Algorithmica (1993) 9:35%381 Algorithmica (C) 1993 Springer-Verlag New YorkInc. Optimal Parallel Algorithms for Multiple Updates of Minimum Spanning Trees Shaunak Pawagi 1 and Owen Kaser 1 Abstract. Parallel updates of minimum spanning trees (MSTs) have been studied in the past. These updates allowed a single change in the underlying graph, such as a change in the cost of an edge or an insertion of a new vertex. Multiple update problems for MSTs are concerned with handling more than one such change. In the sequential case multiple update problems may be solved using repeated applications of an efficient algorithm for a single update. However, for efficiency reasons, parallel algorithms for multiple update problems must consider all changes to the underlying graph simultan- eously. In this paper we describe parallel algorithms for updating an MST when k new vertices are inserted or deleted in the underlying graph, when the costs of k edges are changed, or when k edge insertions and deletions are performed. For multiple vertex insertion update, our algorithm achieves time and processor bounds of O(log n. log k) and nk/(log n. log k), respectively, on a CREW parallel random access machine. These bounds are optimal for dense graphs. A novel feature of this algorithm is a transformation of the previous MST and k new vertices to a bipartite graph which enables us to obtain the above-mentioned bounds. Key Words. CREW PRAM, Parallel algorithms, Optimality, Updates, Minimum spanning tree. 1. Introduction. Incremental graph algorithms deal with recomputing a solution to a computational problem on a graph after an incremental change is made to the graph, such as addition and deletion of vertices and edges, as well as changes in the costs or capacities (if any) associated with the edges of the graph. Such recomputations are also referred to as "updating" graph properties. The problem of updating a minimum spanning tree (MST) involves reconstructing the new MST from the current MST when a vertex, along with all its incident edges, is inserted or deleted from the underlying graph, or when an edge is inserted or deleted, or the cost of an edge (a tree edge or a nontree edge) has changed. These subproblems are referred to as the vertex update and the edge update problem, respectively. Sequential algorithms for updating MSTs have received considerable attention in the past [1], [5], [19]. In particular, Frederickson [5] describes an O(x/m) sequential algorithm for the edge update problem, where m is the number of edges in the graph. Spira and Pan [19] and Chin and Houck [1] present O(n) sequential algorithms for updating the MST of an n vertex graph when a new vertex is inserted into the graph. These algorithms are efficient when compared with the known start-over algorithms for MST construction; O(m log n) time for sparse graphs and O(n 2) time for dense graphs. (A start-over algorithm does not assume existence of any previous MST.) Also, an O(n)-time algorithm for single vertex I Department of Computer Science, State University of New York, Stony Brook, NY 11794, USA. Received August 2, 1990; revised February 15, 1991. Communicated by Kurt Mehlhorno