Sensitivity Analysis of Minimum Spanning Trees and Shortest Path Trees

Let G be a graph, either undirected or directed consisting of a vertex set V and an edge set E. We shall use n to denote the number of vertices and m to denote the number of edges of G. Suppose each edge e E E has an associated real-valued edn cost c(e). A number of network optimization problems call for determining a subgraph of such a graph that is minimum with respect to some function of the edge costs. For example, the minimum qwnniw pee problem is that of determining, for a connected, undirected graph G, a spanning tree of minimum total cdga co& The shortest path tree problem is that of computing, for a @en directed graph G and a given vertex r, a spanning tree rooted at r that contains a minImumcost path from r to every other vertex. In such optimization problems it may be u&al to measure the robustness of the solution. That isi, gWen a minimum subgraph, we would like to know by how much we can perturb each edge cost individually without changing the mMmality of the subgraph. We call this the sensMvity CIMIysis pr0ble.a Shier and Witqall[3) have proposed several algorithms for sensitivity anal* of shorteM path trees. Gusfield [2] has shown that two of their algorithms can be implemented to run in O(m 10s n) time and O(m) gprrcc, urd has noted that his techniques rrlro apply to 8ensiMty analysl8 of minimum spanning trees. In this paper we show how to perform sensitive ity anal* of minimum spanning trees and shortest path trees in q-m, n)) time and O(m) smce, whl re at is a function& inverse of Ackermurn’s function [Irr defined as follows. For integers i, j > 1 we define A(i,j)by