Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques
Volume 1671 of the series Lecture Notes in Computer Science pp 257268
Pfaffian Algorithms for Sampling Routings on Regions with Free Boundary Conditions
 Russell A. MartinAffiliated withSchool of Mathematics, Georgia Institute of Technology
 , Dana RandallAffiliated withCollege of Computing and School of Mathematics, Georgia Institute of Technology
Abstract
Sets of nonintersecting, monotonic lattice paths, or fixed routings, provide a common representation for several combinatorial problems and have been the key element for designing sampling algorithms. Markov chain algorithms based on routings have led to efficient samplers for tilings, Eulerian orientations [8] and triangulations [9], while an algorithm which successively calculates ratios of determinants has led to a very fast method for sampling fixed routings [12]. We extend Wilson’s determinant algorithm [12] to sample free routings where the number of paths, as well as the endpoints, are allowed to vary. The algorithm is based on a technique due to Stembridge for counting free routings by calculating the Pfaffian of a suitable matrix [11] and a method of Colbourn, Myrvold and Neufeld [1] for efficiently calculating ratios of determinants. As an application, we show how to sample tilings on planar lattice regions with free boundary conditions.
 Title
 Pfaffian Algorithms for Sampling Routings on Regions with Free Boundary Conditions
 Book Title
 Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques
 Book Subtitle
 Third International Workshop on Randomization and Approximation Techniques in Computer Science, and Second International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOMAPPROX’99, Berkeley, CA, USA, August 811, 1999.
 Pages
 pp 257268
 Copyright
 1999
 DOI
 10.1007/9783540484134_26
 Print ISBN
 9783540663294
 Online ISBN
 9783540484134
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 1671
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Dorit S. Hochbaum ^{(4)}
 Klaus Jansen ^{(5)}
 José D. P. Rolim ^{(6)}
 Alistair Sinclair ^{(7)}
 Editor Affiliations

 4. Department of Industrial Engineering and Operations Research and Walter A. Haas School of Business, University of California
 5. Institute for Computer Science, University of Kiel
 6. Centre Universitaire d’Informatique
 7. Computer Science Division, University of California
 Authors

 Russell A. Martin ^{(8)}
 Dana Randall ^{(9)}
 Author Affiliations

 8. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA
 9. College of Computing and School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA
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