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.

Let T be a tessellation composed of equilateral triangular regions, where each region has an associated positive weight. We present two methods that discretize the space based on the placement of Steiner points in the cells of T. Using such a discretization, we can use Dijkstra's algorithm for computing the shortest path in the geometric graph obtained. This will lead us to two approximation algorithms for solving the Weighted Region Problem. For a given parameter ε∈(0,1], the first discretization scheme provides an approximate path that is (1+0.428ε) times better than the approximation given by Aleksandrov et al. [Determining approximate shortest paths on weighted polyhedral surfaces. Journal of the ACM, 52(1):25-53, 2005]. The other discretization scheme uses at least (ε+2ε+4ε+4)log2⁡e fewer points per segment of the triangulation with the same approximation factor.

We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path SPw(s,t), which is a shortest path from s to t in the space; a weighted shortest vertex path SVPw(s,t), which is an any-angle shortest path; and a weighted shortest grid path SGPw(s,t), which is a shortest path whose edges are edges of the tessellation. Given any arbitrary weight assignment to the faces of a triangular tessellation, thus extending recent results by Bailey et al. (2021) [6], we prove upper and lower bounds on the ratios ‖SGPw(s,t)‖‖SPw(s,t)‖, ‖SVPw(s,t)‖‖SPw(s,t)‖, ‖SGPw(s,t)‖‖SVPw(s,t)‖, which provide estimates on the quality of the approximation. It turns out, surprisingly, that our worst-case bounds are independent of any weight assignment. Our main result is that ‖SGPw(s,t)‖‖SPw(s,t)‖=23≈1.15 in the worst case, and this is tight. As a corollary, for the weighted any-angle path SVPw(s,t) we obtain the approximation result ‖SVPw(s,t)‖‖SPw(s,t)‖⪅1.15.

Nowadays, theWeb is experiencing a continuous change that is leading to the realization of the Semantic Web. Initiatives such as Linked Data have made a huge amount of structured information publicly available, encouraging the rest of the Internet community to tag their resources with it. Unfortunately, the amount of interlinked domains and information is so big that handling it efficiently has become really difficult for the final users. DBPedia, one of the biggest and most important Linked Data repositories, is a perfect example of this issue.