A Partial Ranking Algorithm for Resource Allocation Problems

We present an algorithm to solve resource allocation problems with a single resource, a convex separable objective function, a convex separable resource-usage constraint and bounded variables.Through evaluation of specific functions in the lower and/or upper bounds, we obtain information on whether or not these bounds are binding.Once this information is available for all variables, the optimum is found through determination of the unique root of a strictly decreasing function.A comparison is made with the currently known most efficient algorithms.Programming;non-linear

Probabilistic embeddings of the Fr\'echet distance

The Fr\'echet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric space. However, its inherent complexity poses considerable computational challenges in practice. To address this problem we study distortion of the probabilistic embedding that results from projecting the curves to a randomly chosen line. Such an embedding could be used in combination with, e.g. locality-sensitive hashing. We show that in the worst case and under reasonable assumptions, the discrete Fr\'echet distance between two polygonal curves of complexity $t$ in $\mathbb{R}^d$, where $d\in\lbrace 2,3,4,5\rbrace$, degrades by a factor linear in $t$ with constant probability. We show upper and lower bounds on the distortion. We also evaluate our findings empirically on a benchmark data set. The preliminary experimental results stand in stark contrast with our lower bounds. They indicate that highly distorted projections happen very rarely in practice, and only for strongly conditioned input curves. Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure

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

A Partial Ranking Algorithm for Resource Allocation Problems

We present an algorithm to solve resource allocation problems with a single resource, a convex separable objective function, a convex separable resource-usage constraint and bounded variables.Through evaluation of specific functions in the lower and/or upper bounds, we obtain information on whether or not these bounds are binding.Once this information is available for all variables, the optimum is found through determination of the unique root of a strictly decreasing function.A comparison is made with the currently known most efficient algorithms.