David Eppstein

David Eppstein

The Erdős-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$δ$, at most $O(δ^2)$ points can have integer distances from all three triangle vertices. We prove the same results for any strictly convex distance function on the plane, and analogous results for every two-dimensional complete Riemannian manifold of bounded genus and for geodesic distance on the boundary of every three-dimensional Euclidean convex set. As a consequence, we resolve a 1983 question of Richard Guy on the equilateral dimension of Riemannian manifolds. Our proofs are based on the properties of additively weighted Voronoi diagrams of these distances.

David Eppstein

In nearest-neighbor classification problems, a set of $d$-dimensional training points are given, each with a known classification, and are used to infer unknown classifications of other points by using the same classification as the nearest training point. A training point is relevant if its omission from the training set would change the outcome of some of these inferences. We provide a simple algorithm for thinning a training set down to its subset of relevant points, using as subroutines algorithms for finding the minimum spanning tree of a set of points and for finding the extreme points (convex hull vertices) of a set of points. The time bounds for our algorithm, in any constant dimension $d\ge 3$, improve on a previous algorithm for the same problem by Clarkson (FOCS 1994).

David Eppstein, Michael T. Goodrich, Jenny Lam et al.

We introduce models and algorithmic foundations for graph watermarking. Our frameworks include security definitions and proofs, as well as characterizations when graph watermarking is algorithmically feasible, in spite of the fact that the general problem is NP-complete by simple reductions from the subgraph isomorphism or graph edit distance problems. In the digital watermarking of many types of files, an implicit step in the recovery of a watermark is the mapping of individual pieces of data, such as image pixels or movie frames, from one object to another. In graphs, this step corresponds to approximately matching vertices of one graph to another based on graph invariants such as vertex degree. Our approach is based on characterizing the feasibility of graph watermarking in terms of keygen, marking, and identification functions defined over graph families with known distributions. We demonstrate the strength of this approach with exemplary watermarking schemes for two random graph models, the classic Erdős-Rényi model and a random power-law graph model, both of which are used to model real-world networks.