EFFECTIVE DYNAMIC PROGRAMMING ALGORITHMS

Abstract

This article is devoted to the analysis of the possibilities of increasing the speed of dynamic programming algorithms in solving applied problems of large dimension. Dynamic programming is considered rather than as an optimization method, but as a methodology that allows developing, from a single theoretical point of view, algorithms for solving problems that can be formalized in the form of multi-stage (multi-step) processes in which similar tasks are solved at all steps. It is shown that traditional dynamic programming algorithms based on preliminary setting of a regular grid of states are ineffective, especially if the parameters defining the states are not integer. The problems are considered, in the solution of which it is advisable to build a set of states in the process of counting, moving from one stage to another. Additional conditions are formulated that must be satisfied by the problem so that deliberately hopeless states do not fall into sets of states at each step. This ensures the rejection of not only the paths leading to each of the states, as in traditional dynamic programming algorithms, but also the unpromising states themselves, which greatly increases the efficiency of dynamic programming. Examples of applied problems are given, for the solution of which traditional dynamic programming algorithms were previously proposed, but which can be more efficiently solved by the proposed algorithm with state rejection. As applied to two-parameter problems, the concrete examples demonstrate the effectiveness of the algorithm with rejecting states in comparison with traditional algorithms, especially with increasing the dimension of the problem. An applied problem is considered, in the solution of which dynamic programming is used to construct recurrent formulas for calculating the optimal solution without enumerating options at all.