Document Open Access Logo

On the Complexity of Computing Strahler Numbers

Authors Moses Ganardi , Markus Lohrey

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2026.41.pdf
  • Filesize: 0.99 MB
  • 22 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Grammars and context-free languages
Keywords
  • Strahler number
  • circuit complexity classes
  • context-free grammars

Metrics

Abstract

It is shown that the problem of computing the Strahler number of a binary tree given as a term is complete for the circuit complexity class uniform NC¹. For several variants, where the binary tree is given by a pointer structure or in a succinct form by a directed acyclic graph or a tree straight-line program, the complexity of computing the Strahler number is determined as well. The problem, whether a given context-free grammar in Chomsky normal form produces a derivation tree (resp., an acyclic derivation tree), whose Strahler number is at least a given number k is shown to be P-complete (resp., PSPACE-complete).

Cite As Get BibTex

Moses Ganardi and Markus Lohrey. On the Complexity of Computing Strahler Numbers. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.41

Author Details

Moses Ganardi
  • University of Kaiserslautern-Landau (RPTU), Germany
Markus Lohrey
  • Universität Siegen, Germany

References

  1. Alex Arenas, Leon Danon, Albert Díaz-Guilera, Pablo M. Gleiser, and Roger Guimerá. Community analysis in social networks. European Physical Journal B, 38(2):373-380, 2004. URL: https://doi.org/10.1140/epjb/e2004-00130-1.
  2. Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. URL: https://doi.org/10.1017/CBO9780511804090.
  3. Mohamed Faouzi Atig and Pierre Ganty. Approximating Petri net reachability along context-free traces. In Proceedings of the IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2011, volume 13 of LIPIcs, pages 152-163. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2011. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2011.152.
  4. David A. Mix Barrington, Neil Immerman, and Howard Straubing. On uniformity within NC¹. Journal of Computer and System Sciences, 41:274-306, 1990. URL: https://doi.org/10.1016/0022-0000(90)90022-D.
  5. Martin Beaudry and Pierre McKenzie. Circuits, matrices, and nonassociative computation. Journal of Computer and System Sciences, 50(3):441-455, 1995. URL: https://doi.org/10.1006/JCSS.1995.1035.
  6. 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.
  7. Clotilde Bizière and Wojciech Czerwinski. Reachability in one-dimensional pushdown vector addition systems is decidable. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, pages 1851-1862. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718149.
  8. Richard P. Brent. The parallel evaluation of general arithmetic expressions. Journal of the ACM, 21(2):201-206, 1974. URL: https://doi.org/10.1145/321812.321815.
  9. Samuel R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Annual Symposium on Theory of Computing, STOC 1987, pages 123-131. ACM Press, 1987. URL: https://doi.org/10.1145/28395.28409.
  10. Samuel R. Buss, Stephen A. Cook, A. Gupta, and V. Ramachandran. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing, 21(4):755-780, 1992. URL: https://doi.org/10.1137/0221046.
  11. Andrew Chiu, George I. Davida, and Bruce E. Litow. Division in logspace-uniform NC^1. RAIRO - Theoretical Informatics and Applications, 35(3):259-275, 2001. URL: https://doi.org/10.1051/ITA:2001119.
  12. Michal P. Chytil and Burkhard Monien. Caterpillars and context-free languages. In Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1990, volume 415 of Lecture Notes in Computer Science, pages 70-81. Springer, 1990. URL: https://doi.org/10.1007/3-540-52282-4.
  13. James Cook and Ian Mertz. Tree evaluation is in space O(log n ⋅ log log n). In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 1268-1278. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649664.
  14. Stephen A. Cook. A taxonomy of problems with fast parallel algorithms. Information and Control, 64(1-3):2-21, 1985. URL: https://doi.org/10.1016/S0019-9958(85)80041-3.
  15. Bruno Courcelle. Graph rewriting: An algebraic and logic approach. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics, pages 193-242. Elsevier and MIT Press, 1990. URL: https://doi.org/10.1016/B978-0-444-88074-1.50010-X.
  16. Laure Daviaud, Marcin Jurdziński, and K. S. Thejaswini. The Strahler number of a parity game. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, volume 168 of LIPIcs, pages 123:1-123:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.123.
