Let S be a set of n points in the plane. We present four different algorithms for finding a pair of points in S such that any disk that contains that pair must contain at least cn points of S , for some constant c > 0. The first is a randomized algorithm that finds a pair in O(n log n) expected time for points in general position and c = 1/2 − 1/√6 ≈ 1/10.9. The second algorithm, also for points in general position, takes O(n 2 ) time but the constant c is improved to 1/2 − 1/√12 ≈ 1/4.7. Using this algorithm and applying binary search, we find the pair that achieves the optimal c in O(n 2 log n ) time. The final algorithm finds in linear time a pair of points such that any disk through them contains at least n/3 of the points of S when S is in convex position. We also adapt these algorithms to find a pair of points of S in a polygon P such that any geodesic disk that contains that pair must contain at least cn points of S for some constant c > 0.

In this article, we present a construction of a spanner on a set of n points in Rd that we call a heavy path WSPD spanner. The construction is parameterized by a constant s>2 called the separation ratio. The size of the graph is O(sdn) and the spanning ratio is at most 1+2/s+2/(s−1). We also show that this graph has a hop spanning ratio of at most 2lg⁡n+1. We present a memoryless local routing algorithm for heavy path WSPD spanners. The routing algorithm requires a vertex v of the graph to store O(deg⁡(v)log⁡n) bits of information, where deg⁡(v) is the degree of v. The routing ratio is at most 1+4/s+1/(s−1) and at least 1+4/s in the worst case. The number of edges on the routing path is bounded by 2lg⁡n+1. We then show that the heavy path WSPD spanner can be constructed in metric spaces of bounded doubling dimension. These metric spaces have been studied in computational geometry as a generalization of Euclidean space. We show that, in a metric space with doubling dimension λ, the heavy path WSPD spanner has size O(sλn) where s is the separation ratio. The spanning ratio and hop spanning ratio are the same as in the Euclidean case. Finally, we show that the local routing algorithm works in the bounded doubling dimension case. The vertices require the same amount of storage, but the routing ratio becomes at most 1+(2+ττ−1)/s+1/(s−1) in the worst case, where τ≥11 is a constant related to the doubling dimension.