Igor Chikalov

Decision tree is a widely used form of representing algorithms and knowledge. Compact data models and fast algorithms require optimization of tree complexity. This book is a research monograph on average time complexity of decision trees. It generalizes several known results and considers a number of new problems. The book contains exact and approximate algorithms for decision tree optimization, and bounds on minimum average time complexity of decision trees. Methods of combinatorics, probability theory and complexity theory are used in the proofs as well as concepts from various branches of discrete mathematics and computer science. The considered applications include the study of average depth of decision trees for Boolean functions from closed classes, the comparison of results of the performance of greedy heuristics for average depth minimization with optimal decision trees constructed by dynamic programming algorithm, and optimization of decision trees for the corner point recognition problem from computer vision. The book can be interesting for researchers working on time complexity of algorithms and specialists in test theory, rough set theory, logical analysis of data and machine learning.

Average Time Complexity of Decision Trees

Mikhail Moshkov

Shahid Hussain

Hassan AbouEisha

Talha Amin

Dynamic programming is an efficient technique for solving optimization problems. It is based on breaking the initial problem down into simpler ones and solving these sub-problems, beginning with the simplest ones. A conventional dynamic programming algorithm returns an optimal object from a given set of objects. This book develops extensions of dynamic programming, enabling us to (i) describe the set of objects under consideration; (ii) perform a multi-stage optimization of objects relative to different criteria; (iii) count the number of optimal objects; (iv) find the set of Pareto optimal points for bi-criteria optimization problems; and (v) to study relationships between two criteria. It considers various applications, including optimization of decision trees and decision rule systems as algorithms for problem solving, as ways for knowledge representation, and as classifiers; optimization of element partition trees for rectangular meshes, which are used in finite element methods for solving PDEs; and multi-stage optimization for such classic combinatorial optimization problems as matrix chain multiplication, binary search trees, global sequence alignment, and shortest paths. The results presented are useful for researchers in combinatorial optimization, data mining, knowledge discovery, machine learning, and finite element methods, especially those working in rough set theory, test theory, logical analysis of data, and PDE solvers. This book can be used as the basis for graduate courses.

Extensions of Dynamic Programming for Combinatorial Optimization and Data Mining

Fawaz Alsolami

Mohammad Azad

The results presented here (including the assessment of a new tool – inhibitory trees) offer valuable tools for researchers in the areas of data mining, knowledge discovery, and machine learning, especially those whose work involves decision tables with many-valued decisions. The authors consider various examples of problems and corresponding decision tables with many-valued decisions, discuss the difference between decision and inhibitory trees and rules, and develop tools for their analysis and design. Applications include the study of totally optimal (optimal in relation to a number of criteria simultaneously) decision and inhibitory trees and rules; the comparison of greedy heuristics for tree and rule construction as single-criterion and bi-criteria optimization algorithms; and the development of a restricted multi-pruning approach used in classification and knowledge representation.

Igor Chikalov

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