Some Results in Computational Topology
Journal of the ACM (JACM)
Iterative Arrays with Small Time Bounds
MFCS '00 Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science
Iterative Arrays with a Wee Bit Alternation
FCT '99 Proceedings of the 12th International Symposium on Fundamentals of Computation Theory
Fast parallel processing array algorithms for some graph problems(Preliminary Version)
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
On Some Open Problems in the Theory of Cellular Automata
IEEE Transactions on Computers
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The pattern of initial states thus introduced represents a computation based on the input figure. If one waits for a specially designated cell to indicate acceptance or rejection of the figure, the array is said to be working on a recognition problem. If one waits for the array to come to a stable configuration representing an output figure, the array is said to be working on a transformation problem. Chapter 2 contains a general theory of recognition. Theorems on the amount of time required to perform recognition and on methods of speeding up recognition are presented. Some properties of the classes of recognizable figures are given. Arrays are compared to other types of figure recognition devices. In the last section the class of linear predicates is studied. A linear predicate is a family of figures which can be recognized in time proportional to the perimeter of the figure. Chapter 3 contains a study of the recognition of some topologically invariant properties of figures. A fundamental transformation of figures is presented and is then used to show that a wide variety of topologically invariant properties form linear predicates including connectivity and maze solvability. Two properties whose linearity is open are discussed. Chapter 4 contains a brief study of transformation problems. Some general theorems are presented as well as discussions of specific transformations. An optimal solution to the two-dimensional firing squad synchronization problem is also presented in chapter 4. In addition to the formal results, several open questions are presented and some iterative programming techniques are considered.