Abstract
One of the most common applications of graphs in everyday life is representing networks for traffic or for data communication. The schematic map of the German motorway system in the official guide, the railroad or bus lines in a public transportation system, and the network of routes an airline offers are routinely represented by graphs. Therefore, it is obviously of great practical interest to study paths in such graphs. In particular, we often look for paths which are good or even best in some respect: sometimes the shortest or the fastest route is required, sometimes we want the cheapest path or the one which is safest—for example, we might want the route where we are least likely to encounter a speed-control installation. Thus we will study shortest paths in graphs and digraphs in this chapter; as we shall see, this is a topic whose interest extends far beyond traffic networks. We will present some useful theoretical concepts (e.g., the Bellman equations, shortest path threes, and path algebras) as well as the most important algorithms for finding shortest paths (in particular, breadth first search, the algorithm of Dijkstra, and the algorithm of Floyd and Warshall). We also discuss two applications, namely to project scheduling and to finding optimal connections in a public transportation system.
So many paths that wind and wind…
Ella Wheeler Wilcox
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Notes
- 1.
- 2.
For nonnegative length functions, the undirected case can be treated by considering the complete orientation \(\stackrel{\rightarrow}{G}\) instead of G. If we want to allow edges of negative length, we need a construction which is considerably more involved, see Sect. 14.6.
- 3.
In particular, we may apply this technique to the distances in G, since they satisfy Bellman’s equations, which again proves the existence of SP-trees (under the stronger assumption that (G,w) contains only cycles of positive length).
- 4.
I owe the material of this section to my former student, Dr. Michael Guckert.
- 5.
Remember the Circle line in the London Underground system!
- 6.
We will neglect the amount of time a train stops at station v i . This can be taken into account by either adding it to the travelling time f(e i ) or by introducing an additional term w L (v i ) which then has to be added to t L (v i−1)+f(e i ).
- 7.
Note that we cannot just put t L (v 0)=0, as different lines may leave their start stations at different times.
- 8.
More precisely, the trains of line L leave station v at these times, that is, they reach v a little bit earlier. We assume that this short time interval suffices for the process of changing trains, so that we can leave this out of our considerations as well.
- 9.
It is obvious how this notion could be applied in the context of traffic or communication networks.
- 10.
This section is included just to provide some more theoretical background. As it will not be used in the rest of the book, it may be skipped.
- 11.
This assumption does not always hold in practice. For instance, gozinto graphs containing cycles are quite common in chemical production processes; see [Mul66].
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Jungnickel, D. (2013). Shortest Paths. In: Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32278-5_3
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