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

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1 Author(s)
Silio, C.B., Jr. ; Department of Electrical Engineering, University of Maryland

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.

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Computers, IEEE Transactions on  (Volume:C-28 ,  Issue: 2 )