Network Probabilistic Connectivity: Exact Calculation with Use of Chains

Abstract

Abstract.The algorithmic techniques which allow high efficiency in the exact calculation of reliability of an undirected graph with absolutely reliable nodes and unreliable edges are considered in this paper. The new variant of the branching algorithm that allow branching by chains is presented along with improvement of series-parallel reduction method that permits the reduction of a long chain by one step.

Network Probabilistic Connectivity: Using Node Cuts

“…For experiments we have used the computer with the processor Intel Celeron 1000 MHz, RAM=320Mb. Average time for a random graph is about 80 sec while the previous algorithm that in [1] was proven to be fastest at that time gives no results in an hour. At the same time the proposed algorithm is slightly slower for complete graphs (for K 9 we have 47 sec and 47.6 sec correspondingly) as a result of unfruitful tests on the Step 3.…”

“…Based on the presented results we propose the modification of the algorithm for strict calculation of the graph reliability that is discussed in [1]. Briefly this modification looks as follows (we assume that a graph under consideration has connected structure).…”

“…Step 1. Substitute all simple chains (if exist) by edges according to formulas from [1], ν is the corresponding factor (if there are no simple chains in a graph then ν = 1).…”

“…Yet we can conduct the exact calculation of reliability for networks with dimension of a practical interest by taking into consideration some special features of real network structures and based on modern high-speed computers. In [1] the technique was proposed that allows faster calculation based on reduction of simple chains and/or branching by them. This technique, in its turn, is based on the well-known factoring (branching) algorithm (often called MooreShannon algorithm [2]) uses branching on the alternative states for an arbitrary edge.…”

“…This technique, in its turn, is based on the well-known factoring (branching) algorithm (often called MooreShannon algorithm [2]) uses branching on the alternative states for an arbitrary edge. Some good approaches are proposed in [3,4] but, as it is shown in [1], it is possible to improve them. In this paper we present new approach to reducing the task dimension by considering 2-node cuts.…”