Testing approximate symmetry in the plane is NP-hard

Iwanowski, Sebastian

doi:10.1007/3-540-51486-4_76

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…Sections 3 and 4 prepared the main complexity results in section 5: polynomial time algorithms for computing and comparing isosets (Theorems 5.2, Corollary 5.3), and approximating the new boundary-tolerant cluster distance d C (Theorem 5.6) and EMD on isosets (Corollary 5.7). Lemma 5.5 is also new because even in R 2 the approximate matching of finite sets needs O(m 5 log m) time [34] and For a given ε and symmetry group in R 2 , approximate symmetry detection is NP-hard for the groups D k , C k for k ≥ 3 [44]. Due to a stable radius α, which is not fixed but determined by a given periodic set S with easy upper bounds in Lemma 3.7, the isoset I(S; α) becomes a complete invariant.…”

Section: By the Above Choice Ofmentioning

confidence: 99%

Algorithms for continuous metrics on periodic crystals

Anosova¹,

Kurlin²

2022

Preprint

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Periodic point sets model all solid crystalline materials (crystals) whose structures are determined in a rigid form. The emergent area of Periodic Geometry classifies periodic point sets up to rigid motion or isometry preserving inter-point distances. The first result is a polynomial-time algorithm distinguishing all periodic point sets up to isometry in any dimension. Noisy measurements motivate the problem to compute a metric satisfying all metric axioms and continuity under perturbations. All past metrics on finite sets including the bottleneck distance are discontinuous for periodic point sets. The second result is a polynomial-time algorithm approximating new continuous metrics on periodic point sets with a small factor in low dimensions. The proofs develop new tools such as a boundary-tolerant metric applicable to other data ambiguity problems. The isometry invariants and metrics detected unexpected duplicates in the Cambridge Structural Database.

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Section: By the Above Choice Ofmentioning

confidence: 99%

Algorithms for continuous metrics on periodic crystals

Anosova¹,

Kurlin²

2022

Preprint

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“…Instead of using the Real-RAM model, it also makes sense to test congruence with an error tolerance ε, but this problem is known to be NP-hard even in the plane [14,20] as mentioned earlier in Section 2. However, the problem becomes polynomial if the input points are sufficiently separated compared to ε.…”

Section: Practical Implementabilitymentioning

confidence: 99%

Congruence Testing of Point Sets in 4 Dimensions

Kim,

Rote

2016

Preprint

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“…Exact symmetries can be found in O(n log n) time: see [24] and the references therein. However, differing definitions of (global) approximate symmetry cause its detection to be NP-hard [15] or to take O(n 8 ) time [1]. Our approach extends the global approximate symmetry detection algorithm in [29], which takes only O(n 3.5 log 4 n) time.…”

Section: Symmetry Detectionmentioning

confidence: 99%

Detecting design intent in approximate CAD models using symmetry

Langbein

Martin

2010

Computer-Aided Design

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Finding design intent embodied as high-level geometric relations between a CAD model's sub-parts facilitates various tasks such as model editing and analysis. This is especially important for boundary-representation models arising from, e.g., reverse engineering or CAD data transfer. These lack explicit information about design intent, and often the intended geometric relations are only approximately present. The novel solution to this problem presented is based on detecting approximate local incomplete symmetries, in a hierarchical decomposition of the model into simpler, more symmetric sub-parts. Design intent is detected as congruencies, symmetries and symmetric arrangements of the leaf-parts in this decomposition. All elementary 3D symmetry types and common symmetric arrangements are considered. They may be present only locally in subsets of the leaf-parts, and may also be incomplete, i.e. not all elements required for a symmetry need be present. Adaptive tolerance intervals are detected automatically for matching interpoint distances, enabling efficient, robust and consistent detection of approximate symmetries. Doing so avoids finding many spurious relations, reliably resolves ambiguities between relations, and reduces inconsistencies. Experiments show that detected relations reveal significant design intent.

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Testing approximate symmetry in the plane is NP-hard

Cited by 4 publications

References 9 publications

Algorithms for continuous metrics on periodic crystals

Algorithms for continuous metrics on periodic crystals

Congruence Testing of Point Sets in 4 Dimensions

Detecting design intent in approximate CAD models using symmetry

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