Split-Order Distance for Clustering and Classification Hierarchies

Abstract

Abstract. Clustering and classification hierarchies are organizational structures of a set of objects. Multiple hierarchies may be derived over the same set of objects, which makes distance computation between hierarchies an important task for summarization and similarity search of hierarchical patterns. In this paper, we model the classification and clustering hierarchies as rooted, leaf-labeled, unordered trees. We propose a novel distance metric Split-Order distance to evaluate the organizational structure difference between two hierarchies over the same set of leaf objects. The Split-Order distance reflects the order in which subsets of the tree leaves are differentiated from each other and can be used to explain the relationships between the leaf objects. We also propose an efficient algorithm for computing Split-Order distance between two trees in O(n 2 d 4 ) time, where n is the number of leaves, and d is the maximum number of children of any node. Our experiments on both real and synthetic data demonstrate the efficiency and effectiveness of our algorithm.

Split-Jaccard Distance of Hierarchical Decompositions for Software Architecture

“…Most previous work on comparing software clustering results has focused on only flat decompositions [6]. Our work most closely relates to recent work by Shtern et al [6], [15] and Zhang et al [7]. …”

“…P(u 1 , u 2 ) denotes a unique path that consists of a sequence of consecutive vertices connecting u 1 and u 2 . In the above, u and v are typically called splits [7] because each of them corresponds to the nearest common ancestor of the leaves with the labels l 1 and l 2 . Given trees…”

“…1. As indicated in [7], the general tree edit distance algorithms may generate answers contradicting our understanding of the hierarchies, when we consider unordered leaf-labeled trees as ordered fully-labeled trees. Figure 3 illustrates such examples.…”

“…The proposed distance is influenced by the split-order distance [7] does not suffer from an anomaly shown in Sect. 3.…”

“…Unfortunately, most of them are designed to handle only flat decompositions [6]; i.e., a cluster cannot contain another cluster, hence they cannot compare hierarchical decompositions although most software clustering algorithms produce hierarchical ones. It should be noted that tree edit distance algorithms cannot be directly applied to the problem of comparing software decompositions, which are typically represented as unordered leaf-labeled trees [7].…”