A distributed algorithm is presented that constructs the minimum-weight spanning tree in a connected undirected graph with distinct edge weights. A processor exists at each node of the graph, knowing initially only the weights of the adjacent edges. The processors obey the same algorithm and exchange messages with neighbors until the tree is constructed. The total number of messages required for a graph of N nodes and E edges is at most 5N log2N + 2E, and a message contains at most one edge weight plus log28N bits. The algorithm can be initiated spontaneously at any node or at any subset of nodes.

Abstract. We consider one origin shortest path and minimum spanning tree computations in weighted graphs. We give a lower bound on the number of analytic functions of the input computed by a tree program which solves either of these problems equal to half the number of worst-case comparisons which well-known algorithms attain. We consider the work necessary to update spanning tree and shortest path solutions when the graph is altered after the computation has terminated. Optimal or near-optimal algorithms are attained for the cases considered. The most notable result is that a spanning tree solution can be updated in O(n) when a new node is added to an n-node graph whose minimum spanning tree is known.Key words, spanning trees, shortest paths, lower bounds on computation, graph computations 1. Synopsis of results. Dijkstra [2] has given an algorithm to find all shortest paths from a single origin in a directed graph with positive arc weights and Prim has given an algorithm t6 find a minimal spanning tree in an undirected graph. We discuss the optimality of these algorithms in the sequel and show that no program whose unit operation is the evaluation and testing for positivity of an analytic function ofthe weights can better these algorithms by more than a factor of two. We then consider the problem ofupdating previous shortest path and minimum spanning tree solutions when parameters of the graph are changed. We consider what must be done when nodes are added or deleted and when weights on arcs are increased or decreased. We obtain lower bounds and optimal or near optimal algorithms for these problems in terms of how many analytic functions of the weights must be considered.

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S c i e n t i f i c Report No. 24 Prepared under N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o nGrant N s G 377 R a d i o s c i e n c e Laboratory S t a n f o r d U n i v e r s i t y S t a n f o r d , C a l i f o r n i a S t a n f o r d E l e c t r o n i c s L a b o r a t o r i e s