Variations of Stack Sorting and Pattern Avoidance

University of Technology Sydney. Faculty of Science.Sorting is a process of arranging certain objects into an ordered sequence. Real world problems such as sorting using switchyard networks, genome arrangement, and delivery of network data packets can be realised as sorting problems. The study of these problems can be translated into the study of sorting permutations using a system of data structures which can store and output data. For instance, the sorting problem in certain switchyard networks can be formulated as the problem of sorting permutations using . The results of this thesis address the following research questions related to sorting permutations. Open research questions 1. In a sorting process with a finite stack followed by an infinite stack in series, what is the depth of the finite stack at which the basis becomes infinite? 2. Is there a pattern avoidance characterisation for k-pass pop stack sortable permutations? Apart from answering these questions, we develop a new notion of barred pattern avoidance to accommodate some of the limitations in the existing barred pattern avoidance definition. With the new notion of barred pattern avoidance, a proof can be established to answer question 2 above. The organisation of this thesis is as follows. Chapter 1 gives a detailed introduction to the research in permutation patterns and stack sorting. It includes some history and some major research outcomes. Moreover, several variations of sorting machines are described. Finally, a few types of non-classical patterns are explained. Chapter 2 answers the first question above. Based on some experimental data, a conjecture was made that the basis changes from finite to infinite when the depth of the finite stack is 3. We found an infinite antichain in the form of extendable sequence of numbers to prove the conjecture. Chapter 3 answers the second question and introduces a new notion of barred pattern avoidance to characterise permutations sortable by a k-pass pop-stack. Then, we finish the chapter by proving the number of forbidden patterns that the permutations sortable by k-pass pop-stack must avoid is finite. The set of forbidden patterns can be algorithmically constructed. Chapter 4 concludes the thesis with some directions for future research

Novel Broadcasting Algorithm of Recursive Networks

[[abstract]]The interconnection network considered in this paper is the complete WK-Recursive network that demonstrates many attractive properties, such as high degree of regularity, symmetry and efficient communication. Chen and Duh have proposed a distributed stack-base broadcasting algorithm for the complete WK-Recursive networks [Networks, 24 (1994) 303-317]. To perform this algorithm, a stack of O(log N) elements, where N is the number of nodes, to keep the labels of links is included in each message. Moreover, as a node receives the message, a series of O(log N) pop and push operations on the stack is required. In this paper, we present a novel broadcasting algorithm for the complete WKRecursive network, which is much simpler and requires only constant data included in each message and constant time to determine the neighbors to forward the message

Sorting with a forklift

A fork stack is a generalised stack which allows pushes and pops of several items at a time. We consider the problem of determining which input streams can be sorted using a single forkstack, or dually, which permutations of a fixed input stream can be produced using a single forkstack. An algorithm is given to solve the sorting problem and the minimal unsortable sequences are found. The results are extended to fork stacks where there are bounds on how many items can be pushed and popped at one time. In this context we also establish how to enumerate the collection of sortable sequences.Comment: 24 pages, 2 figure

Towards a Uniform Theory of Effectful State Machines

Using recent developments in coalgebraic and monad-based semantics, we present a uniform study of various notions of machines, e.g. finite state machines, multi-stack machines, Turing machines, valence automata, and weighted automata. They are instances of Jacobs' notion of a T-automaton, where T is a monad. We show that the generic language semantics for T-automata correctly instantiates the usual language semantics for a number of known classes of machines/languages, including regular, context-free, recursively-enumerable and various subclasses of context free languages (e.g. deterministic and real-time ones). Moreover, our approach provides new generic techniques for studying the expressivity power of various machine-based models.Comment: final version accepted by TOC

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

Two Vignettes On Full Rook Placements

Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234-avoiding permutations, obtained recently by Callan via a complicated decomposition. The second is a streamlined bijection between 1342-avoiding permutations and permutations which can be sorted by two increasing stacks in series, originally due to Atkinson, Murphy, and Ru\v{s}kuc.Comment: 9 pages, 4 figure

Operational semantics for signal handling

Signals are a lightweight form of interprocess communication in Unix. When a process receives a signal, the control flow is interrupted and a previously installed signal handler is run. Signal handling is reminiscent both of exception handling and concurrent interleaving of processes. In this paper, we investigate different approaches to formalizing signal handling in operational semantics, and compare them in a series of examples. We find the big-step style of operational semantics to be well suited to modelling signal handling. We integrate exception handling with our big-step semantics of signal handling, by adopting the exception convention as defined in the Definition of Standard ML. The semantics needs to capture the complex interactions between signal handling and exception handling.Comment: In Proceedings EXPRESS/SOS 2012, arXiv:1208.244