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. 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. 2. Definitions and preliminaries. Let G be an n-node with d the distance from node to node j so that G is undirected if d d for all and j. DEFINITION 2.1. An analytic tree program T is one defined by a rooted tree. Each internal node and the root are labeled by analytic functions, and each leaf is labeled by an answerthe output of the program. Computation begins at the root. At each node the analytic function is evaluated and the next node visited is the left or right successor of the present node. Computation terminates when a terminal node is reached. The depth of T, d (T), is the length of the longest branch.

No abstract

No abstract

No abstract

No abstract