**Superpatterns and universal point sets**.

M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein.

arXiv:1308.0403.

*21st Int. Symp. Graph Drawing*, Bordeaux, France, 2013.

Springer,*Lecture Notes in Comp. Sci.*8242, 2013, pp. 208–219.

Bannister's talk on this paper won the GD2013 best presentation award.

*J. Graph Algorithms & Applications*18 (2): 177–209, 2014 (special issue for GD'13).A superpattern of a set of permutations is a permutation that contains as a pattern every permutation in the set. Previously superpatterns had been considered for all permutations of a given length; we generalize this to sets of permutations defined by forbidden patterns; we show that the 213-avoiding permutations have superpatterns half the length of the known bound for all permutations, and that any proper permutation subclass of the 213-avoiding permutations has near-linear superpatterns. We apply these results to the construction of universal point sets, sets of points that can be used as the vertices of drawings of all n-vertex planar graphs. We use our 213-avoiding superpatterns to construct universal sets of size approximately

*n*^{2}/4, and we also construct near-linear universal sets for graphs of bounded pathwidth.**Linear-time algorithms for proportional apportionment**.

Z. Cheng and D. Eppstein.

arXiv:1409.2603.

*Proc. 25th Int. Symp. Algorithms and Computation (ISAAC 2014)*, Jeonju, Korea, 2014.

Springer,*Lecture Notes in Comp. Sci.*8889, 2014, pp. 581–592.

We consider a broad class of highest averages methods for proportional allocation (the problems of allocating seats to parties after a parliamentary election, or of allocating congressmen to states based on total population). We show that these methods can be simulated by an algorithm whose running time is proportional only to the number of parties or states, independent of the number of seats allocated or the number of voters.

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

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