Randomized Speedup of the Bellman-Ford Algorithm

We describe a variant of the Bellman-Ford algorithm for single-source shortest paths in graphs with negative edges but no negative cycles that randomly permutes the vertices and uses this randomized order to process the vertices within each pass of the algorithm. The modification reduces the worst-case expected number of relaxation steps of the algorithm, compared to the previously-best variant by Yen (1970), by a factor of 2/3 with high probability. We also use our high probability bound to add negative cycle detection to the randomized algorithm.Comment: 12 Pages, 6 Figures, ANALCO 201

Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth

We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bounded treewidth.Comment: Graph Drawing 201

Fixed parameter tractability of crossing minimization of almost-trees

We investigate exact crossing minimization for graphs that differ from trees by a small number of additional edges, for several variants of the crossing minimization problem. In particular, we provide fixed parameter tractable algorithms for the 1-page book crossing number, the 2-page book crossing number, and the minimum number of crossed edges in 1-page and 2-page book drawings.Comment: Graph Drawing 201

Small Superpatterns for Dominance Drawing

We exploit the connection between dominance drawings of directed acyclic graphs and permutations, in both directions, to provide improved bounds on the size of universal point sets for certain types of dominance drawing and on superpatterns for certain natural classes of permutations. In particular we show that there exist universal point sets for dominance drawings of the Hasse diagrams of width-two partial orders of size O(n^{3/2}), universal point sets for dominance drawings of st-outerplanar graphs of size O(n\log n), and universal point sets for dominance drawings of directed trees of size O(n^2). We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}), riffle permutations (321-, 2143-, and 2413-avoiding permutations) have superpatterns of size O(n), and the concatenations of sequences of riffles and their inverses have superpatterns of size O(n\log n). Our analysis includes a calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of the 321-superpattern siz

Col-OSSOS: Colors of the Interstellar Planetesimal 1I/`Oumuamua

The recent discovery by Pan-STARRS1 of 1I/2017 U1 (`Oumuamua), on an unbound and hyperbolic orbit, offers a rare opportunity to explore the planetary formation processes of other stars, and the effect of the interstellar environment on a planetesimal surface. 1I/`Oumuamua's close encounter with the inner Solar System in 2017 October was a unique chance to make observations matching those used to characterize the small-body populations of our own Solar System. We present near-simultaneous g$^\prime$, r$^\prime$, and J photometry and colors of 1I/`Oumuamua from the 8.1-m Frederick C. Gillett Gemini North Telescope, and $gri$ photometry from the 4.2 m William Herschel Telescope. Our g$^\prime$r$^\prime$J observations are directly comparable to those from the high-precision Colours of the Outer Solar System Origins Survey (Col-OSSOS), which offer unique diagnostic information for distinguishing between outer Solar System surfaces. The J-band data also provide the highest signal-to-noise measurements made of 1I/`Oumuamua in the near-infrared. Substantial, correlated near-infrared and optical variability is present, with the same trend in both near-infrared and optical. Our observations are consistent with 1I/`Oumuamua rotating with a double-peaked period of $8.10 \pm 0.42$ hours and being a highly elongated body with an axial ratio of at least 5.3:1, implying that it has significant internal cohesion. The color of the first interstellar planetesimal is at the neutral end of the range of Solar System $g-r$ and $r-J$ solar-reflectance colors: it is like that of some dynamically excited objects in the Kuiper belt and the less-red Jupiter Trojans.Comment: Accepted to ApJ