9,832 research outputs found
A Probabilistic Analysis of the Power of Arithmetic Filters
The assumption of real-number arithmetic, which is at the basis of
conventional geometric algorithms, has been seriously challenged in recent
years, since digital computers do not exhibit such capability.
A geometric predicate usually consists of evaluating the sign of some
algebraic expression. In most cases, rounded computations yield a reliable
result, but sometimes rounded arithmetic introduces errors which may invalidate
the algorithms. The rounded arithmetic may produce an incorrect result only if
the exact absolute value of the algebraic expression is smaller than some
(small) varepsilon, which represents the largest error that may arise in the
evaluation of the expression. The threshold varepsilon depends on the structure
of the expression and on the adopted computer arithmetic, assuming that the
input operands are error-free.
A pair (arithmetic engine,threshold) is an "arithmetic filter". In this paper
we develop a general technique for assessing the efficacy of an arithmetic
filter. The analysis consists of evaluating both the threshold and the
probability of failure of the filter.
To exemplify the approach, under the assumption that the input points be
chosen randomly in a unit ball or unit cube with uniform density, we analyze
the two important predicates "which-side" and "insphere". We show that the
probability that the absolute values of the corresponding determinants be no
larger than some positive value V, with emphasis on small V, is Theta(V) for
the which-side predicate, while for the insphere predicate it is Theta(V^(2/3))
in dimension 1, O(sqrt(V)) in dimension 2, and O(sqrt(V) ln(1/V)) in higher
dimensions. Constants are small, and are given in the paper.Comment: 22 pages 7 figures Results for in sphere test inproved in
cs.CG/990702
Further Results on Arithmetic Filters for Geometric Predicates
An efficient technique to solve precision problems consists in using exact
computations. For geometric predicates, using systematically expensive exact
computations can be avoided by the use of filters. The predicate is first
evaluated using rounding computations, and an error estimation gives a
certificate of the validity of the result. In this note, we studies the
statistical efficiency of filters for cosphericity predicate with an assumption
of regular distribution of the points. We prove that the expected value of the
polynomial corresponding to the in sphere test is greater than epsilon with
probability O(epsilon log 1/epsilon) improving the results of a previous paper
by the same authors.Comment: 7 pages 2 figures presented at the 15th European Workshop Comput.
Geom., 113--116, 1999 improve previous results (in other paper
On Deletion in Delaunay Triangulation
This paper presents how the space of spheres and shelling may be used to
delete a point from a -dimensional triangulation efficiently. In dimension
two, if k is the degree of the deleted vertex, the complexity is O(k log k),
but we notice that this number only applies to low cost operations, while time
consuming computations are only done a linear number of times.
This algorithm may be viewed as a variation of Heller's algorithm, which is
popular in the geographic information system community. Unfortunately, Heller
algorithm is false, as explained in this paper.Comment: 15 pages 5 figures. in Proc. 15th Annu. ACM Sympos. Comput. Geom.,
181--188, 199
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