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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| Published in: | IEEE transactions on computers Vol. C-28; no. 2; pp. 109 - 120 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
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IEEE
01.02.1979
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| ISSN: | 0018-9340, 1557-9956 |
| Online Access: | Get full text |
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| 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. |
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| AbstractList | 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. |
| Author | Silio |
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| Cites_doi | 10.1109/SWAT.1966.20 10.1109/T-C.1970.222898 10.1016/0020-0190(73)90020-3 10.1109/PGEC.1967.264724 10.1007/BF01694007 10.1016/0020-0190(72)90045-2 10.1109/T-C.1974.223848 10.1016/S0019-9958(67)90465-2 |
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| References | aho (ref1) 1974 uspensky (ref15) 1948 grunbaum (ref6) 1967 ref11 ref10 bancilhon (ref2) 1974; c 23 silo (ref14) 1972 ref8 ref7 silo (ref13) 1970 ref9 ref3 ref5 gantmacher (ref4) 1960 paz (ref12) 1971 |
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| Snippet | The problem of determining the existence of a simplex which covers a given convex polytope inside another given convex polytope in two-dimensional Euclidean... |
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| StartPage | 109 |
| SubjectTerms | 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 |
| Title | An Efficient Simplex Coverability Algorithm in E2 with Application to Stochastic Sequential Machines |
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