[+] Shortest Paths with Negative Lengths 
We start with a recap of the MSSP algorithm and discuss how the data structure can be used to report shortest paths and distances. The bulk of this lecture is devoted to a nearlylinear time algorithm for shortest paths in the presence of negative lengths. The heart of the algorithm is a procedure to implement the BellmanFord algorithm on the dense distance graph  the complete graph on vertices of a cycle separator, where the length of an edge in the DDG corresponds to the length of a shortest path between its endpoints in the underlying planar graph. We show that implementing an iteration of BellmanFord can be cast into the problem of finding all column minima in the weighted incidence matrix of the DDG. We then introduce the Monge property, and show that we can find all column minima of a Monge matrix in time that is nearly linear in the number of rows. Finally we show that the weighted incidence matrix of the DDG can be decomposed into a small number of Monge matrices. 
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