Document Open Access Logo

Frontier Space-Time Algorithms Using Only Full Memory

Authors Petr Chmel , Aditi Dudeja , Michal Koucký , Ian Mertz , Ninad Rajgopal

PDF

File

Thumbnail PDF
PDF
LIPIcs.CCC.2026.29.pdf
  • Filesize: 0.88 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Complexity classes
Keywords
  • Catalytic computation
  • Connectivity
  • Edit distance
  • Discrete Fréchet distance

Metrics

Abstract

We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only 𝒪(log(n)) workspace, and use sublinear catalytic space matching the best-known space bounds of non-catalytic algorithms running in polynomial time.
First, we design a polynomial time algorithm for directed s-t connectivity using n / 2^{Θ(√{log n})} catalytic space, which matches the state-of-the-art time-space bounds in the non-catalytic setting [Barnes et al., 1998], and improves the catalytic space usage of the best known algorithm [James Cook and Edward Pyne, 2026]. Furthermore, using only 𝒪(log(n)) random bits we get a randomized algorithm whose running time nearly matches the fastest time bounds known for space-unrestricted algorithms.
Second, we design polynomial time algorithms for the problems of computing Edit Distance, Longest Common Subsequence, and the Discrete Fréchet Distance, again using n / 2^{Θ(√{log n})} catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].

Cite As Get BibTex

Petr Chmel, Aditi Dudeja, Michal Koucký, Ian Mertz, and Ninad Rajgopal. Frontier Space-Time Algorithms Using Only Full Memory. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 29:1-29:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.29

Author Details

Petr Chmel
  • Computer Science Institute of Charles University, Prague, Czech Republic
Aditi Dudeja
  • The School of Data Science, The Chinese University of Hong Kong, Shenzhen, China
Michal Koucký
  • Computer Science Institute of Charles University, Prague, Czech Republic
Ian Mertz
  • Computer Science Institute of Charles University, Prague, Czech Republic
Ninad Rajgopal
  • Computer Science Institute of Charles University, Prague, Czech Republic

Funding

Michal Koucký, Ian Mertz, Ninad Rajgopal: Partially supported by the Grant Agency of the Czech Republic under the grant agreement no. 24-10306S and by Charles Univ. project UNCE 24/SCI/008.
  • Chmel, Petr: Partially supported by the Grant Agency of the Czech Republic under the grant agreement no. 24-10306S, by GAUK project No. 70924 and by Charles Univ. project UNCE 24/SCI/008.
  • Dudeja, Aditi: This work was initiated while the author was affiliated with the University of Salzburg. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 947702).

References

  1. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015), pages 59-78, 2015. URL: https://doi.org/10.1109/FOCS.2015.14.
  2. Amir Abboud, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Ryan Williams. Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made. In 48th Annual ACM Symposium on Theory of Computing (STOC 2016), pages 375-388, 2016. URL: https://doi.org/10.1145/2897518.2897653.
  3. Amir Abboud and Aviad Rubinstein. Fast and deterministic constant factor approximation algorithms for LCS imply new circuit lower bounds. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of Leibniz International Proceedings in Informatics (LIPIcs), pages 35:1-35:14, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.35.
  4. Pankaj K. Agarwal, Rinat Ben Avraham, Haim Kaplan, and Micha Sharir. Computing the discrete Fréchet distance in subquadratic time. SIAM Journal on Computing, 43(2):429-449, 2014. URL: https://doi.org/10.1137/130920526.
  5. Aryan Agarwala, Yaroslav Alekseev, and Antoine Vinciguerra. Linear matroid intersection is in catalytic logspace. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026), volume 362 of Leibniz International Proceedings in Informatics (LIPIcs), pages 3:1-3:23, 2026. URL: https://doi.org/10.4230/LIPIcs.ITCS.2026.3.
  6. Aryan Agarwala and Ian Mertz. Bipartite matching is in catalytic logspace. In 66th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2025), 2025. Google Scholar
  7. Miklós Ajtai, János Komlós, and Endre Szemerédi. Deterministic simulation in LOGSPACE. In 19th Annual ACM Symposium on Theory of Computing (STOC 1987), pages 132-140, 1987. URL: https://doi.org/10.1145/28395.28410.
  8. Yaroslav Alekseev, Yuval Filmus, Ian Mertz, Alexander Smal, and Antoine Vinciguerra. Catalytic computing and register programs beyond log-depth. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025), volume 345 of Leibniz International Proceedings in Informatics (LIPIcs), pages 6:1-6:18, 2025. URL: https://doi.org/10.4230/LIPIcs.MFCS.2025.6.
  9. Eric Allender, David A. Mix Barrington, Tanmoy Chakraborty, Samir Datta, and Sambuddha Roy. Grid graph reachability problems. In 21st Annual IEEE Conference on Computational Complexity (CCC 2006), pages 299-313, 2006. URL: https://doi.org/10.1109/CCC.2006.22.
  10. Eric Allender and Klaus-Jörn Lange. RUSPACE(log n) ⊆ DSPACE(log² n / log log n). Theory of Computing Systems, 31(5):539-550, 1998. URL: https://doi.org/10.1007/S002240000102.
  11. Carlos E. R. Alves, Edson N. Caceres, and Siang Wun Song. A coarse-grained parallel algorithm for the all-substrings longest common subsequence problem. Algorithmica, 45(3):301-335, 2006. URL: https://doi.org/10.1007/s00453-006-1216-z.
  12. Alexandr Andoni, Michel Deza, Anupam Gupta, Piotr Indyk, and Sofya Raskhodnikova. Lower bounds for embedding edit distance into normed spaces. In 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), pages 523-526, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644196.
  13. Alexandr Andoni, Assaf Goldberger, Andrew McGregor, and Ely Porat. Homomorphic fingerprints under misalignments: Sketching edit and shift distances. In 45th Annual ACM Symposium on Theory of Computing (STOC 2013), pages 931-940, 2013. URL: https://doi.org/10.1145/2488608.2488726.
  14. Alexandr Andoni and Robert Krauthgamer. The computational hardness of estimating edit distance [Extended abstract]. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), pages 667-676, 2007. URL: https://doi.org/10.1109/FOCS.2007.60.
  15. Alexandr Andoni and Robert Krauthgamer. The smoothed complexity of edit distance. In 35th International Colloquium on Automata, Languages, and Programming (ICALP 2008), volume 5125 of Lecture Notes in Computer Science, pages 357-369, 2008. URL: https://doi.org/10.1007/978-3-540-70575-8_30.
  16. Alexandr Andoni, Robert Krauthgamer, and Krzysztof Onak. Polylogarithmic approximation for edit distance and the asymmetric query complexity. In 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pages 377-386, 2010. URL: https://doi.org/10.1109/FOCS.2010.43.
  17. Alexandr Andoni and Krzysztof Onak. Approximating edit distance in near-linear time. In 41st Annual ACM Symposium on Theory of Computing (STOC 2009), pages 199-204, 2009. URL: https://doi.org/10.1145/1536414.1536444.
  18. Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In 47th Annual ACM Symposium on Theory of Computing (STOC 2015), pages 51-58, 2015. URL: https://doi.org/10.1145/2746539.2746612.
  19. Nikhil Bansal, Moshe Lewenstein, Bin Ma, and Kaizhong Zhang. On the longest common rigid subsequence problem. Algorithmica, 56(2):270-280, 2010. URL: https://doi.org/10.1007/s00453-008-9175-1.
  20. Greg Barnes, Jonathan F. Buss, Walter L. Ruzzo, and Baruch Schieber. A sublinear space, polynomial time algorithm for directed st-connectivity. SIAM Journal on Computing, 27(5):1273-1282, 1998. URL: https://doi.org/10.1137/S0097539793283151.
  21. Tuğkan Batu, Funda Ergun, and Cenk Sahinalp. Oblivious string embeddings and edit distance approximations. In 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pages 792-801, 2006. URL: https://dl.acm.org/doi/10.5555/1109557.1109644.
  22. Michael Ben-Or and Richard Cleve. Computing algebraic formulas using a constant number of registers. SIAM Journal on Computing, 21(1):54-58, 1992. URL: https://doi.org/10.1137/0221006.
  23. Sagar Bisoyi, Krishnamoorthy Dinesh, and Jayalal Sarma. On pure space vs catalytic space. Theoretical Computer Science, 921:112-126, 2022. URL: https://doi.org/10.1016/J.TCS.2022.04.005.
  24. Sagar Bisoyi, Krishnamoothy Dinesh, Bhabya Deep Rai, and Jayalal Sarma. Almost-catalytic computation. In 14th International Conference on Algorithms and Complexity (CIAC 2025), volume 15680 of Lecture Notes in Computer Science, pages 35-51, 2025. URL: https://doi.org/10.1007/978-3-031-92935-9_3.
  25. Mahdi Boroujeni, Soheil Ehsani, Mohammad Ghodsi, MohammadTaghi HajiAghayi, and Saeed Seddighin. Approximating edit distance in truly subquadratic time: Quantum and MapReduce. In 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018), 2018. URL: https://doi.org/10.1137/1.9781611975031.76.
  26. Mahdi Boroujeni and Saeed Seddighin. Improved MPC algorithms for edit distance and ulam distance. In 31st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2019), pages 31-40, 2019. URL: https://doi.org/10.1145/3323165.3323205.
  27. Karl Bringmann, Fabrizio Grandoni, Barna Saha, and Virginia Vassilevska Williams. Truly sub-cubic algorithms for language edit distance and RNA-folding via fast bounded-difference min-plus product. In 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016), pages 375-384, 2016. URL: https://doi.org/10.1109/FOCS.2016.48.
  28. Karl Bringmann and Marvin Künnemann. Quadratic conditional lower bounds for string problems and dynamic time warping. In 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015), pages 79-97, 2015. URL: https://doi.org/10.1109/FOCS.2015.15.
  29. Karl Bringmann and Marvin Künnemann. Multivariate fine-grained complexity of longest common subsequence. In 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018), pages 1216-1235, 2018. URL: https://doi.org/10.1137/1.9781611975031.79.
  30. Kevin Buchin, Maike Buchin, and Joachim Gudmundsson. Detecting single file movement. In 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (GIS 2008), pages 1-10, 2008. URL: https://doi.org/10.1145/1463434.1463476.
  31. Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Maarten Löffler, and Jun Luo. Detecting commuting patterns by clustering subtrajectories. International Journal of Computational Geometry & Applications, 21(3):253-282, 2011. URL: https://doi.org/10.1142/S0218195911003652.
  32. Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff, and Florian Speelman. Computing with a full memory: Catalytic space. In 46th Annual ACM Symposium on Theory of Computing (STOC 2014), pages 857-866, 2014. URL: https://doi.org/10.1145/2591796.2591874.
  33. Harry Buhrman, Marten Folkertsma, Ian Mertz, Florian Speelman, Sergii Strelchuk, Sathyawageeswar Subramanian, and Quinten Tupker. Quantum catalytic space. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025), volume 350 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:24, 2025. URL: https://doi.org/10.4230/LIPIcs.TQC.2025.11.
  34. Harry Buhrman, Michal Koucký, Bruno Loff, and Florian Speelman. Catalytic space: Non-determinism and hierarchy. Theory of Computing Systems, 62(1):116-135, 2018. URL: https://doi.org/10.1007/S00224-017-9784-7.
