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Computing the Flip Distance Between Triangulations

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Abstract

Let \(\mathcal{{T}}\) be a triangulation of a set \(\mathcal{{P}}\) of n points in the plane, and let e be an edge shared by two triangles in \(\mathcal{{T}}\) such that the quadrilateral Q formed by these two triangles is convex. A flip of e is the operation of replacing e by the other diagonal of Q to obtain a new triangulation of \(\mathcal{{P}}\) from \(\mathcal{{T}}\). The flip distance between two triangulations of \(\mathcal{{P}}\) is the minimum number of flips needed to transform one triangulation into the other. The Flip Distance problem asks if the flip distance between two given triangulations of \(\mathcal{{P}}\) is at most k, for some given \(k \in \mathbb {N}\). It is a fundamental and a challenging problem. We present an algorithm for the Flip Distance problem that runs in time \(\mathcal {O}(n + k \cdot c^{k})\), for a constant \(c \le 2 \cdot 14^{11}\), which implies that the problem is fixed-parameter tractable. We extend our results to triangulations of polygonal regions with holes, and to labeled triangulated graphs.

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Notes

  1. If an action \(\sigma \) involves flipping an edge e, then \(\sigma \) is performed only when a flip of e is admissible.

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Authors and Affiliations

  1. School of Computing, DePaul University, 243 S. Wabash Avenue, Chicago, IL, 60604, USA

    Iyad Kanj & Eric Sedgwick

  2. Department of Computer Science, Lafayette College, 730 High St, Easton, PA, 18042, USA

    Ge Xia

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  1. Iyad Kanj
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  2. Eric Sedgwick
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  3. Ge Xia
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Correspondence to Iyad Kanj.

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Editor in Charge: Günter M. Ziegler

An extended abstract of this work (without the complete proofs and the extension to labeled triangulated graphs) appears in [14].

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Kanj, I., Sedgwick, E. & Xia, G. Computing the Flip Distance Between Triangulations. Discrete Comput Geom 58, 313–344 (2017). https://doi.org/10.1007/s00454-017-9867-x

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  • DOI: https://doi.org/10.1007/s00454-017-9867-x

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