Paper by Erik D. Demaine
- Ilya Baran and Erik D. Demaine, “Optimal Adaptive Algorithms for Finding the Nearest and Farthest Point on a Parametric Black-Box Curve”, in Proceedings of the 20th Annual ACM Symposium on Computational Geometry (SoCG 2004), Brooklyn, New York, June 9–11, 2004, pages 220–229.
We consider a general model for representing and manipulating parametric
curves, in which a curve is specified by a black box mapping a parameter
value between 0 and 1 to a point in Euclidean d-space.
In this model, we consider the nearest-point-on-curve and
given a curve C and a point p, find a point on C nearest to p
or farthest from p.
In the general black-box model, no algorithm can solve these problems.
Assuming a known bound on the speed of the curve (a Lipschitz condition),
the answer can be estimated up to an additive error of ε
using O(1/ε) samples, and this bound is tight in the
worst case. However, many instances can be solved with
substantially fewer samples, and we give algorithms that adapt to
the inherent difficulty of the particular instance,
up to a logarithmic factor. More precisely, if OPT(C, p, ε)
is the minimum number of samples of C that every correct
algorithm must perform to achieve tolerance ε, then our
O(OPT(C, p, ε) log (ε−1 / OPT(C, p, ε)))
samples. Furthermore, any
Ω(k log (ε−1 / k))
samples for some instance C' with
OPT(C', p, ε) = k;
except that, for the nearest-point-on-curve problem when the distance
between C and p is less than ε, OPT is 1
but the upper and lower bounds on the number of samples
are both Θ(1/ε).
When bounds on relative error are desired, we give algorithms that
O(OPT log (2 + (1 + ε−1) m−1 / OPT))
samples (where m is the exact minimum or maximum distance from p
to C) and prove that Ω(OPT log (1/ε)) samples
are necessary on some problem instances.
- This paper is also available from the ACM Digital Library.
- The paper is 10 pages.
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- Related papers:
- Curves_IJCGA (Optimal Adaptive Algorithms for Finding the Nearest and Farthest Point on a Parametric Black-Box Curve)
See also other papers by Erik Demaine.
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