An improvement on the complexity of factoring read-once Boolean functions

Read-once functions have gained recent, renewed interest in the fields of theory and algorithms of Boolean functions, computational learning theory and logic design and verification. In an earlier paper [M.C. Golumbic, A. Mintz, U. Rotics, Factoring and recognition of read-once functions using cographs and normality, and the readability of functions associated with partial k-trees, Discrete Appl. Math. 154 (2006) 1465-1677], we presented the first polynomial-time algorithm for recognizing and factoring read-once functions, based on a classical characterization theorem of Gurvich which states that a positive Boolean function is read-once if and only if it is normal and its co-occurrence graph is P₄-free. In this note, we improve the complexity bound by showing that the method can be modified slightly, with two crucial observations, to obtain an O (n | f |) implementation, where | f | denotes the length of the DNF expression of a positive Boolean function f, and n is the number of variables in f. The previously stated bound was O (n² k), where k is the number of prime implicants of the function. In both cases, f is assumed to be given as a DNF formula consisting entirely of the prime implicants of the function.