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Better Bounds for Online Line Chasing

Authors Marcin Bienkowski , Jarosław Byrka , Marek Chrobak , Christian Coester , Łukasz Jeż , Elias Koutsoupias

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ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • convex body chasing
  • line chasing
  • competitive analysis

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Abstract

We study online competitive algorithms for the line chasing problem in Euclidean spaces R^d, where the input consists of an initial point P_0 and a sequence of lines X_1, X_2, ..., X_m, revealed one at a time. At each step t, when the line X_t is revealed, the algorithm must determine a point P_t in X_t. An online algorithm is called c-competitive if for any input sequence the path P_0, P_1 , ..., P_m it computes has length at most c times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets X_t are arbitrary convex sets.
To date, the best competitive ratio for the line chasing problem was 28.1, even in the plane. We improve this bound by providing a simple 3-competitive algorithm for any dimension d. We complement this bound by a matching lower bound for algorithms that are memoryless in the sense of our algorithm, and a lower bound of 1.5358 for arbitrary algorithms. The latter bound also improves upon the previous lower bound of sqrt{2}~=1.412 for convex body chasing in 2 dimensions.

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Marcin Bienkowski, Jarosław Byrka, Marek Chrobak, Christian Coester, Łukasz Jeż, and Elias Koutsoupias. Better Bounds for Online Line Chasing. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.MFCS.2019.8

Author Details

Marcin Bienkowski
  • Institute of Computer Science, University of Wrocław, Poland
Jarosław Byrka
  • Institute of Computer Science, University of Wrocław, Poland
Marek Chrobak
  • University of California at Riverside, CA, USA
Christian Coester
  • University of Oxford, United Kingdom
Łukasz Jeż
  • Institute of Computer Science, University of Wrocław, Poland
Elias Koutsoupias
  • University of Oxford, United Kingdom

Funding

Research supported by NSF grant CCF-1536026, by Polish National Science Centre grants 2016/22/E/ST6/00499 and 2015/18/E/ST6/0045, and by ERC Advanced Grant 321171 (ALGAME).

References

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