David Eppstein - Publications

Publications with Maarten Löffler

• Listing all maximal cliques in sparse graphs in near-optimal time.
  D. Eppstein, M. Löffler, and D. Strash.
  arXiv:1006.5440.
  Workshop on Exact Algorithms for NP-Hard Problems, Dagstuhl, Germany, 2010.
  Proc. 21st International Symposium on Algorithms and Computation (ISAAC 2010), Jeju, Korea, 2010.
  Springer, Lecture Notes in Comp. Sci. 6506, 2010, pp. 403–414.
  

  We describe an algorithm for finding all maximal cliques in a graph, in time O(dn3^d/3) where n is the number of vertices and d is the degeneracy of the graph, a standard measure of its sparsity. This time bound matches the worst-case output size for these parameters. The algorithm modifies the Bron-Kerbosch algorithm for maximal cliques by ordering the vertices by degree in the outer recursive call of the algorithm.

  For the journal version, see "Listing all maximal cliques in large sparse real-world graphs," which combines results from this and a different conference paper.

• Drawing graphs in the plane with a prescribed outer face and polynomial area.
  E. Chambers, D. Eppstein, M. T. Goodrich, and M. Löffler.
  Proc. 18th Int. Symp. Graph Drawing, Konstanz, Germany, 2010.
  Springer, Lecture Notes in Comp. Sci. 6502, 2011, pp. 129–140.
  arXiv:1009.0088.
  J. Graph Algorithms and Applications 16 (2): 243–259, 2012.

  Tutte's method of spring embeddings allows any triangulated planar graph to be drawn so that the outer face has any pre-specified convex shape, but it may place vertices exponentially close to each other. Alternative graph drawing methods provide polynomial-area straight line drawings but do not allow the outer face shape to be specified. We describe a drawing method that combines both properties: it has polynomial area, and can match any pre-specified shape of the outer face, even a shape in which some of the vertices have 180 degree angles. We apply our results to drawing polygonal schemas for graphs embedded on surfaces of positive genus.

• Optimal 3D angular resolution for low-degree graphs.
  D. Eppstein, M. Löffler, E. Mumford, and M. Nöllenburg.
  Proc. 18th Int. Symp. Graph Drawing, Konstanz, Germany, 2010.
  Springer, Lecture Notes in Comp. Sci. 6502, 2011, pp. 208–219.
  arXiv:1009.0045.
  J. Graph Algorithms and Applications 17 (3): 173–200, 2013.

  We show how to draw any graph of maximum degree three in three dimensions with 120 degree angles at each vertex or bend, and any graph of maximum degree four in three dimensions with the angles of the diamond lattice at each vertex or bend. In each case there are no crossings and the number of bends per edge is a small constant.

• Bounds on the complexity of halfspace intersections when the bounded faces have small dimension.
  D. Eppstein and M. Löffler.
  Proc. 27th ACM Symp. on Computational Geometry, Paris, 2011, pp. 361–368.
  arXiv:1103.2575.
  Discrete Comput. Geom. 50 (1): 1–21, 2013.

  Suppose that P is the intersection of n halfspaces in D dimensions, but that the bounded faces of P are at most d-dimensional, for some d that is much smaller than D. Then in this case we show that the number of vertices of P is O(n^d), independent of D. We also investigate related bounds on the number of bounded faces of all dimensions of P, and algorithms for efficiently listing the vertices and bounded faces of P.

• Adjacency-preserving spatial treemaps.
  K. Buchin, D. Eppstein, M. Löffler, M. Nöllenburg, and R. I. Silveira.
  arXiv:1105.0398.
  12th International Symposium on Algorithms and Data Structures (WADS 2011), New York, 2011.
  Springer, Lecture Notes in Comp. Sci. 6844, 2011, pp. 159–170.
  J. Computational Geometry 7 (1): 100–122, 2016.

  We study the recursive partitions of rectangles into sets of rectangles, and partitions of those rectangles into smaller rectangles, to form stylized visualizations of hierarchically subdivided geographic regions. There are several variations of varying difficulty depending on how much of the geographic information from the input we require the output to preserve.

