Clustering in Trees: Optimizing Cluster Sizes and Number of Subtrees
Susanne E. Hambrusch, Chuan-Ming Liu, Hyeong-Seok Lim · Journal of Graph Algorithms and Applications · 2000
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 di#er 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 di#er from the ideal size of c vertices per cluster, n = cp. The last criteria is controlled by a constant #, 0 # # < 1, such that cluster C i satisfies (1 - # 2 )c # |C i | # c(1 +#), 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, #, and the maximum degree of a vertex. We present experimental results which give insight into how parameters c, #, and the maximum degree of a v...