Davide Cherubini - PhD Thesis - UniCA Eprints

6.2 Graphs and Network Flows6.2.3 The Shortest Path ProblemLet G = (N, A) be a directed and weighted graph, where a cost c ij is associatedat each arc (i, j). For each path P in G, the total cost C(P ) is the sum of thecosts of the arcs constituting P :C(P ) = ∑c ij (6.19)(i,j)∈PLet us consider two nodes r and t, and let P be the set of paths connecting rto t, then the corresponding shortest path problem is defined as:min{C(P ) : P ∈ P} (6.20)It is possible to formulate such problem as a Minimum Cost Flow Problem,with arc capacity = +∞ and the costs taken from the original shortest pathproblem. Furthermore, the source node r sends a single unit of flow receivedfrom the destination node t.where E is the incidence matrix, andmin cx (6.21)Ex = b (6.22)x ≥ 0 x integer (6.23)⎧⎨ −1 if i = rb i = 1 if i = t⎩0 otherwiseThe described problem (single-source shortest path) can be extended to findthe shortest paths for every pair of nodes in the network (all-pairs shortest pathproblem).The most important algorithms for solving this problem are listed in Table 6.2.The shortest path algorithms are used in telecommunication in order to findthe best route for the traffic flow.As mentioned in chapter 2, the Dijkstra’salgorithm is applied when the IS-IS or the OSPF routing protocol is used, whilethe Bellman-Ford’s algorithm is used in distance-vector routing protocols such as48