Efficient Algorithm for the Partitioning of Trees

This paper describes an algorithm for partitioning a graph that is in the form of a tree. The algorithm has a growth in computation time and storage requirements that is directly proportional to the number of nodes in the tree. Several applications of the algorithm are briefly described. In particular it is shown that the tree partitioning problem frequently arises in the allocation of computer information to blocks of storage. Also, a heuristic method of partitioning a general graph based on this algorithm is suggested. Introduction Consider a graph G whose nodes have nonnegative integer weights and whose edges have positive values. A familiar combinatorial problem is the partitioning of G into subgraphs such that the sum of the node weights in any subgraph does not exceed a given maximum and the sum of the values of the edges joining the different subgraphs is minimal. This type of graph partitioning problem arises in a variety of forms in computer systems, e.g., the clustering of logic circuits onto integrated circuit chips and the mapping of computer information onto physical blocks of storage. No computationally efficient algorithm is known to exist for the partitioning of a general graph. However, partitioning algorithms have been described [ 1-81 that can be computationally efficient, given special features of the graph to be partitioned. Here we restrict our attention to connected, acyclic graphs, or trees, and describe a dynamic programming algorithm for the partitioning of a tree. This partitioning algorithm has a growth in computation time and storage requirements directly proportional to the number of nodes in the graph. The ability to partition a tree with integer-weighted nodes and multivalued edges has not been considered in the literature. An algorithm [ 2 ] has been reported that partitions a special type of tree with a growth in computation of n(log, n ) for an n-node tree; the edges of the tree must, however, assume a rather restricted set of values. In this paper we first define the partitioning problem, as well as some terms and concepts useful in its characterization. Next we describe the tree partitioning algorithm and illustrate it by an example. We then consider possible applications of the algorithm. In particular we show that a tree partitioning problem of the type described here can arise in the allocation of data in hierarchical files to physical blocks of storage. Also, we suggest a possible heuristic procedure based on the algorithm for the partitioning of a general graph. Definitions and basic concepts Assume a tree T = ( V , E ) with node set V and edge set E , as shown in Fig. 1. A partition of T is defined as a collection of k clusters of nodes {ci), i= 1,2 , . . ., k, such that