On Finding and Updating Spanning Trees and Shortest Paths

Abstract

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.

Maintenance of a minimum spanning forest in a dynamic plane graph

“…Several algorithms have been proposed for maintaining fundamental structural information about dynamic graphs, such as connectivity [10,11,16,24,26], transitive closure [18,19,20,21,22,33,23], and shortest paths [1,9,25,28,33]. Dynamic planar graphs arise in communication networks, graphics, and VLSI design, and they occur in algorithms that build planar sul,,'i 'sions such as Voronoi diagrams.…”

“…Algorithms have been proposed for ma-i--ing the embedding of a planar graph [29] and for incremental planarity besting [2,3]. The dynamic minimum spanning tree problem has been considered by Spira and Pan [28], Chin and Houck [7], Frederickson [11], and Gabow and Stallmann [12]. The best result is that of Frederickson, who gave an algorithm based on "topology trees" that runs in O(v/m) time per operation on general graphs, and O((log n) 2 ) time on planar graphs.…”

Parallel Update of Minimum Spanning Trees in Logarithmic Time.

“…Spira and Pan [12] update the MST in O(n) time when a vertex is inserted in the graph. Their algorithm constructs the MST all over again by examining the n-I edges in the old MST and the new edges (there can be at most n of them) brought in by the…”

“…In particular, Spira and Pan [12], Chin Parallel algorithms for updating an MST have not been studied so far. In this paper we present parallel algorithms for updating an MST.…”

Combinatorial Optimization: What Is the State of the Art?

“…The in teres ted rea der should consul t Kerr (1970) , Sp ira andPan (1975), andYao et al (1977) concerning shortest paths , Spira and Pan (1975 ) and Shamo s and Hoey (1975) concern ing minimum spanning trees , Harper and Savage (1972) concerning maximum matching , Harper and Savage (1972) and Rabin (1972) concern in g shor tes t tours , Holt and Reingold (1972) …”