In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary structures from
data sampled around them. We prove that they can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and we provide an efficient and easy to implement algorithm to compute such approximations in almost linear time. We illustrate the performances of our algorithm on a few synthetic and real data sets.
The phase diagram and equation of state of dense nitrogen are of interest in understanding the fundamental physics and chemistry under extreme conditions, including planetary processes, and in discovering new materials. We predict several stable phas
es of nitrogen at multi-TPa pressures, including a P4/nbm structure consisting of partially charged N2 pairs and N5 tetrahedra, which is stable in the range 2.5-6.8 TPa. This is followed by a modulated layered structure between 6.8 and 12.6 TPa, which also exhibits a significant charge transfer. The P4/nbm metallic nitrogen salt and the modulated structure are stable at high pressures and temperatures, and they exhibit strongly ionic features and charge density distortions, which is unexpected in an element under such extreme conditions and could represent a new class of nitrogen materials. The P-T phase diagram of nitrogen at TPa pressures is investigated using quasiharmonic phonon calculations and ab initio molecular dynamics simulations.
