Document Open Access Logo

Hardness of Frequency-Related Queries on Compressed Strings

Authors Rajat De , Dominik Kempa

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2026.143.pdf
  • Filesize: 0.92 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Data compression
  • Information systems → Information retrieval
Keywords
  • compressed indexing
  • grammar compression
  • Lempel-Ziv 78 compression
  • conditional lower bounds
  • rank queries
  • range distinct count queries
  • range mode frequency queries

Metrics

Abstract

Compressed indexing is a recent trend in the design of data structures that aims to support fundamental string queries in space proportional to the size of the data in compressed form. One of the most popular compression frameworks in this field is grammar compression. A length-n string T ∈ Σⁿ (where Σ is any finite set of size up to |Σ| = |T|^𝒪(1)) represented using a context-free grammar of size |G| can be augmented to support random access queries (given any i ∈ [1..n], return T[i]) in 𝒪(|G| log^𝒪(1) n) space and 𝒪(log^𝒪(1) n) time. Numerous other queries, including pattern matching, longest common extension, lexicographical predecessor/successor, Burrows-Wheeler Transform, suffix array, and even suffix tree queries, can also be supported within the same bounds.
Despite this progress, one fundamental class of queries has remained elusive: frequency-related queries, such as reporting the number of occurrences of a symbol c ∈ Σ in a substring T(b..e] (the so-called rank query), or simply checking whether c occurs in T(b..e] (the symbol occurrence query). To date, no fully general structure achieving 𝒪(|G| log^𝒪(1) n) space and 𝒪(log^𝒪(1) n) query time is known. In this work, we establish new conditional lower bounds for frequency-related problems: 
- We prove that answering rank and symbol occurrence queries on grammar-compressed texts in polylogarithmic time using a 𝒪(|G| log^𝒪(1) n)-space structure that is constructible from the input grammar in 𝒪(|G| log^𝒪(1) n) time would imply an 𝒪(n² log^𝒪(1) n)-time algorithm for Boolean Matrix Multiplication (BMM), where the best known algorithms achieve 𝒪(n^{2.371339}) time. Our result is achieved using a more general lower bound for efficiently answering a batch of rank and symbol occurrence queries.
- We generalize the above result, showing that even LZ78-compressed strings cannot support efficient rank queries. Since LZ78 is provably weaker than grammar compression, this yields a stronger result: rank and symbol occurrence queries remain hard for a wider class of compressors. We further show that achieving even additive approximations of rank queries would imply faster BMM algorithms.
- After establishing hardness of rank and symbol occurrence queries, we consider a broader class of frequency-related queries and show that, under the popular Orthogonal Vectors (OV) conjecture, other problems, including range distinct counting and range mode frequency queries, also cannot be efficiently supported in compressed space. 
In summary, we develop new techniques for reasoning about computation over compressed data, and establish tight connections between compressed indexing and long-standing problems in fine-grained complexity. This sheds new light on compressed indexing by isolating a new class of frequency-related queries whose complexity hinges on known hard problems.

Cite As Get BibTex

Rajat De and Dominik Kempa. Hardness of Frequency-Related Queries on Compressed Strings. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 143:1-143:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.143

Author Details

Rajat De
  • Department of Computer Science, Stony Brook University, NY, USA
Dominik Kempa
  • Department of Computer Science, Stony Brook University, NY, USA

Funding

Partially funded by the NSF CAREER Award 2337891.

