All-pair shortest path

Input direct $G=(V, E)$

Output: $(u, v)$ for each $u, v$

Flyod-Warshall Algorithm

$D[u, v, k]$

Case 1: $v_k$ is not in the path for $D[u, v, k]$

$\{u\cdots, S, \cdots,v\}, v_k\notin S$

then,

$D[u, v, k]=D[u, v, k-1]$

Case 2: $v_k$ is on the path

$\{u\cdots,v_k,\cdots,k\}$

$D[u, v, k]=D[u, v_k, k-1]+D[v_k,v,k-1]$

$$