Parallel Computation of Alpha Complex for Biomolecules

Talha Bin Masood; Tathagata Ray; Vijay Natarajan

REVIEW 3 major objections 4 minor 14 references

Parallel Computation of Alpha Complex for Biomolecules

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Three conditions on a witness point decide alpha-complex membership, so the Delaunay triangulation can be skipped.

desk verdict Useful GPU engineering, mostly sound proofs for tetrahedra/triangles/edges, but the vertex shortcut is genuinely unsound and the 'new characterization' is standard. read the letter →

arxiv 1908.05944 v2 pith:VY2DNP5G submitted 2019-08-16 cs.CG cs.DC

Talha Bin Masood , Tathagata Ray , Vijay Natarajan This is my paper · ORCID

classification cs.CGcs.DC MSC 68U0568W10

keywords alphacomplexweightedshapeDelaunaytriangulationpointsbiomoleculesGPUparallelalgorithmcomputationalgeometryorthocenter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that membership in the $\alpha$ complex—the shape representation used for biomolecules—can be decided by a direct three-condition test on one witness point, so the full weighted Delaunay triangulation does not have to be built first. The test requires equal power distance to the simplex's balls, no other ball closer in power distance, and that this distance be at most the parameter $\alpha$. The paper turns this characterization into a five-step algorithm that first enumerates all tetrahedra, triangles, edges, and vertices whose orthocenter distance is within $\alpha$, then prunes false positives downward in dimension by checking the no-closer-ball condition. On biomolecular data, the resulting GPU implementation computes $\alpha$ complexes for millions of atoms in seconds, with speedups up to a factor of 50 over prior biomolecule-optimized methods.

What carries the argument

The central object is the orthocenter of a simplex, the point that minimizes the equal power distance to the simplex's balls, together with the OrthoSize (that minimum distance). The key move is to test the $\alpha$-complex condition AC2 at the orthocenter rather than at the closest witness; since OrthoSize is a lower bound on the Size of a simplex, a simplex whose orthocenter passes the AC2 test is automatically within the required $\alpha$. The proof for edges and triangles shows that a free simplex failing the test is actually a face of a higher-dimensional accepted simplex, which justifies the top-down pruning. Faces are further exploited because OrthoSize is monotone with respect to faces, allowing candidates to be generated bottom-up from edges.

What would settle it

Construct a weighted point set that violates the packing assumption—for example, a small ball nested inside a much larger ball, with $\alpha=0$—and compare the algorithm's output with the exact $\alpha$ complex defined by the union of grown balls. If the algorithm reports the nested ball's center as a vertex, the claim that this implementation computes $K_\alpha$ exactly on biomolecular-style inputs is falsified.

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Extended reading notes

Core claim

The paper's central claim is Definition 2: a $d$-dimensional simplex $\sigma^d$ belongs to the $\alpha$ complex $K_\alpha$ of a weighted ball set $B$ if and only if there exists a point $p\in\mathbb{R}^3$ with $\pi(p,b_{\sigma_0})=\cdots=\pi(p,b_{\sigma_d})$ and $\pi(p,b_{\sigma_0})\le\pi(p,b_i)$ for every $b_i\in B-\sigma^d$, and $\pi(p,b_{\sigma_0})\le\alpha$, where $\pi(p,b)=\|p-c\|^2-r^2$ is the power distance. The algorithm is exact with respect to this condition: tetrahedra that pass the second condition at their orthocenter are accepted, free triangles and edges are accepted if and only if they pass it, and faces of accepted simplices are inherited. The paper also reports that the resulting GPU implementation computes $\alpha$ complexes for a 10-million-atom dataset in about ten seconds, up to a factor of 50 faster than methods optimized for biomolecules.

Load-bearing premise

The implementation assumes biomolecular packing—atom radii around $1$-$2$ Å, centers at least about $1$ Å apart, and tiny alpha values such as $0$ or $1.4$ Å—which guarantees that each atom owns some region of space and that only a bounded number of nearby atoms matter; if an input violates that packing, the skipped vertex checks and fixed grid can make the method inexact.

Editorial extensions

If this is right

The weighted Delaunay triangulation is no longer a required preprocessing step, removing the main bottleneck for large biomolecular inputs.
Candidate generation can be organized bottom-up (edges, then triangles, then tetrahedra) because OrthoSize is monotone with respect to faces, so only local grid neighborhoods need to be searched.
The output-sensitive nature keeps memory proportional to the alpha complex itself; the paper demonstrates chunked processing of datasets up to 16 million atoms on a GPU with 2 GB of memory.
Reported timings include about 3 seconds for 5.9-million-atom biomolecules at $\alpha=0$, and about 10 seconds for 10 million atoms, with up to 22x speedup over a parallel Delaunay code and up to 50x over biomolecule-optimized existing software.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same orthocenter test may extend to unweighted point clouds or to higher dimensions, but the constant-incidence assumption would need a separate check; a natural stress test is a point set with widely varying radii and separations.
Because pruning tetrahedra dominates runtime, an implementation that sorts candidates by distance or uses exact rational predicates only for near-ties could remove the reported numerical misclassifications without changing typical performance.
The characterization suggests an incremental route to entire alpha filtrations: start from the computed $K_\alpha$ and update only simplices whose size crosses the next threshold, instead of recomputing from scratch.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: the paper's real contribution is the algorithm pipeline—enumerate candidate simplices bottom-up via OrthoSize, prune with AC2, and run it on a GPU so you never build the full Delaunay triangulation. For biomolecular inputs that is genuinely useful and the reported speedups are plausible. The correctness arguments for tetrahedra, triangles, and edges are solid under exact arithmetic, and monotonicity of OrthoSize justifies the bottom-up enumeration. They share code and are honest about numerical errors in Table 3.

The soft spots are real. Definition 2 is not a new characterization. It is the standard weighted Delaunay witness condition plus the alpha size filter, so the theoretical novelty is thin. More importantly, the vertex case is not handled correctly. Step 5 of the proved algorithm checks AC2 for free vertices at p=p_i. Section 4.5 skips that check for all vertices, justified by non-empty Voronoi regions. Non-empty Voronoi region is not enough. At alpha=0, AC2 and AC3 require a point whose power distance to the atom is at most 0 and where that atom is closest among all balls—meaning the atom's ball must intersect its own power cell. Non-empty power cell alone does not give you that. A concrete counterexample satisfying their packing assumptions: atom i at the origin with radius 1; three atoms of radius 1.5 at distance 1.1 in directions 120 degrees apart. The inequalities for i to win at any point are inconsistent, so i's power cell is empty. The correct alpha complex at alpha=0 excludes i, but their implementation inserts it. Table 3 only counts edges, triangles, and tetrahedra, so the validation never sees this divergence.

There is also the numerical correctness issue. Section 5.4 admits misclassifications, so the abstract's claim that the implementation computes the correct alpha complex overstates things. For the intended application, a 0.1% error rate might be tolerable, but the claim needs qualification.

This paper deserves a serious referee. The algorithmic idea is useful, and the high-dimensional parts are mostly proven. But the vertex shortcut needs to be fixed or the claims narrowed to conditions where the shortcut is valid, and the exactness language needs softening. Send it to peer review with the expectation of heavy revision.

Referee Report

3 major / 4 minor

Summary. The paper proposes a GPU-based parallel algorithm for computing the alpha complex of a set of weighted balls representing biomolecules, without first constructing the full weighted Delaunay triangulation. The authors introduce a direct characterization (Definition 2, conditions AC1–AC3) of simplices in the alpha complex and use it in a five-step algorithm: enumerate all simplices with OrthoSize at most alpha, then prune tetrahedra, triangles, edges, and vertices by checking condition AC2. Correctness claims are proved for tetrahedra, triangles, and edges, with a sketched argument for vertices. A CUDA implementation uses a uniform grid and biomolecular packing assumptions to achieve reported runtimes of a few seconds for multi-million-atom molecules, with speedups up to 22x against the gReg3D Delaunay code and a stated 50x over biomolecule-optimized methods. Section 5.4 and Table 3 report small numbers of misclassified edges, triangles, and tetrahedra due to numerical precision.

