Dartmouth logo Dartmouth College Computer Science
Technical Report series
CS home
TR home
TR search TR listserv
By author: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
By number: 2017, 2016, 2015, 2014, 2013, 2012, 2011, 2010, 2009, 2008, 2007, 2006, 2005, 2004, 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992, 1991, 1990, 1989, 1988, 1987, 1986

Methods of Computing Deque Sortable Permutations Given Complete and Incomplete Information
Daniel J. Denton
Dartmouth TR2012-719

Abstract: The problem of determining which permutations can be sorted using certain switchyard networks is a venerable problem in computer science dating back to Knuth in 1968. In this work, we are interested in permutations which are sortable on a double-ended queue (called a deque), or on two parallel stacks. In 1982, Rosenstiehl and Tarjan presented an O(n) algorithm for testing whether a given permutation was sortable on parallel stacks. In the same paper, they also presented a modification giving O(n) test for sortability on a deque. We demonstrate a slight error in the version of their algorithm for testing deque sortability, and present a fix for this problem.

The general enumeration problem for both of these classes of permutations remains unsolved. What is known is that the growth rate of both classes is approximately Theta(8^n), so computing the number of sortable permutations of length n, even for small values of n, is difficult to do using any method that must evaluate each sortable permutation individually. As far as we know, the number of deque sortable permutations was known only up to n=14. This was computed using algorithms which effectively generate all sortable permutations. By using the symmetries inherent in the execution of Tarjan's algorithm, we have developed a new dynamic programming algorithm which can count the number of sortable permutations in both classes in O(n^5 2^n) time, allowing the calculation of the number of deque and parallel stack sortable permutation for much higher values of n than was previously possible.

Finally, we have examined the problem of trying to sort a permutation on a deque when the input elements are only revealed at the time when they are pushed to the deque. (Instead of having an omniscient view of the input permutation, this corresponds to encoding the input permutation as a deck of cards which must be drawn and pushed onto the deque without looking at the remaining cards in the deck.) We show that there are some sortable permutations which cannot necessarily be sorted correctly on a deque using only this imperfect information.

Note:

Senior Honors Thesis. Advisors: Peter Doyle and Scot Drysdale.


PDF PDF (848KB)

Bibliographic citation for this report: [plain text] [BIB] [BibTeX] [Refer]

Or copy and paste:
   Daniel J. Denton, "Methods of Computing Deque Sortable Permutations Given Complete and Incomplete Information." Dartmouth Computer Science Technical Report TR2012-719, June 2012.


Notify me about new tech reports.

Search the technical reports.

To receive paper copy of a report, by mail, send your address and the TR number to reports AT cs.dartmouth.edu


Copyright notice: The documents contained in this server are included by the contributing authors as a means to ensure timely dissemination of scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may not be reposted without the explicit permission of the copyright holder.

Technical reports collection maintained by David Kotz.