7 research outputs found
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
One of the longest-standing open problems in computational geometry is to
bound the lower envelope of univariate functions, each pair of which
crosses at most times, for some fixed . This problem is known to be
equivalent to bounding the length of an order- Davenport-Schinzel sequence,
namely a sequence over an -letter alphabet that avoids alternating
subsequences of the form with length
. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let be the maximum length of an order- DS sequence over
letters. What is asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when is even or . However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order . Our results reveal that,
contrary to one's intuition, behaves essentially like
when is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201
Querying two boundary points for shortest paths in a polygonal domain
AbstractWe consider a variant of two-point Euclidean shortest path query problem: given a polygonal domain, build a data structure for two-point shortest path query, provided that query points always lie on the boundary of the domain. As a main result, we show that a logarithmic-time query for shortest paths between boundary points can be performed using O˜(n5) preprocessing time and O˜(n5) space where n is the number of corners of the polygonal domain and the O˜-notation suppresses the polylogarithmic factor. This is realized by observing a connection between Davenport–Schinzel sequences and our problem in the parameterized space. We also provide a tradeoff between space and query time; a sublinear time query is possible using O(n3+ϵ) space. Our approach also extends to the case where query points should lie on a given set of line segments