research-article

Optimal Orthogonal Graph Drawing with Convex Bend Costs

Authors:

Thomas Bläsius,

Dorothea WagnerAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 12, Issue 3

Article No.: 33, Pages 1 - 32

https://doi.org/10.1145/2838736

Published: 25 April 2016 Publication History

Abstract

Traditionally, the quality of orthogonal planar drawings is quantified by the total number of bends or the maximum number of bends per edge. However, this neglects that, in typical applications, edges have varying importance. We consider the problem OptimalFlexDraw that is defined as follows. Given a planar graph G on n vertices with maximum degree 4 (4-planar graph) and for each edge e a cost function cost_e : N₀ → R defining costs depending on the number of bends e has, compute a planar orthogonal drawing of G of minimum cost.

In this generality OptimalFlexDraw is NP-hard. We show that it can be solved efficiently if (1) the cost function of each edge is convex and (2) the first bend on each edge does not cause any cost. Our algorithm takes time O(n, ⋅, T_flow(n) and O(n², ⋅, T^flow(n)) for biconnected and connected graphs, respectively, where T_flow(n) denotes the time to compute a minimum-cost flow in a planar network with multiple sources and sinks. Our result is the first polynomial-time bend-optimization algorithm for general 4-planar graphs optimizing over all embeddings. Previous work considers restricted graph classes and unit costs.

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Cited By

Barth LNiedermann BRutter IWolf M(2023)A Topology-Shape-Metrics Framework for Ortho-Radial Graph DrawingDiscrete & Computational Geometry10.1007/s00454-023-00593-y70:4(1292-1355)Online publication date: 1-Nov-2023
https://doi.org/10.1007/s00454-023-00593-y
Brückner GKrisam NMchedlidze T(2022)Level-Planar Drawings with Few SlopesAlgorithmica10.1007/s00453-021-00884-x84:1(176-196)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1007/s00453-021-00884-x
Angelini PRutter ISandhya T(2020)Extending Partial Orthogonal DrawingsGraph Drawing and Network Visualization10.1007/978-3-030-68766-3_21(265-278)Online publication date: 16-Sep-2020
https://dl.acm.org/doi/10.1007/978-3-030-68766-3_21
Show More Cited By

Index Terms

Optimal Orthogonal Graph Drawing with Convex Bend Costs
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
    2. Graph theory
2. Theory of computation
  1. Design and analysis of algorithms

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 12, Issue 3

June 2016

408 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2930058

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 25 April 2016

Accepted: 01 October 2015

Revised: 01 September 2015

Received: 01 August 2013

Published in TALG Volume 12, Issue 3

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Cited By

Barth LNiedermann BRutter IWolf M(2023)A Topology-Shape-Metrics Framework for Ortho-Radial Graph DrawingDiscrete & Computational Geometry10.1007/s00454-023-00593-y70:4(1292-1355)Online publication date: 1-Nov-2023
https://doi.org/10.1007/s00454-023-00593-y
Brückner GKrisam NMchedlidze T(2022)Level-Planar Drawings with Few SlopesAlgorithmica10.1007/s00453-021-00884-x84:1(176-196)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1007/s00453-021-00884-x
Angelini PRutter ISandhya T(2020)Extending Partial Orthogonal DrawingsGraph Drawing and Network Visualization10.1007/978-3-030-68766-3_21(265-278)Online publication date: 16-Sep-2020
https://dl.acm.org/doi/10.1007/978-3-030-68766-3_21
Niedermann BRutter I(2020)An Integer-Linear Program for Bend-Minimization in Ortho-Radial DrawingsGraph Drawing and Network Visualization10.1007/978-3-030-68766-3_19(235-249)Online publication date: 16-Sep-2020
https://dl.acm.org/doi/10.1007/978-3-030-68766-3_19
Angelini PBekos M(2019)Hierarchical Partial PlanarityAlgorithmica10.1007/s00453-018-0530-681:6(2196-2221)Online publication date: 1-Jun-2019
https://dl.acm.org/doi/10.1007/s00453-018-0530-6
Brückner GHimmel MRutter I(2019)An SPQR-Tree-Like Embedding Representation for Upward PlanarityGraph Drawing and Network Visualization10.1007/978-3-030-35802-0_39(517-531)Online publication date: 17-Sep-2019
https://dl.acm.org/doi/10.1007/978-3-030-35802-0_39
Didimo WLiotta GPatrignani M(2018)Bend-Minimum Orthogonal Drawings in Quadratic TimeGraph Drawing and Network Visualization10.1007/978-3-030-04414-5_34(481-494)Online publication date: 26-Sep-2018
https://dl.acm.org/doi/10.1007/978-3-030-04414-5_34
Alam MKobourov SMondal D(2017)Orthogonal layout with optimal face complexityComputational Geometry10.1016/j.comgeo.2017.02.00563(40-52)Online publication date: Jun-2017
https://doi.org/10.1016/j.comgeo.2017.02.005

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