Efficient enumeration of stereoisomers of tree structured molecules using dynamic programming

Abstract

Nonredundant and exhaustive generation of stereoisomers of a chemical compound with a specified constitution is one of the important tools for molecular structure elucidation and molecular design. In this paper, we deal with chemical compounds composed of carbon, hydrogen, oxygen and nitrogen atoms whose graphical structures are tree-like graphs because these compounds are most fundamental, and consider stereoisomers that can be generated by asymmetric carbon atoms and double bonds between two adjacent carbon atoms. Based on dynamic programming, we propose an algorithm of generating all stereoisomers without duplication. We treat a given tree-like graph as a tree rooted at its structural center. Our algorithm first computes recursively the numbers of stereoisomers of the subgraphs induced by the descendants of each vertex, and then constructs each stereoisomer by backtracking the process of computing the numbers of stereoisomers. Our algorithm correctly counts the number of stereoisomers in O(n) time and space, and correctly enumerates all the stereoisomers in O(n) space and in O(n) time per stereoisomer, where n is the number of atoms in a given structure. The source code of the program implementing the proposed algorithm is freely available for academic use upon request.

Efficient Enumeration of Stereoisomers of Outerplanar Chemical Graphs Using Dynamic Programming

“…and h(v) for v which is not in any block are given in our previous study [11]. In the following, we consider the case when v is in some block.…”

“…The center of G consists of a bridge (v 1 , v 2 ): We compute f * (G) in the similar way to that of computing f * (G) of tree structured molecules G with the bicentroid [11]. …”

“…and h(v) as described in Section 3.1.1 or our previous study [11]; Let v be "scanned"; /* Let u be the parent of v. */ if all the children of u are "scanned" and u is not in the center of G then ENQUE(Q, u) end if end while; Compute f * (G) as described in Section 3.1.2.…”

“…We can find the centroid of BC(G) in O(n) time by Jordan's Theorem [12]. We can compute signatures of all rooted-subgraphs Sections 3.1.1, 3.1.2 and our previous study [11] take all the cases into consideration, and hence Counting phase computes the number of stereoisomers f * (G) correctly.…”

“…Similarly to our previous work [11], our algorithm is based on dynamic programming. For this, we treat a given outerplanar chemical graph as a graph rooted at its structural center, and derive recursive formulas for the number of stereoisomers of rooted subgraphs.…”