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A linear algorithm for Brick Wang tiling

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Alexandre Derouet-Jourdan, Shizuo Kaji, Yoshihiro Mizoguchi

Discrete Mathematics · Submitted 14 Mar 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

how it is stated · asserted 1.00 read the record →

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arXiv ID1603.04292 (version 2)
Pith IDIGHU4IV7, Pith's permanent record of this paper and its review
CitationAlexandre Derouet-Jourdan, Shizuo Kaji, Yoshihiro Mizoguchi. "A linear algorithm for Brick Wang tiling." arXiv:1603.04292 (2016). https://pith.science/paper/IGHU4IV7
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,122 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.15 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.42 · compares with prior results 0.71 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2015 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The Wang tiling is a classical problem in combinatorics. background 1.00 0.03 0.01 asserted 1.00
s1 A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. background 0.99 0.04 0.02 asserted 0.99
s2 In this case, resulting tilings are rather restrictive. background 0.96 0.07 0.02 asserted 1.00
s3 On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. background 1.00 0.04 0.02 asserted 1.00
s4 In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. background 1.00 0.05 0.02 asserted 0.97
s5 With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. contribution 1.00 0.92 0.09 asserted 0.99
s6 We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2015 to model wall patterns. background 0.40 0.46 0.05 asserted 0.96
s7 We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes. contribution 0.82 0.95 0.88 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2015 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

Human review

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How the literature uses it

1 arXiv paper cite this in 4 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 3 sentences without a confident label

Contests it 1 paper
  • A problem of the previous algorithms in [ref] is that it sometimes produces long traversal edges; horizontally consecutive occurrence of tiles with $c_1=c_3$ or vertically consecutive occurrence of tiles with $c_2=c_4$.

    Dappled tiling · Example: Brick Wang Tiles · contest 0.76

What it cites

Cites as background 1 paper

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