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DOI: 10.4230/LIPIcs.ISAAC.2020.60
URN: urn:nbn:de:0030-drops-134042
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de Berg, Mark ; Markovic, Aleksandar ; Umboh, Seeun William

The Online Broadcast Range-Assignment Problem

LIPIcs-ISAAC-2020-60.pdf (0.5 MB)


Let P = {pā‚€,ā€¦,p_{n-1}} be a set of points in ā„^d, modeling devices in a wireless network. A range assignment assigns a range r(p_i) to each point p_i āˆˆ P, thus inducing a directed communication graph š’¢_r in which there is a directed edge (p_i,p_j) iff dist(p_i, p_j) ā©½ r(p_i), where dist(p_i,p_j) denotes the distance between p_i and p_j. The range-assignment problem is to assign the transmission ranges such that š’¢_r has a certain desirable property, while minimizing the cost of the assignment; here the cost is given by āˆ‘_{p_i āˆˆ P} r(p_i)^Ī±, for some constant Ī± > 1 called the distance-power gradient.
We introduce the online version of the range-assignment problem, where the points p_j arrive one by one, and the range assignment has to be updated at each arrival. Following the standard in online algorithms, resources given out cannot be taken away - in our case this means that the transmission ranges will never decrease. The property we want to maintain is that š’¢_r has a broadcast tree rooted at the first point pā‚€. Our results include the following.
- We prove that already in ā„Ā¹, a 1-competitive algorithm does not exist. In particular, for distance-power gradient Ī± = 2 any online algorithm has competitive ratio at least 1.57.
- For points in ā„Ā¹ and ā„Ā², we analyze two natural strategies for updating the range assignment upon the arrival of a new point p_j. The strategies do not change the assignment if p_j is already within range of an existing point, otherwise they increase the range of a single point, as follows: Nearest-Neighbor (NN) increases the range of NN(p_j), the nearest neighbor of p_j, to dist(p_j, NN(p_j)), and Cheapest Increase (CI) increases the range of the point p_i for which the resulting cost increase to be able to reach the new point p_j is minimal. We give lower and upper bounds on the competitive ratio of these strategies as a function of the distance-power gradient Ī±. We also analyze the following variant of NN in ā„Ā² for Ī± = 2: 2-Nearest-Neighbor (2-NN) increases the range of NN(p_j) to 2ā‹… dist(p_j,NN(p_j)),
- We generalize the problem to points in arbitrary metric spaces, where we present an O(log n)-competitive algorithm.

BibTeX - Entry

  author =	{Mark de Berg and Aleksandar Markovic and Seeun William Umboh},
  title =	{{The Online Broadcast Range-Assignment Problem}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{60:1--60:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Yixin Cao and Siu-Wing Cheng and Minming Li},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-134042},
  doi =		{10.4230/LIPIcs.ISAAC.2020.60},
  annote =	{Keywords: Computational geometry, online algorithms, range assignment, broadcast}

Keywords: Computational geometry, online algorithms, range assignment, broadcast
Collection: 31st International Symposium on Algorithms and Computation (ISAAC 2020)
Issue Date: 2020
Date of publication: 04.12.2020

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