Decision trees for geometric models

Arkin, Esther M.; Meijer, Henk; Mitchell, Joseph S. B.; Rappaport, David; Skiena, Steven

doi:10.1145/160985.161167

Cited by 25 publications

(29 citation statements)

References 22 publications

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“…Here the notation "±" means "+" in case (1) and "−" in case (2) and the notation "∓" means just the opposite. Several steps of manipulation on the above equations plus a detailed subsequent analysis on the ranges of angles reveal the locus of p 0 as described below.…”

Section: A Triangle Sliding In a Conementioning

confidence: 99%

“…More recent related work includes Belleville and Shermer (1993) and Arkin et al (1993a). Belleville and Shermer (1993) show that the problem of deciding whether k line probes are sufficient to distinguish a convex polygon from a collection of n convex polygons is NPcomplete.…”

Section: Related Workmentioning

confidence: 99%

“…Finger probing is closely related to sensing by point sampling, 2 This can be implemented in multiple ways, such as placing light detectors in the table; probing at the points; or if the robot has a vision system, taking a scene image and checking the corresponding pixel values.…”

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confidence: 99%

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Geometric Sensing of Known Planar Shapes

Jia

Erdmann

1996

The International Journal of Robotics Research

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Industrial assembly involves sensing the pose (orientation and position) of a part. Efficient and reliable sensing strategies can be developed for an assembly task if the shape of the part is known in advance. In this article we investigate two problems of determining the pose of a polygonal part of known shape for the cases of a continuum and a finite number of possible poses respectively.The first problem, named sensing by inscription, involves determining the pose of a convex n-gon from a set of m supporting cones. An algorithm with running time O(nm) that almost always reduces to O(n + m log n) is presented to solve for all possible poses of the polygon. We prove that the number of possible poses cannot exceed 6n, given m ≥ 2 supporting cones with distinct vertices. Simulation experiments demonstrate that two supporting cones are sufficient to determine the real pose of the n-gon in most cases. Our results imply that sensing in practice can be carried out by obtaining viewing angles of a planar part at multiple exterior sites in the plane.On many occasions a parts feeder will have reduced the number of possible poses of a part to a small finite set. Our second problem, named sensing by point sampling, is concerned with a more general version: finding the minimum number of sensing points required to distinguish between n polygonal shapes with a total of m edges. In practice this can be implemented by embedding a series of point light detectors in a feeder tray or by using a set of mechanical probes that touch the feeder at a finite number of predetermined points. We show that this problem is equivalent to an NP-complete set-theoretic problem introduced as Discriminating Set, and present an O(n 2 m 2 ) approximation algorithm to solve it with a ratio of 2 ln n. Furthermore, we prove that one can use an algorithm for Discriminating Set with ratio c log n to construct an algorithm for Set Covering with ratio c log n + O(log log n). Thus the ratio 2 ln n is asymptotically optimal unless NP ⊂ DTIME(n poly log n ), a consequence of known results on approximating Set Covering. The complexity of subproblems of Discriminating Set is also analyzed, based on their relationship to a generalization of Independent Set called 3-Independent Set. Finally, simulation results suggest that sensing by point sampling is mostly applicable when poses are densely distributed in the plane.

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Section: A Triangle Sliding In a Conementioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Geometric Sensing of Known Planar Shapes

Jia

Erdmann

1996

The International Journal of Robotics Research

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“…In this work, our function classes are meant to model a set of experts. Hence we assume only that we have a finite class F and try to derive algorithms that perform well relative to opt(F), the optimal worst-case bound on the number of membership queries for learning arbitrary elements of F. This particular problem was studied in Arkin et al (1998) (see Angluin, 2001). They showed that the very simple and fast "query-by-committee" (Seung, Opper, & Sompolinsky, 1992) algorithm, which maintains a list of possible targets, and chooses the query for which the remaining possibilities are most evenly divided, learns any class F with an approximately optimal number of membership queries in the worst case.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2003

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Abstract. We consider the problem of determining which of a set of experts has tastes most similar to a given user by asking the user questions about his likes and dislikes. We describe a simple algorithm for generating queries for a theoretical model of this problem. We show that the algorithm requires at most opt(F)(ln(|F|/opt(F)) + 1) + 1 queries to find the correct expert, where opt(F) is the optimal worst-case bound on the number of queries for learning arbitrary elements of the set of experts F. The algorithm runs in time polynomial in |F| and |X | (where X is the domain) and we prove that no polynomial-time algorithm can have a significantly better bound on the number of queries unless all problems in NP have n O(log log n) time algorithms. We also study a more general case where the user ratings come from a finite set Y and there is an integer-valued loss function on Y that is used to measure the distance between the ratings. Assuming that the loss function is a metric and that there is an expert within a distance η from the user, we give a polynomial-time algorithm that is guaranteed to find such an expert after at most 2opt(F, η) ln |F| 1+deg(F,η) + 2(η + 1)(1 + deg (F, η)) queries, where deg(F, η) is the largest number of experts in F that are within a distance 2η of any f ∈ F.

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“…Then, we choose x 3 as the second variable to read. In this case, we will pay a cost of 3 to evaluate the function, regardless of the value of x 3 .…”

Section: Introductionmentioning

confidence: 99%

On the Simultaneous Minimization of Worst Testing Cost and Expected Testing Cost With Decision Trees

Saettler¹

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Decision trees for geometric models

Cited by 25 publications

References 22 publications

Geometric Sensing of Known Planar Shapes

Geometric Sensing of Known Planar Shapes

Untitled

On the Simultaneous Minimization of Worst Testing Cost and Expected Testing Cost With Decision Trees

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