  35. Diptarka Chakraborty, Elazar Goldenberg, and Michal Koucký. Streaming algorithms for embedding and computing edit distance in the low distance regime. In 48th Annual ACM Symposium on Theory of Computing (STOC 2016), pages 712-725, 2016. URL: https://doi.org/10.1145/2897518.2897577.
  36. Diptarka Chakraborty and Raghunath Tewari. An O(n^ε) space and polynomial time algorithm for reachability in directed layered planar graphs. ACM Transactions on Computation Theory, 9(4):1-11, 2017. URL: https://doi.org/10.1145/3154857.
  37. Kuan Cheng, Alireza Farhadi, MohammadTaghi Hajiaghayi, Zhengzhong Jin, Xin Li, Aviad Rubinstein, Saeed Seddighin, and Yu Zheng. Streaming and small space approximation algorithms for edit distance and longest common subsequence. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 54:1-54:20, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.54.
  38. James Cook, Surendra Ghentiyala, Ian Mertz, Edward Pyne, and Nathan S. Sheffield. The structure of in-place space-bounded computation. arXiv preprint, arXiv:2510.12005, 2025. URL: https://doi.org/10.48550/arXiv.2510.12005.
  39. James Cook, Jiatu Li, Ian Mertz, and Edward Pyne. The structure of catalytic space: Capturing randomness and time via compression. In 57th Annual ACM Symposium on Theory of Computing (STOC 2025), pages 554-564, 2025. URL: https://doi.org/10.1145/3717823.3718112.
  40. James Cook and Ian Mertz. Catalytic approaches to the tree evaluation problem. In 52nd Annual ACM Symposium on Theory of Computing (STOC 2020), pages 752-760, 2020. URL: https://doi.org/10.1145/3357713.3384316.
  41. James Cook and Ian Mertz. Trading time and space in catalytic branching programs. In 37th Computational Complexity Conference (CCC 2022), volume 234 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:21, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.8.
  42. James Cook and Ian Mertz. Tree evaluation is in space O(log n ⋅ log log n). In 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 1268-1278, 2024. URL: https://doi.org/10.1145/3618260.3649664.
  43. James Cook and Edward Pyne. Efficient catalytic graph algorithms. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026), volume 362 of Leibniz International Proceedings in Informatics (LIPIcs), pages 43:1-43:22, 2026. URL: https://doi.org/10.4230/LIPIcs.ITCS.2026.43.
  44. Samir Datta, Chetan Gupta, Rahul Jain, Vimal Raj Sharma, and Raghunath Tewari. Randomized and symmetric catalytic computation. In 15th International Computer Science Symposium in Russia (CSR 2020), volume 12159 of Lecture Notes in Computer Science, pages 211-223, 2020. URL: https://doi.org/10.1007/978-3-030-50026-9_15.
  45. Dean Doron, Edward Pyne, and Roei Tell. Opening up the distinguisher: A hardness to randomness approach for BPL=L that uses properties of BPL. In 56th Annual ACM Symposium on Theory of Computing (STOC 2024), pages 2039-2049, 2024. URL: https://doi.org/10.1145/3618260.3649772.
  46. Dean Doron, Edward Pyne, Roei Tell, and R. Ryan Williams. When connectivity is hard, random walks are easy with non-determinism. In 57th Annual ACM Symposium on Theory of Computing (STOC 2025), pages 1108-1117, 2025. URL: https://doi.org/10.1145/3717823.3718303.
  47. Roman Edenhofer. A space-space trade-off for directed st-connectivity. arXiv preprint, arXiv:2602.21088, 2026. URL: https://doi.org/10.48550/arXiv.2602.21088.
  48. Jeff Edmonds, Chung Keung Poon, and Dimitris Achlioptas. Tight lower bounds for st-connectivity on the NNJAG model. SIAM Journal on Computing, 28(6):2257-2284, 1999. URL: https://doi.org/10.1137/S0097539795295948.
  49. Thomas Eiter and Heikki Mannila. Computing discrete Fréchet distance. Technical Report CD-TR 94/64, Christian Doppler Laboratory for Expert Systems, TU Vienna, 1994. Google Scholar
  50. Marten Folkertsma, Ian Mertz, Florian Speelman, and Quinten Tupker. Fully characterizing lossy catalytic computation. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025), volume 325 of Leibniz International Proceedings in Informatics (LIPIcs), pages 50:1-50:13, 2025. URL: https://doi.org/10.4230/LIPIcs.ITCS.2025.50.
