Data Structures for Maintaining Set Partitions

Each test or feature in a classification system defines a set partition on a class of objects. Adding new features refines the classification, whereas deleting features may result in merging previously distinguished classes. As an illustration, consider the set of automobile types { VW Beetle, Toyota, Lexus, Cadillac }. The feature size partitions the cars into sets of small and large cars, {{ VW Beetle, Toyota}, { Lexus, Cadillac }}. The feature domestic-origin partitions the cars into {{ VW Beetle, Toyota, Lexus }, { Cadillac }}. The feature ugly-shape distinguishes { VW Beetle, Cadillac } from { Toyota, Lexus }. Incorporating both size and origin induces the refined partition {{ VW Beetle, Toyota}, { Lexus }, { Cadillac }}, whereas the union of all three features completely distinguishes the types of cars. In fact, size and ugly-shape are sufficient for complete identification, so domestic-origin could be deleted from the set of features without affecting the induced partition. Efficiently maintaining the partition induced by a set of features is an important problem in building decision tree classifiers. For example, in building an optical character recognition (OCR) system [15,16] based on point-probe decision trees [1], each of the 1500-plus pixels in each character-sized window of the image may be evaluated as a possible feature. An important goal is to find a small, robust set of probe points sufficient to distinguish among the 70-plus characters in a font, a process that may require repeatedly inserting and deleting features to see the impact on the final classification. In this paper, we introduce techniques to speed up this process of feature identification. We propose a series of data structures for maintaining a collection of set partitions on elements U = {1, . . . , n}. The data structures efficiently support the following three operations: