Verification of partitions of 2d and 3d objects

We consider the problems of deciding whether a given collection of polygons (polyhedra resp.) forms (i) a partition or (ii) a cell complex decomposition of a given polygon (polyhedron resp.). We describe simple O(n log n)-time and O(n)-space algorithms for these problems, where n is the total description size of the input. If, in the input, vertices are referenced by means of indices to an array of distinct vertices, then our cell complex decomposition verification algorithms run in O(n) time

A CASE STUDY OF TRAUMA-INFORMED PRACTICE AND IMPLEMENTATION TO SUPPORT MENTAL HEALTH AND LEARNING IN PUBLIC SCHOOL IN SUFFOLK COUNTY, NEW YORK

The purpose of this study is to examine the readiness of school districts in Suffolk County, New York, to implement a trauma-informed system to address the growing needs of mental health interventions in student populations. A review of the literature will show a historical prevalence of mental health providers and individual student interventions within the school building, or in partnership with community agencies. Recent literature shows an increase in school-related issues have origins in student trauma or adverse childhood experiences. The study will examine the issue by conducting a mixed method analysis, using a survey instrument and focus group interviews, of members of the Suffolk Directors of Guidance. Significance of the study will help districts who want to implement a systematic and districtwide approach to mitigating trauma-related student issues by identifying current readiness and examining gaps in preparation to implement the National Dropout Prevention Center’s Trauma-Skilled Schools Model

Join-Reachability Problems in Directed Graphs

For a given collection G of directed graphs we define the join-reachability graph of G, denoted by J(G), as the directed graph that, for any pair of vertices a and b, contains a path from a to b if and only if such a path exists in all graphs of G. Our goal is to compute an efficient representation of J(G). In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for G. In the implicit version we wish to build an efficient data structure (in terms of space and query time) such that we can report fast the set of vertices that reach a query vertex in all graphs of G. This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problem. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs

Illuminating the x-Axis by ?-Floodlights

Given a set S of regions with piece-wise linear boundary and a positive angle α < 90°, we consider the problem of computing the locations and orientations of the minimum number of α-floodlights positioned at points in S which suffice to illuminate the entire x-axis. We show that the problem can be solved in O(n log n) time and O(n) space, where n is the number of vertices of the set S