  51. Chetan Gupta, Rahul Jain, Vimal Raj Sharma, and Raghunath Tewari. Unambiguous catalytic computation. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019), volume 150 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:13, 2019. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2019.16.
  52. Chetan Gupta, Rahul Jain, Vimal Raj Sharma, and Raghunath Tewari. Lossy catalytic computation. arXiv preprint, arXiv:2408.14670, 2024. URL: https://doi.org/10.48550/arXiv.2408.14670.
  53. Chetan Gupta, Raghunath Tewari, and Vimal Raj Sharma. Efficient isolation of perfect matching in o(log n) genus bipartite graphs. arXiv preprint, arXiv:2511.21217, 2025. URL: https://doi.org/10.48550/arXiv.2511.21217.
  54. William Hesse, Eric Allender, and David A. Mix Barrington. Uniform constant-depth threshold circuits for division and iterated multiplication. Journal of Computer and System Sciences, 65(4):695-716, 2002. URL: https://doi.org/10.1016/S0022-0000(02)00025-9.
  55. Sampath Kannan, Sanjeev Khanna, and Sudeepa Roy. STCON in directed unique-path graphs. In 28th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2008), volume 2 of Leibniz International Proceedings in Informatics (LIPIcs), pages 256-267, 2008. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2008.1758.
  56. Eamonn J. Keogh and Michael J. Pazzani. Scaling up dynamic time warping to massive datasets. In 3rd European Conference on Principles of Data Mining and Knowledge Discovery (PKDD 1999), volume 1704 of Lecture Notes in Computer Science, pages 1-11, 1999. URL: https://doi.org/10.1007/978-3-540-48247-5_1.
  57. Man-Soon Kim, Sang-Wook Kim, and Miyoung Shin. Optimization of subsequence matching under time warping in time-series databases. In 2005 ACM Symposium on Applied Computing (SAC 2005), pages 581-586, 2005. URL: https://doi.org/10.1145/1066677.1066814.
  58. Masashi Kiyomi, Takashi Horiyama, and Yota Otachi. Longest common subsequence in sublinear space. Information Processing Letters, 168:106084, 2021. URL: https://doi.org/10.1016/j.ipl.2020.106084.
  59. Michal Koucký. Catalytic computation. Bulletin of the EATCS, 118, 2016. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/400.
  60. Michal Koucký, Ian Mertz, Edward Pyne, and Sasha Sami. Collapsing catalytic classes. In 66th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2025), 2025. Google Scholar
  61. Jiatu Li, Edward Pyne, and Roei Tell. Distinguishing, predicting, and certifying: On the long reach of partial notions of pseudorandomness. In 65th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2024), pages 1-13, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00095.
  62. Ian Mertz. Reusing space: Techniques and open problems. Bulletin of the EATCS, 141, 2023. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/780.
  63. Edward Pyne. Derandomizing logspace with a small shared hard drive. Computational Complexity, 34:13, 2025. URL: https://doi.org/10.1007/S00037-025-00275-6.
  64. Omer Reingold. Undirected connectivity in log-space. Journal of the ACM, 55(4):17:1-17:24, 2008. URL: https://doi.org/10.1145/1391289.1391291.
  65. Omer Reingold, Luca Trevisan, and Salil Vadhan. Pseudorandom walks on regular digraphs and the RL vs. L problem. In 38th Annual ACM Symposium on Theory of Computing (STOC 2006), pages 457-466, 2006. URL: https://doi.org/10.1145/1132516.1132583.
  66. Michael Saks and C. Seshadhri. Space efficient streaming algorithms for the distance to monotonicity and asymmetric edit distance. In 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), pages 1698-1709, 2013. URL: https://doi.org/10.1137/1.9781611973105.122.
  67. Walter J. Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences, 4(2):177-192, 1970. URL: https://doi.org/10.1016/S0022-0000(70)80006-X.
  68. Yakov Shalunov. Improved bounds on the space complexity of circuit evaluation. arXiv preprint, arXiv:2504.20950, 2025. URL: https://doi.org/10.48550/arXiv.2504.20950.
  69. Salil Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
  70. R. Ryan Williams. Simulating time with square-root space. In 57th Annual ACM Symposium on Theory of Computing (STOC 2025), pages 13-23, 2025. URL: https://doi.org/10.1145/3717823.3718225.
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