References

  1. Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. More asymmetry yields faster matrix multiplication. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 ACM-SIAM Symposium on Discrete Algorithms (SODA 2025), pages 2005-2039. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.63.
  2. Djamal Belazzougui, Manuel Cáceres, Travis Gagie, Pawel Gawrychowski, Juha Kärkkäinen, Gonzalo Navarro, Alberto Ordóñez Pereira, Simon J. Puglisi, and Yasuo Tabei. Block trees. Journal of Computer and System Sciences, 117:1-22, 2021. URL: https://doi.org/10.1016/j.jcss.2020.11.002.
  3. Djamal Belazzougui, Patrick Hagge Cording, Simon J. Puglisi, and Yasuo Tabei. Access, rank, and select in grammar-compressed strings. In Nikhil Bansal and Irene Finocchi, editors, Proceedings of the 23rd Annual European Symposium on Algorithms (ESA 2015), pages 142-154. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_13.
  4. Djamal Belazzougui and Gonzalo Navarro. Alphabet-independent compressed text indexing. ACM Transactions on Algorithms, 10(4):23:1-23:19, 2014. URL: https://doi.org/10.1145/2635816.
  5. Djamal Belazzougui and Gonzalo Navarro. Optimal lower and upper bounds for representing sequences. ACM Transactions on Algorithms, 11(4):31:1-31:21, 2015. URL: https://doi.org/10.1145/2629339.
  6. Oren Ben-Kiki, Philip Bille, Dany Breslauer, Leszek Ga̧sieniec, Roberto Grossi, and Oren Weimann. Towards optimal packed string matching. Theoretical Computer Science, 525:111-129, 2014. URL: https://doi.org/10.1016/j.tcs.2013.06.013.
  7. Philip Bille, Inge Li Gørtz, and Frederik Rye Skjoldjensen. Deterministic indexing for packed strings. In Juha Kärkkäinen, Jakub Radoszewski, and Wojciech Rytter, editors, Proceedings of the 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017), pages 6:1-6:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CPM.2017.6.
  8. Philip Bille, Gad M. Landau, Rajeev Raman, Kunihiko Sadakane, Srinivasa Rao Satti, and Oren Weimann. Random access to grammar-compressed strings and trees. SIAM Journal on Computing, 44(3):513-539, 2015. URL: https://doi.org/10.1137/130936889.
  9. Michael Burrows and David J. Wheeler. A block-sorting lossless data compression algorithm. Technical Report 124, Digital Equipment Corporation, Palo Alto, California, 1994. URL: https://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR- 124.pdf.
  10. Moses Charikar, Eric Lehman, Ding Liu, Rina Panigrahy, Manoj Prabhakaran, Amit Sahai, and Abhi Shelat. The smallest grammar problem. IEEE Transactions on Information Theory, 51(7):2554-2576, 2005. URL: https://doi.org/10.1109/TIT.2005.850116.
  11. Anders Roy Christiansen, Mikko Berggren Ettienne, Tomasz Kociumaka, Gonzalo Navarro, and Nicola Prezza. Optimal-time dictionary-compressed indexes. ACM Transactions on Algorithms, 17(1):8:1-8:39, 2021. URL: https://doi.org/10.1145/3426473.
  12. Francisco Claude and Gonzalo Navarro. Self-indexed grammar-based compression. Fundamenta Informaticae, 111(3):313-337, 2011. URL: https://doi.org/10.3233/FI-2011-565.
  13. Francisco Claude and Gonzalo Navarro. Improved grammar-based compressed indexes. In Liliana Calderón-Benavides, Cristina N. González-Caro, Edgar Chávez, and Nivio Ziviani, editors, Proceedings of the 19th International Symposium on String Processing and Information Retrieval (SPIRE 2012), pages 180-192. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-34109-0_19.
  14. Francisco Claude, Gonzalo Navarro, and Alejandro Pacheco. Grammar-compressed indexes with logarithmic search time. Journal of Computer and System Sciences, 118:53-74, 2021. URL: https://doi.org/10.1016/j.jcss.2020.12.001.
  15. Vincent Cohen-Addad, Laurent Feuilloley, and Tatiana Starikovskaya. Lower bounds for text indexing with mismatches and differences. In Timothy M. Chan, editor, Proceedings of the 2019 ACM-SIAM Symposium on Discrete Algorithms (SODA 2019), pages 1146-1164. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.70.
  16. European Commission. 1+ Million Genomes Initiative. URL: https://digital-strategy.ec.europa.eu/en/policies/1-million-genomes.
  17. Maxime Crochemore, Christophe Hancart, and Thierry Lecroq. Algorithms on strings. Cambridge University Press, Cambridge, UK, 2007. URL: https://doi.org/10.1017/cbo9780511546853.
  18. Rajat De and Dominik Kempa. Grammar boosting: A new technique for proving lower bounds for computation over compressed data. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms (SODA 2024), pages 3376-3392. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.121.
  19. Rajat De and Dominik Kempa. Optimal random access and conditional lower bounds for 2D compressed strings. In Kasper Green Larsen and Barna Saha, editors, Proceedings of the 2026 ACM-SIAM Symposium on Discrete Algorithms (SODA 2026), pages 1903-1915. SIAM, 2026. URL: https://doi.org/10.1137/1.9781611978971.69.
  20. Akashnil Dutta, Reut Levi, Dana Ron, and Ronitt Rubinfeld. A simple online competitive adaptation of Lempel-Ziv compression with efficient random access support. In Ali Bilgin, Michael W. Marcellin, Joan Serra-Sagristà, and James A. Storer, editors, Proceedings of the 2013 Data Compression Conference (DCC 2013), pages 113-122. IEEE, 2013. URL: https://doi.org/10.1109/DCC.2013.19.
  21. Paolo Ferragina and Giovanni Manzini. Indexing compressed text. Journal of the ACM, 52(4):552-581, 2005. URL: https://doi.org/10.1145/1082036.1082039.
  22. José Fuentes-Sepúlveda, Juha Kärkkäinen, Dmitry Kosolobov, and Simon J. Puglisi. Run compressed rank/select for large alphabets. In Ali Bilgin, Michael W. Marcellin, Joan Serra-Sagristà, and James A. Storer, editors, Proceedings of the 2018 Data Compression Conference (DCC 2018), pages 315-324. IEEE, 2018. URL: https://doi.org/10.1109/DCC.2018.00040.
  23. Travis Gagie, Pawel Gawrychowski, Juha Kärkkäinen, Yakov Nekrich, and Simon J. Puglisi. A faster grammar-based self-index. In Adrian-Horia Dediu and Carlos Martín-Vide, editors, Proceedings of the 6th International Conference on Language and Automata Theory and Applications (LATA 2012), pages 240-251. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-28332-1_21.
  24. Travis Gagie, Pawel Gawrychowski, Juha Kärkkäinen, Yakov Nekrich, and Simon J. Puglisi. LZ77-based self-indexing with faster pattern matching. In Alberto Pardo and Alfredo Viola, editors, Proceedings of the 11th Latin American Symposium on Theoretical Informatics (LATIN 2014), pages 731-742. Springer, 2014. URL: https://doi.org/10.1007/978-3-642-54423-1_63.
  25. Travis Gagie, Gonzalo Navarro, and Nicola Prezza. On the approximation ratio of Lempel-Ziv parsing. In Michael A. Bender, Martin Farach-Colton, and Miguel A. Mosteiro, editors, Proceedings of the 13th Latin American Symposium on Theoretical Informatics (LATIN 2018), pages 490-503. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-77404-6_36.
  26. Travis Gagie, Gonzalo Navarro, and Nicola Prezza. Fully functional suffix trees and optimal text searching in BWT-runs bounded space. Journal of the ACM, 67(1):1-54, 2020. URL: https://doi.org/10.1145/3375890.
  27. Moses Ganardi, Artur Jez, and Markus Lohrey. Balancing straight-line programs. Journal of the ACM, 68(4):27:1-27:40, 2021. URL: https://doi.org/10.1145/3457389.
  28. Pawel Gawrychowski, Adam Karczmarz, Tomasz Kociumaka, Jakub Lacki, and Piotr Sankowski. Optimal dynamic strings. In Artur Czumaj, editor, Proceedings of the 2018 ACM-SIAM Symposium on Discrete Algorithms (SODA 2018), pages 1509-1528. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.99.
  29. Genomics England. The 100,000 Genomes Project. URL: https://www.genomicsengland.co.uk/about-genomics-england/the-100000-genomes-project/.
  30. Alexander Golynski, J. Ian Munro, and S. Srinivasa Rao. Rank/select operations on large alphabets: A tool for text indexing. In Proceedings of the 2006 ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pages 368-373. ACM Press, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109599.
  31. Rodrigo González and Gonzalo Navarro. Rank/select on dynamic compressed sequences and applications. Theoretical Computer Science, 410(43):4414-4422, 2009. URL: https://doi.org/10.1016/j.tcs.2009.07.022.
  32. Roberto Grossi, Ankur Gupta, and Jeffrey Scott Vitter. High-order entropy-compressed text indexes. In Proceedings of the 2003 ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), pages 841-850. ACM/SIAM, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644250.
  33. Roberto Grossi, Ankur Gupta, and Jeffrey Scott Vitter. High-order entropy-compressed text indexes. In Proceedings of the 2003 ACM-SIAM Symposium on Discrete Algorithms, SODA 2003, pages 841-850, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644250.
  34. Roberto Grossi and Jeffrey Scott Vitter. Compressed suffix arrays and suffix trees with applications to text indexing and string matching (extended abstract). In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC 2000), pages 397-406. ACM, 2000. URL: https://doi.org/10.1145/335305.335351.
  35. Dan Gusfield. Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology. Cambridge University Press, 1997. URL: https://doi.org/10.1017/cbo9780511574931.
  36. Torben Hagerup. Sorting and searching on the word RAM. In Michel Morvan, Christoph Meinel, and Daniel Krob, editors, Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science (STACS 1998), volume 1373 of LNCS, pages 366-398. Springer, 1998. URL: https://doi.org/10.1007/BFb0028575.
  37. Tomohiro I. Longest common extensions with recompression. In Juha Kärkkäinen, Jakub Radoszewski, and Wojciech Rytter, editors, Proceedings of the 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017), pages 18:1-18:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CPM.2017.18.
  38. Artur Jeż. A really simple approximation of smallest grammar. Theoretical Computer Science, 616:141-150, 2016. URL: https://doi.org/10.1016/j.tcs.2015.12.032.
  39. Haim Kaplan, Natan Rubin, Micha Sharir, and Elad Verbin. Counting colors in boxes. In Nikhil Bansal, Kirk Pruhs, and Clifford Stein, editors, Proceedings of the 2007 ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pages 785-794. SIAM, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283467.
  40. Toru Kasai, Gunho Lee, Hiroki Arimura, Setsuo Arikawa, and Kunsoo Park. Linear-time longest-common-prefix computation in suffix arrays and its applications. In Proceedings of the 18th Annual Symposium on Combinatorial Pattern Matching (CPM 2001), pages 181-192, 2001. URL: https://doi.org/10.1007/3-540-48194-X_17.
  41. Dominik Kempa and Tomasz Kociumaka. String synchronizing sets: Sublinear-time BWT construction and optimal LCE data structure. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019), pages 756-767. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316368.
  42. Dominik Kempa and Tomasz Kociumaka. Resolution of the Burrows-Wheeler transform conjecture. In Sandy Irani, editor, Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2020), pages 1002-1013. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00097.
  43. Dominik Kempa and Tomasz Kociumaka. Collapsing the hierarchy of compressed data structures: Suffix arrays in optimal compressed space. In Proceedings of the 64th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2023), pages 1877-1886. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00114.
  44. Dominik Kempa and Tomasz Kociumaka. Tight lower bounds for central string queries in compressed space. In Kasper Green Larsen and Barna Saha, editors, Proceedings of the 2026 ACM-SIAM Symposium on Discrete Algorithms (SODA 2026), pages 1824-1840. SIAM, 2026. URL: https://doi.org/10.1137/1.9781611978971.65.
  45. Dominik Kempa and Nicola Prezza. At the roots of dictionary compression: String attractors. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018), pages 827-840. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188814.
  46. Dominik Kempa and Barna Saha. An upper bound and linear-space queries on the LZ-End parsing. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms (SODA 2022), pages 2847-2866. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.111.
  47. Tomasz Kociumaka, Gonzalo Navarro, and Francisco Olivares. Near-optimal search time in δ-optimal space. In Armando Castañeda and Francisco Rodríguez-Henríquez, editors, Proceedings of the 15th Latin American Symposium on Theoretical Informatics (LATIN 2022), pages 88-103. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-20624-5_6.
  48. Tomasz Kociumaka, Gonzalo Navarro, and Nicola Prezza. Toward a definitive compressibility measure for repetitive sequences. IEEE Transactions on Information Theory, 69(4):2074-2092, 2023. URL: https://doi.org/10.1109/TIT.2022.3224382.
  49. Veli Mäkinen. Compact suffix array: a space-efficient full-text index. Fundamenta Informaticae, 56(1,2):191-210, October 2002. URL: http://content.iospress.com/articles/fundamenta-informaticae/fi56-1-2-12.
  50. Veli Mäkinen, Djamal Belazzougui, Fabio Cunial, and Alexandru I. Tomescu. Genome-Scale Algorithm Design: Bioinformatics in the Era of High-Throughput Sequencing (2nd edition). Cambridge University Press, 2023. URL: http://www.genome-scale.info/.
  51. Veli Mäkinen and Gonzalo Navarro. Position-restricted substring searching. In José R. Correa, Alejandro Hevia, and Marcos A. Kiwi, editors, Proceedings of the 7th Latin American Symposium on Theoretical Informatics (LATIN 2006), pages 703-714. Springer, 2006. URL: https://doi.org/10.1007/11682462_64.
  52. Udi Manber and Eugene W. Myers. Suffix arrays: A new method for on-line string searches. SIAM Journal on Computing, 22(5):935-948, 1993. URL: https://doi.org/10.1137/0222058.
  53. J. Ian Munro, Gonzalo Navarro, and Yakov Nekrich. Text indexing and searching in sublinear time. In Proceedings of the 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020), pages 24:1-24:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CPM.2020.24.
  54. Gonzalo Navarro. Wavelet trees for all. Journal of Discrete Algorithms, 25:2-20, 2014. URL: https://doi.org/10.1016/j.jda.2013.07.004.
  55. Gonzalo Navarro. Compact data structures: A practical approach. Cambridge University Press, Cambridge, UK, 2016. URL: https://doi.org/10.1017/cbo9781316588284.
  56. Gonzalo Navarro. Indexing highly repetitive string collections, part I: Repetitiveness measures. ACM Computing Surveys, 54(2):29:1-29:31, 2021. URL: https://doi.org/10.1145/3434399.
  57. Gonzalo Navarro. Indexing highly repetitive string collections, part II: Compressed indexes. ACM Computing Surveys, 54(2):26:1-26:32, 2021. URL: https://doi.org/10.1145/3432999.
  58. Takaaki Nishimoto, Tomohiro I, Shunsuke Inenaga, Hideo Bannai, and Masayuki Takeda. Fully dynamic data structure for LCE queries in compressed space. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, Proceedings of the 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), pages 72:1-72:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.MFCS.2016.72.
  59. Enno Ohlebusch. Bioinformatics Algorithms: Sequence Analysis, Genome Rearrangements, and Phylogenetic Reconstruction. Oldenbusch Verlag, 2013. URL: http://www.oldenbusch-verlag.de/.
  60. Nicola Prezza. Optimal rank and select queries on dictionary-compressed text. In Nadia Pisanti and Solon P. Pissis, editors, Proceedings of the 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019), pages 4:1-4:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CPM.2019.4.
  61. Wojciech Rytter. Application of Lempel-Ziv factorization to the approximation of grammar-based compression. Theoretical Computer Science, 302(1-3):211-222, 2003. URL: https://doi.org/10.1016/S0304-3975(02)00777-6.
  62. Elad Verbin and Wei Yu. Data structure lower bounds on random access to grammar-compressed strings. In Johannes Fischer and Peter Sanders, editors, Proceedings of the 24th Annual Symposium on Combinatorial Pattern Matching (CPM 2013), pages 247-258. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-38905-4_24.
  63. Peter Weiner. Linear pattern matching algorithms. In Proceedings of the 14th Annual Symposium on Switching and Automata Theory (SWAT/FOCS 1973), pages 1-11. IEEE Computer Society, 1973. URL: https://doi.org/10.1109/SWAT.1973.13.
  64. Ryan Williams. The orthogonal vectors conjecture and non-uniform circuit lower bounds. In Proceedings of the 65th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2024), pages 1372-1387. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00088.
  65. Jacob Ziv and Abraham Lempel. A universal algorithm for sequential data compression. IEEE Transactions on Information Theory, 23(3):337-343, 1977. URL: https://doi.org/10.1109/TIT.1977.1055714.
  66. Jacob Ziv and Abraham Lempel. Compression of individual sequences via variable-rate coding. IEEE Transactions on Information Theory, 24(5):530-536, 1978. URL: https://doi.org/10.1109/TIT.1978.1055934.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact