19 research outputs found
Methods of computing deque sortable permutations given complete and incomplete information
The problem of determining which permutations can be sorted using certain
switchyard networks dates 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.Comment: dartmouth senior honors thesis advised by Peter Doyle and Scot
Drysdale 45 pages, 9 figure
Sorting by shuffling methods and a queue
We consider sorting by a queue that can apply a permutation from a given set
over its content. This gives us a sorting device
corresponding to any shuffling method since every such method is
associated with a set of permutations. Two variations of these devices are
considered - and
. These require the entire content of the
device to be unloaded after a permutation is applied or unloaded by each pop
operation, respectively.
First, we show that sorting by a deque is equivalent to sorting by a queue
that can reverse its content. Next, we focus on sorting by cuts. We prove that
the set of permutations that one can sort by using
is the set of the -avoiding separable
permutations. We give lower and upper bounds to the maximum number of times the
device must be used to sort a permutation. Furthermore, we give a formula for
the number of -permutations, , that one
can sort by using , for any shuffling method
, such that the permutations associated with it are irreducible.
Next, we prove a generalization of the fact that
can sort all permutations. We also show
that is given by the odd indexed
Fibonacci numbers , for any shuffling method having a
specific back-front property. The rest of the work is dedicated to a surprising
conjecture inspired by Diaconis and Graham which states that one can sort the
same number of permutations of any given size by using the devices
and
, corresponding to the popular
In-shuffle and Monge shuffling methods.Comment: 29 pages, 7 figure
2-stack pushall sortable permutations
In the 60's, Knuth introduced stack-sorting and serial compositions of
stacks. In particular, one significant question arise out of the work of Knuth:
how to decide efficiently if a given permutation is sortable with 2 stacks in
series? Whether this problem is polynomial or NP-complete is still unanswered
yet. In this article we introduce 2-stack pushall permutations which form a
subclass of 2-stack sortable permutations and show that these two classes are
closely related. Moreover, we give an optimal O(n^2) algorithm to decide if a
given permutation of size n is 2-stack pushall sortable and describe all its
sortings. This result is a step to the solve the general 2-stack sorting
problem in polynomial time.Comment: 41 page
Sorting Pattern-Avoiding Permutations via 0-1 Matrices Forbidding Product Patterns
We consider the problem of comparison-sorting an -permutation that
avoids some -permutation . Chalermsook, Goswami, Kozma, Mehlhorn, and
Saranurak prove that when is sorted by inserting the elements into the
GreedyFuture binary search tree, the running time is linear in the extremal
function . This is the maximum number
of 1s in an 0-1 matrix avoiding , where
is the permutation matrix of , the Kronecker
product, and . The
same time bound can be achieved by sorting with Kozma and Saranurak's
SmoothHeap.
In this paper we give nearly tight upper and lower bounds on the density of
-free matrices in terms of the inverse-Ackermann
function . \mathrm{Ex}(P_\pi\otimes \text{hat},n) =
\left\{\begin{array}{ll} \Omega(n\cdot 2^{\alpha(n)}), & \mbox{for most
$\pi$,}\\ O(n\cdot 2^{O(k^2)+(1+o(1))\alpha(n)}), & \mbox{for all $\pi$.}
\end{array}\right. As a consequence, sorting -free sequences can be
performed in time. For many corollaries of the
dynamic optimality conjecture, the best analysis uses forbidden 0-1 matrix
theory. Our analysis may be useful in analyzing other classes of access
sequences on binary search trees
Permutation classes
This is a survey on permutation classes for the upcoming book Handbook of
Enumerative Combinatorics