Lecture Notes Discrete Optimization - Applied Mathematics

definition, the interior nodes of P belong to the set {1,...,k}. There are two cases that
can occur: Either node k is an interior node of P or not.
First, assume that k is not an interior node of P. Then all interior nodes of P must belong
to the set {1,...,k−1}. That is, P is a shortest (u,v,k−1)-path and thus δ k (u,v) =
δ k−1 (u,v).
Next, suppose k is an interior node of P, i.e., P=〈u=v 1 ,...,k,...,v l = v〉. We can then
break P into two paths P 1 =〈u,...,k〉 and P 2 =〈k,...,v〉. Note that the interior nodes of
P 1 and P 2 are contained in{1,...,k−1} because P is simple. Moreover, because subpaths
of shortest paths are shortest paths, we conclude that P 1 is a shortest(u,k,k−1)-path and
P 2 is a shortest(k,v,k−1)-path. Therefore, δ k (u,v)=δ k−1 (u,k)+δ k−1 (k,v).
The above observations lead to the following recursive definition of δ k (u,v):
⎧
⎪⎨ 0 if u=v
δ 0 (u,v)= c(u,v) if (u,v)∈E
⎪⎩
∞ otherwise.
and
δ k (u,v)=min{δ k−1 (u,v), δ k−1 (u,k)+δ k−1 (k,v)}
if k≥1
The Floyd-Warshall algorithm simply computes δ k (u,v) in a bottom-up manner. The
algorithm is given in Algorithm 6.
Theorem 4.3. The Floyd-Warshall algorithm solves the APSP problem without negative
cycles in time Θ(n 3 ).
References
The presentation of the material in this section is based on [3, Chapters 25 and 26].
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