Invariant manifold

Min-plus matrix multiplication, also known as the distance product, is an operation on matrices.

Given two $n\times n$ matrices $A=(a_{ij})$ and $B=(b_{ij})$, their distance product $C=(c_{ij})=A\star B$ is defined as an $n\times n$ matrix such that $c_{ij}=\min _{k=1}^{n}\{a_{ik}+b_{kj}\}$.

This operation is closely related to the shortest path problem. If $W$ is an $n\times n$ matrix containing the edge weights of a graph, then $W^{k}$ gives the distances between vertices using paths of length at most $k$ edges, and $W^{n}$ is the distance matrix of the graph.

References

• Uri Zwick. 2002. All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM 49, 3 (May 2002), 289–317.
• Liam Roditty and Asaf Shapira. 2008. All-Pairs Shortest Paths with a Sublinear Additive Error. ICALP '08, Part I, LNCS 5125, pp. 622–633, 2008.

See also

• Floyd–Warshall algorithm
• Tropical geometry: The distance product is equivalent to standard matrix multiplication in the tropical semi-ring.