摘要：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.
关键词：convex hull; diameter; monotone path; sequence of points; trajectory
