than half the size of the game, so the costliest callwith the full size of the game as parameter needs to be called just once.Our simpliﬁcation, in Algorithm 3, replaces each of the two while-loops witha single recursive call that also halves a precision parameter, but, unlike Parys’salgorithm, operates on the whole input game arena at once (or what is left of it,in the case of the last call), rather than on a series of subgames of lower maximalcolour. In brief, our algorithm computes three regions W1, W2 and W3 one afterthe other which together contain all small Odd dominions and no small Evendominion, in order to return the complement of their union. W1 and W3 arebased on calls that only identify very small dominions; the correctness of thealgorithm hinges on proving that W1 and W2 together already account for overhalf of any small Odd dominion, and hence the last call will correctly handlewhat is left.For both algorithms, the dual, SolveO is deﬁned by replacing E with O andvice-versa, although in our case SolveO returns ∅, rather than G, if h = 0.4 K. Lehtinen, S. Schewe, and D. WojtczakAlgorithm 2 SolveE(G, h, pE , pO)1: if G = ∅ then2: return ∅;3: end if4: if pO = 0 or h = 0 then5: return G6: end if7: W = G \ SolveE(G,h, pE , ⌊pO/2⌋);8: W1 = AttrO(W′O, G);9: G1 := G \W1;10: Nh := {v ∈ G1|pi(v) = h};11: G2 := G1 \AttrE(Nh, G1)12: W ′ := SolveO(G2, h− 1, pO, pE);13: W2 := AttrO(W′, G1)14: G3 := G1 \W2;15: W3 := G3 \ SolveE(G3, h, pE, ⌊pO/2⌋);16: G := G \ (W1 +W2 +W3);17: return G4 CorrectnessWe prove the following lemma, which guarantees that SolveE(G, h, pE , pO) andSolveO(G, h, pE , pO) partition a parity game G of maximal priority at most hinto a region that contains all odd dominions up to size up to pO and a regionthat contains all even dominions of size up to pE . Then SolveE(G, h, |G|, |G|)solves G of maximal priority h.Lemma 4. SolveE(G, h, pE , pO), where h is even and no smaller than the max-imal priority in G, returns a set that:i) contains all even dominions up to size pE, andii) does not intersect with an odd dominion with size up to pO.Similarly, SolveO(G, h, pO, pE), where h is odd and no smaller than the maximalpriority in G, returns a set that:i) contains all odd dominions up to size pO, andii) does not intersect with an even dominion with size up to pE.Proof. We show this by induction over the sum h+ pE + pO.Base case h+ pE + pO = 0. Then pE = pO = 0 and any set will do.Induction step We consider the case of SolveE ; the case of SolveO is similar.If h = 0, then G is a dominion for Even; we are done. We proceed with h > 0.We ﬁrst show i) that SolveE(G, h, pE , pO) returns all even dominions up tosize pE . Let D be such a dominion. According to the IH, D does not intersectwith W and therefore it does not intersect with W1 either. It is thereforecontained in G1.The intersection D′ of D and G2 is an even dominion in G2Improving the complexity of Parys’ recursive algorithm 5(Lemma 2) and therefore, from the IH, it does not intersect with W ′ norwith its Odd attractor W2 in G1 (Lemma 1). Then, D does not intersectwith W2 either. D′ is also contained in G3 and by IH does not intersect withW3 and therefore neither does D. Since D does not intersect with W1,W2nor W3 it is contained in the returned G.We proceed with showing ii) that SolveE(G, h, pE , pO) returns a set thatdoes not intersect with odd dominions of size up to pO. Let D be such adominion, let S be the union of odd dominions up to size ⌊pO/2⌋ containedin D and let A be its Odd-attractor in D.S is contained in W by IH, and therefore A is contained in W1 and does notintersect with G1. If A = D then D is contained in W1 and we are done.We consider the case of A 6= D. D \A is non-empty and a dominion in G\A(Lemma 2). It contains an odd dominion C of G \ A in which h does notoccur (Lemma 3). Observe that since C is an odd dominion in G \ A andA is an odd dominion in G, C ∪ A is an odd dominion in G. Since it is notincluded in S, it is larger than pO/2. We now show that C ∪ A is includedin W1 +W2.Since W1 is an odd attractor, G1 ∩ C is an odd dominion in G1, and sinceC contains no h, also in G2. By IH, it is contained in W2 and C is thereforecontained in W1 ∪W2, as is A ∪ C.Then, since A∪C is larger than pO/2, D \ (W1∪W2) is not only a dominionof G3 (Lemma 2), it is also of size up to ⌊pO/2⌋ and by IH contained in W3.Hence D is does not intersect with the returned G.5 AnalysisLet f(h, l) be the number of calls to SolveE and SolveO of SolveE(G, h, pE , pO)(or of SolveO(G, h, pE , pO), if it is greater) where l = ⌊log(pE)⌋+ ⌊log(pO)⌋.An induction on l+h shows that f(h, l) ≤ 2l(h+ll). If h+ l = 0 then h = 0 soSolveE(G, h, pE , pO) and SolveO(G, h, pE , pO) return immediately. For h+ l ≥1, we have:f(h, l) ≤ 2f(h, l− 1) + f(h− 1, l)≤ 2l−1(h+ l − 1l− 1)+ 2l(h+ l− 1l)≤ 2l(h+ ll) (1)Then, as l = 2⌊˙ logn⌋, this bring the complexity of the simpliﬁed algorithmdown by a quasi-polynomial factor from Parys’ version.Remark 1. A (n, d)-universal tree is a tree into which all trees of height d withn leaves can be embedded while preserving the ordering of children. These com-binatorial objects have emerged as a unifying thread among quasi-polynomial6 K. Lehtinen, S. Schewe, and D. Wojtczaksolutions to parity games and have therefore been the object of a recent spree ofattention [CDF+19,FGO18,CF]. In particular, the size of a universal trees is atleast quasi-polynomial, making this a potentially promising direction for lowerbounds. We observe that the call tree where the node SolveE(G, h, pE , pO) hasfor children its calls to SolveE and SolveO with parameter h − 1 takes theshape of a universal (n, d)-tree where n is the size of the parity game and d itsmaximal colour. The recursive approach therefore does not seem to be free fromuniversal trees either.6 ConclusionThis improvement brings the complexity of solving parity games recursivelydown to almost match the complexity to the algorithms based on Calude etal.’s method [CJK+17,FJS+17] and Jurdzin´ski and Lazic´’s algorithm [JL17].In particular it is ﬁxed-parameter tractable, and polynomial when the num-ber of colours is logarithmic. However, since the recursion solves the gamesymmetrically—that is, it goes through every colour, rather than just everyother colour—and since the size of only the guarantees for the even or odd do-minions are halved, in the(ab)notation both a (c vs. c/2) and b (2 logn vs. logn)double compared to Jurdzin´ski and Lazic´’s algorithm [JL17].Whether this simpliﬁcation to the recursion scheme makes this algorithmusable in practice remains to be seen.AcknowledgementsWe thank Nathanae¨l Fijalkow for enlightening discussions and Pawe l Parys forhis comments.ReferencesCDF+19. Wojciech Czerwin´ski, Laure Daviaud, Nathanae¨l Fijalkow, Marcin Jur-dzin´ski, Ranko Lazic´, and Pawe l Parys. Universal trees grow inside sep-arating automata: Quasi-polynomial lower bounds for parity games. InProceedings of the Thirtieth Annual ACM-SIAM Symposium on DiscreteAlgorithms, pages 2333–2349. SIAM, 2019.CF. Thomas Colcombet and Nathanae¨l Fijalkow. Universal graphs and good forsmall games automata: New tools for inﬁnite duration games. to appear inproceedings of FOSSACS 2019.CJK+17. Cristian S. Calude, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, and FrankStephan. Deciding parity games in quasipolynomial time. In Proceedings ofthe 49th Annual ACM SIGACT Symposium on Theory of Computing, pages252–263. ACM, 2017.FGO18. Nathanae¨l Fijalkow, Pawe l Gawrychowski, and Pierre Ohlmann. The com-plexity of mean payoﬀ games using universal graphs. arXiv preprintarXiv:1812.07072, 2018.Improving the complexity of Parys’ recursive algorithm 7FJS+17. John Fearnley, Sanjay Jain, Sven Schewe, Frank Stephan, and Dominik Wo-jtczak. An ordered approach to solving parity games in quasi polynomialtime and quasi linear space. In Proceedings of the 24th ACM SIGSOFT In-ternational SPIN Symposium on Model Checking of Software, pages 112–121.ACM, 2017.JL17. Marcin Jurdzin´ski and Ranko Lazic. Succinct progress measures for solvingparity games. In 2017 32nd Annual ACM/IEEE Symposium on Logic inComputer Science (LICS), volume 00, pages 1–9, June 2017.Leh18. Karoliina Lehtinen. A modal µ perspective on solving parity games in quasi-polynomial time. In Proceedings of the 33rd Annual ACM/IEEE Symposiumon Logic in Computer Science, pages 639–648. ACM, 2018.Mar75. Donald A Martin. Borel determinacy. Annals of Mathematics, pages 363–371, 1975.Par19. Pawe l Parys. Parity games: Zielonka’s algorithm in quasi-polynomial time.Available online: https://arxiv.org/abs/1904.12446, 2019.Zie98. Wieslaw Zielonka. Inﬁnite games on ﬁnitely coloured graphs with applica-tions to automata on inﬁnite trees. Theoretical Computer Science, 200(1):135– 183, 1998.

Improving the complexity of Parys' recursive algorithm

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Improving the complexity of Parys' recursive algorithm

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