A parallel algorithm for the enumeration of benzenoid hydrocarbons

arXiv:0808.0963v2 [cond-mat.stat-mech] 6 Feb 2009 A parallel algorithm for the enumeration of benzenoid hydrocarbons Iwan Jensen Department of Mathematics and Statistics The University of Melbourne, Vic. 3010, Australia October 31, 2018 Abstract We present an improved parallel algorithm for the enumeration of fixed benzenoids Bh containing h hexagonal cells. We can thus extend the enumeration of Bh from the previous best h = 35 up to h = 50. Analysis of the associated generating function confirms to a very high degree of certainty that Bh ∼Aκh/h and we estimate that the growth constant κ = 5.161930154(8) and the amplitude A = 0.2808499(1). Keywords: Benzenoids, hexagonal polygons, exact enumerations, par- allel processing, series analysis 1 Introduction A benzenoid or planar polyhex is a special type of hydrocarbon molecule. Its hexagonal system is obtained by deleting all carbon-hydrogen bonds, leaving clusters of hexagons joined at an edge (a carbon-carbon bond). They thus appear as clusters of identical hexagons in the plane. The interior of the clus- ters are filled with hexagons so there are no internal holes. These structures have appeared independently in the chemical and mathematical literature. In the mathematics literature they are discussed as self-avoiding polygons on the hexagonal lattice [1] and a distinction is made between fixed and free embeddings. Fixed polygons are considered distinct up to a translation while 1 free polygons are considered equivalent under translations, rotations and re- flections. Polygons are typically enumerated according to their perimeter or area. In the chemistry literature the number of free polygons [2] has been universally considered. The number of benzenoids or planar polyhexes is equal to the number of free hexagonal self-avoiding polygons enumerated by area. The enumeration of the number bh of benzenoids of h cells remains an important topic in computational and theoretical chemistry. The monograph by Gutman and Cyvin [2] provides a comprehensive review of all aspects. Until a few years ago progress was slow and incremental as calculations were based on direct counting of benzenoids. As the number of these grows as bh ∼κh, where the growth constant κ ≃5.16, it is clear that, to obtain one further term one needs more than 5 times the computing power. Up to 1989, the number of benzenoids up to h = 12 was known [2]. Ten years later this had been improved to h = 21 [3], while more recently, the number of benzenoids up to h = 24 was obtained [4]. In 2002 [5] a major break-through was obtained using a different type of algorithm that enabled the number of fixed benzenoids Bh to be enumerated for h ≤35 and bh was then obtained to the same size by using direct counting algorithms to enumerate benzenoids possessing certain symmetries, e.g. they may be symmetric with respect to an axis of reflection or certain rotations. For direct counting algorithms the CPU time taken to enumerate Bh grows as κh, whereas for our algorithm time consumption grows approximately as 1.65h, since 1.65 < κ ≃5.16 we may say that the new algorithm is exponentially faster than direct counting. Its drawbacks are that it is much more memory intensive (memory grows exponentially with h) than direct counting, for which memory requirements are negligible, as well as being much more difficult to implement. In [5] it was shown that there exists a growth constant κ such that lim h→∞B1/h h = κ (1) and the universally accepted, but as yet unproved, conjecture Bh ∼Aκhhθ as h →∞ (2) for the asymptotic form for Bh was confirmed to a high degree of certainty. It is widely accepted that for models such as benzenoids, other self-avoiding polygon models enumerated by area and polyominoes (or lattice animals) the exponent θ is given by the Lee-Yang edge singularity exponent [6] and thus 2 θ = −1 for benzenoids. Numerical analysis [5] confirmed this conjecture to a very high degree of certitude and yielded the estimate κ = 5.16193016(8) for the growth constant and A = 0.2808491(1) for the critical amplitude. In this paper we describe an efficient parallel version of the algorithm used in [5] and extend the count for fixed benzenoids up to h = 50. We do not attempt to count bh since asymptotically Bh = 12bh so any results regarding the asymptotic behaviour of Bh and bh are essentially the same (and the ratio of the two sequences Bh/bh converge rapidly to its asymptotic limit as evidenced by the fact that 12 −B35/b35 ≃1.355 × 10−10). Further- more the direct counting algorithms for benzenoids with a symmetry have computational complexity λh where λ = κ1/k if enumerating benzenoids with a k-fold symmetry so that in the worst case we have λ = √κ ≃2.27, which is a much worse asymptotic growth than that achieved with the algorithm for fixed benzenoids. Our analysis of the extended data yields the even more precise estimates κ = 5.161930154(8) and a revised estimate for the critical amplitude A = 0.2808499(1). 2 Computer algorithm A

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