On the 2-Central Path Problem

  • Yongding Zhu
  • Jinhui Xu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

In this paper we consider the following 2-Central Path Problem (2CPP): Given a set of m polygonal curves \(\mathcal{P} =\{P_1,P_2,\ldots,P_m\}\) in the plane, find two curves P u and P l , called 2-central paths, that best represent all curves in \(\mathcal{P}\). Despite its theoretical interest and wide range of practical applications, 2CPP has not been well studied. In this paper, we first establish criteria that P u and P l ought to meet in order for them to best represent \(\mathcal{P}\). In particular, we require that there exists parametrizations f u (t) and f l (t) (t ∈ [a,b]) of P u and P l respectively such that the maximum distance from {f u (t), f l (t)} to curves in \(\mathcal{P}\) is minimized. Then an efficient algorithm is presented to solve 2CPP under certain realistic assumptions. Our algorithm constructs P u and P l in O(nmlog4 n 2 α(n) α(n)) time, where n is the total complexity of \(\mathcal{P}\) (i.e., the total number of vertices and edges), m is the number of curves in \(\mathcal{P}\), and α(n) is the inverse Ackermann function.Our algorithm uses the parametric search technique and is faster than arrangement-related algorithms (i.e. Ω(n 2)) when m ≪ n as in most real applications.

Keywords

Event Point Total Complexity Input Curve Parallel Step Input Trajectory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yongding Zhu
    • 1
  • Jinhui Xu
    • 1
  1. 1.Department of Computer Science and EngineeringState University of New York at BuffaloBuffaloUSA

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