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Optimizing Symbol Visibility Through Displacement

Authors Bernd Gärtner, Vishwas Kalani, Meghana M. Reddy , Wouter Meulemans, Bettina Speckmann , Miloš Stojaković

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Author Details

Bernd Gärtner
  • Department of Computer Science, ETH Zürich, Switzerland
Vishwas Kalani
  • Department of Computer Science and Engineering, I.I.T. Delhi, India
Meghana M. Reddy
  • Department of Computer Science, ETH Zürich, Switzerland
Wouter Meulemans
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Bettina Speckmann
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Miloš Stojaković
  • Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia

Funding

  • M. Reddy, Meghana: Supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681.
  • Meulemans, Wouter: Partially supported by the Dutch Research Council (NWO) under project number VI.Vidi.223.137.
  • Stojaković, Miloš: Partly supported by Ministry of Science, Technological Development and Innovation of Republic of Serbia (Grants 451-03-66/2024-03/200125 & 451-03-65/2024-03/200125). Partly supported by Provincial Secretariat for Higher Education and Scientific Research, Province of Vojvodina (Grant No. 142-451-2686/2021).

Acknowledgements

This research was initiated at the 19th Gremo’s Workshop on Open Problems (GWOP), Binn, Switzerland, June 13-17, 2022.

Cite As Get BibTex

Bernd Gärtner, Vishwas Kalani, Meghana M. Reddy, Wouter Meulemans, Bettina Speckmann, and Miloš Stojaković. Optimizing Symbol Visibility Through Displacement. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.24

Abstract

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the symbols.
Specifically, we consider unit square symbols that need to be placed at specified y-coordinates. We optimize the drawing order of the symbols as well as their x-displacement, constrained within a rectangular container, to maximize the minimum visible perimeter over all squares. If the container has width and height at most 2, there is a point that stabs all squares. In this case, we prove that a staircase layout is arbitrarily close to optimality and can be computed in O(nlog n) time. If the width is at most 2, there is a vertical line that stabs all squares, and in this case, we give a 2-approximation algorithm (assuming fixed container height) that runs in O(nlog n) time. As a minimum visible perimeter of 2 is always trivially achievable, we measure this approximation with respect to the visible perimeter exceeding 2. We show that, despite its simplicity, the algorithm gives asymptotically optimal results for certain instances.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • symbol placement
  • visibility
  • jittering
  • stacking order

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