Abstract
We present efficient algorithms for approximately answering distance queries in disk graphs. Let G be a disk graph with n vertices and m edges. For any fixed ε> 0, we show that G can be preprocessed in \(O(m\sqrt{n}\epsilon^{-1}+m\epsilon^{-2}\log S)\) time, constructing a data structure of size O(n 3/2 ε − 1+nε − 2logS), such that any subsequent distance query can be answered approximately in \(O(\sqrt{n}\epsilon^{-1}+\epsilon^{-2}\log S)\) time. Here S is the ratio between the largest and smallest radius. The estimate produced is within an additive error which is only ε times the longest edge on some shortest path.
The algorithm uses an efficient subdivision of the plane to construct a sparse graph having many of the same distance properties as the input disk graph. Additionally, the sparse graph has a small separator decomposition, which is then used to answer distance queries. The algorithm extends naturally to the higher dimensional ball graphs.
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Fürer, M., Kasiviswanathan, S.P. (2007). Approximate Distance Queries in Disk Graphs. In: Erlebach, T., Kaklamanis, C. (eds) Approximation and Online Algorithms. WAOA 2006. Lecture Notes in Computer Science, vol 4368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11970125_14
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DOI: https://doi.org/10.1007/11970125_14
Publisher Name: Springer, Berlin, Heidelberg
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