Building Voronoi Diagrams for Convex Polygons in Linear Expected Time

Let P be a list of points in the plane such that the points of P taken in order form the vertices of a convex polygon. We introduce a simple, linear expected-time algorithm for finding the Voronoi diagram of the points in P. Unlike previous results on expected-time algorithms for Voronoi diagrams, this method does not require any assumptions about the distribution of points. With minor modifications, this method can be used to design fast algorithms for certain problems involving unrestricted sets of points. For example, fast expected-time algorithms can be designed to delete a point from a Voronoi diagram, to build an order k Voronoi diagram for an arbitrary set of points, and to determine the smallest enclosing circle for points at the vertices of a convex hull

There is a Planar Graph Almost as Good as the Complete Graph

Given a set S of points in the plane, there is a triangulation of S such that a path found within this triangulation has length bounded by a constant times the straight-line distance between the endpoints of the path. Specifically, for any two points a and b of S there is a path along edges of the triangulation with length less that sqrt(10) times [ab], where [ab] is the straight-line Euclidean distance between a and b. The triangulation that has this property is the L1 metric Delauney triangulation for the set S. This result can be applied to motion planning in the plane. Given a source, a destination, and a set of polygonal obstacles of size n, an O(n) size data structure can be used to find a reasonable approximation to the shortest path between the source and the destination in O (n log n) time

Planar Graphs and Sparse Graphs from Efficient Motion Planning in the Plane

Given a source, a destination, and a number of obstacles in the plane, the Motion Planning Program is to determine the best path to move an object (a robot) from the source to the destination without colliding with any of the obstacles. For us, motion is restricted to the plane, the robot is represented by a point, and the obstacles are represented by a set of polygons with a total of n vertices among all the polygonal obstacles

Term Reduction Using Directed Congruence Closure

Many problems in computer science can be described in terms of reduction rules that tell how to transform terms. Problems that can be handled in this way include interpreting programs, implementing abstract data types, and proving certain kinds of theorems. A terms is said to have a normal form if it can be transformed, using the reduction rules, into a term to which no further reduction rules apply. In this paper, we extend the Congruence Closure Algorithm, an algorithm for finding the consequences of a finite set of equations, to develop Directed Congruence Closure, a technique for finding the normal form of a term provided the reduction rules satisfy the conditions for a regular term rewriting system. This technique is particularly efficient because it inherits, from the Congruence Closure Algorithm, the ability to remember all objects that have already been proved equivalent

Finding Largest Empty Circles with Location Constraints

Let S be a set of n points in the plane and let CH(S) represent the convex hull of S. The Largest Empty Circle (LEC) problem is the problem of finding the largest circle centered with CH(S) such that no point of S lies within the circle. Shamos and Hoey (SH75) outlined an algorithm for solving this problem in time O(n log n) by first computing the Voronoi diagram, V(S), in time O(n log n), then using V(S) and CH(S) to compute the largest empty circle in time O(n). In a recent paper [Tou83], Toussaint pointed out some problems with the algorithm as outlined by Shamos and presented an algorithm which, given V(S) and CH(S), solves the LEC problem in time O(n log n). In this note we show that Shamos\u27 original claim was correct: given V(S) and CH(S), the LEC problem can be solved in time O(n). More generally, given V(S) and a convex k-gon P, the LEC centered within P can be found in time O(k+n). We also improve on an algorithm given by Toussaint for computing the LEC when the center is constrained to lie within an arbitrary simple polygon. Given a set S of n points and an arbitrary simple k-gon P, the largest empty circle centered within P can be found in time O(kn + n log n). This becomes O(kn) if the Voronoi diagram of S is already given

Modeling the Pyrolysis and Combustion Behaviors of Non-Charring and Intumescent-Protected Polymers Using “FiresCone”

