Evaluating Search Methods Analytically

A unified approach to analyzing search algorithms is presented. Each algorithm is characterized by the types of random problems that it can solve rapidly. The results are displayed in a way that clearly indicabes the strengths and weaknesses of each algorithm. Many interesting problems can, at present, best be solved by search methods [7]. In the worst case, searching requires exponent ial time. Several clever techniques have been developed to speed up searching (see, for example, 11, 3, 6, 8, 12, 13, 14, 15, 20, 21, 221). While each of these techniques is clearly helpful for some class of problems, it is difficult to evaluate the importance of each method (and of each combination of methods). Analytical studies have been done on several search methods [2, 4, 5, 9, 10, 11, 171. Each analysis was parameterixed in a way that emphasized search problems for which the method under consideration had an interesting behavior. Thus, Goldberg (9, lo] studied the pure literal rule using conjunctive normal form (CNF) predicates with long clauses, while Brown and Purdom (2, 17] studied backtracking using CNF predicates with short clauses. Since each algorithm has its own region of interesting behavior, the results of the analyses are difficult to compare with each other. In this paper we describe a unified approach to analyzing search algorithms, one that indicates the strengths and weaknesses of each algorithm in a way that makes comparisons straightforward. \Ve first analyze the average time behavior of each algorithm on random CNF problems characterized by u the number of variables, t the number of clauses, and p the probability a given literal is in a clause (so that the average length of a clause is 2pu ), This step is similar to the initial step of previous approaches, which continued by choosing particular functions p(u) and t(v) and studying the resulting asymptotic behavior. We continue by letting p(u) and t(v) be arbitrary functions of v , and finding the asymptotic behavior of the algorithm as u approaches infinity. Finally, we find the relation between PC’(J) and t(u) that charact,erizes the boundary bet,ween exponential and polynomial average time for the algorithm. The results can be displayed with a diagram of p(u),t(u) space which shows the exponential vs. polynomial contour for each algorithm. Fig. 1 shows the results for several basic algorit,hms. Contours are drawn for ordinary backtracking i:ese.srcb reported herein was supported in part by the Nrtionkl Science Foundation ur,der grant number hfCS 7008110. where all solutions are found, using fixed search order (the results are derived in [18]); searching using the pure literal rule and a fixed search order [lo, IS]; pattern of occurrence (an algorithm developed for use with problems that have only g few clauses) [18]; and elimination of unused variables followed by exhaustive search (this paper). Fig. 2. shows the results of using t.he same approach to obtain a more detailed comparison between ordinary backt,racking and simple search rearrangement [16]. From these figures it is clear that the pure liberal rule is useful for problems with large clauses (constraints that are easy to satisfy), that backtracking is useful for problems with many short clauses (large numbers of constraints that are not easy to satisfy), and that specialized methods are useful for problems with a small number of clauses (constraints). As more sophisticated algorithms are analyzed, the same approach can be used to identify the types of problems for which they are most suitable. The approach is even more useful for identifying which elementary techniques should be combined to produce powerful general purpose algorit,hms.