Trees with Convex Faces and Optimal Angles

We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons. Among all such drawings we seek the one maximizing the angular resolution of the drawing. We find linear time algorithms for solving this problem, both for plane trees and for trees without a fixed embedding. In any such drawing, the edge lengths may be set independently of the angles, without crossing; we describe multiple strategies for setting these lengths.Comment: 12 pages, 10 figures. To appear at 14th Int. Symp. Graph Drawing, 200

Witness (Delaunay) Graphs

Proximity graphs are used in several areas in which a neighborliness relationship for input data sets is a useful tool in their analysis, and have also received substantial attention from the graph drawing community, as they are a natural way of implicitly representing graphs. However, as a tool for graph representation, proximity graphs have some limitations that may be overcome with suitable generalizations. We introduce a generalization, witness graphs, that encompasses both the goal of more power and flexibility for graph drawing issues and a wider spectrum for neighborhood analysis. We study in detail two concrete examples, both related to Delaunay graphs, and consider as well some problems on stabbing geometric objects and point set discrimination, that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200

Efficient Computation of Expected Hypervolume Improvement Using Box Decomposition Algorithms

In the field of multi-objective optimization algorithms, multi-objective Bayesian Global Optimization (MOBGO) is an important branch, in addition to evolutionary multi-objective optimization algorithms (EMOAs). MOBGO utilizes Gaussian Process models learned from previous objective function evaluations to decide the next evaluation site by maximizing or minimizing an infill criterion. A common criterion in MOBGO is the Expected Hypervolume Improvement (EHVI), which shows a good performance on a wide range of problems, with respect to exploration and exploitation. However, so far it has been a challenge to calculate exact EHVI values efficiently. In this paper, an efficient algorithm for the computation of the exact EHVI for a generic case is proposed. This efficient algorithm is based on partitioning the integration volume into a set of axis-parallel slices. Theoretically, the upper bound time complexities are improved from previously $O (n^2)$ and $O(n^3)$, for two- and three-objective problems respectively, to $\Theta(n\log n)$, which is asymptotically optimal. This article generalizes the scheme in higher dimensional case by utilizing a new hyperbox decomposition technique, which was proposed by D{\"a}chert et al, EJOR, 2017. It also utilizes a generalization of the multilayered integration scheme that scales linearly in the number of hyperboxes of the decomposition. The speed comparison shows that the proposed algorithm in this paper significantly reduces computation time. Finally, this decomposition technique is applied in the calculation of the Probability of Improvement (PoI)

Improving Orbit Estimates for Incomplete Orbits with a New Approach to Priors -- with Applications from Black Holes to Planets

We propose a new approach to Bayesian prior probability distributions (priors) that can improve orbital solutions for low-phase-coverage orbits, where data cover less than approximately 40% of an orbit. In instances of low phase coverage such as with stellar orbits in the Galactic center or with directly-imaged exoplanets, data have low constraining power and thus priors can bias parameter estimates and produce under-estimated confidence intervals. Uniform priors, which are commonly assumed in orbit fitting, are notorious for this. We propose a new observable-based prior paradigm that is based on uniformity in observables. We compare performance of this observable-based prior and of commonly assumed uniform priors using Galactic center and directly-imaged exoplanet (HR 8799) data. The observable-based prior can reduce biases in model parameters by a factor of two and helps avoid under-estimation of confidence intervals for simulations with less than about 40% phase coverage. Above this threshold, orbital solutions for objects with sufficient phase coverage such as S0-2, a short-period star at the Galactic center with full phase coverage, are consistent with previously published results. Below this threshold, the observable-based prior limits prior influence in regions of prior dominance and increases data influence. Using the observable-based prior, HR 8799 orbital analyses favor lower eccentricity orbits and provide stronger evidence that the four planets have a consistent inclination around 30 degrees to within 1-sigma. This analysis also allows for the possibility of coplanarity. We present metrics to quantify improvements in orbital estimates with different priors so that observable-based prior frameworks can be tested and implemented for other low-phase-coverage orbits.Comment: Published in AJ. 23 pages, 14 figures. Monte Carlo chains are available in the published article, or are available upon reques

Accuracy in strategy imitations promotes the evolution of fairness in the spatial ultimatum game

Spatial structure has a profound effect on the outcome of evolutionary games. In the ultimatum game, it leads to the dominance of much fairer players than those predicted to evolve in well-mixed settings. Here we show that spatiality leads to fair ultimatums only if the intervals from which the players are able to choose how much to offer and how little to accept are sufficiently fine-grained. Small sets of discrete strategies lead to the stable coexistence of the two most rational strategies in the set, while larger sets lead to the dominance of a single yet not necessarily the fairest strategy. The fairest outcome is obtained for the most accurate strategy imitation, that is in the limit of a continuous strategy set. Having a multitude of choices is thus crucial for the evolution of fairness, but not necessary for the evolution of empathy.Comment: 6 two-column pages, 5 figures; accepted for publication in Europhysics Letter