Notation

Algorithms use an adjacency matrix w to represent graphs, where

$\begin{displaymath}w_{ij} \;=\; \left\{ \begin{array}{ll} 0 & {\rm if} \;i=j \\ ... ...;i \neq j \;{\rm and} \;(i,j) \not\in E \\ \end{array} \right.\end{displaymath}$

Negative weights are allowed, but not negative cycles.

Algorithms output a distance matrix D, where d_ij is the weight of the shortest path from i to j.

For example, $d_{ij} \;=\; \delta(i,j)$ .

Algorithms may also output predecessor matrix $\Pi$ , where $\pi_{ij}$ is NIL if either i=j or there is no path from i to j; otherwise, $\pi_{ij}$ is the predecessor of j on a shortest path from i.

From the predecessor matrix $\Pi$ , we can derive the predecessor subgraphs $G_{\pi, i} \;=\; (V_{\pi, i}, E_{\pi, i})$ for each vertex i $\in$ V as

$\begin{displaymath}V_{\pi, i} \;=\; \{j \in V \;\mid\; \pi_{i,j} \neq NIL\} \;\cup\; \{i\}\end{displaymath}$

$\begin{displaymath}E_{\pi, i} \;=\; \{(\pi_{i,j}, j) \;\mid\; j \in V_{\pi, i} \;{\rm and} \; \pi_{i,j} \neq NIL\}\end{displaymath}$