Small Superpatterns for Dominance Drawing

Bannister, Michael J.; Devanny, William E.; Eppstein, David

doi:10.1137/1.9781611973204.9

Cited by 10 publications

(10 citation statements)

References 29 publications

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“…For this reason, the permutations that define planar dominance drawings have no forbidden patterns. However, in later research, we have shown that the dominance drawings of some other classes of graphs have forbidden patterns, leading to smaller universal sets for these drawings [24].…”

Section: Dominance Drawingmentioning

confidence: 82%

Superpatterns and Universal Point Sets

et al. 2013

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Abstract. An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n 2 /4 + Θ(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n 2 /4 − Θ(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(n log O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.

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Section: Dominance Drawingmentioning

confidence: 82%

Superpatterns and Universal Point Sets

et al. 2013

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“…A similar but simpler reduction uses S n -superpatterns to construct universal point sets for dominance drawings of transitively reduced st-planar graphs. In subsequent work [4], we study dominance drawings based on superpatterns for 321-avoiding permutations and their subclasses, and we relate these subclasses to natural classes of st-planar graphs and nonplanar Hasse diagrams of width-2 partial orders.…”

Section: Introductionmentioning

confidence: 99%

Superpatterns and Universal Point Sets

Bannister¹,

Cheng²,

Devanny³

et al. 2014

JGAA

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An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n 2 /4 + Θ(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n 2 /4−Θ(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(n log O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.

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“…The width of a partial order is the maximum number of elements in an antichain, a set of mutually-incomparable elements. Low-width partial orders arise, for instance, in the edit histories of version control repositories [5]. The treewidth of a partial order is less than twice its width, 1 but partial orders of width w have O(n w ) downsets, tighter than the bound that would be obtained by using treewidth.…”

Section: New Resultsmentioning

confidence: 99%

“…Partially ordered sets of bounded width arise, for instance, in the version histories of a distributed version control repository controlled by a small set of developers (with the assumption that each developer maintains only a single branch of the version history). In this application, there may be many elements of the partially ordered set (versions of the repository) but the width may be bounded by the number of developers [5]. A downset in this application is a set of versions that could possibly describe the simultaneous states of all the developers at some past moment in the history of the repository.…”

Section: Bounded Widthmentioning

confidence: 99%

The Parametric Closure Problem

Eppstein

2017

ACM Trans. Algorithms

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Abstract. We define the parametric closure problem, in which the input is a partially ordered set whose elements have linearly varying weights and the goal is to compute the sequence of minimum-weight downsets of the partial order as the weights vary. We give polynomial time solutions to many important special cases of this problem including semiorders, reachability orders of bounded-treewidth graphs, partial orders of bounded width, and series-parallel partial orders. Our result for series-parallel orders provides a significant generalization of a previous result of Carlson and Eppstein on bicriterion subtree problems.

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Small Superpatterns for Dominance Drawing

Cited by 10 publications

References 29 publications

Superpatterns and Universal Point Sets

Superpatterns and Universal Point Sets

Superpatterns and Universal Point Sets

The Parametric Closure Problem

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