1,784 research outputs found
Implementing path coloring algorithms on planar graphs
Master's Project (M.S.) University of Alaska Fairbanks, 2017A path coloring of a graph partitions its vertex set into color classes such that each class induces a disjoint union of paths. In this project we implement several algorithms to compute path colorings of graphs embedded in the plane. We present two algorithms to path color plane graphs with 3 colors based on a proof by Poh in 1990. First we describe a naive algorithm that directly follows Poh's procedure, then we give a modified algorithm that runs in linear time. Independent results of Hartman and Skrekovski describe a procedure that takes a plane graph G and a list of 3 colors for each vertex, and computes a path coloring of G such that each vertex receives a color from its list. We present a linear time implementation based on Hartman and Skrekovski's proofs. A C++ implementation is provided for all three algorithms, utilizing the Boost Graph Library. Instructions are given on how to use the implementation to construct colorings for plane graphs represented by Boost data structures
Triangulated surfaces in triangulated categories
For a triangulated category A with a 2-periodic dg-enhancement and a
triangulated oriented marked surface S we introduce a dg-category F(S,A)
parametrizing systems of exact triangles in A labelled by triangles of S. Our
main result is that F(S,A) is independent on the choice of a triangulation of S
up to essentially unique Morita equivalence. In particular, it admits a
canonical action of the mapping class group. The proof is based on general
properties of cyclic 2-Segal spaces.
In the simplest case, where A is the category of 2-periodic complexes of
vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya
category of the surface S. Therefore, our construction can be seen as
implementing a 2-dimensional instance of Kontsevich's program on localizing the
Fukaya category along a singular Lagrangian spine.Comment: 55 pages, v2: references added and typos corrected, v3: expanded
version, comments welcom
Transversal structures on triangulations: a combinatorial study and straight-line drawings
This article focuses on a combinatorial structure specific to triangulated
plane graphs with quadrangular outer face and no separating triangle, which are
called irreducible triangulations. The structure has been introduced by Xin He
under the name of regular edge-labelling and consists of two bipolar
orientations that are transversal. For this reason, the terminology used here
is that of transversal structures. The main results obtained in the article are
a bijection between irreducible triangulations and ternary trees, and a
straight-line drawing algorithm for irreducible triangulations. For a random
irreducible triangulation with vertices, the grid size of the drawing is
asymptotically with high probability up to an additive
error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for
these triangulations only guarantees a grid size .Comment: 42 pages, the second version is shorter, focusing on the bijection
(with application to counting) and on the graph drawing algorithm. The title
has been slightly change
Hand Keypoint Detection in Single Images using Multiview Bootstrapping
We present an approach that uses a multi-camera system to train fine-grained
detectors for keypoints that are prone to occlusion, such as the joints of a
hand. We call this procedure multiview bootstrapping: first, an initial
keypoint detector is used to produce noisy labels in multiple views of the
hand. The noisy detections are then triangulated in 3D using multiview geometry
or marked as outliers. Finally, the reprojected triangulations are used as new
labeled training data to improve the detector. We repeat this process,
generating more labeled data in each iteration. We derive a result analytically
relating the minimum number of views to achieve target true and false positive
rates for a given detector. The method is used to train a hand keypoint
detector for single images. The resulting keypoint detector runs in realtime on
RGB images and has accuracy comparable to methods that use depth sensors. The
single view detector, triangulated over multiple views, enables 3D markerless
hand motion capture with complex object interactions.Comment: CVPR 201
Regularity of Edge Ideals and Their Powers
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals
of graphs and their powers. Our focus is on bounds and exact values of and the asymptotic linear function , for in terms of combinatorial data of the given graph Comment: 31 pages, 15 figure
- …