  17. Luc Devroye and Paul Kruszewski. A note on the Horton-Strahler number for random trees. Information Processing Letters, 56(2):95-99, 1995. URL: https://doi.org/10.1016/0020-0190(95)00114-R.
  18. Luc Devroye and Paul Kruszewski. On the Horton-Strahler number for random tries. RAIRO - Theoretical Informatics and Applications, 30(5):443-456, 1996. URL: https://doi.org/10.1051/ita/1996300504431.
  19. Peter J. Downey, Ravi Sethi, and Robert Endre Tarjan. Variations on the common subexpression problem. Journal of the ACM, 27(4):758-771, 1980. URL: https://doi.org/10.1145/322217.322228.
  20. Andrzej Ehrenfeucht, Grzegorz Rozenberg, and Dirk Vermeir. On ETOL systems with finite tree-rank. SIAM Journal on Computing, 10(1):40-58, 1981. URL: https://doi.org/10.1137/0210004.
  21. Michael Elberfeld, Andreas Jakoby, and Till Tantau. Logspace versions of the theorems of Bodlaender and Courcelle. In Proceedings of the 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, pages 143-152. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/FOCS.2010.21.
  22. Michael Elberfeld, Andreas Jakoby, and Till Tantau. Algorithmic meta theorems for circuit classes of constant and logarithmic depth. In Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, volume 14 of LIPIcs, pages 66-77. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPIcs.STACS.2012.66.
  23. Andrey Petrovich Ershov. On programming of arithmetic operations. Communications of the ACM, 1(8):3-6, 1958. URL: https://doi.org/10.1145/368892.368907.
  24. Javier Esparza, Stefan Kiefer, and Michael Luttenberger. Newtonian program analysis. Journal of the ACM, 57(6), 2010. URL: https://doi.org/10.1145/1857914.1857917.
  25. Javier Esparza, Michael Luttenberger, and Maximilian Schlund. A brief history of Strahler numbers. In Proceedings of the 8th International Conference on Language and Automata Theory and Applications, LATA 2014, volume 8370 of Lecture Notes in Computer Science, pages 1-13. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-04921-2_1.
  26. Philippe Flajolet, Jean-Claude Raoult, and Jean Vuillemin. The number of registers required for evaluating arithmetic expressions. Theoretical Computer Science, 9(1):99-125, 1979. URL: https://doi.org/10.1016/0304-3975(79)90009-4.
  27. Philippe Flajolet, Paolo Sipala, and Jean-Marc Steyaert. Analytic variations on the common subexpression problem. In Proceedings of the 17th International Colloquium on Automata, Languages and Programming, ICALP 1990, volume 443 of Lecture Notes in Computer Science, pages 220-234. Springer, 1990. URL: https://doi.org/10.1007/BFB0032034.
  28. Moses Ganardi and Markus Lohrey. A universal tree balancing theorem. ACM Transactions on Computation Theory, 11(1):1:1-1:25, 2019. URL: https://doi.org/10.1145/3278158.
  29. Moses Ganardi and Markus Lohrey. On the complexity of computing Strahler numbers, 2025. URL: https://arxiv.org/abs/2512.19060.
  30. Moses Ganardi, Irmak Saglam, and Georg Zetzsche. Directed regular and context-free languages. In Proceedings of the 41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024, volume 289 of LIPIcs, pages 36:1-36:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.STACS.2024.36.
  31. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, 1979. Google Scholar
  32. Adrià Gascón, Markus Lohrey, Sebastian Maneth, Carl Philipp Reh, and Kurt Sieber. Grammar-based compression of unranked trees. Theory of Computing Systems, 64(1):141-176, 2020. URL: https://doi.org/10.1007/S00224-019-09942-Y.
  33. Seymour Ginsburg and Edwin H. Spanier. Derivation-bounded languages. Journal of Computer and System Sciences, 2(3):228-250, 1968. URL: https://doi.org/10.1016/S0022-0000(68)80009-1.
  34. Jozef Gruska. A few remarks on the index of context-free grammars and languages. Information and Control, 19(3):216-223, 1971. URL: https://doi.org/10.1016/S0019-9958(71)90095-7.
  35. 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, 65:695-716, 2002. URL: https://doi.org/10.1016/S0022-0000(02)00025-9.
  36. John E. Hopcroft and Jeffrey D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading, MA, 1979. Google Scholar
  37. Robert E. Horton. Erosional development of streams and their drainage basins: hydro-physical approach to quantitative morphology. Geological Society of America Bulletin, 56(3):275-370, 1945. URL: https://doi.org/10.1130/0016-7606(1945)56[275:EDOSAT]2.0.CO;2.
  38. Andreas Jakoby and Till Tantau. Computing shortest paths in series-parallel graphs in logarithmic space. In Complexity of Boolean Functions, volume 06111 of Dagstuhl Seminar Proceedings. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany, 2006. URL: https://doi.org/10.4230/DagSemProc.06111.6.
  39. Rainer Kemp. The average number of registers needed to evaluate a binary tree optimally. Acta Informatica, 11(4):363-372, 1979. URL: https://doi.org/10.1007/BF00289094.
  40. S. Rao Kosaraju. On parallel evaluation of classes of circuits. In Proceedings of the 10th Conference on Foundations of Software Technology and Theoretical Computer Science, volume 472 of Lecture Notes in Computer Science, pages 232-237. Springer, 1990. URL: https://doi.org/10.1007/3-540-53487-3_48.
  41. Andreas Krebs, Nutan Limaye, and Michael Ludwig. A unified method for placing problems in polylogarithmic depth. In Proceedings of the 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, volume 93 of LIPIcs, pages 36:36-36:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2017.36.
  42. Paul Kruszewski. A note on the Horton-Strahler number for random binary search trees. Information Processing Letters, 69(1):47-51, 1999. URL: https://doi.org/10.1016/S0020-0190(98)00192-6.
  43. Markus Lohrey. On the parallel complexity of tree automata. In Proceedings of the 12th International Conference on Rewrite Techniques and Applications, RTA 2001, number 2051 in Lecture Notes in Computer Science, pages 201-215. Springer, 2001. URL: https://doi.org/10.1007/3-540-45127-7_16.
  44. Markus Lohrey. Grammar-based tree compression. In Proceedings of the 19th International Conference on Developments in Language Theory, DLT 2015, number 9168 in Lecture Notes in Computer Science, pages 46-57. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21500-6_3.
  45. Markus Lohrey, Andreas Rosowski, and Georg Zetzsche. Membership problems in finite groups. Journal of Algebra, 675:23-58, 2025. URL: https://doi.org/10.1016/j.jalgebra.2025.03.011.
  46. Carine Pivoteau, Bruno Salvy, and Michèle Soria. Algorithms for combinatorial structures: Well-founded systems and Newton iterations. Journal of Combinatorial Theory, Series A, 119(8):1711-1773, 2012. URL: https://doi.org/10.1016/j.jcta.2012.05.007.
  47. Klaus Reinhardt and Eric Allender. Making nondeterminism unambiguous. SIAM Journal on Computing, 29(4):1118-1131, 2000. URL: https://doi.org/10.1137/S0097539798339041.
  48. Walter L. Ruzzo. On uniform circuit complexity. Journal of Computer and System Sciences, 22(3):365-383, 1981. URL: https://doi.org/10.1016/0022-0000(81)90038-6.
  49. Petra Scheffler. A linear algorithm for the pathwidth of trees. In Rainer Bodendiek and Rudolf Henn, editors, Topics in Combinatorics and Graph Theory: Essays in Honour of Gerhard Ringel, pages 613-620. Physica-Verlag HD, 1990. URL: https://doi.org/10.1007/978-3-642-46908-4_70.
  50. Yousef Shakiba, Henry Sinclair-Banks, and Georg Zetzsche. A complexity dichotomy for semilinear target sets in automata with one counter. In Proceedings of the 40th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2025, pages 594-608. IEEE, 2025. URL: https://doi.org/10.1109/LICS65433.2025.00051.
  51. Ekaterina Shemetova, Alexander Okhotin, and Semyon Grigorev. Rational index of languages defined by grammars with bounded dimension of parse trees. Theory of Computing Systems, 68(3):487-511, 2023. URL: https://doi.org/10.1007/s00224-023-10159-3.
  52. Philip M. Spira. On time-hardware complexity tradeoffs for boolean functions. In Proceedings of the 4th Hawaii Symposium on System Sciences, pages 525-527, 1971. Google Scholar
  53. Arthur N. Strahler. Hypsometric (area-altitude) analysis of erosional topology. Geological Society of America Bulletin, 63(11):1117-1142, 1952. URL: https://doi.org/10.1130/0016-7606(1952)63[1117:HAAOET]2.0.CO;2.
  54. Ivan Hal Sudborough. On the tape complexity of deterministic context-free languages. Journal of the ACM, 25(3):405-414, 1978. URL: https://doi.org/10.1145/322077.322083.
  55. Joseph Swernofsky and Michael Wehar. On the complexity of intersecting regular, context-free, and tree languages. In Proceedings of the 42nd International Colloquium Automata, Languages, and Programming, Part II, ICALP 2015, volume 9135 of Lecture Notes in Computer Science, pages 414-426. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47666-6_33.
  56. Xavier Viennot. Trees. In Mots, mélanges offert á M.P. Schützenberger. Hermes, Paris, 1990. Available online at URL: http://www.xavierviennot.org.
  57. Heribert Vollmer. Introduction to Circuit Complexity - A Uniform Approach. Texts in Theoretical Computer Science. An EATCS Series. Springer, 1999. URL: https://doi.org/10.1007/978-3-662-03927-4.
  58. R. Ryan Williams. Simulating time with square-root space. In Michal Koucký and Nikhil Bansal, editors, Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC 2025, pages 13-23. ACM, 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact