Replacement Paths via Row Minima of Concise Matrices

Matrix $M$ is {\em $k$-concise} if the finite entries of each column of $M$ consist of $k$ or less intervals of identical numbers. We give an $O(n+m)$-time algorithm to compute the row minima of any $O(1)$-concise $n\times m$ 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.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete Mathematic

An Efficient Algorithm for Row Minima Computations on Basic Reconfigurable Meshes

A matrix A of size m \Theta n containing items from a totally ordered universe is termed monotone if for every i; j; 1 i! j m, the minimum value in row j lies below or to the right of the minimum in row i. Monotone matrices and variations thereof are known to have many important applications. In particular, the problem of computing the row minima of a monotone matrix is of import in image processing, pattern recognition, text editing, facility location, optimization, and VLSI. Our first main contribution is to show that the task of computing the row minima of an m \Theta n monotone matrix, 1 m n, pretiled onto a Basic Reconfigurable Mesh of the same size can be performed in O(log n) time if m = 1; 2 and in O( log n log m log log m) time if m? 2. Our second contribution is to exhibit a number of non-trivial lower bounds for matrix search problems. These lower bound results hold for arbitrary infinite two-dimensional reconfigurable meshes as long as the input is pretiled onto an n \Theta n submesh thereof. Specifically, in this context we show that every algorithm that solves the problem of computing the minimum of an n \Theta n matrix must take\Omega\Gamma218 log n) time. The same lower bound is shown to hold for the problem of computing the minimum in each row of an arbitrary n \Theta n matrix. As a byproduct, we obtain an \Omega\Gamma/15 log n) time lower bound for the problem of selecting the k-th smallest item in a monotone matrix, thus extending the best previously-known lower bound for selection on the reconfigurable mesh. Finally, we show an almost tight \Omega\Gamma p log log n) time lower bound for the task of computing the row minima of a monotone n \Theta n matrix