Aggregate-Max Nearest Neighbor Searching in the Plane

We study the aggregate nearest neighbor searching for the Max operator in the plane. For a set P of n points and a query set Q of m points, the query asks for a point of P whose maximum distance to the points in Q is minimized. We present data structures for answering such queries for both L1 and L2 distance measures. Previously, only heuristic and approximation algorithms were given for both versions. For the L1 version, we build a data structure of O(n) size in O(n log n) time, such that each query can be answered in O(m + log n) time. For the L2 version, we build a data structure of O(n log log n) size in O(n log n) time, such that each query can be answered in O(m √ n log n) time, and alternatively, we build a data structure in O(n ) time and space for any > 0, such that each query can be answered in O(m log n) time.