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Applying standard dimensionality reduction techniques, we show how to perform approximate range searching in higher dimension while avoiding the curse of dimensionality. Given n points in a unit ball in R d, an approximate halfspace range query counts (or reports) the points in a query halfspace; the qualifier “approximate ” indicates that points within distance ε of the boundary of the halfspace might be misclassified. Allowing errors near the boundary has a dramatic effect on the complexity of the problem. We give a solution with Õ(d/ε 2) query time and dn O(ε−2) storage. For an exact solution with comparable query time, one needs roughly Ω(n d) storage. In other words, an approximate answer to a range query lowers the storage requirement from exponential to polynomial. We generalize our solution to polytope/ball range searching. 1

We study the complexity of the dynamic partial sum problem in the cell-probe model. We give the model access to nondeterministic queries and prove that the problem remains hard. We give the model access to the right answer as an oracle and prove that the problem remains hard. This suggests which kind of information is hard to maintain. From these results, we derive a number of lower bounds for dynamic algorithms and data structures: We prove lower bounds for dynamic algorithms for existential range queries, reachability in directed graphs, planarity testing, planar point location, incremental parsing, and fundamental data structure problems like maintaining the majority of the prefixes of a string of bits. We prove a lower bound for reachability in grid graphs in terms of the graph's width. We characterize the complexity of maintaining the value of any symmetric function on the prefixes of a bit string. Keywords. cell-probe model, partial sum, dynamic algorithm, data structure AMS subject classifications. 68Q17, 68Q10, 68Q05, 68P05

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We consider a Delaunay triangulation defined on n points distributed independently and uniformly on a planar compact convex set of positive volume. Let the stabbing number be the maximal number of intersections between a line and edges of the triangulation. We show that the stabbing number Sn is Θ ( √ n) in the mean, and provide tail bounds for P{Sn ≥ t √ n}. Applications to planar point location, nearest neighbor searching, range queries, planar separator determination, approximate shortest paths, and the diameter of the Delaunay triangulation are discussed.

Abstract not found

union of colorful simplices spanned by a colored point set

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union of colorful simplices spanned by a colored point set