## An Efficient Simplex Coverability Algorithm in E2 with Application to Stochastic Sequential Machines

Issue No. 02 - February (1979 vol. 28)

ISSN: 0018-9340

pp: 109-120

C.B., Jr. Silio , Department of Electrical Engineering, University of Maryland

ABSTRACT

The problem of determining the existence of a simplex which covers a given convex polytope inside another given convex polytope in two-dimensional Euclidean space is shown to be efficiently solvable, and an effective procedure is derived to find a suitable covering simplex or show that none exists. This solution provides a finite algorithm for satisfying a necessary condition in the search for a simplicial (fewest states) stochastic sequential machine (SSM) which either covers or is covered by a given SSM of rank three. Theorems are proved which establish necessary and sufficient conditions restricting the class of corresponding simplexes through which a search must proceed to those whose vertices lie in the boundary of the bounding polytope and whose facet-supporting flats contain certain specified vertices of the polytope to be covered. The problem is then reduced to testing roots in the finite solution tree of a set of second degree algebraic equations against a finite table of linear constraints. An algorithm to generate the constraint sets for the equations to be solved is also presented along with examples of its application.

INDEX TERMS

simplex covering, Computational complexity, covering problems, convex polytopes, finite search procedure, geometric programming, probabilistic automata

CITATION

C.B., Jr. Silio, "An Efficient Simplex Coverability Algorithm in E2 with Application to Stochastic Sequential Machines",

*IEEE Transactions on Computers*, vol. 28, no. , pp. 109-120, February 1979, doi:10.1109/TC.1979.1675300