Steiner trees in probabilistic networks

An Efficient Parallel Strategy for ComputingK-Terminal Reliability and Finding Most Vital Edges in 2-Trees and Partial 2-Trees

“…Wald and Colbourn [26] developed a linear time sequential algorithm to solve the K-terminal reliability problem in partial 2-trees. They first transform an arbitrary partial 2-tree into a 2-tree by adding dummy edges with probability measure zero.…”

“…In this subsection, we will present a parallel algorithm modified from the sequential algorithm in [26] to transform a connected partial 2-tree into a biconnected one by adding dummy edges each with a probability measure of zero. Before presenting the algorithm, we need the following definitions.…”

“…When a series reduction is applied to two edges, e 1 =x Ä y and e 2 = y Ä z, to form the edge e 3 =x Ä z, we use Rule A (described below) to summarize the probability measures (containing eight terms which are derived from the sequential algorithm in [26]) in e 3 . When a parallel reduction is applied to two edges e 1 =e 2 =x Ä y, we use Rule B to summarize the probability measures for the reduced edge x Ä y.…”

An efficient parallel strategy for computing K-terminal reliability and finding most vital edge in 2-trees and partial 2-trees

“…Wald and Colbourn [9] have developed a linear time sequential algorithm to solve the I<-terminal reliability in 2-trees: when a vertex with degree two is removed from G, they summarize the information about the triangle { z, y, z } , where y is the vertex with degree two, on the arcs (x, z ) and ( z , z) prior to deleting y (note that two arcs (z, z ) and ( z , x) may have different associated probability measures). With each arc Q = ( x , z ) , there are six associated probability measures which summarize the probabilities examined in a subgraph which has so far been reduced to the edge {z, z } .…”

“…Once a canonical tree is constructed, we can compute R ( G K ) of a 2-tree by dividing each 2-tree reduction into one series reduction and followed by one parallel reduction by using the rules modified from [9]. Moreover, it can be shown that those rules satisfy the computation requirements of applying tree contraction [7].…”