Construction of linear tree-layouts which are optimal with respect to vertex separation in linear time

We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers the open questions raised in [Ellis et al., Inform. Comput. 113 (1994) 50-79, Megiddo, J. Assoc. Comput. Mach. 35 (1) (1988) 18-44].

[1]  T. D. Parsons,et al.  Pursuit-evasion in a graph , 1978 .

[2]  Rolf H. Möhring,et al.  The Pathwidth and Treewidth of Cographs , 1993, SIAM J. Discret. Math..

[3]  Mihalis Yannakakis,et al.  A polynomial algorithm for the MIN CUT linear arrangement of trees , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[4]  Ton Kloks,et al.  Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs , 1993, J. Algorithms.

[5]  Paul D. Seymour,et al.  Graph minors. I. Excluding a forest , 1983, J. Comb. Theory, Ser. B.

[6]  Ernst W. Mayr,et al.  Computational Graph Theory , 2002, Computing Supplementum.

[7]  Dieter Kratsch,et al.  Treewidth and Pathwidth of Permutation Graphs , 1995, SIAM J. Discret. Math..

[8]  Fillia Makedon,et al.  On minimizing width in linear layouts , 1989, Discret. Appl. Math..

[9]  Ivan Hal Sudborough,et al.  Min Cut is NP-Complete for Edge Weigthed Trees , 1986, ICALP.

[10]  Fillia Makedon,et al.  Polynomial time algorithms for the MIN CUT problem on degree restricted trees , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[11]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[12]  Thomas Lengauer.  Black-white pebbles and graph separation , 2004, Acta Informatica.

[13]  Christos H. Papadimitriou,et al.  The complexity of searching a graph , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[14]  D. R. Lick,et al.  Theory and Applications of Graphs , 1978 .

[15]  Stephen A. Cook,et al.  Storage Requirements for Deterministic Polynomial Time Recognizable Languages , 1976, J. Comput. Syst. Sci..

[16]  Christos H. Papadimitriou,et al.  Searching and Pebbling , 1986, Theor. Comput. Sci..

[17]  Nancy G. Kinnersley,et al.  The Vertex Separation Number of a Graph equals its Path-Width , 1992, Inf. Process. Lett..

[18]  Neil Robertson,et al.  Disjoint Paths—A Survey , 1985 .

[19]  Christos H. Papadimitriou,et al.  Interval graphs and seatching , 1985, Discret. Math..

[20]  Jens Gustedt,et al.  On the Pathwidth of Chordal Graphs , 1993, Discret. Appl. Math..

[21]  Andrea S. LaPaugh,et al.  Recontamination does not help to search a graph , 1993, JACM.

[22]  Ivan Hal Sudborough,et al.  The Vertex Separation and Search Number of a Graph , 1994, Inf. Comput..

[23]  Rolf H. Möhring,et al.  Graph Problems Related to Gate Matrix Layout and PLA Folding , 1990 .