Kevin Cattell

Synthesis of one-dimensional linear hybrid cellular automata

This paper presents a method for the synthesis of a one-dimensional linear hybrid cellular automaton (CA) from a given irreducible polynomial. A detailed description of the algorithm is given, together with an outline of the theoretical background. It is shown that two CA exist for each irreducible polynomial, solving the previously open CA existence conjecture. An in-depth example of the synthesis is presented, along with timing benchmarks and an operation count. The algorithm solves the previously open problem of synthesizing CA for all practical applications.

Fast Algorithms to Generate Necklaces, Unlabeled Necklaces, and Irreducible Polynomials over GF(2)

Many applications call for exhaustive lists of strings subject to various constraints, such as inequivalence under group actions. A k-ary necklace is an equivalence class of k-ary strings under rotation (the cyclic group). A k-ary unlabeled necklace is an equivalence class of k-ary strings under rotation and permutation of alphabet symbols. We present new, fast, simple, recursive algorithms for generating (i.e., listing) all necklaces and binary unlabeled necklaces. These algorithms have optimal running times in the sense that their running times are proportional to the number of necklaces produced. The algorithm for generating necklaces can be used as the basis for efficiently generating many other equivalence classes of strings under rotation and has been applied to generating bracelets, fixed density necklaces, and chord diagrams. As another application, we describe the implementation of a fast algorithm for listing all degree n irreducible and primitive polynomials over GF(2).

Minimal cost one-dimensional linear hybrid cellular automata of degree through 500

This letter is a supplement of the table of the minimal cost one-dimensional linear hybrid cellular automata with the maximum length cycle by Zhang, Miller, and Muzio [IEE Electronics Letters, 27(18):1625–1627, August 1991].

Analysis of one-dimensional linear hybrid cellular automata over GF(q)

The paper studies theoretical aspects of one dimensional linear hybrid cellular automata over a finite (Galois) field. General results concerning the characteristic polynomials of such automata are presented. A probabilistic synthesis algorithm for determining such a linear hybrid cellular automaton with a specific characteristic polynomial is given, along with empirical results and a theoretical analysis. Cyclic boundary cellular automata are defined and related to the more common null boundary cellular automate. An explicit similarity transform between a cellular automaton and its corresponding linear feedback shift register is derived.

On computing graph minor obstruction sets

Abstract The Graph Minor Theorem of Robertson and Seymour establishes nonconstructively that many natural graph properties are characterized by a finite set of forbidden substructures, the obstructions for the property. We prove several general theorems regarding the computation of obstruction sets from other information about a family of graphs. The methods can be adapted to other partial orders on graphs, such as the immersion and topological orders. The algorithms are in some cases practical and have been implemented. Two new technical ideas are introduced. The first is a method of computing a stopping signal for search spaces of increasing pathwidth. This allows obstruction sets to be computed without the necessity of a prior bound on maximum obstruction width. The second idea is that of a second order congruence for a graph property. This is an equivalence relation defined on finite sets of graphs that generalizes the recognizability congruence that is defined on single graphs. It is shown that the obstructions for a graph ideal can be effectively computed from an oracle for the canonical second-order congruence for the ideal and a membership oracle for the ideal. It is shown that the obstruction set for a union F = F 1 ∪ F 2 of minor ideals can be computed from the obstruction sets for F 1 and F 2 if there is at least one tree that does not belong to the intersection of F 1 and F 2 . As a corollary, it is shown that the set of intertwines of an arbitrary graph and a tree are effectively computable.

A Characterization of Graphs with Vertex Cover up to Five

For the family of graphs with fixed-size vertex cover k, we present all of the forbidden minors (obstructions), for k up to five. We derive some results, including a practical finite-state recognition algorithm, needed to compute these obstructions.

Forbidden Minors to Graphs with Small Feedback Sets

Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the graph minor theorem. In this paper we characterize several families of graphs with small feedback sets, namely k1-FEEDBACK VERTEX SET, k2-FEEDBACK EDGE SET and (k1;k2)FEEDBACK VERTEX=EDGE SET, for small integer parameters k1 and k2. Our constructive methods can compute obstruction sets for any minor-closed family of graphs, provided the pathwidth (or treewidth) of the largest obstruction is known. c 2001 Elsevier Science B.V. All rights reserved.

Obstructions to Within a Few Vertices or Edges of Acyclic

Finite obstruction sets for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. It has been known for several years that, in principle, obstruction sets can be mechanically computed for most natural lower ideals. In this paper, we describe a general-purpose method for finding obstructions by using a bounded treewidth (or pathwidth) search. We illustrate this approach by characterizing certain families of cycle-cover graphs based on the two well-known problems: k-Feedback Vertex Set and k-Feedback Edge Set. Our search is based on a number of algorithmic strategies by which large constants can be mitigated, including a randomized strategy for obtaining proofs of minimality.

A simple linear-time algorithm for finding path-decompositions of small width

Abstract We described a simple algorithm running in linear time for each fixed constant, k , that either establishes that the pathwidth of a graph G is greater than k , or finds a path-decomposition of G of width at most O (2 k ). This provides a simple proof of the result by Bodlaender that many families of graphs of bounded pathwidth can be recognized in linear time.

The analysis of one dimensional multiple-valued linear cellular automata

The authors present an analysis of multiple-valued linear cellular automata (CA) and their properties over GF(q). An application for pseudorandom pattern generation over a finite alphabet is discussed. For these cellular automata, the legal computational rules are defined and classified. The desired cellular automata must also have a maximal-length cycle in their state transition graph. An efficient recurrence relation and similarity transformations are presented. Three separate methods are outlined to produce minimal-cost CA with the above properties.<<ETX>>