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5

One can think about skip quadtree as a skip-list implementation of a data-structure storing the points according to their z-order. It is (arguably) at least conceptually simpler... See chapter 2 here: http://goo.gl/pLiEO . And yes, assuming you can perform some basic z-order operations in constant time, you can definitely do ANN in logarithmic time. The ...

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This can be reduced to rectangular RMQ in $O(n^2)$ time and space. Create a new array $H$ where $H[i][i+j] := G[i][j]$, padding entries $H[i][k]$ with $k < i$ or $k \ge i + n$ with $\infty$. Run $RMQ(a, b, c, d)$ on $H$ whenever you would run $RMQ_P(a, b, c, d)$ on $G$. Now run your favorite 2D-RMQ algorithm such as this. Perhaps you want to be very ...

2

In some of these bounds, the $\varepsilon$ doesn't come from an approximation, but rather from the degree to which you recurse in a partition tree. Roughly speaking, the idea goes like this. Find a small sample of the input objects that form an arrangement so that the remaining objects are "spread out nicely". In particular, each cell of the arrangement is ...

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