Compressed graphs representation has become an attractive research topic because of its applications in the manipulation of huge Web graphs in main memory. By far the best current result is the technique by Boldi and Vigna, which takes advantage of several particular properties of Web graphs. In this paper we show that the same properties can be exploited with a different and elegant technique, built on Re-Pair compression, which achieves about the same space but much faster navigation of the graph. Moreover, the technique has the potential of adapting well to secondary memory. In addition, we introduce an approximate Re-Pair version that works efficiently with limited main memory.

The recent Sparse Voronoi Refinement (SVR) Algorithm for mesh generation has the fastest theoretical bounds for runtime and memory usage. We present a robust practical software implementation of the SVR for meshing a piecewise linear complex in 3 dimensions. Our software is competitive in runtime with state of the art freely available packages on generic inputs, and on pathological worse cases inputs, we show SVR indeed leverages its theoretical guarantees to produce vastly superior runtime and memory usage. The theoretical algorithm description of SVR leaves open several data structure design options, especially with regard to point location strategies. We show that proper strategic choices can greatly effect constant factors involved in runtime.

Mesh refinement is the problem to produce a triangulation (typically Delaunay) of an input set of points augmented by Steiner points, such that every triangle or tetrahedron has good quality (no small angles). The requirement arises from the applications: in scientific computing and in graphics, meshes are often used to discretely represent the value of a function over space. In addition to the quality requirement, the user often has input segments or polygons (generally, a piecewise linear complex) they would like see retained in the mesh; the mesh must respect these constraints. Finally, the mesh should be size-conforming: the size of mesh elements should be related to a particular sizing function based on the distance between input features. The static meshing problem is increasingly well-understood: one can download software with provable guarantees that on reasonable input, the meshes will have good quality, will respect the input, and will be size-conforming; more recently, these algorithms have started to come with optimal runtimes of O(n lg(L/s) +m), where L/s is the spread of the input. As a first result, I

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