arXiv subjects

Taylor Gordon

Publications and source records attributed to Taylor Gordon.

4 recordsLinked to original sources

arXiv2026

Almost Linear Universal Point Sets for Planar Graphs

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

cs.CG↗

arXiv2020

Accelerating 3D Deep Learning with PyTorch3D

Deep learning has significantly improved 2D image recognition. Extending into 3D may advance many new applications including autonomous vehicles, virtual and augmented reality, authoring 3D content, and even improving 2D recognition. However despite growing interest, 3D deep learning remains relatively underexplored. We believe that some of this disparity is due to the engineering challenges involved in 3D deep learning, such as efficiently processing heterogeneous data and reframing graphics operations to be differentiable. We address these challenges by introducing PyTorch3D, a library of modular, efficient, and differentiable operators for 3D deep learning. It includes a fast, modular differentiable renderer for meshes and point clouds, enabling analysis-by-synthesis approaches. Compared with other differentiable renderers, PyTorch3D is more modular and efficient, allowing users to more easily extend it while also gracefully scaling to large meshes and images. We compare the PyTorch3D operators and renderer with other implementations and demonstrate significant speed and memory improvements. We also use PyTorch3D to improve the state-of-the-art for unsupervised 3D mesh and point cloud prediction from 2D images on ShapeNet. PyTorch3D is open-source and we hope it will help accelerate research in 3D deep learning.

cs.CV↗

arXiv2014

The Minimum Bends in a Polyline Drawing with Fixed Vertex Locations

We consider embeddings of planar graphs in $R^2$ where vertices map to points and edges map to polylines. We refer to such an embedding as a polyline drawing, and ask how few bends are required to form such a drawing for an arbitrary planar graph. It has long been known that even when the vertex locations are completely fixed, a planar graph admits a polyline drawing where edges bend a total of $O(n^2)$ times. Our results show that this number of bends is optimal. In particular, we show that $\Omega(n^2)$ total bends is required to form a polyline drawing on any set of fixed vertex locations for almost all planar graphs. This result generalizes all previously known lower bounds, which only applied to convex point sets, and settles 2 open problems.

cs.CG↗

arXiv2012

Simultaneous Embeddings with Vertices Mapping to Pre-Specified Points

We discuss the problem of embedding graphs in the plane with restrictions on the vertex mapping. In particular, we introduce a technique for drawing planar graphs with a fixed vertex mapping that bounds the number of times edges bend. An immediate consequence of this technique is that any planar graph can be drawn with a fixed vertex mapping so that edges map to piecewise linear curves with at most $3n + O(1)$ bends each. By considering uniformly random planar graphs, we show that $2n + O(1)$ bends per edge is sufficient on average. To further utilize our technique, we consider simultaneous embeddings of $k$ uniformly random planar graphs with vertices mapping to a fixed, common point set. We explain how to achieve such a drawing so that edges map to piecewise linear curves with $O(n^{1-1/k})$ bends each, which holds with overwhelming probability. This result improves upon the previously best known result of O(n) bends per edge for the case where $k \geq 2$. Moreover, we give a lower bound on the number of bends that matches our upper bound, proving our results are optimal.

cs.CG↗