Version 5.0 The LEDA User Manual

12.2 Shortest Path Algorithms ( shortest path )
Let G be a graph, s a node in G, and c a cost function on the edges of G. Edge costs
may be positive or negative. For a node v let µ(v) be the length of a shortest path from
s to v (more precisely, the infimum of the lengths of all paths from s to v). If v is not
reachable from s then µ(v) = +∞ and if v is reachable from s through a cycle of negative
cost then µ(v) = −∞. Let V + , V f , and V − be the set of nodes v with µ(v) = +∞,
−∞ < µ(v) < +∞, and µ(v) = −∞, respectively.
The solution to a single source shortest path problem (G, s, c) is a pair (dist, pred) where
dist is a node array and pred is a node array with the following properties.
Let P = {pred[v] ; v ∈ V and pred[v] ≠ nil }. A P -cycle is a cycle all of whose edges
belong to P and a P -path is a path all of whose edges belong to P .
• v ∈ V + iff v ≠ s and pred[v] = nil and v ∈ V f ∪ V − iff v = s or pred[v] ≠ nil.
• s ∈ V f if pred[s] = nil and s ∈ V − otherwise.
• v ∈ V f if v is reachable from s by a P -path and s ∈ V f . P restricted to V f forms
a shortest path tree and dist[v] = µ(s, v) for v ∈ V f .
• All P -cycles have negative cost and v ∈ V − iff v lies on a P -cycle or is reachable
from a P -cycle by a P -path.
Most functions in this section are template functions. The template parameter NT can be
instantiated with any number type. In order to use the template version of the function
the .h-file
#include
must be included. The functions are pre-instantiated with int and double. The function
names of the pre-instantiated versions are without the suffix _T.
Special care should be taken when using the functions with a number type NT that
can incur rounding error, e.g., the type double. The functions perform correctly if all
arithmetic performed is without rounding error. This is the case if all numerical values
in the input are integers (albeit stored as a number of type NT ) and if none of the
intermediate results exceeds the maximal integer representable by the number type (2 52
in the case of doubles). All intermediate results are sums and differences of input values,
in particular, the algorithms do not use divisions and multiplications. All intermediate
values are bounded by nC where n is the number of nodes and C is the maximal absolute
value of any edge cost.