Clustering in Trees: Optimizing Cluster Sizes and Number of Subtrees

Abstract

This paper considers partitioning the vertices of an n-vertex tree into p disjoint sets C1, C2, . . . , Cp, called clusters so that the number of vertices in a cluster and the number of subtrees in a cluster are minimized. For this NP-hard problem we present greedy heuristics which differ in (i) how subtrees are identified (using either a best-fit, good-fit, or first-fit selection criteria), (ii) whether clusters are filled one at a time or simultaneously, and (iii) how much cluster sizes can differ from the ideal size of c vertices per cluster, n = cp. The last criteria is controlled by a constant α, 0 ≤ α < 1, such that cluster Ci satisfies (1− α2 )c ≤ |Ci| ≤ c(1 + α), 1 ≤ i ≤ p. For algorithms resulting from combinations of these criteria we develop worst-case bounds on the number of subtrees in a cluster in terms of c, α, and the maximum degree of a vertex. We present experimental results which give insight into how parameters c, α, and the maximum degree of a vertex impact the number of subtrees and the cluster sizes. Communicated by G. Liotta: submitted November 1999, revised August 2000. 1. Hambrusch’s research supported in part by the National Science Foundation under Grant 9988339-CCR. 2. Lim’s research supported in part by Korea Science and Engineering Foundation under Contract No. 98-0102-07-01-3. S. E. Hambrusch et al., Clustering in Trees, JGAA, 4(4) 1–26 (2000) 2