Searching edges in the overlap of two plane graphs

Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains one of which is convex in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n) time and O(n+m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n axis-aligned rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure

Parallel searching in the plane

AbstractIn this paper we investigate parallel searching in the plane using robots as searchers. We show that a huge number of robots are not necessary for several problems and under different assumptions. This corresponds to real situations since, actually, the number of processors of parallel machines is fixed and independent of the dimension of the problem to be solved

Searching for Inflow Towards Massive Starless Clump Candidates Identified in the Bolocam Galactic Plane Survey

Recent Galactic plane surveys of dust continuum emission at long wavelengths have identified a population of dense, massive clumps with no evidence for on-going star formation. These massive starless clump candidates are excellent sites to search for the initial phases of massive star formation before the feedback from massive star formation effects the clump. In this study, we search for the spectroscopic signature of inflowing gas toward starless clumps, some of which are massive enough to form a massive star. We observed 101 starless clump candidates identified in the Bolocam Galactic Plane Survey (BGPS) in HCO+ J = 1-0 using the 12m Arizona Radio Observatory telescope. We find a small blue excess of E = (Nblue - Nred)/Ntotal = 0.03 for the complete survey. We identified 6 clumps that are good candidates for inflow motion and used a radiative transfer model to calculate mass inflow rates that range from 500 - 2000 M /Myr. If the observed line profiles are indeed due to large-scale inflow motions, then these clumps will typically double their mass on a free fall time. Our survey finds that massive BGPS starless clump candidates with inflow signatures in HCO+ J = 1-0 are rare throughout our Galaxy.Comment: 14 pages, 9 figure

Competitive Online Searching for a Ray in the Plane

We consider the problem of a searcher that looks, for example, for a lost flashlight in a dusty environment. The searcher finds the flashlight as soon as it crosses the ray emanating from the flashlight. In order to pick it up, the searcher moves to the origin of the light beam. We compare the length of the path of the searcher to the shortest path to the goal. First, we give a search strategy for a special case of the ray search---the window shopper problem---, where the ray we are looking for is perpendicular to a known ray. Our strategy achieves a competitive factor of $1.059ldots$, which is optimal. Then, we consider rays in arbitrary position in the plane. We present an online strategy that achieves a factor of $22.513ldots$, and give a lower bound of $2pi,e=17.079ldots$