Approximating the Diameter of Planar Graphs in Near Linear Time

We present a (1 + ϵ)-approximation algorithm running in <i>O</i>(<i>f</i>(ϵ) · <i>n</i>log <sup>4</sup><i>n</i>) time for finding the diameter of an undirected planar graph with <i>n</i> vertices and with nonnegative edge lengths.

[1]  Ittai Abraham,et al.  Object location using path separators , 2006, PODC '06.

[2]  Piotr Indyk,et al.  Fast estimation of diameter and shortest paths (without matrix multiplication) , 1996, SODA '96.

[3]  Greg N. Frederickson,et al.  Fast Algorithms for Shortest Paths in Planar Graphs, with Applications , 1987, SIAM J. Comput..

[4]  Robert E. Tarjan,et al.  Better Approximation Algorithms for the Graph Diameter , 2014, SODA.

[5]  Donald B. Johnson,et al.  Efficient Algorithms for Shortest Paths in Sparse Networks , 1977, J. ACM.

[6]  Philip N. Klein,et al.  Multiple-source shortest paths in planar graphs , 2005, SODA '05.

[7]  Raphael Yuster,et al.  Approximating the Diameter of Planar Graphs in Near Linear Time , 2013, ICALP.

[8]  Piotr Berman,et al.  Faster Approximation of Distances in Graphs , 2007, WADS.

[9]  Stephan Olariu,et al.  A simple linear-time algorithm for computing the center of an interval graph , 1990, Int. J. Comput. Math..

[10]  Yijie Han,et al.  An O(n 3 loglogn/log2 n) Time Algorithm for All Pairs Shortest Paths , 2012, SWAT.

[11]  Arthur M. Farley,et al.  Computation of the center and diameter of outerplanar graphs , 1980, Discret. Appl. Math..

[12]  Ryan Williams,et al.  Faster all-pairs shortest paths via circuit complexity , 2013, STOC.

[13]  Philip N. Klein,et al.  Preprocessing an undirected planar network to enable fast approximate distance queries , 2002, SODA '02.

[14]  David Hartvigsen,et al.  The All-Pairs Min Cut Problem and the Minimum Cycle Basis Problem on Planar Graphs , 1994, SIAM J. Discret. Math..

[15]  Feodor F. Dragan,et al.  A Linear-Time Algorithm for Finding a Central Vertex of a Chordal Graph , 1994, ESA.

[16]  Philip N. Klein,et al.  Faster Shortest-Path Algorithms for Planar Graphs , 1997, J. Comput. Syst. Sci..

[17]  Liam Roditty,et al.  Fast approximation algorithms for the diameter and radius of sparse graphs , 2013, STOC '13.

[18]  Timothy M. Chan.  More algorithms for all-pairs shortest paths in weighted graphs , 2007, STOC '07.

[19]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[20]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[21]  Karlis Freivalds,et al.  Fast and Simple Approximation of the Diameter and Radius of a Graph , 2006, WEA.

[22]  David Eppstein,et al.  The Polyhedral Approach to the Maximum Planar Subgraph Problem: New Chances for Related Problems , 1994, GD.

[23]  Mikkel Thorup.  Compact oracles for reachability and approximate distances in planar digraphs , 2004, JACM.

[24]  Christian Wulff-Nilsen.  Wiener Index, Diameter, and Stretch Factor of a Weighted Planar Graph in Subquadratic Time , 2015 .

[25]  Seth Pettie,et al.  A Faster All-Pairs Shortest Path Algorithm for Real-Weighted Sparse Graphs , 2002, ICALP.

[26]  Feodor F. Dragan,et al.  LexBFS-Orderings and Power of Graphs , 1996, WG.