Variable breeding phenology affects the exposure of amphibian embryos to ultraviolet radiation and optical characteristics of natural waters protect amphibians from UV-B in the U.S. Pacific Northwest: Comment

Peer Reviewe

Approximating Nearest Neighbor Distances

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we consider the most natural metric with this property, which we call the nearest neighbor metric. Given a point set P and a path γ, our metric charges each point of γ with its distance to P. The total charge along γ determines its nearest neighbor length, which is formally defined as the integral of the distance to the input points along the curve. We describe a (3+ε)-approximation algorithm and a (1+ε)-approximation algorithm to compute the nearest neighbor metric. Both approximation algorithms work in near-linear time. The former uses shortest paths on a sparse graph using only the input points. The latter uses a sparse sample of the ambient space, to find good approximate geodesic paths.</p

Solving 1-Laplacians in Nearly Linear Time: Collapsing and Expanding a Topological Ball

We present an efficient algorithm for solving a linear system arising from the 1-Laplacian corresponding to a collapsible simplicial complex with a known collapsing sequence. When combined with a result of Chillingworth, our algorithm is applicable to convex simplicial complexes embedded in ℝ3. The running time of our algorithm is nearly-linear in the size of the complex and is logarithmic on its numerical properties. Our algorithm is based on projection operators and combinatorial steps for transferring between them. The former relies on decomposing flows into circulations and potential flows using fast solvers for graph Laplacians, and the latter relates Gaussian elimination to topological properties of simplicial complexes.</p