Generality and Computational Cost

The purpose of this note is pedagogi-cal. It discusses how one can reduce the computational cost of applying a set of operators (or predicates) by breaking them up into combinations of commonly occurring, simpler ones. This can be thought of as a process of generalization, in the sense that the common, simple operators are more "general" than the original, more complex ones. We are thus suggesting that even when one has a priori knowledge of a specialized nature (i.e., that the complex operators are applicable) , it may still be desirable to use generalized operators in order to reduce computational cost. To illustrate this idea, suppose that we want to apply a set of predicates P-,,...,P to an input I, and suppose that the cost of applying predicate Pi is (proportional to) the cardinality of its set of support Thus the total cost of applying the P's is For example, applying P. might involve a template-matching process, where Pi is true iff. a perfect match to the template is found in I. Here I could be an image, or a string (where the "template" is the right-hand side of a rule in a grammar), or a graph (where the "template" is a subgraph). In what follows, we will use the image/template metaphor. Suppose now that there exists a set of subtemplates Qj, ,Q m such that, for a concatenation of n. of X 1 the The cost of applying the 's support. If we store the match positions in a new array I 1 , then to test for P., we need only apply a template of cardin-ality n i to I'. Thus testing for all the and the total cost of the two-step matching process is Under what circumstances is the two-step cost less than the brute-force cost of applying the directly? We claim that this depends on the degree to which the Q's "generalize" the P's — i.e., on how few Q's are needed to construct all the P's. For concreteness, suppose that all the Q.'s have the same support size | and that each P. consists of the same number of Thus each P. has support size and the costs of the brute-force and two-step approaches are nrs and mr+ns, respectively. If there are few Q's, they must be used repeatedly, and we have m << ns (m=ns would mean that each Q is used only once); …