Algorithm 97 computes every shortest path length

Let $w$ assign a real length to every direct link in a directed network of $n$ points, with $\infty$ where no direct link exists. Assume every closed path in the network has nonnegative length. For all points $i$ and $j$, Floyd's Algorithm 97 leaves in entry $(i,j)$ exactly the shortest path length:

The equality also says that the final entry is $\infty$ when no path exists. It characterizes the complete output matrix, including diagonal entries and negative individual links.

Formalization Note The no-negative-cycle condition is added because the paper omits a necessary premise: on a one-point network with self-link length $-1$, the procedure produces $-2$ instead of the shortest simple closed-path length $-1$. The paper's finite sentinel ₁₀10 is represented as $\infty$, indices are zero-based, and arithmetic is exact real arithmetic. No nonnegativity of individual links or zero diagonal is assumed.