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### Finding All Maximal Subsequences with Hereditary Properties

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### 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.

### BibTeX - Entry

```@InProceedings{bokal_et_al:LIPIcs:2015:5113,
author =	{Drago Bokal and Sergio Cabello and David Eppstein},
title =	{{Finding All Maximal Subsequences with Hereditary Properties}},
booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
pages =	{240--254},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-83-5},
ISSN =	{1868-8969},
year =	{2015},
volume =	{34},
editor =	{Lars Arge and J{\'a}nos Pach},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},