Relating Graph Thickness to Planar Layers and Bend Complexity

The thickness of a graph $G=(V,E)$ with $n$ vertices is the minimum number of planar subgraphs of $G$ whose union is $G$. A polyline drawing of $G$ in $\mathbb{R}^2$ is a drawing $\Gamma$ of $G$, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of $\Gamma$ is the maximum number of bends per edge in $\Gamma$, and the layer complexity of $\Gamma$ is the minimum integer $r$ such that the set of polygonal chains in $\Gamma$ can be partitioned into $r$ disjoint sets, where each set corresponds to a planar polyline drawing. Let $G$ be a graph of thickness $t$. By F\'{a}ry's theorem, if $t=1$, then $G$ can be drawn on a single layer with bend complexity $0$. A few extensions to higher thickness are known, e.g., if $t=2$ (resp., $t>2$), then $G$ can be drawn on $t$ layers with bend complexity 2 (resp., $3n+O(1)$). However, allowing a higher number of layers may reduce the bend complexity, e.g., complete graphs require $\Theta(n)$ layers to be drawn using 0 bends per edge. In this paper we present an elegant extension of F\'{a}ry's theorem to draw graphs of thickness $t>2$. We first prove that thickness-$t$ graphs can be drawn on $t$ layers with $2.25n+O(1)$ bends per edge. We then develop another technique to draw thickness-$t$ graphs on $t$ layers with bend complexity, i.e., $O(\sqrt{2}^{t} \cdot n^{1-(1/\beta)})$, where $\beta = 2^{\lceil (t-2)/2 \rceil }$. Previously, the bend complexity was not known to be sublinear for $t>2$. Finally, we show that graphs with linear arboricity $k$ can be drawn on $k$ layers with bend complexity $\frac{3(k-1)n}{(4k-2)}$.Comment: A preliminary version appeared at the 43rd International Colloquium on Automata, Languages and Programming (ICALP 2016

Linear-Space Data Structures for Range Mode Query in Arrays

A mode of a multiset $S$ is an element $a \in S$ of maximum multiplicity; that is, $a$ occurs at least as frequently as any other element in $S$. Given a list $A[1:n]$ of $n$ items, we consider the problem of constructing a data structure that efficiently answers range mode queries on $A$. Each query consists of an input pair of indices $(i, j)$ for which a mode of $A[i:j]$ must be returned. We present an $O(n^{2-2\epsilon})$-space static data structure that supports range mode queries in $O(n^\epsilon)$ time in the worst case, for any fixed $\epsilon \in [0,1/2]$. When $\epsilon = 1/2$, this corresponds to the first linear-space data structure to guarantee $O(\sqrt{n})$ query time. We then describe three additional linear-space data structures that provide $O(k)$, $O(m)$, and $O(|j-i|)$ query time, respectively, where $k$ denotes the number of distinct elements in $A$ and $m$ denotes the frequency of the mode of $A$. Finally, we examine generalizing our data structures to higher dimensions.Comment: 13 pages, 2 figure

Rapid evolution of metabolic traits explains thermal adaptation in phytoplankton

Understanding the mechanisms that determine how phytoplankton adapt to warming will substantially improve the realism of models describing ecological and biogeochemical effects of climate change. Here, we quantify the evolution of elevated thermal tolerance in the phytoplankton, Chlorella vulgaris. Initially, population growth was limited at higher temperatures because respiration was more sensitive to temperature than photosynthesis meaning less carbon was available for growth. Tolerance to high temperature evolved after ≈ 100 generations via greater down-regulation of respiration relative to photosynthesis. By down-regulating respiration, phytoplankton overcame the metabolic constraint imposed by the greater temperature sensitivity of respiration and more efficiently allocated fixed carbon to growth. Rapid evolution of carbon-use efficiency provides a potentially general mechanism for thermal adaptation in phytoplankton and implies that evolutionary responses in phytoplankton will modify biogeochemical cycles and hence food web structure and function under warming. Models of climate futures that ignore adaptation would usefully be revisited

Toward the Rectilinear Crossing Number of KnK_n: New Drawings, Upper Bounds, and Asymptotics

Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersect in a point unless that point is an endpoint of all three. The rectilinear crossing number of K_n is the fewest number of edge crossings attainable over all rectilinear drawings of K_n. For each n we construct a rectilinear drawing of K_n that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of drawings of K_n with good asymptotics. Finally, we mention some old and new open problems.Comment: 13 Page

L'ADN: dans l'enquête et au tribunal - Étude de cas du tueur en série Robert William Pickton

Travail dirigé présenté à la Faculté des Études supérieures en vue de l’obtention du grade de Maîtrise en Criminologie, option Criminalistique et informationsPlusieurs éléments peuvent avoir un impact sur le cheminement des traces ADN dans le processus judiciaire. Le présent travail est une étude de cas portant sur l’affaire Pickton, un tueur en série de Vancouver. Cette étude de cas s’intéresse précisément à la motivation derrière la sélection des traces ADN tout au long de l’enquête ainsi qu’au niveau de l’admissibilité des preuves ADN au procès. La revue de littérature fait un survol des études allant du rôle de la trace ADN depuis la collecte des traces jusqu’à l’admissibilité des preuves ADN à la Cour. L’analyse se base sur les éléments de traces ADN collectés, analysés et présentés à la Cour, des données obtenues dans quatre sources de données différentes. Cette analyse permet de démontrer que la plupart des traces ADN analysées ont été présentées à la Cour. Dans le cas de l’affaire Pickton, celles qui ont été jugés inadmissibles l’ont été pour des raisons procédurales.Several factors can affect the flow of DNA traces in the judicial process. This work is a case study on the Pickton case, a serial killer from Vancouver. This case study is specifically interested in the motivation behind the selection of DNA traces throughout the investigation and at the level of admissibility of DNA evidence at trial. The literature review provides an overview of studies ranging from the role of DNA traces from collection traces to the admissibility of DNA evidence in court. The analysis is based on DNA traces collected, analyzed and presented to the Court, the data were obtained in four different data sources. This analysis demonstrated that most of the analyzed DNA traces were presented to the Court. In the Pickton case, those that have been deemed ineligible were for procedural reasons