In the context of Floyd's algorithm, the value of d$($i$,$j$,$k$)$ is the shortest distance from i to j when the intermediate vertices are chosen from $\{0,1,...,$ k$\}.$ In Floyd's algorithm, this corresponds to the value at row i and column j of the distance matrix after the $($k$+1)-$th iteration of the outer loop is complete. We use k $=-1$ when the set of possible intermediate vertices is empty.