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Finding All Maximal Subsequences with Hereditary Properties

Authors Drago Bokal, Sergio Cabello, David Eppstein

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Drago Bokal
Sergio Cabello
David Eppstein

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Drago Bokal, Sergio Cabello, and David Eppstein. Finding All Maximal Subsequences with Hereditary Properties. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 240-254, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.SOCG.2015.240

Abstract

Consider a sequence s_1,...,s_n of points in the plane. We want to find all maximal subsequences with a given hereditary property P: find for all indices i the largest index j^*(i)  such that s_i,...,s_{j^*(i)} has property P. We provide a general methodology that leads to the following specific results:

- In O(n log^2 n) time we can find all maximal subsequences with diameter at most 1. 

- In O(n log n loglog n) time we can find all maximal subsequences whose convex hull has area at most 1.

- In O(n) time we can find all maximal subsequences that define monotone paths in some (subpath-dependent) direction.

The same methodology works for graph planarity, as follows. Consider a sequence of edges e_1,...,e_n over a vertex set V. In O(n log n) time we can find, for all indices i, the largest index j^*(i) such that (V,{e_i,..., e_{j^*(i)}}) is planar.

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Keywords
  • convex hull
  • diameter
  • monotone path
  • sequence of points
  • trajectory

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