Chapter

Algorithms

Volume 450 of the series Lecture Notes in Computer Science pp 338-347

Date:

Planar separators and the Euclidean norm

  • Hillel GazitAffiliated withDepartment of Computer Science, Duke University
  • , Gary L. MillerAffiliated withSchool of Computer Science Carnegie Mellon University & Dept of Computer Science, University of Southern California

* Final gross prices may vary according to local VAT.

Get Access

Abstract

In this paper we show that every 2-connected embedded planar graph with faces of sizes d 1.....d f has a simple cycle separator of size 1.58 \(\sqrt {d_1^2 + \cdots + d_f^2 }\)and we give an almost linear time algorithm for finding these separators, O(no(n,n)). We show that the new upper bound expressed as a function of ‖G‖=\(\sqrt {d_1^2 + \cdots + d_f^2 }\)is no larger, up to a constant factor than previous bounds that where expressed in terms of \(\sqrt {d \cdot v}\)where d is the maximum face size and ν is the number of vertices and is much smaller for many graphs. The algorithms developed are simpler than earlier algorithms in that they work directly with the planar graph and its dual. They need not construct or work with the face-incidence graph as in [Mil86, GM87, GM].