Seidel's algorithm

The core of the algorithm is a procedure that computes the length of the shortest-paths between any pair of vertices. In the worst case this can be done in ${\displaystyle O(V^{\omega }\log V)}$ time. Once the lengths are computed, the paths can be reconstructed using a Las Vegas algorithm whose expected running time is ${\displaystyle O(V^{\omega }\log V)}$ for ${\displaystyle \omega >2}$ and ${\displaystyle O(V^{2}\log ^{2}V)}$ for ${\displaystyle \omega =2}$.

The Python code below assumes the input graph is given as a ${\displaystyle n\times n}$ ${\displaystyle 0}$-${\displaystyle 1}$ adjacency matrix ${\displaystyle A}$ with zeros on the diagonal. It defines the function APD which returns a matrix with entries ${\displaystyle D_{i,j}}$ such that ${\displaystyle D_{i,j}}$ is the length of the shortest path between the vertices ${\displaystyle i}$ and ${\displaystyle j}$. The matrix class used can be any matrix class implementation supporting the multiplication, exponentiation, and indexing operators (for example numpy.matrix).

def apd(A, n: int):
    """Compute the shortest-paths lengths."""
    if all(A[i][j] for i in range(n) for j in range(n) if i != j):
        return A
    Z = A**2
    B = matrix(
        [
            [1 if i != j and (A[i][j] == 1 or Z[i][j] > 0) else 0 for j in range(n)]
            for i in range(n)
        ]
    )
    T = apd(B, n)
    X = T * A
    degree = [sum(A[i][j] for j in range(n)) for i in range(n)]
    D = matrix(
        [
            [
                2 * T[i][j] if X[i][j] >= T[i][j] * degree[j] else 2 * T[i][j] - 1
                for j in range(n)
            ]
            for i in range(n)
        ]
    )
    return D

The base case tests whether the input adjacency matrix describes a complete graph, in which case all shortest paths have length ${\displaystyle 1}$.

Algorithms for undirected and directed graphs with weights from a finite universe ${\displaystyle \{1,\ldots ,M,+\infty \}}$ also exist. The best known algorithm for the directed case is in time ${\displaystyle {\tilde {O}}(M^{1/(4-\omega )}V^{2+1/(4-\omega )})}$ by Zwick in 1998.^[2] This algorithm uses rectangular matrix multiplication instead of square matrix multiplication. Better upper bounds can be obtained if one uses the best rectangular matrix multiplication algorithm available instead of achieving rectangular multiplication via multiple square matrix multiplications. The best known algorithm for the undirected case is in time ${\displaystyle {\tilde {O}}(MV^{\omega })}$ by Shoshan and Zwick in 1999.^[3] The original implementation of this algorithm was erroneous and has been corrected by Eirinakis, Williamson, and Subramani in 2016.^[4]