• Tracking moving objects with few handovers.
  D. Eppstein, M. T. Goodrich, and M. Löffler.
  arXiv:1105.0392.
  12th International Symposium on Algorithms and Data Structures (WADS 2011), New York, 2011.
  Springer, Lecture Notes in Comp. Sci. 6844, 2011, pp. 362–373.

  We apply competitive analysis to the problem of deciding online which cell phone tower to change to when a phone moves out of the coverage region of the tower it is connected to. We show that, when the coverage regions have constant ply (at most a constant number of them overlap any point of the plane) it is possible to get within a constant factor of the minimum possible number of handovers that an offline algorithm could achieve.

• Category-based routing in social networks: membership dimension and the small-world phenomenon.
  D. Eppstein, M. T. Goodrich, M. Löffler, D. Strash, and L. Trott.
  Workshop on Graph Algorithms and Applications, Zürich, Switzerland, July 2011.
  International Conference on Computational Aspects of Social Networks (CASoN 2011), Salamanca, Spain, October 2011.
  arXiv:1108.4675.
  Theor. Comput. Sci. 514: 96–104, 2013. (Special issue on Graph Algorithms and Applications: in Honor of Professor Giorgio Ausiello)
  

  We investigate greedy routing schemes for social networks, in which participants know categorical information about some other participants and use it to guide message delivery by forwarding messages to neighbors that have more categories in common with the eventual destination. We define the membership dimension of such a scheme to be the maximum number of categories that any individual belongs to, a natural measure of the cognitive load of greedy routing on its participants. And we show that membership dimension is closely related to the small world phenomenon: a social network can be given a category system with polylogarithmic membership dimension that supports greedy routing if, and only if, the network has polylogarithmic diameter.

• Planar and poly-arc Lombardi drawings.
  C. Duncan, D. Eppstein, M. T. Goodrich, S. Kobourov, and M. Löffler, and M. Nöllenburg.
  Proc. 19th Int. Symp. Graph Drawing, Eindhoven, The Netherlands, 2011.
  Springer, Lecture Notes in Comp. Sci. 7034, 2012, pp. 308–319.
  arXiv:1109.0345.
  J. Computational Geometry 9 (1): 328–355, 2018.

  We extend Lombardi drawing (in which each edge is a circular arc and the edges incident to a vertex must be equally spaced around it) to drawings in which edges are composed of multiple arcs, and we investigate the graphs that can be drawn in this more relaxed framework.

• Strict confluent drawing.
  D. Eppstein, D. Holten, M. Löffler, M. Nöllenburg, and B. Speckmann, and K. Verbeek.
  arXiv:1308.6824.
  21st Int. Symp. Graph Drawing, Bordeaux, France, 2013.
  Springer, Lecture Notes in Comp. Sci. 8242, 2013, pp. 352–363.
  J. Computational Geometry 7 (1): 22–46, 2016.

  A confluent drawing of a graph is a set of points and curves in the plane with the property that two vertices are adjacent in the graph if and only if the corresponding points can be connected to each other by smooth paths in the drawing. We define a variant of confluent drawing, strict confluent drawing, in which a smooth path between two vertices (if it exists) must be unique. We show that it is NP-complete to test whether such drawings exist, in contrast to unrestricted confluence for which the complexity remains open. Additionally, we show that finding outerplanar drawings (in which the points are on the boundary of a disk and the curves are interior to it) with a fixed cyclic vertex ordering can be performed in polynomial time. We use circle packings to find nice versions of these drawings in which all tracks are represented by piecewise-circular curves.

• Listing all maximal cliques in large sparse real-world graphs.
  D. Eppstein, M. Löffler, and D. Strash.
  J. Experimental Algorithmics 18 (3): 3.1, 2013 (special issue for SEA).

  This paper combines our theoretical results on clique-finding algorithms from ISAAC 2010 with our experimental results on the same algorithms from SEA 2011. We show how to list all maximal cliques in graphs of bounded degeneracy in time that is linear in the size of the graph and near-optimal in the degeneracy, and we show that low degeneracy explains the good behavior of the algorithm in our experiments on large real-world social networks.