Relative neighborhood graphs in three dimensions
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
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\indent We present a randomized algorithm of expected time complexity \[ O(m^{2/3} n^{2/3}\, \mbox {log} ^ {4/3} m + m \,\mbox {log}^{2}\, m + n \, \mbox{log}^{2}\, n) \] for computing bi-chromatic farthest neighbors between $n$ red points and $m$ blue points in ${\cal E} ^{3}$. The algorithm can also be used to compute all farthest neighbors or external farthest neighbors of $n$ points in ${\cal E } ^{3}$ in $O(n^{4/3} \, \mbox {log}^{4/3} n)$ expected time. Using these procedures as building blocks, we can compute a Euclidean maximum spanning tree or a minimum-diameter two-partition of $n$ points in ${\cal E } ^{3}$ in $O(n^{4/3} \, \mbox {log} ^{7/3}\, n)$ expected time. The previous best bound for these problems was $O(n ^{3/2} \, \mbox {log} ^ {1/2} \, n)$. Our algorithms can be extended to higher dimensions. We also propose fast and simple approximation algorithms for these problems. These approximation algorithms produce solutions that approximate the true value with a relative accuracy $\varepsilon$ and run in time $O(n\varepsilon ^{(1-k)/2} \, \mbox {log} \, n)$ or $O(n\varepsilon ^{(1-k)/2} \, \mbox {log} ^{2} \, n)$ in $k$-dimensional space.