Enumeration of nonequivalent substitutional structures using advanced data structure of binary decision diagram

Abstract

A derivative structure is a nonequivalent substitutional atomic configuration derived from a given primitive cell. The enumeration of derivative structures plays an essential role in searching for the ground states in multicomponent systems. However, it is computationally difficult to enumerate derivative structures if the number of derivative structures of a target system becomes huge. In this study, we introduce a novel compact data structure of the zero-suppressed binary decision diagram (ZDD) for enumerating derivative structures much more efficiently. We show its simple applications to the enumeration of structures derived from the face-centered cubic and hexagonal close-packed lattices in binary, ternary, and quaternary systems. The present ZDD-based procedure should contribute to computational approaches based on derivative structures in physics and materials science.

Finding well-optimized special quasirandom structures with decision diagram

“…We apply the procedure to represent nonequivalent labelings using the ZDD, which the authors proposed in Ref. 7. The procedure is summarized in Sec.…”

“…The ZDDs for Cconc and C conc can be constructed with a procedure similar to that for constructing a ZDD representing labelings with a fixed concentration, as introduced in Ref. 7. Figure 4 (a) shows the ZDD corresponding to the binary equiatomic substitutional structures of Fig.…”

“…Recently, the authors have proposed the frontier-based ZDD method for efficiently enumerating derivative structures [7]. The algorithm for eliminating symmetrically equivalent labelings is based on Ref.…”

“…Recently, a compact data structure of the zerosuppressed binary decision diagram (ZDD) [6] has been applied to the efficient enumeration of derivative structures [7]. The ZDD enables us to store many structures in a compressed manner and enumerate more than 10 15 derivative structures in a reasonable time.…”

“…The ZDD enables us to store many structures in a compressed manner and enumerate more than 10 15 derivative structures in a reasonable time. Another advantage of using the ZDD is that a set of structures satisfying given constraints can be extracted from the ZDD efficiently, and the extraction of only nonequivalent structures is such a constraint [7]. In the field of discrete algorithms, the ZDD has been used to enumerate subgraphs of a given graph with a specific property, such as all self-avoiding paths between two vertices (s-t paths) [8,9].…”