# New Results on Path Approximation

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DOI: 10.1007/s00453-003-1046-1

- Cite this article as:
- Daescu, O. Algorithmica (2004) 38: 131. doi:10.1007/s00453-003-1046-1

- 4 Citations
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## Abstract

In this paper we give bounds on the complexity of some algorithms for
approximating 2-D and 3-D polygonal paths with the
infinite beam measure of error. While the time/space complexities
of the algorithms known for other error measures are well understood,
path approximation with infinite beam measure seems to be harder due to the
complexity of some geometric structures that arise in the known approaches.
Our results answer some open problems left in previous work.
We also present a more careful analysis of the time complexity of the
general iterative algorithm for infinite beam measure and show that it
could perform much better in practice than in the worst case. We then
propose a new approach for path approximation that consists of a
breadth first traversal of the path approximation graph,
without constructing the graph. This approach
can be applied to any error criterion in any constant dimension.
The running time of the new algorithm depends on the size of the output:
if the optimal approximating path has *m* vertices, the algorithm performs
*F*(*m*) iterations rather than *n*–1 in the previous approaches, where *F*(*m*) \le *n*–1
is the number of vertices of the path approximation graph that can be reached
with at most *m*–2 edges. This is the first output sensitive path approximation algorithm.