# Min-plus matrix and Shortest path variation

I was solving a problem in which given a directed weighted graph with no self loops (adjacency matrix),I had to find minimum path of length at least K between ever pair of nodes.

One method is :

let say we want path from $$u$$ to $$v$$, let $$w$$ be the vertex in which we are after exactly $$k$$ edges. If the walk is shortest, we must get from $$u$$ to $$w$$ using a shortest walk with exactly $$k$$ edges (compute $$D^k$$ using Min-plus matrix multiplication) and then from $$w$$ to $$v$$ using Floyd-Warshall algorithm to compute all pair shortest distance.

Other method is what I don't understand and needed help with:

Calculate all pair shortest distance matrix using Floyd-Warshall algorithm and then compute $$D^k$$ using Min-plus matrix multiplication from that matrix, and every entry of it corresponds to the least path of length atleast $$k$$.

Needed some hints regarding correctness of second method also is doing such thing is a well known algorithm/method?

Thanks.