Impact of boundaries on fully connected random geometric networks

Many complex networks exhibit a percolation transition involving a macroscopic connected component, with universal features largely independent of the microscopic model and the macroscopic domain geometry. In contrast, we show that the transition to full connectivity is strongly influenced by details of the boundary, but observe an alternative form of universality. Our approach correctly distinguishes connectivity properties of networks in domains with equal bulk contributions. It also facilitates system design to promote or avoid full connectivity for diverse geometries in arbitrary dimension.Comment: 6 pages, 3 figure

Embedding Four-directional Paths on Convex Point Sets

A directed path whose edges are assigned labels "up", "down", "right", or "left" is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used. A \emph{direction-consistent embedding} of an \mbox{$n$-vertex} four-directional path $P$ on a set $S$ of $n$ points in the plane is a straight-line drawing of $P$ where each vertex of $P$ is mapped to a distinct point of $S$ and every edge points to the direction specified by its label. We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.Comment: 11 pages, full conference version including all proof

Dynamic Range Majority Data Structures

Given a set $P$ of coloured points on the real line, we study the problem of answering range $\alpha$-majority (or "heavy hitter") queries on $P$. More specifically, for a query range $Q$, we want to return each colour that is assigned to more than an $\alpha$-fraction of the points contained in $Q$. We present a new data structure for answering range $\alpha$-majority queries on a dynamic set of points, where $\alpha \in (0,1)$. Our data structure uses O(n) space, supports queries in $O((\lg n) / \alpha)$ time, and updates in $O((\lg n) / \alpha)$ amortized time. If the coordinates of the points are integers, then the query time can be improved to $O(\lg n / (\alpha \lg \lg n) + (\lg(1/\alpha))/\alpha))$. For constant values of $\alpha$, this improved query time matches an existing lower bound, for any data structure with polylogarithmic update time. We also generalize our data structure to handle sets of points in d-dimensions, for $d \ge 2$, as well as dynamic arrays, in which each entry is a colour.Comment: 16 pages, Preliminary version appeared in ISAAC 201

KINETIC ANALYSIS OF SEVERAL VARIATIONS OF PUSH-UPS

Push-ups are a common and practical exercise though the kinetic characteristics of this exercise and its variations have yet to be quantified. This study assessed the peak ground reaction forces (GRF) of push-up variations including the regular push-up and those performed with bent knee, feet elevated on a 30.48 cm box and a 60.96 cm box, hands elevated on a 30.48 cm box and a 60.96 cm box. Peak GRF and peak GRF expressed as a coefficient of subject body mass were obtained with a force platform. Push-ups with the feet elevated produced higher GRF than all other push-up variations (p ≤ 0.05). Push-ups with hands elevated and from the bent knee position produced lower GRF than all other push-up variations (p ≤ 0.05). These data can be used to progress the intensity of push-ups in a program with loads that are quantified as a percentage of body mass

The Complexity of Drawing Graphs on Few Lines and Few Planes

It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. - For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $\rho^1_d(G)\le k$ is NP-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. - Let $\rho^2_3(G)$ be the minimum number of planes in $\mathbb{R}^3$ needed to cover a straight-line drawing of a graph $G$. We prove that deciding whether $\rho^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$

Encodings of Range Maximum-Sum Segment Queries and Applications

Given an array A containing arbitrary (positive and negative) numbers, we consider the problem of supporting range maximum-sum segment queries on A: i.e., given an arbitrary range [i,j], return the subrange [i' ,j' ] \subseteq [i,j] such that the sum of the numbers in A[i'..j'] is maximized. Chen and Chao [Disc. App. Math. 2007] presented a data structure for this problem that occupies {\Theta}(n) words, can be constructed in {\Theta}(n) time, and supports queries in {\Theta}(1) time. Our first result is that if only the indices [i',j'] are desired (rather than the maximum sum achieved in that subrange), then it is possible to reduce the space to {\Theta}(n) bits, regardless the numbers stored in A, while retaining the same construction and query time. We also improve the best known space lower bound for any data structure that supports range maximum-sum segment queries from n bits to 1.89113n - {\Theta}(lg n) bits, for sufficiently large values of n. Finally, we provide a new application of this data structure which simplifies a previously known linear time algorithm for finding k-covers: i.e., given an array A of n numbers and a number k, find k disjoint subranges [i_1 ,j_1 ],...,[i_k ,j_k ], such that the total sum of all the numbers in the subranges is maximized.Comment: 19 pages + 2 page appendix, 4 figures. A shortened version of this paper will appear in CPM 201

CDC5 Inhibits the Hyperphosphorylation of the Checkpoint Kinase Rad53, Leading to Checkpoint Adaptation

The mechanistic role of the yeast kinase CDC5, in allowing cells to adapt to the presence of irreparable DNA damage and continue to divide, is revealed