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Maarten Löffler

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13 papers
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13

AAAI Conference 2023 Conference Paper

The Influence of Dimensions on the Complexity of Computing Decision Trees

  • Stephen G. Kobourov
  • Maarten Löffler
  • Fabrizio Montecchiani
  • Marcin Pilipczuk
  • Ignaz Rutter
  • Raimund Seidel
  • Manuel Sorge
  • Jules Wulms

A decision tree recursively splits a feature space \mathbb{R}^d and then assigns class labels based on the resulting partition. Decision trees have been part of the basic machine-learning toolkit for decades. A large body of work considers heuristic algorithms that compute a decision tree from training data, usually aiming to minimize in particular the size of the resulting tree. In contrast, little is known about the complexity of the underlying computational problem of computing a minimum-size tree for the given training data. We study this problem with respect to the number d of dimensions of the feature space \mathbb{R}^d, which contains n training examples. We show that it can be solved in O(n^(2d + 1)) time, but under reasonable complexity-theoretic assumptions it is not possible to achieve f(d) * n^o(d / log d) running time. The problem is solvable in (dR)^O(dR) * n^(1+o(1)) time, if there are exactly two classes and R is an upper bound on the number of tree leaves labeled with the first class.

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MFCS Conference 2021 Conference Paper

Uncertain Curve Simplification

  • Kevin Buchin
  • Maarten Löffler
  • Aleksandr Popov 0001
  • Marcel Roeloffzen

We study the problem of polygonal curve simplification under uncertainty, where instead of a sequence of exact points, each uncertain point is represented by a region which contains the (unknown) true location of the vertex. The regions we consider are disks, line segments, convex polygons, and discrete sets of points. We are interested in finding the shortest subsequence of uncertain points such that no matter what the true location of each uncertain point is, the resulting polygonal curve is a valid simplification of the original polygonal curve under the Hausdorff or the Fréchet distance. For both these distance measures, we present polynomial-time algorithms for this problem.

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