Document Open Access Logo

The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds

Authors Karl Bringmann, Allan Grønlund, Marvin Künnemann, Kasper Green Larsen

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2024.22.pdf
  • Filesize: 0.77 MB
  • 25 pages

Document Identifiers

Author Details

Karl Bringmann
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Allan Grønlund
  • Aarhus University, Denmark
  • Kvantify, Aarhus, Denmark
Marvin Künnemann
  • Karlsruhe Institute of Technology, Germany
Kasper Green Larsen
  • Aarhus University, Denmark

Funding

  • Bringmann, Karl: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).
  • Künnemann, Marvin: Research partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 462679611.
  • Larsen, Kasper Green: Supported by Independent Research Fund Denmark (DFF) Sapere Aude Research Leader grant No 9064-00068B.

Acknowledgements

We thank Virginia Vassilevska Williams, Thatchaphol Saranurak, and Rupak Majumdar for helpful discussions on the NFA hypothesis. We also thank Thatchaphol Saranurak for popularizing this hypothesis by tweeting the open problem on Colored Walk and its implication for OMv, crediting one of the authors of this paper.

Cite AsGet BibTex

Karl Bringmann, Allan Grønlund, Marvin Künnemann, and Kasper Green Larsen. The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 22:1-22:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.22

Abstract

We pose the fine-grained hardness hypothesis that the textbook algorithm for the NFA Acceptance problem is optimal up to subpolynomial factors, even for dense NFAs and fixed alphabets. We show that this barrier appears in many variations throughout the algorithmic literature by introducing a framework of Colored Walk problems. These yield fine-grained equivalent formulations of the NFA Acceptance problem as problems concerning detection of an s-t-walk with a prescribed color sequence in a given edge- or node-colored graph. For NFA Acceptance on sparse NFAs (or equivalently, Colored Walk in sparse graphs), a tight lower bound under the Strong Exponential Time Hypothesis has been rediscovered several times in recent years. We show that our hardness hypothesis, which concerns dense NFAs, has several interesting implications: - It gives a tight lower bound for Context-Free Language Reachability. This proves conditional optimality for the class of 2NPDA-complete problems, explaining the cubic bottleneck of interprocedural program analysis. - It gives a tight (n+nm^{1/3})^{1-o(1)} lower bound for the Word Break problem on strings of length n and dictionaries of total size m. - It implies the popular OMv hypothesis. Since the NFA acceptance problem is a static (i.e., non-dynamic) problem, this provides a static reason for the hardness of many dynamic problems. Thus, a proof of the NFA Acceptance hypothesis would resolve several interesting barriers. Conversely, a refutation of the NFA Acceptance hypothesis may lead the way to attacking the current barriers observed for Context-Free Language Reachability, the Word Break problem and the growing list of dynamic problems proven hard under the OMv hypothesis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Fine-grained complexity theory
  • non-deterministic finite automata

