Replacement Paths via Row Minima of Concise Matrices

Matrix M is k-concise if the finite entries of each column of M consist of k or fewer intervals of identical numbers. We give an O(n + m)-time algorithm to compute the row minima of any O(1)-concise n×m matrix. Our algorithm yields the first O(n+m)-time reductions from the replacement-paths problem on an n-node m-edge undirected graph (respectively, directed acyclic graph) to the single-source shortest-paths problem on an O(n)-node O(m)-edge undirected graph (respectively, directed acyclic graph). That is, we prove that the replacement-paths problem is no harder than the single-source shortest-paths problem on undirected graphs and directed acyclic graphs. Moreover, our linear-time reductions lead to the first O(n + m)-time algorithms for the replacement-paths problem on the following classes of n-node m-edge graphs: (1) undirected graphs in the word-RAM model of computation, (2) undirected planar graphs, (3) undirected minor-closed graphs, and (4) directed acyclic graphs.