ACM Transactions on Algorithms
Testing bipartiteness of geometric intersection graphs.
D. Eppstein.
arXiv:cs.CG/0307023.
15th ACM-SIAM Symp. Discrete Algorithms, New Orleans, 2004, pp. 853–861.
ACM Trans. Algorithms 5(2):15, 2009.We consider problems of partitioning sets of geometric objects into two subsets, such that no two objects within the same subset intersect each other. Typically, such problems can be solved in quadratic time by constructing the intersection graph and then applying a graph bipartiteness testing algorithm; we achieve subquadratic times for general objects, and O(n log n) times for balls in R^d or simple polygons in the plane, by using geometric data structures, separator based divide and conquer, and plane sweep techniques, respectively. We also contrast the complexity of bipartiteness testing with that of connectivity testing, and provide evidence that for some classes of object, connectivity is strictly harder due to a computational equivalence with Euclidean minimum spanning trees.
Deterministic sampling and range counting in geometric data streams.
A. Bagchi, A. Chaudhary, D. Eppstein, and M. T. Goodrich.
arXiv:cs.CG/0307027.
20th ACM Symp. Comp. Geom., Brooklyn, 2004, pp. 144–151.
ACM Trans. Algorithms 3(2):A16, 2007.We describe an efficient streaming-model construction of epsilon-nets and epsilon-approximations, and use it to find deterministic streaming-model approximation algorithms for iceberg range queries and for various robust statistics problems.
All maximal independent sets and dynamic dominance for sparse graphs.
D. Eppstein.
arXiv:cs.DS/0407036.
16th ACM-SIAM Symp. Discrete Algorithms, Vancouver, 2005, pp. 451–459.
ACM Trans. Algorithms 5(4):A38, 2009.We show how to apply reverse search to list all maximal independent sets in bounded-degree graphs in constant time per set, in graphs from minor closed families in linear time per set, and in sparse graphs in subquadratic time per set. The latter two results rely on new data structures for maintaining a dynamic vertex set in a graph and quickly testing whether the set dominates all other vertices.
Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition.
D. Eppstein.
arXiv:cs.CG/0604034.
18th ACM-SIAM Symp. Discrete Algorithms, New Orleans, 2007, pp. 29–38.
ACM Trans. Algorithms 5(3): Article 29, 2009.We find efficient constant factor approximation algorithms for hierarchically clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition with approximately minimum total length.
The parametric closure problem.
D. Eppstein.
arXiv:1504.04073.
14th Algorithms and Data Structures Symp. (WADS 2015), Victoria, BC.
Springer, Lecture Notes in Comp. Sci. 9214 (2015), pp. 327–338.
ACM Trans. Algorithms 14 (1): Article 2, 2018.We consider the minimum weight closure problem for a partially ordered set whose elements have weights that vary linearly as a function of a parameter. For several important classes of partial orders the number of changes to the optimal solution as the parameter varies is near-linear, and the sequence of optimal solutions can be found in near-linear time.