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Amir Abboud, Arturs Backurs, Karl Bringmann, and Marvin Künnemann. Fine-grained complexity of analyzing compressed data: Quantifying improvements over decompress-and-solve. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 192-203. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.26.
  2. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the current clique algorithms are optimal, so is valiant’s parser. In 56th Annual IEEE Symposium on Foundations of Computer Science, 2015. Google Scholar
  3. Amir Abboud and Karl Bringmann. Tighter connections between formula-sat and shaving logs. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 8:1-8:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.8.
  4. Amir Abboud and Søren Dahlgaard. Popular conjectures as a barrier for dynamic planar graph algorithms. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 477-486. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.58.
  5. Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. Time and tape complexity of pushdown automaton languages. Inf. Control., 13(3):186-206, 1968. URL: https://doi.org/10.1016/S0019-9958(68)91087-5.
  6. Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster matrix multiplication. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 522-539. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.32.
  7. Amihood Amir, Moshe Lewenstein, and Noa Lewenstein. Pattern matching in hypertext. J. Algorithms, 35(1):82-99, 2000. URL: https://doi.org/10.1006/jagm.1999.1063.
  8. Marcelo Arenas, Luis Alberto Croquevielle, Rajesh Jayaram, and Cristian Riveros. Counting the answers to a query. SIGMOD Rec., 51(3):6-17, 2022. URL: https://doi.org/10.1145/3572751.3572753.
  9. Arturs Backurs, Nishanth Dikkala, and Christos Tzamos. Tight hardness results for maximum weight rectangles. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 81:1-81:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.81.
  10. Arturs Backurs and Piotr Indyk. Which regular expression patterns are hard to match? In 57th Annual IEEE Symposium on Foundations of Computer Science, 2016. Google Scholar
  11. Arturs Backurs and Christos Tzamos. Improving viterbi is hard: Better runtimes imply faster clique algorithms. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, volume 70 of Proceedings of Machine Learning Research, pages 311-321. PMLR, 2017. URL: http://proceedings.mlr.press/v70/backurs17a.html.
  12. Pablo Barceló. Querying graph databases. In Richard Hull and Wenfei Fan, editors, Proceedings of the 32nd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2013, New York, NY, USA - June 22 - 27, 2013, pages 175-188. ACM, 2013. URL: https://doi.org/10.1145/2463664.2465216.
  13. Christoph Berkholz, Jens Keppeler, and Nicole Schweikardt. Answering conjunctive queries under updates. In Emanuel Sallinger, Jan Van den Bussche, and Floris Geerts, editors, Proceedings of the 36th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2017, Chicago, IL, USA, May 14-19, 2017, pages 303-318. ACM, 2017. URL: https://doi.org/10.1145/3034786.3034789.
  14. Karl Bringmann, Allan Grønlund, and Kasper Green Larsen. A dichotomy for regular expression membership testing. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 307-318. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.36.
  15. Marco L. Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mihajlin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In Madhu Sudan, editor, Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 261-270. ACM, 2016. URL: https://doi.org/10.1145/2840728.2840746.
  16. Katrin Casel and Markus L. Schmid. Fine-grained complexity of regular path queries. In Ke Yi and Zhewei Wei, editors, 24th International Conference on Database Theory, ICDT 2021, March 23-26, 2021, Nicosia, Cyprus, volume 186 of LIPIcs, pages 19:1-19:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICDT.2021.19.
  17. Timothy M. Chan, Virginia Vassilevska Williams, and Yinzhan Xu. Algorithms, reductions and equivalences for small weight variants of all-pairs shortest paths. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 47:1-47:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.47.
  18. Krishnendu Chatterjee, Bhavya Choudhary, and Andreas Pavlogiannis. Optimal dyck reachability for data-dependence and alias analysis. Proc. ACM Program. Lang., 2(POPL):30:1-30:30, 2018. URL: https://doi.org/10.1145/3158118.
  19. Swarat Chaudhuri. Subcubic algorithms for recursive state machines. In George C. Necula and Philip Wadler, editors, Proceedings of the 35th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2008, San Francisco, California, USA, January 7-12, 2008, pages 159-169. ACM, 2008. URL: https://doi.org/10.1145/1328438.1328460.
  20. Dmitry Chistikov, Rupak Majumdar, and Philipp Schepper. Subcubic certificates for CFL reachability. Proc. ACM Program. Lang., 6(POPL):1-29, 2022. URL: https://doi.org/10.1145/3498702.
  21. Raphaël Clifford, Pawel Gawrychowski, Tomasz Kociumaka, Daniel P. Martin, and Przemyslaw Uznanski. The dynamic k-mismatch problem. In Hideo Bannai and Jan Holub, editors, 33rd Annual Symposium on Combinatorial Pattern Matching, CPM 2022, June 27-29, 2022, Prague, Czech Republic, volume 223 of LIPIcs, pages 18:1-18:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CPM.2022.18.
  22. Raphaël Clifford, Allan Grønlund, Kasper Green Larsen, and Tatiana Starikovskaya. Upper and lower bounds for dynamic data structures on strings. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, volume 96 of LIPIcs, pages 22:1-22:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.22.
  23. Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. Journal on Symbolic Computation, 9(3):251-280, 1990. URL: https://doi.org/10.1016/S0747-7171(08)80013-2.
  24. Artur Czumaj and Andrzej Lingas. Finding a heaviest vertex-weighted triangle is not harder than matrix multiplication. SIAM J. Comput., 39(2):431-444, 2009. URL: https://doi.org/10.1137/070695149.
  25. Søren Dahlgaard. On the hardness of partially dynamic graph problems and connections to diameter. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 48:1-48:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.48.
  26. Justin Dallant and John Iacono. Conditional lower bounds for dynamic geometric measure problems. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pages 39:1-39:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.39.
  27. Massimo Equi, Roberto Grossi, Veli Mäkinen, and Alexandru I. Tomescu. On the complexity of string matching for graphs. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 55:1-55:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.55.
  28. Pawel Gawrychowski and Przemyslaw Uznanski. Towards unified approximate pattern matching for hamming and l_1 distance. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 62:1-62:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.62.
  29. Daniel Gibney, Gary Hoppenworth, and Sharma V. Thankachan. Simple reductions from formula-sat to pattern matching on labeled graphs and subtree isomorphism. In Hung Viet Le and Valerie King, editors, 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 232-242. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.26.
  30. Daniel Gibney and Sharma V. Thankachan. Text indexing for regular expression matching. Algorithms, 14(5):133, 2021. URL: https://doi.org/10.3390/a14050133.
  31. Nevin Heintze and David A. McAllester. On the cubic bottleneck in subtyping and flow analysis. In Proceedings, 12th Annual IEEE Symposium on Logic in Computer Science, Warsaw, Poland, June 29 - July 2, 1997, pages 342-351. IEEE Computer Society, 1997. URL: https://doi.org/10.1109/LICS.1997.614960.
  32. Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 21-30. ACM, 2015. URL: https://doi.org/10.1145/2746539.2746609.
  33. Monika Henzinger and Xiaowei Wu. Upper and lower bounds for fully retroactive graph problems. In Anna Lubiw and Mohammad R. Salavatipour, editors, Algorithms and Data Structures - 17th International Symposium, WADS 2021, Virtual Event, August 9-11, 2021, Proceedings, volume 12808 of Lecture Notes in Computer Science, pages 471-484. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-83508-8_34.
  34. Jianyu Huang, Tyler M. Smith, Greg M. Henry, and Robert A. van de Geijn. Strassen’s algorithm reloaded. In John West and Cherri M. Pancake, editors, Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC 2016, Salt Lake City, UT, USA, November 13-18, 2016, pages 690-701. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/SC.2016.58.
  35. Chirag Jain, Haowen Zhang, Yu Gao, and Srinivas Aluru. On the complexity of sequence-to-graph alignment. J. Comput. Biol., 27(4):640-654, 2020. URL: https://doi.org/10.1089/cmb.2019.0066.
  36. Shunhua Jiang, Binghui Peng, and Omri Weinstein. Dynamic least-squares regression. CoRR, abs/2201.00228, 2022. URL: https://arxiv.org/abs/2201.00228.
  37. Ahmet Kara, Milos Nikolic, Dan Olteanu, and Haozhe Zhang. Conjunctive queries with free access patterns under updates. In Floris Geerts and Brecht Vandevoort, editors, 26th International Conference on Database Theory, ICDT 2023, March 28-31, 2023, Ioannina, Greece, volume 255 of LIPIcs, pages 17:1-17:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ICDT.2023.17.
  38. Dominik Kempa and Tomasz Kociumaka. Dynamic suffix array with polylogarithmic queries and updates. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 1657-1670. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520061.
  39. Paraschos Koutris and Shaleen Deep. The fine-grained complexity of CFL reachability. Proc. ACM Program. Lang., 7(POPL):1713-1739, 2023. URL: https://doi.org/10.1145/3571252.
  40. Marvin Künnemann. A tight (non-combinatorial) conditional lower bound for klee’s measure problem in 3d. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 555-566. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00059.
  41. Kasper Green Larsen and R. Ryan Williams. Faster online matrix-vector multiplication. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2182-2189. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.142.
