---
_id: '13182'
abstract:
- lang: eng
  text: "We characterize critical points of 1-dimensional maps paired in persistent
    homology\r\ngeometrically and this way get elementary proofs of theorems about
    the symmetry\r\nof persistence diagrams and the variation of such maps. In particular,
    we identify\r\nbranching points and endpoints of networks as the sole source of
    asymmetry and\r\nrelate the cycle basis in persistent homology with a version
    of the stable marriage\r\nproblem. Our analysis provides the foundations of fast
    algorithms for maintaining a\r\ncollection of sorted lists together with its persistence
    diagram."
acknowledgement: Open access funding provided by Austrian Science Fund (FWF). This
  project has received funding from the European Research Council (ERC) under the
  European Union’s Horizon 2020 research and innovation programme, grant no. 788183,
  from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and
  from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry
  and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35. The authors of
  this paper thank anonymous reviewers for their constructive criticism and Monika
  Henzinger for detailed comments on an earlier version of this paper.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Sebastiano
  full_name: Cultrera Di Montesano, Sebastiano
  id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
  last_name: Cultrera Di Montesano
  orcid: 0000-0001-6249-0832
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Morteza
  full_name: Saghafian, Morteza
  id: f86f7148-b140-11ec-9577-95435b8df824
  last_name: Saghafian
citation:
  ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization
    of the persistence of 1D maps. <i>Journal of Applied and Computational Topology</i>.
    2023. doi:<a href="https://doi.org/10.1007/s41468-023-00126-9">10.1007/s41468-023-00126-9</a>
  apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M.
    (2023). Geometric characterization of the persistence of 1D maps. <i>Journal of
    Applied and Computational Topology</i>. Springer Nature. <a href="https://doi.org/10.1007/s41468-023-00126-9">https://doi.org/10.1007/s41468-023-00126-9</a>
  chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
    and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.”
    <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2023. <a
    href="https://doi.org/10.1007/s41468-023-00126-9">https://doi.org/10.1007/s41468-023-00126-9</a>.
  ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric
    characterization of the persistence of 1D maps,” <i>Journal of Applied and Computational
    Topology</i>. Springer Nature, 2023.
  ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2023. Geometric
    characterization of the persistence of 1D maps. Journal of Applied and Computational
    Topology.
  mla: Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D
    Maps.” <i>Journal of Applied and Computational Topology</i>, Springer Nature,
    2023, doi:<a href="https://doi.org/10.1007/s41468-023-00126-9">10.1007/s41468-023-00126-9</a>.
  short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal
    of Applied and Computational Topology (2023).
date_created: 2023-07-02T22:00:44Z
date_published: 2023-06-17T00:00:00Z
date_updated: 2023-10-18T08:13:10Z
day: '17'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00126-9
ec_funded: 1
file:
- access_level: open_access
  checksum: 697249d5d1c61dea4410b9f021b70fce
  content_type: application/pdf
  creator: alisjak
  date_created: 2023-07-03T09:41:05Z
  date_updated: 2023-07-03T09:41:05Z
  file_id: '13185'
  file_name: 2023_Journal of Applied and Computational Topology_Biswas.pdf
  file_size: 487355
  relation: main_file
  success: 1
file_date_updated: 2023-07-03T09:41:05Z
has_accepted_license: '1'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
  grant_number: I4887
  name: Discretization in Geometry and Dynamics
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: The Wittgenstein Prize
publication: Journal of Applied and Computational Topology
publication_identifier:
  eissn:
  - 2367-1734
  issn:
  - 2367-1726
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Geometric characterization of the persistence of 1D maps
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '10773'
abstract:
- lang: eng
  text: The Voronoi tessellation in Rd is defined by locally minimizing the power
    distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined
    by locally maximizing the negative power distance to other such points. We prove
    that the average of the two piecewise quadratic functions is piecewise linear,
    and that all three functions have the same critical points and values. Discretizing
    the two piecewise quadratic functions, we get the alpha shapes as sublevel sets
    of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel
    sets of the discrete function on the Voronoi tessellation. For the same non-critical
    value, the corresponding shapes are disjoint, separated by a narrow channel that
    contains no critical points but the entire level set of the piecewise linear function.
acknowledgement: Open access funding provided by the Institute of Science and Technology
  (IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Sebastiano
  full_name: Cultrera Di Montesano, Sebastiano
  id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
  last_name: Cultrera Di Montesano
  orcid: 0000-0001-6249-0832
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Morteza
  full_name: Saghafian, Morteza
  last_name: Saghafian
citation:
  ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Continuous
    and discrete radius functions on Voronoi tessellations and Delaunay mosaics. <i>Discrete
    and Computational Geometry</i>. 2022;67:811-842. doi:<a href="https://doi.org/10.1007/s00454-022-00371-2">10.1007/s00454-022-00371-2</a>
  apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian, M.
    (2022). Continuous and discrete radius functions on Voronoi tessellations and
    Delaunay mosaics. <i>Discrete and Computational Geometry</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00454-022-00371-2">https://doi.org/10.1007/s00454-022-00371-2</a>
  chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
    and Morteza Saghafian. “Continuous and Discrete Radius Functions on Voronoi Tessellations
    and Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1007/s00454-022-00371-2">https://doi.org/10.1007/s00454-022-00371-2</a>.
  ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Continuous
    and discrete radius functions on Voronoi tessellations and Delaunay mosaics,”
    <i>Discrete and Computational Geometry</i>, vol. 67. Springer Nature, pp. 811–842,
    2022.
  ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2022. Continuous
    and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete
    and Computational Geometry. 67, 811–842.
  mla: Biswas, Ranita, et al. “Continuous and Discrete Radius Functions on Voronoi
    Tessellations and Delaunay Mosaics.” <i>Discrete and Computational Geometry</i>,
    vol. 67, Springer Nature, 2022, pp. 811–42, doi:<a href="https://doi.org/10.1007/s00454-022-00371-2">10.1007/s00454-022-00371-2</a>.
  short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete
    and Computational Geometry 67 (2022) 811–842.
date_created: 2022-02-20T23:01:34Z
date_published: 2022-04-01T00:00:00Z
date_updated: 2023-08-02T14:31:25Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-022-00371-2
external_id:
  isi:
  - '000752175300002'
file:
- access_level: open_access
  checksum: 9383d3b70561bacee905e335dc922680
  content_type: application/pdf
  creator: dernst
  date_created: 2022-08-02T06:07:55Z
  date_updated: 2022-08-02T06:07:55Z
  file_id: '11718'
  file_name: 2022_DiscreteCompGeometry_Biswas.pdf
  file_size: 2518111
  relation: main_file
  success: 1
file_date_updated: 2022-08-02T06:07:55Z
has_accepted_license: '1'
intvolume: '        67'
isi: 1
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 811-842
publication: Discrete and Computational Geometry
publication_identifier:
  eissn:
  - 1432-0444
  issn:
  - 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Continuous and discrete radius functions on Voronoi tessellations and Delaunay
  mosaics
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 67
year: '2022'
...
---
_id: '11658'
abstract:
- lang: eng
  text: The depth of a cell in an arrangement of n (non-vertical) great-spheres in
    Sd is the number of great-spheres that pass above the cell. We prove Euler-type
    relations, which imply extensions of the classic Dehn–Sommerville relations for
    convex polytopes to sublevel sets of the depth function, and we use the relations
    to extend the expressions for the number of faces of neighborly polytopes to the
    number of cells of levels in neighborly arrangements.
acknowledgement: This project has received funding from the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation programme,
  grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant
  no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization
  in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35.
article_processing_charge: No
author:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Sebastiano
  full_name: Cultrera di Montesano, Sebastiano
  id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
  last_name: Cultrera di Montesano
  orcid: 0000-0001-6249-0832
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Morteza
  full_name: Saghafian, Morteza
  id: f86f7148-b140-11ec-9577-95435b8df824
  last_name: Saghafian
citation:
  ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements:
    Dehn–Sommerville–Euler relations with applications. <i>Leibniz International Proceedings
    on Mathematics</i>.'
  apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian,
    M. (n.d.). Depth in arrangements: Dehn–Sommerville–Euler relations with applications.
    <i>Leibniz International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz
    Zentrum für Informatik.'
  chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
    and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations
    with Applications.” <i>Leibniz International Proceedings on Mathematics</i>. Schloss
    Dagstuhl - Leibniz Zentrum für Informatik, n.d.'
  ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth
    in arrangements: Dehn–Sommerville–Euler relations with applications,” <i>Leibniz
    International Proceedings on Mathematics</i>. Schloss Dagstuhl - Leibniz Zentrum
    für Informatik.'
  ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in
    arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International
    Proceedings on Mathematics.'
  mla: 'Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations
    with Applications.” <i>Leibniz International Proceedings on Mathematics</i>, Schloss
    Dagstuhl - Leibniz Zentrum für Informatik.'
  short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Leibniz
    International Proceedings on Mathematics (n.d.).
date_created: 2022-07-27T09:27:34Z
date_published: 2022-07-27T00:00:00Z
date_updated: 2022-07-28T07:57:48Z
day: '27'
ddc:
- '510'
department:
- _id: GradSch
- _id: HeEd
ec_funded: 1
file:
- access_level: open_access
  checksum: b2f511e8b1cae5f1892b0cdec341acac
  content_type: application/pdf
  creator: scultrer
  date_created: 2022-07-27T09:25:53Z
  date_updated: 2022-07-27T09:25:53Z
  file_id: '11659'
  file_name: D-S-E.pdf
  file_size: 639266
  relation: main_file
file_date_updated: 2022-07-27T09:25:53Z
has_accepted_license: '1'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Submitted Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: The Wittgenstein Prize
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: Leibniz International Proceedings on Mathematics
publication_status: submitted
publisher: Schloss Dagstuhl - Leibniz Zentrum für Informatik
quality_controlled: '1'
status: public
title: 'Depth in arrangements: Dehn–Sommerville–Euler relations with applications'
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2022'
...
---
_id: '11660'
abstract:
- lang: eng
  text: 'We characterize critical points of 1-dimensional maps paired in persistent
    homology geometrically and this way get elementary proofs of theorems about the
    symmetry of persistence diagrams and the variation of such maps. In particular,
    we identify branching points and endpoints of networks as the sole source of asymmetry
    and relate the cycle basis in persistent homology with a version of the stable
    marriage problem. Our analysis provides the foundations of fast algorithms for
    maintaining collections of interrelated sorted lists together with their persistence
    diagrams. '
acknowledgement: 'This project has received funding from the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation programme,
  grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant
  no. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization
  in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35. '
alternative_title:
- LIPIcs
article_processing_charge: No
author:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Sebastiano
  full_name: Cultrera di Montesano, Sebastiano
  id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
  last_name: Cultrera di Montesano
  orcid: 0000-0001-6249-0832
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Morteza
  full_name: Saghafian, Morteza
  last_name: Saghafian
citation:
  ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window to
    the persistence of 1D maps. I: Geometric characterization of critical point pairs.
    <i>LIPIcs</i>.'
  apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian,
    M. (n.d.). A window to the persistence of 1D maps. I: Geometric characterization
    of critical point pairs. <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für
    Informatik.'
  chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
    and Morteza Saghafian. “A Window to the Persistence of 1D Maps. I: Geometric Characterization
    of Critical Point Pairs.” <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für
    Informatik, n.d.'
  ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “A
    window to the persistence of 1D maps. I: Geometric characterization of critical
    point pairs,” <i>LIPIcs</i>. Schloss Dagstuhl - Leibniz-Zentrum für Informatik.'
  ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. A window
    to the persistence of 1D maps. I: Geometric characterization of critical point
    pairs. LIPIcs.'
  mla: 'Biswas, Ranita, et al. “A Window to the Persistence of 1D Maps. I: Geometric
    Characterization of Critical Point Pairs.” <i>LIPIcs</i>, Schloss Dagstuhl - Leibniz-Zentrum
    für Informatik.'
  short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, LIPIcs
    (n.d.).
date_created: 2022-07-27T09:31:15Z
date_published: 2022-07-25T00:00:00Z
date_updated: 2022-07-28T08:05:34Z
day: '25'
ddc:
- '510'
department:
- _id: GradSch
- _id: HeEd
ec_funded: 1
file:
- access_level: open_access
  checksum: 95903f9d1649e8e437a967b6f2f64730
  content_type: application/pdf
  creator: scultrer
  date_created: 2022-07-27T09:30:30Z
  date_updated: 2022-07-27T09:30:30Z
  file_id: '11661'
  file_name: window 1.pdf
  file_size: 564836
  relation: main_file
file_date_updated: 2022-07-27T09:30:30Z
has_accepted_license: '1'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Submitted Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: The Wittgenstein Prize
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: I02979-N35
  name: Persistence and stability of geometric complexes
publication: LIPIcs
publication_status: submitted
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
status: public
title: 'A window to the persistence of 1D maps. I: Geometric characterization of critical
  point pairs'
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2022'
...
---
_id: '9604'
abstract:
- lang: eng
  text: Generalizing Lee’s inductive argument for counting the cells of higher order
    Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse
    theoretic quantities for piecewise constant functions on planar arrangements.
    Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number
    of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for
    1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s
    first k-1 sublevel sets. We get similar expressions for the vertices, edges, and
    polygons of the order-k Voronoi tessellation.
alternative_title:
- LIPIcs
article_number: '16'
article_processing_charge: No
author:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Sebastiano
  full_name: Cultrera di Montesano, Sebastiano
  id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
  last_name: Cultrera di Montesano
  orcid: 0000-0001-6249-0832
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Morteza
  full_name: Saghafian, Morteza
  last_name: Saghafian
citation:
  ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Counting cells
    of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. In: <i>Leibniz
    International Proceedings in Informatics</i>. Vol 189. Schloss Dagstuhl - Leibniz-Zentrum
    für Informatik; 2021. doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2021.16">10.4230/LIPIcs.SoCG.2021.16</a>'
  apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., &#38; Saghafian,
    M. (2021). Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with
    morse theory. In <i>Leibniz International Proceedings in Informatics</i> (Vol.
    189). Online: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2021.16">https://doi.org/10.4230/LIPIcs.SoCG.2021.16</a>'
  chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
    and Morteza Saghafian. “Counting Cells of Order-k Voronoi Tessellations in ℝ<sup>3</sup>
    with Morse Theory.” In <i>Leibniz International Proceedings in Informatics</i>,
    Vol. 189. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2021.16">https://doi.org/10.4230/LIPIcs.SoCG.2021.16</a>.
  ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Counting
    cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory,” in
    <i>Leibniz International Proceedings in Informatics</i>, Online, 2021, vol. 189.
  ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2021. Counting
    cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse theory. Leibniz
    International Proceedings in Informatics. SoCG: International Symposium on Computational
    Geometry, LIPIcs, vol. 189, 16.'
  mla: Biswas, Ranita, et al. “Counting Cells of Order-k Voronoi Tessellations in
    ℝ<sup>3</sup> with Morse Theory.” <i>Leibniz International Proceedings in Informatics</i>,
    vol. 189, 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, doi:<a
    href="https://doi.org/10.4230/LIPIcs.SoCG.2021.16">10.4230/LIPIcs.SoCG.2021.16</a>.
  short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, in:,
    Leibniz International Proceedings in Informatics, Schloss Dagstuhl - Leibniz-Zentrum
    für Informatik, 2021.
conference:
  end_date: 2021-06-11
  location: Online
  name: 'SoCG: International Symposium on Computational Geometry'
  start_date: 2021-06-07
date_created: 2021-06-27T22:01:48Z
date_published: 2021-06-02T00:00:00Z
date_updated: 2023-02-23T14:02:28Z
day: '02'
ddc:
- '516'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2021.16
ec_funded: 1
file:
- access_level: open_access
  checksum: 22b11a719018b22ecba2471b51f2eb40
  content_type: application/pdf
  creator: asandaue
  date_created: 2021-06-28T13:11:39Z
  date_updated: 2021-06-28T13:11:39Z
  file_id: '9611'
  file_name: 2021_LIPIcs_Biswas.pdf
  file_size: 727817
  relation: main_file
  success: 1
file_date_updated: 2021-06-28T13:11:39Z
has_accepted_license: '1'
intvolume: '       189'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: Z00342
  name: The Wittgenstein Prize
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
  grant_number: I4887
  name: Discretization in Geometry and Dynamics
publication: Leibniz International Proceedings in Informatics
publication_identifier:
  isbn:
  - '9783959771849'
  issn:
  - '18688969'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Counting cells of order-k voronoi tessellations in ℝ<sup>3</sup> with morse
  theory
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: conference
user_id: D865714E-FA4E-11E9-B85B-F5C5E5697425
volume: 189
year: '2021'
...