Abstract
Let S be a set of points in ℝd. Given a geometric spanner graph, G = (S,E), with constant stretch factor t, and a positive constant ε, we show how to construct a (1+ε)-spanner of G with \(\mathcal{O}(|S|)\) edges in time \(\mathcal{O}(|E|+|S|{\rm log}|S|)\). Previous algorithms require a preliminary step in which the edges are sorted in non-decreasing order of their lengths and, thus, have running time Ω(|E| log |S|). We obtain our result by designing a new algorithm that finds the pair in a well-separated pair decomposition separating two given query points. Previously, it was known how to answer such a query in \(\mathcal{O}({\rm log}|S|)\) time. We show how a sequence of such queries can be answered in \(\mathcal{O}(1)\) amortized time per query, provided all query pairs are from a polynomially bounded range.
J.G. was supported by the Netherlands Organisation for Scientific Research (NWO) and M.S. was supported by NSERC.
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Gudmundsson, J., Narasimhan, G., Smid, M. (2005). Fast Pruning of Geometric Spanners. In: Diekert, V., Durand, B. (eds) STACS 2005. STACS 2005. Lecture Notes in Computer Science, vol 3404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31856-9_42
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