Significance. If the correctness gap identified below is fixed, the paper makes a useful contribution: it provides a direct Delaunay-free route to alpha complexes for biomolecular data, with plausible correctness proofs for dimensions 1–3 and a reproducible CUDA implementation with no fitted parameters. The bottom-up candidate generation is justified by monotonicity of OrthoSize, and the runtime results are impressive. However, the claimed new characterization in Definition 2 is logically equivalent to the standard definition of the alpha complex (Delaunay membership plus a size filter), so the novelty lies in the algorithmic and implementation strategy rather than in the characterization itself. The main reservations concern the vertex handling in the shipped implementation, which is not justified by the stated packing assumptions, and the absence of vertex errors in the reported validation.

major comments (3)

[Section 4.4, 'Pruning'] The statement that all vertices in Sigma_0_ortho are directly inserted into K_0_alpha without the AC2 check 'because for biomolecular data we assume that Voronoi regions of all the atoms are non-empty' is not justified. For a 0-simplex, AC2 requires a point p with pi(p,b_i) <= pi(p,b_j) for every j, and AC3 requires pi(p,b_i) <= alpha; a non-empty weighted Voronoi region does not imply that the atom center p_i itself lies in the region or that any point of the region lies in the alpha-grown ball. This is not a merely theoretical concern: the packing assumptions of Section 4.1 permit a concrete counterexample. Place atom i at the origin with radius 1 Å, and place three atoms of radius 1.5 Å at distance 1.1 Å from the origin in directions separated by 120 degrees in a plane. No atom contains another and all center distances are at least 1 Å, so the stated assumptions hold. Yet the three inequalities pi(p,b_i) <= pi(p,b_j) sum to 0 <= -3 epsilon for epsilon > 0, so the power cell of i is empty; at alpha = 0 atom i is not in the alpha complex, but the implementation inserts it. Table 3 does not count vertex errors, so the reported experimental validation does not detect this divergence.
[Section 3.2, Claim 8] Claim 8 is only sketched. The sentence 'Similar to Lemmas 4 and 6, it is easy to prove that checking for AC2 for p = p_i is necessary and sufficient...' is not a correctness proof, and the assertion that a free vertex in the alpha complex that fails AC2 at p_i would be incident on an edge in K_alpha is precisely the load-bearing point. A formal proof is needed: starting from a witness p in the power cell with pi(p,b_i) <= alpha and following the segment from p to p_i, one must show that if p_i is not in the power cell, the segment exits the power cell at a point q on a Voronoi face with pi(q,b_i) <= alpha, yielding an incident edge in K_alpha. As written, the correctness proof of the full algorithm is incomplete.
[Section 5.4 and Table 3] The paper claims that the pruning stage removes false positives to obtain the correct alpha complex, but Table 3 reports nonzero misclassifications for edges, triangles, and tetrahedra (error rates up to 0.001). This is disclosed as a numerical issue, but it should be reflected in the abstract and in the summary of results: the shipped implementation computes an approximate alpha complex, not the exact one, and the reported timings do not separate verification or correction costs. Moreover, Table 3 has no vertex column, even though the vertex shortcut in Section 4.4 is the part least supported by the experiments. Please state clearly that the experimental implementation is approximate, and either add vertex counts to the validation or explicitly justify why vertex errors cannot occur.

minor comments (4)

[Section 1.1] The sentence 'Simplices that belong to the alpha complex are identified using a size filtration in a second step' is repeated verbatim twice in the related-work paragraph.
[Section 5.1 and reference [5]] The name of the comparison code is inconsistent: the related work introduces gDel3D, while Table 1 and the text refer to gReg3D. Please harmonize the notation.
[Section 5.4] There is an unresolved citation placeholder '[?]' where a multi-precision library is mentioned. Please supply the reference.
[Abstract and Section 6] The abstract and conclusions state a speedup of up to a factor of 50 over methods optimized for biomolecules, but no direct experimental comparison with the UnionBall method is shown in the paper. Please add the comparison data or state explicitly where the 50x figure is established.

Circularity Check

1 steps flagged · score 1.0 of 10

Mild definitional restatement only; no load-bearing circularity.

self definitional [Section 1.2 (Summary of results); Section 2.4, Definition 2]
"A new characterization of the alpha complex – a set of conditions necessary and sufficient for a simplex to be a part of the alpha complex."

Definition 2 is not derived from independent first principles; it is the standard alpha-complex condition restated. AC1 and AC2 are exactly the witness conditions DT1 and DT2 of Definition 1 for weighted Delaunay membership, and AC3 is the same size filter used in the definition Kα={σd∈D such that Size(σd)≤α} given just before. Hence the 'new characterization' is equivalent to the paper's own input definition by construction, and the correctness proofs verify the algorithm against that restatement. This is a mild presentational overclaim; it is not a fitted parameter or a self-citation chain, and the algorithm's pruning lemmas have independent geometric content.

full rationale

The paper's central derivation is not substantively circular. The five-step algorithm is proved through radical-axis and radical-plane arguments (Lemmas 4 and 6, Claims 3, 5, 7, and 8), which are genuine geometric proofs rather than algebraic tautologies. The only definitional circularity is that Definition 2 restates the standard Delaunay-witness-plus-size-filter characterization of the alpha complex; the paper labels it an 'equivalent definition', so this is a novelty overclaim rather than a hidden reduction of the algorithm to its input. Performance claims are benchmarked against external implementations (gReg3D, AlphaVol, UnionBall), and no parameter is fitted to the target output. The implementation's vertex shortcut in Section 4.4, which inserts all vertices in Σ0_ortho without the AC2 check under a non-empty-Voronoi-region assumption, is a correctness-risk issue rather than a circular step, because it is an unproven domain assumption rather than a definitional equivalence. The self-citations in the paper are background and are not load-bearing.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The algorithm introduces no new physical entities and fits no free parameters: α is an input, and the grid cell size is derived as sqrt(r_max^2 + α). The correctness proof uses standard geometric facts plus explicit biomolecular packing assumptions, and the efficiency claim rests on the cited constant bound on incident simplices. The numerical error rate is an implementation limitation, not a fitted parameter.

assumptions (4)

standard math Input points are in general position (no two coincide, no three collinear, no four coplanar, no five cospherical).
Stated in Section 2; used to ensure unique orthocenters and witnesses; degeneracies are said to be handled by symbolic perturbation [17].
domain assumption For biomolecular data, atom radii are bounded between about 1 Å and 2 Å, and distances between atom centers have a lower bound of about 1 Å (van der Waals contact).
Used in Section 4.1 to argue no atom is contained in another, so every weighted Voronoi region is non-empty, and to bound the number of nearby atoms.
domain assumption The number of alpha-complex simplices incident on a single atom is bounded by a constant independent of total atom count (cited from Halperin and Overmars [22]).
This is the key locality assumption that makes the grid-based candidate generation and pruning near-linear; stated in Section 4.1.
standard math For a free simplex in Kα that violates AC2 at the orthocenter, the closest witness lies on the boundary of the intersection of half-intervals (triangles) or half-planes (edges) corresponding to a single other ball.
Used in Lemma 4 and Lemma 6 proofs; the minimum of the convex power-distance function over a convex intersection region is attained on the boundary, and the text asserts it lies on one bounding line/plane, which is argued but not fully formalized.

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Cite this review

Pith. "Pith review of Parallel Computation of Alpha Complex for Biomolecules." pith.science (2026). https://pith.science/paper/VY2DNP5G

@misc{pith2026190805944,
  author       = {Pith},
  title        = {Pith review of: Parallel Computation of Alpha Complex for Biomolecules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VY2DNP5G}},
  note         = {Machine review of arXiv:1908.05944}
}

read the original abstract

The alpha complex, a subset of the Delaunay triangulation, has been extensively used as the underlying representation for biomolecular structures. We propose a GPU-based parallel algorithm for the computation of the alpha complex, which exploits the knowledge of typical spatial distribution and sizes of atoms in a biomolecule. Unlike existing methods, this algorithm does not require prior construction of the Delaunay triangulation. The algorithm computes the alpha complex in two stages. The first stage proceeds in a bottom-up fashion and computes a superset of the edges, triangles, and tetrahedra belonging to the alpha complex. The false positives from this estimation stage are removed in a subsequent pruning stage to obtain the correct alpha complex. Computational experiments on several biomolecules demonstrate the superior performance of the algorithm, up to a factor of 50 when compared to existing methods that are optimized for biomolecules.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 8 canonical work pages

[1]

World Scientiﬁc, 2013.doi:10.1142/8685

1 Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee.Voronoi Diagrams and Delaunay Triangu- lations. World Scientiﬁc, 2013.doi:10.1142/8685. 2 Ulrich Bauer and Herbert Edelsbrunner. The Morse theory of Čech and Delaunay ﬁltrations. In Siu-Wing Cheng and Olivier Devillers, editors,30th Annual Symposium on Computational Geometry, SOCG’14, Kyoto, Japan, June 08...

doi:10.1142/8685 2013
[5]

9 Tran Kai Frank Da, Sébastien Loriot, and Mariette Yvinec

[Online; accessed 17-March-2020]. 9 Tran Kai Frank Da, Sébastien Loriot, and Mariette Yvinec. 3D alpha shapes. InCGAL User and Reference Manual. CGAL Editorial Board, 4.11 edition,

work page 2020
[484]

3 A Bondi

ACM, 2014.doi:10.1145/ 2582112.2582167. 3 A Bondi. van der Waals volumes and radii.The Journal of Physical Chemistry, 68(3):441–451,

arXiv 2014
[1964]

Computing Dirichlet tessellations.Comput

4 Adrian Bowyer. Computing Dirichlet tessellations.Comput. J., 24(2):162–166, 1981.doi: 10.1093/comjnl/24.2.162. 5 Thanh-Tung Cao, Ashwin Nanjappa, Mingcen Gao, and Tiow Seng Tan. A GPU accelerated algorithm for 3d Delaunay triangulation. In John Keyser and Pedro V. Sander, editors, Symposium on Interactive 3D Graphics and Games, I3D ’14, San Francisco, C...

work page arXiv 1981
[1990]

18 Herbert Edelsbrunner and Ernst P

doi:10.1145/77635.77639. 18 Herbert Edelsbrunner and Ernst P. Mücke. Three-dimensional alpha shapes.ACM Trans. Graph., 13(1):43–72, 1994.doi:10.1145/174462.156635. 19 Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements.Discret. Comput. Geom., 1:25–44, 1986.doi:10.1007/BF02187681. 20 Herbert Edelsbrunner and Nimish R. Shah. Increment...

arXiv 1994
[1992]

22 Dan Halperin and Mark H

doi: 10.1007/BF01758770. 22 Dan Halperin and Mark H. Overmars. Spheres, molecules, and hidden surface removal.Comput. Geom., 11(2):83–102, 1998.doi:10.1016/S0925-7721(98)00023-6. 23 Michael Krone, Barbora Kozlíková, Norbert Lindow, Marc Baaden, Daniel Baum, Július Parulek, Hans-Christian Hege, and Ivan Viola. Visual analysis of biomolecular cavities: Stat...

doi:10.1007/bf01758770 1998
[1998]

Geometric measures of large biomolecules: Surface, volume, and pockets

27 Paul Mach and Patrice Koehl. Geometric measures of large biomolecules: Surface, volume, and pockets. Journal of Computational Chemistry, 32(14):3023–3038, 2011.doi:10.1002/ jcc.21884. 28 Talha Bin Masood and Vijay Natarajan. An integrated geometric and topological approach to connecting cavities in biomolecules. In Chuck Hansen, Ivan Viola, and Xiaoru ...

arXiv 2011
[2006]

doi:10.1093/nar/gkl282. 11 H. Edelsbrunner. Weighted alpha shapes. University of Illinois at Urbana-Champaign, Depart- ment of Computer Science,

work page doi:10.1093/nar/gkl282

Show all 14 references

[2010]

16 Herbert Edelsbrunner, David G

URL:http://www.ams.org/bookstore-getitem/item=MBK-69. 16 Herbert Edelsbrunner, David G. Kirkpatrick, and Raimund Seidel. On the shape of a set of points in the plane.IEEE Trans. Inf. Theory, 29(4):551–558, 1983.doi:10.1109/TIT.1983. 1056714. 17 Herbert Edelsbrunner and Ernst P...

