Open Problems from CCCG 2014.

This report provides the problems posed by the participants at the open problem session of the 26 Canadian Conference on Computational Geometry. This well-attended session was held Tuesday, August 12, 2014, as a scheduled session of the conference. Six participants presented a total of seven problems. All presenters kindly agreed to provide written versions of their problems, including references and attributions. The problems appear in the sections below. The references appear at the end. The text is essentially the same, modulo minor editing, as the text provided by the presenters. This material is not refereed. 1 Guarding Orthogonal Terrains presented by: Giovanni Viglietta Partition the plane into finitely many (possibly unbounded) orthogonal polygons, and extrude them in 3D, obtaining a set of “orthogonal skyscrapers” of different heights. Let n be the total number of vertices of the orthogonal polygons. We ask to find the minimum number (as a function of n) of vertex guards for the terrain induced by the skyscrapers. In other words, we seek to select a minimum number of “guards” among the vertices of the skyscrapers such that each point in 3-space lying “above” some skyscraper is visible to some guard, where lines of sight must not intersect a skyscraper’s top face or a side face. The best known lower bound is given by a row of k equal cuboidal skyscrapers, where n = 8k. In this case k + 1 vertex guards are needed, which yields a lower bound of (n/8) + 1 vertex guards. We conjecture n/8 + O(1) guards to be sufficient for all orthogonal terrains on n vertices (observe that an L-shaped skyscraper on 12 vertices needs three guards). To our knowledge, the problem is open even in the case of a single “tower” made of nested orthogonal prisms of increasing height, or a single “well”. For background, see [1]. ∗Professor, Department of Computer Science, U. of Victoria, Canada; email: sue@uvic.ca 1Postdoctoral Fellow, U. of Ottawa and School of Computer Science, Carleton U., Canada; email: viglietta@gmail.com 2 Flows on Terrains presented by: Jack Snoeyink What local actions can make a general difference for flow of water, nutrients, and pollutants in a terrain? This is more of an open application area for computational geometry techniques than an open problem. Consider a real-world terrain with patches having different soil types (e.g., different absorbency properties) together with a network of streams, house gutters, parking lot drains, and underground sewers. There are rain gauges reporting rainfall in cm/hr at some points and flow meters reporting liters/min profiles on some waterways. (These are increasingly common in the “internet of things.”) If we model a rainfall, do we see the measured flows? If not, can we suggest where our information about the flow network is incomplete or inaccurate? If we don’t like, say, the surge of flow in the sewers from a rainfall, can we suggest where rain gardens could most effectively delay the flow? At what scale should these questions be asked based on the sensors we have? There are many simulations that are used [2, 3], but the ideas of computational geometry (like continuous Dijkstra for paths in weighted regions [4], or partitioning terrain into catchments and capturing flow in equilibrium [5]) can be used to preprocess the terrain for more efficient exploration of modifications that would produce the observed or desired flow profiles. 2Professor, Dept. of Computer Science, U. North Carolina; email: snoeyink@cs.unc.edu 27 Canadian Conference on Computational Geometry, 2015

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