Abstract
The enumeration of all minimal paths between a terminal pair of a given graph is widely used in a lot of applications such as network reliability assessment. In this paper, we present a new and efficient algorithm to generate all minimal paths in a graph G(V, E). The algorithm proposed builds the set of minimal paths gradually, starting from the source nodes. We present two versions of our algorithm; the first version determines all feasible paths between a pair of terminals in a directed graph without cycle, and this version runs in linear time O(|V| + |E|). The second version determines all minimal paths in a general graph (directed and undirected graph). In order to show the process and the effectiveness of our method, an illustrative example is presented for each case.
Highlights
The evaluation of the reliability of a system that can be modeled as a network can be made in terms of either minimal cuts (MCs) or minimal paths (MPs) [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]
The use of MPs and MCs for reliability assessment is well documented in [15], [16], and for details on the use of MPs and MCs in reliability evaluation, we refer to these papers
1) The principle of the algorithm: Starting from the fact that the minimal paths linking a source node s to another node x can be obtained from the lists of minimal paths linking s to the predecessors of x by applying a simple increase of the paths eq 1, the main idea of our method is to build, little by little, all minimal paths Pst
Summary
The evaluation of the reliability of a system that can be modeled as a network can be made in terms of either minimal cuts (MCs) or minimal paths (MPs) [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Another family of algorithms for MPs enumeration is called, according to Chen [21], direct search-based algorithms [21], [23], [24], [25], [20], [10]. We present a new method to enumerate all minimal paths in an oriented graph with no cycles. We will, first, introduce a directed graph reduction algorithm to eliminate nodes that cannot appear in the set of minimal paths. We will conclude with some suggestions for future research in the field of minimal paths’ enumeration
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