1983 doi
[2012]

32 Donald R. Sheehy. An output-sensitive algorithm for computing weightedα-complexes. In Proceedings of the 27th Canadian Conference on Computational Geometry, CCCG 2015, Kingston, Ontario, Canada, August 10-12,

2015
[2015]

31 Ashwin Nanjappa.Delaunay triangulation inR3 on the GPU

doi:10.1186/s12859-015-0545-9. 31 Ashwin Nanjappa.Delaunay triangulation inR3 on the GPU. PhD thesis, National University of Singapore,

doi
[2016]

34 David F Watson

doi:10.1007/978-3-319-24523-2\_15. 34 David F Watson. Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. The Computer Journal, 24(2):167–172, 1981

1981 doi
[2019]

URL:http://arxiv.org/abs/1908.05944, arXiv:1908.05944. T.B. Masood, T. Ray, and V. Natarajan 23 30 Talha Bin Masood, Sankaran Sandhya, Nagasuma R. Chandra, and Vijay Natarajan. CH- EXVIS: a tool for molecular channel extraction and visualization.BMC Bioinform., 16:119:1– 119:19,

1908 arXiv
[2020]

8 Nvidia Corporation

[Online; accessed 17-March-2020]. 8 Nvidia Corporation. Thrust.https://developer.nvidia.com/thrust,

2020

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

World Scientiﬁc, 2013.doi:10.1142/8685

1 Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee.Voronoi Diagrams and Delaunay Triangu- lations. World Scientiﬁc, 2013.doi:10.1142/8685. 2 Ulrich Bauer and Herbert Edelsbrunner. The Morse theory of Čech and Delaunay ﬁltrations. In Siu-Wing Cheng and Olivier Devillers, editors,30th Annual Symposium on Computational Geometry, SOCG’14, Kyoto, Japan, June 08...

doi:10.1142/8685 2013

[2] [5]

9 Tran Kai Frank Da, Sébastien Loriot, and Mariette Yvinec

[Online; accessed 17-March-2020]. 9 Tran Kai Frank Da, Sébastien Loriot, and Mariette Yvinec. 3D alpha shapes. InCGAL User and Reference Manual. CGAL Editorial Board, 4.11 edition,

work page 2020

[3] [484]

3 A Bondi

ACM, 2014.doi:10.1145/ 2582112.2582167. 3 A Bondi. van der Waals volumes and radii.The Journal of Physical Chemistry, 68(3):441–451,

arXiv 2014

[4] [1964]

Computing Dirichlet tessellations.Comput

4 Adrian Bowyer. Computing Dirichlet tessellations.Comput. J., 24(2):162–166, 1981.doi: 10.1093/comjnl/24.2.162. 5 Thanh-Tung Cao, Ashwin Nanjappa, Mingcen Gao, and Tiow Seng Tan. A GPU accelerated algorithm for 3d Delaunay triangulation. In John Keyser and Pedro V. Sander, editors, Symposium on Interactive 3D Graphics and Games, I3D ’14, San Francisco, C...

work page arXiv 1981

[5] [1990]

18 Herbert Edelsbrunner and Ernst P

doi:10.1145/77635.77639. 18 Herbert Edelsbrunner and Ernst P. Mücke. Three-dimensional alpha shapes.ACM Trans. Graph., 13(1):43–72, 1994.doi:10.1145/174462.156635. 19 Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements.Discret. Comput. Geom., 1:25–44, 1986.doi:10.1007/BF02187681. 20 Herbert Edelsbrunner and Nimish R. Shah. Increment...

arXiv 1994

[6] [1992]

22 Dan Halperin and Mark H

doi: 10.1007/BF01758770. 22 Dan Halperin and Mark H. Overmars. Spheres, molecules, and hidden surface removal.Comput. Geom., 11(2):83–102, 1998.doi:10.1016/S0925-7721(98)00023-6. 23 Michael Krone, Barbora Kozlíková, Norbert Lindow, Marc Baaden, Daniel Baum, Július Parulek, Hans-Christian Hege, and Ivan Viola. Visual analysis of biomolecular cavities: Stat...

doi:10.1007/bf01758770 1998

[7] [1998]

Geometric measures of large biomolecules: Surface, volume, and pockets

27 Paul Mach and Patrice Koehl. Geometric measures of large biomolecules: Surface, volume, and pockets. Journal of Computational Chemistry, 32(14):3023–3038, 2011.doi:10.1002/ jcc.21884. 28 Talha Bin Masood and Vijay Natarajan. An integrated geometric and topological approach to connecting cavities in biomolecules. In Chuck Hansen, Ivan Viola, and Xiaoru ...

arXiv 2011

[8] [2006]

doi:10.1093/nar/gkl282. 11 H. Edelsbrunner. Weighted alpha shapes. University of Illinois at Urbana-Champaign, Depart- ment of Computer Science,

work page doi:10.1093/nar/gkl282