A mathematical model, named FiresCone, was developed to simulate the pyrolysis and combustion processes of different types of combustible materials, which also took into account both gas and solid phases. In the present study, some non-charring and intumescent-protected polymer samples were investigated regarding their combustion behaviors in response to pre-determined external heat fluxes. The modeling results were validated against the experimental outcomes obtained from a cone calorimeter. The predicted mass loss rates of the samples were found to fit reasonably well with the experimental data collected under various levels of external irradiation. Both the experimental and modeling results showed that the peak mass loss rate of the non-charring polymer material occurred near the end of burning, whereas for the intumescent-protected polymer it happed shortly after the start of the experiment. “FiresCone” is expected to act as a practical tool for the investigation of fire behavior of combustible materials. It is also expected to model fire scenarios under complicated conditions

The anti-apoptotic activity of XIAP is retained upon mutation of both the caspase 3– and caspase 9–interacting sites

The X-linked mammalian inhibitor of apoptosis protein (XIAP) has been shown to bind several partners. These partners include caspase 3, caspase 9, DIABLO/Smac, HtrA2/Omi, TAB1, the bone morphogenetic protein receptor, and a presumptive E2 ubiquitin-conjugating enzyme. In addition, we show here that XIAP can bind to itself. To determine which of these interactions are required for it to inhibit apoptosis, we generated point mutant XIAP proteins and correlated their ability to bind other proteins with their ability to inhibit apoptosis. ∂RING point mutants of XIAP were as competent as their full-length counterparts in inhibiting apoptosis, although impaired in their ability to oligomerize with full-length XIAP. Triple point mutants, unable to bind caspase 9, caspase 3, and DIABLO/HtrA2/Omi, were completely ineffectual in inhibiting apoptosis. However, point mutants that had lost the ability to inhibit caspase 9 and caspase 3 but retained the ability to inhibit DIABLO were still able to inhibit apoptosis, demonstrating that IAP antagonism is required for apoptosis to proceed following UV irradiation

Fungal Levels in the Home and Allergic Rhinitis by 5 Years of Age

Studies have repeatedly demonstrated that sensitization to fungi, such as Alternaria, is strongly associated with allergic rhinitis and asthma in children. However, the role of exposure to fungi in the development of childhood allergic rhinitis is poorly understood. In a prospective birth cohort of 405 children of asthmatic/allergic parents from metropolitan Boston, Massachusetts, we examined in-home high fungal concentrations (> 90th percentile) measured once within the first 3 months of life as predictors of doctor-diagnosed allergic rhinitis in the first 5 years of life. In multivariate Cox regression analyses, predictors of allergic rhinitis included high levels of dust-borne Aspergillus [hazard ratio (HR) = 3.27; 95% confidence interval (CI), 1.50–7.14], Aureobasidium (HR = 3.04; 95% CI, 1.33–6.93), and yeasts (HR = 2.67; 95% CI, 1.26–5.66). The factors controlled for in these analyses included water damage or mild or mildew in the building during the first year of the child’s life, any lower respiratory tract infection in the first year, male sex, African-American race, fall date of birth, and maternal IgE to Alternaria > 0.35 U/mL. Dust-borne Alternaria and non-sporulating and total fungi were also predictors of allergic rhinitis in models excluding other fungi but adjusting for all of the potential confounders listed above. High measured fungal concentrations and reports of water damage, mold, or mildew in homes may predispose children with a family history of asthma or allergy to the development of allergic rhinitis

Tissue sodium excess is not hypertonic and reflects extracellular volume expansion

Our understanding of Na+ homeostasis has recently been reshaped by the notion of skin as a depot for Na+ accumulation in multiple cardiovascular diseases and risk factors. The proposed water-independent nature of tissue Na+ could induce local pathogenic changes, but lacks firm demonstration. Here, we show that tissue Na+ excess upon high Na+ intake is a systemic, rather than skin-specific, phenomenon reflecting architectural changes, i.e. a shift in the extracellular-to-intracellular compartments, due to a reduction of the intracellular or accumulation of water-paralleled Na+ in the extracellular space. We also demonstrate that this accumulation is unlikely to justify the observed development of experimental hypertension if it were water-independent. Finally, we show that this isotonic skin Na+ excess, reflecting subclinical oedema, occurs in hypertensive patients and in association with aging. The implications of our findings, questioning previous assumptions but also reinforcing the importance of tissue Na+ excess, are both mechanistic and clinical