The principle of optimality in the design of efficient algorithms

Abstract

The influence of Richard Bellman is seen in algorithms throughout the computer science literature. Bellman’s principle of optimality has been used to develop highly efficient dynamic programming solutions to many important and difficult problems. The paradigm is now well entrenched as one of the most successful algorithm design tools employed by computer scientists. The optimality principle was given a broad and general statement by Bellman [23, making it applicable to problems of diverse types. Since computer programs are often employed to implement solutions based on the principle of optimality, Bellman’s impact on computing in general has been immense. In this paper we wish to focus in particular on the influence of Bellman’s work on the area of computer science known as algorithm design and analysis. A primary goal of algorithm design and analysis is to discover theoretical properties of classes of algorithms (e.g., how efficient they are, when they are applicable) and thus learn how to better apply the algorithms to new problems. From the perspective of algorithm design and analysis, combinatorial optimization problems form the class of problems on which the principle of optimality has had its greatest impact. Problem decomposition is a basic technique for attacking problems of this type-the solution to a large problem is obtained by combining solutions to smaller subproblems. The trick of this approach, of course, is to define an efficient decomposition procedure which assures that combining optimal solutions to subproblems will result in an optimal solution to the larger problem. As a standard course of action, computer scientists attempt to define a decomposition 97 0022-247X/86 $3.00