# Invariant manifold

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

Given two matrices and , their distance product is defined as an matrix such that .

This operation is closely related to the shortest path problem. If is an matrix containing the edge weights of a graph, then gives the distances between vertices using paths of length at most edges, and 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.