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

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....

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Veröffentlicht in:IEEE transactions on computers Jg. C-28; H. 2; S. 109 - 120
1. Verfasser: Silio
Format: Journal Article
Sprache:Englisch
Veröffentlicht: IEEE 01.02.1979
Schlagworte:
Complexity theory
Computational complexity
Computational efficiency
Computer architecture
Computers
convex polytopes
covering problems
finite search procedure
geometric programming
High level languages
Indexes
Mathematical models
probabilistic automata
Search problems
Silicon
simplex covering
Testing
ISSN:0018-9340, 1557-9956
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Zusammenfassung: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.
ISSN:0018-9340
1557-9956
DOI:10.1109/TC.1979.1675300