How to find the farthest point (from a set of points) from a given point efficiently?

Question

I'm looking for an algorithm or data structure to solve the following problem: You are given a set of points S. And you are given Q queries in form of another point. For every query, find the farthest point in the set from the given point.

There are at most 10^5 points in the set and 10^5 queries. All the coordinates for points are in range from 0 to 10^5.

I am wondering if there is a way to store the set of points such that we can answer the queries in O(log n) or O(log^2 n) if necessary.

Answer 1

A quote from "Approximate Furthest Neighbor with Application to Annulus Query":

In two dimensions the furthest neighbor problem can be solved in linear space and logarithmic query time using point location in a furthest point Voronoi diagram (see, for example, de Berg et al. [5]).

where [5] refers to the "Marks textbook":

Berg, Mark de, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational geometry: algorithms and applications . Springer-Verlag TELOS, 2008.

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^{Farthest-Neighbor Voronoi diagram for a set of eight points.
Image from Jacometti, Welson. "A Note on Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams." (1992).}

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