**Models and algorithms for graph watermarking**.

D. Eppstein, M. T. Goodrich, J. Lam, N. Mamano, M. Mitzenmacher, and M. Torres.

arXiv:1605.09425.

*Proc. 19th Information Security Conference (ISC 2016)*, Honolulu, Hawaii.

Springer,*Lecture Notes in Comp. Sci.*9866 (2016), pp. 283–301.We show how to modify a small number of edges in a large social network in order to make the modified copy easy to identify, even if an adversary tries to hide the modification by permuting the vertices and flipping a much larger number of edges. The result depends on the random fluctuation of vertex degrees in such networks, and the ability to uniquely identify vertices by their adjacencies to a small number of high-degree landmark vertices. This paper won the best student paper award at ISC for its student co-authors Lam, Mamano, and Torres.

**Algorithms for stable matching and clustering in a grid**.

D. Eppstein, M. T. Goodrich, and N. Mamano.

arXiv:1704.02303

*Proc. 18th International Workshop on Combinatorial Image Analysis (IWCIA 2017)*, Plovdiv, Bulgaria, 2017.

Springer,*Lecture Notes in Comp. Sci.*10256 (2017), pp. 117–131.Motivated by redistricting, we consider geometric variants of the stable matching problem in which points (such as the pixels of a discretization of the unit square) are to be matched to a smaller number of centers such that each center has the same number of matches and no match is unstable with respect to Euclidean distances. We show how to solve such problems in polylogarithmic time per matched point, experiment with practical heuristics for solving these problems, and test methods for moving the centers to improve the shape of the matched regions.

**Defining equitable geographic districts in road networks via stable matching**.

D. Eppstein, M. T. Goodrich, D. Korkmaz, and N. Mamano.

arXiv:1706.09593

*Proc. 25th ACM SIGSPATIAL Int. Conf. Advances in Geographic Information Systems (ACM SIGSPATIAL 2017)*, Redondo Beach, California, to appear.

We cluster road networks (modeled as planar graphs, or more generally as graphs obeying a separator theorem) with a given set of cluster centers, by matching graph vertices to centers stably according to distance: no unmatched vertex and center should have smaller distances than the matched pairs for the same points. We provide a separator-based data structure for dynamic nearest neighbor queries in planar or separated graphs, which allows the optimal stable clustering to be constructed in time

*O*(*n*^{3/2}log*n*). We also experiment with heuristics for fast practical construction of this clustering.**Reactive proximity data structures for graphs**.

D. Eppstein, M. T. Goodrich, and N. Mamano.

arXiv:1803.04555.

*Proc. 13th Latin American Theoretical Informatics Symposium (LATIN 2018)*, Buenos Aires, Argentina.

Springer,*Lecture Notes in Comp. Sci.*10807 (2018), pp. 777–789.We develop data structures for solving nearest neighbor queries for dynamic subsets of vertices in a planar graph, or more generally for a graph in any graph class with small separators (polynomial expansion).

**Stable-matching Voronoi diagrams: combinatorial complexity and algorithms**.

G. Barequet, D. Eppstein, M. T. Goodrich, and N. Mamano.

arXiv:1804.09411

*Proc. 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)*, Prague, to appear.The stable-matching Voronoi diagram of a collection of point sites in the plane, each with a specified area, is a collection of disjoint regions of the plane, one for each site and having the specified area, so that no pair of a point and a site are closer to each other than to the farthest other site and point that they may be matched to. We prove nearly-matching upper and lower bounds on the combinatorial complexity of these diagrams and provide algorithms that can compute them in a polynomial number of primitive steps.

Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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