Worst-Case Efficient Sorting with QuickMergesort

The two most prominent solutions for the sorting problem are Quicksort and Mergesort. While Quicksort is very fast on average, Mergesort additionally gives worst-case guarantees, but needs extra space for a linear number of elements. Worst-case efficient in-place sorting, however, remains a challenge: the standard solution, Heapsort, suffers from a bad cache behavior and is also not overly fast for in-cache instances. In this work we present median-of-medians QuickMergesort (MoMQuickMergesort), a new variant of QuickMergesort, which combines Quicksort with Mergesort allowing the latter to be implemented in place. Our new variant applies the median-of-medians algorithm for selecting pivots in order to circumvent the quadratic worst case. Indeed, we show that it uses at most $n \log n + 1.6n$ comparisons for $n$ large enough. We experimentally confirm the theoretical estimates and show that the new algorithm outperforms Heapsort by far and is only around 10% slower than Introsort (std::sort implementation of stdlibc++), which has a rather poor guarantee for the worst case. We also simulate the worst case, which is only around 10% slower than the average case. In particular, the new algorithm is a natural candidate to replace Heapsort as a worst-case stopper in Introsort

Plane-Based Optimization of Geometry and Texture for RGB-D Reconstruction of Indoor Scenes

We present a novel approach to reconstruct RGB-D indoor scene with plane primitives. Our approach takes as input a RGB-D sequence and a dense coarse mesh reconstructed by some 3D reconstruction method on the sequence, and generate a lightweight, low-polygonal mesh with clear face textures and sharp features without losing geometry details from the original scene. To achieve this, we firstly partition the input mesh with plane primitives, simplify it into a lightweight mesh next, then optimize plane parameters, camera poses and texture colors to maximize the photometric consistency across frames, and finally optimize mesh geometry to maximize consistency between geometry and planes. Compared to existing planar reconstruction methods which only cover large planar regions in the scene, our method builds the entire scene by adaptive planes without losing geometry details and preserves sharp features in the final mesh. We demonstrate the effectiveness of our approach by applying it onto several RGB-D scans and comparing it to other state-of-the-art reconstruction methods.Comment: in International Conference on 3D Vision 2018; Models and Code: see https://github.com/chaowang15/plane-opt-rgbd. arXiv admin note: text overlap with arXiv:1905.0885

A top-down scenario for the formation of massive Tidal Dwarf Galaxies

Among those objects formed out of collisional debris during galaxy mergers, the prominent gaseous accumulations observed near the tip of some long tidal tails are the most likely to survive long enough to form genuine recycled galaxies. Using simple numerical models, Bournaud, Duc & Masset (2003) claimed that tidal objects as massive as 10^9 Msun could only form, in these simulations, within extended dark matter (DM) haloes. We present here a new set of simulations of galaxy collisions to further investigate the structure of tidal tails. First of all, we checked that massive objects are still produced in full N-body codes that include feedback and a large number of particles. Using a simpler N-body code with rigid haloes, we noticed that dissipation and self-gravity in the tails, although important, are not the key factors. Exploiting toy models, we found that, for truncated DM haloes, material is stretched along the tail, while, within extended haloes, the tidal field can efficiently carry away from the disk a large fraction of the gas, while maintaining its surface density to a high value. This creates a density enhancement near the tip of the tail. Only later-on, self-gravity takes over; the gas clouds collapse and start forming stars. Thus, such objects were fundamentally formed following a kinematical process, according to a top-down scenario, contrary to the less massive Super Star Clusters that are also present around mergers. This conclusion leads us to introduce a restrictive definition for Tidal Dwarf Galaxies (TDGs) and their progenitors, considering only the most massive ones, initially mostly made of gas, that were able to pile up in the tidal tails. More simulations will be necessary to precisely determine the fate of these proto--TDGs and estimate their number.Comment: 13 pages, 11 figures, accepted for publication in A&