Time and space complexity of deterministic and nondeterministic decision trees

TL;DR

This paper analyzes the complexity of deterministic and nondeterministic decision trees over infinite binary information systems, classifying systems into five classes based on growth patterns of depth and size, and exploring time-space trade-offs.

Contribution

It provides a comprehensive classification of infinite binary information systems based on decision tree complexity and examines associated time-space trade-offs.

Findings

Deterministic decision tree depth grows logarithmically or linearly.

Nondeterministic decision tree depth is bounded or grows linearly.

Number of nodes in decision trees grows polynomially or exponentially.

Abstract

In this paper, we study arbitrary infinite binary information systems each of which consists of an infinite set called universe and an infinite set of two-valued functions (attributes) defined on the universe. We consider the notion of a problem over information system which is described by a finite number of attributes and a mapping corresponding a decision to each tuple of attribute values. As algorithms for problem solving, we use deterministic and nondeterministic decision trees. As time and space complexity, we study the depth and the number of nodes in the decision trees. In the worst case, with the growth of the number of attributes in the problem description, (i) the minimum depth of deterministic decision trees grows either almost as logarithm or linearly, (ii) the minimum depth of nondeterministic decision trees either is bounded from above by a constant or grows linearly, (iii) the minimum number of nodes in deterministic decision trees has either polynomial or exponential growth, and (iv) the minimum number of nodes in nondeterministic decision trees has either polynomial or exponential growth. Based on these results, we divide the set of all infinite binary information systems into five complexity classes, and study for each class issues related to time-space trade-off for decision trees.