  42. Joshua Lau and Angus Ritossa. Algorithms and hardness for multidimensional range updates and queries. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 35:1-35:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.35.
  43. François Le Gall. Powers of tensors and fast matrix multiplication. In Proc. 39th International Symposium on Symbolic and Algebraic Computation (ISSAC'14), pages 296-303, 2014. URL: https://doi.org/10.1145/2608628.2608664.
  44. Lillian Lee. Fast context-free grammar parsing requires fast boolean matrix multiplication. J. ACM, 49(1):1-15, 2002. URL: https://doi.org/10.1145/505241.505242.
  45. Andrea Lincoln, Adam Polak, and Virginia Vassilevska Williams. Monochromatic triangles, intermediate matrix products, and convolutions. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 53:1-53:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.53.
  46. Andrea Lincoln, Virginia Vassilevska Williams, and R. Ryan Williams. Tight hardness for shortest cycles and paths in sparse graphs. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1236-1252. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.80.
  47. Udi Manber and Sun Wu. Approximate string matching with arbitrary costs for text and hypertext. In Proc. International Workshop on Advances in Structural and Syntactic Pattern Recognition, pages 22-33, 1992. URL: https://doi.org/10.1142/9789812797919_0002.
  48. David Melski and Thomas W. Reps. Interconvertibility of a class of set constraints and context-free-language reachability. Theor. Comput. Sci., 248(1-2):29-98, 2000. URL: https://doi.org/10.1016/S0304-3975(00)00049-9.
  49. Alberto O. Mendelzon and Peter T. Wood. Finding regular simple paths in graph databases. SIAM J. Comput., 24(6):1235-1258, 1995. URL: https://doi.org/10.1137/S009753979122370X.
  50. Gonzalo Navarro. Improved approximate pattern matching on hypertext. Theor. Comput. Sci., 237(1-2):455-463, 2000. URL: https://doi.org/10.1016/S0304-3975(99)00333-3.
  51. Radford Neal. The computational complexity of taxonomic inference. Unpublished manuscript, 1989. Google Scholar
  52. Jaroslav Nešetřil and Svatopluk Poljak. On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae, 026(2):415-419, 1985. URL: http://eudml.org/doc/17394.
  53. Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 603-610. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806772.
  54. Aaron Potechin and Jeffrey O. Shallit. Lengths of words accepted by nondeterministic finite automata. Inf. Process. Lett., 162:105993, 2020. URL: https://doi.org/10.1016/j.ipl.2020.105993.
  55. Wojciech Rytter. Fast recognition of pushdown automaton and context-free languages. Inf. Control., 67(1-3):12-22, 1985. URL: https://doi.org/10.1016/S0019-9958(85)80024-3.
  56. Philipp Schepper. Fine-grained complexity of regular expression pattern matching and membership. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 80:1-80:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.80.
  57. Volker Strassen. Gaussian elimination is not optimal. Numerische Mathematik, 13(4):354-356, August 1969. URL: https://doi.org/10.1007/BF02165411.
  58. Leslie G. Valiant. General context-free recognition in less than cubic time. J. Comput. Syst. Sci., 10(2):308-315, 1975. URL: https://doi.org/10.1016/S0022-0000(75)80046-8.
  59. Jan van den Brand, Danupon Nanongkai, and Thatchaphol Saranurak. Dynamic matrix inverse: Improved algorithms and matching conditional lower bounds. In David Zuckerman, editor, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, Maryland, USA, November 9-12, 2019, pages 456-480. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00036.
  60. Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proc. 44th Annual ACM Symposium on Theory of Computing Conference (STOC'12), pages 887-898, 2012. URL: https://doi.org/10.1145/2213977.2214056.
  61. Virginia Vassilevska Williams. Fine-grained algorithms and complexity. In Proc. 21st International Conference on Database Theory (ICDT'18), pages 1:1-1:1, 2018. URL: https://doi.org/10.4230/LIPIcs.ICDT.2018.1.
  62. Virginia Vassilevska Williams and Yinzhan Xu. Monochromatic triangles, triangle listing and APSP. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 786-797. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00078.
  63. Andrew J. Viterbi. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans. Inf. Theory, 13(2):260-269, 1967. URL: https://doi.org/10.1109/TIT.1967.1054010.
  64. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1-27:38, 2018. URL: https://doi.org/10.1145/3186893.
  65. Mihalis Yannakakis. Graph-theoretic methods in database theory. In Daniel J. Rosenkrantz and Yehoshua Sagiv, editors, Proceedings of the Ninth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, April 2-4, 1990, Nashville, Tennessee, USA, pages 230-242. ACM Press, 1990. URL: https://doi.org/10.1145/298514.298576.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact