A cluster-based cylindrical algebraic decomposition algorithm
Given a set A of r-variate integral polynomials, an A-invariant cylindrical algebraic decomposition (cad) of euclidean r-space Er is a certain partition of Er into connected subsets, such that each subset is an i-cell for some i, 0≤i≤r (an i-cell is a homeomorph of Ei), and such that each element of A is sign-invariant (i.e. either negative, vanishing, or positive) on each cell. A cluster of a cad D is a collection of cells of D whose union is connected. We give a new cad construction algorithm which, as it extends a cad of Ei−1 to a cad of Ei (2≤i≤r), partitions the cells of the Ei−1 cad into clusters, and performs various computations only once for each cluster, rather than once for each cell as previous cad algorithms do. Preliminary experiments suggest that this new algorithm can be significantly more efficient in practice than previous cad algorithms. The clusters, which are part of the new algorithm's output, can be chosen to have useful mathematical properties. For example, if r≤3, each cluster (i.e. the union of its cells) can be a maximal connected A-invariant subset of Er.
Unable to display preview. Download preview PDF.
- 1. [ACM84a]D. S. Arnon, G. E. Collins, S. McCallum Cylindrical algebraic decomposition I: the basic algorithm, SIAM J. Comp, 13, (1984), pp. 865–877.Google Scholar
- 2. [ACM84b]— —, Cylindrical algebraic decomposition II: an adjacency algorithm for the plane, SIAM J. Comp, 13, (1984), pp. 878–889.Google Scholar
- 3. [ACM85]— —, An adjacency algorithm for cylindrical algebraic decompositions of three-dimensional space, Manuscript, November 1984 (submitted to EUROCAL '85).Google Scholar
- 4. [ARN79]D. S. Arnon, A cellular decomposition algorithm for semi-algebraic sets, Proc. EUROSAM '79, Lecture Notes in Computer Scienc No. 72, Springer-Verlag, Berlin, 1979, pp. 301–315.Google Scholar
- 5. [ARN81]— —, Algorithms for the geometry of semi-algebraic sets, Ph.D. Dissertation, Technical Report #436, Computer Sciences Department, University of Wisconsin — Madison, 1981.Google Scholar
- 6. [ARM84]D. S. Arnon and S. McCallum, A polynomial-time algorithm for the topological type of a real algebraic curve — extended abstract, Rocky Mountain J. Math., 14 (1984), pp. 849–852.Google Scholar
- 7. [COL75]G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, in Second GI Conference on Automata Theory and Formal Languages, vol. 33 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1975, pp 134–183.Google Scholar
- 8. [LOO82a]R. G. K. Loos, Computing in algebraic extensions, in Computing, Supplementum 4: Computer Algebra-Symbolic and Algebraic Computation, Springer-Verlag, Vienna and New York, 1982.Google Scholar
- 10. [SCH83]