Constructing Small Tree Grammars and Small Circuits for Formulas

Authors Danny Hucke, Markus Lohrey, Eric Noeth

Thumbnail PDF


  • Filesize: 480 kB
  • 12 pages

Document Identifiers

Author Details

Danny Hucke
Markus Lohrey
Eric Noeth

Cite AsGet BibTex

Danny Hucke, Markus Lohrey, and Eric Noeth. Constructing Small Tree Grammars and Small Circuits for Formulas. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 457-468, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


It is shown that every tree of size n over a fixed set of sigma different ranked symbols can be decomposed into O(n/log_sigma(n)) = O((n * log(sigma))/ log(n)) many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size O(n/log_sigma(n)), which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of O(n/log^{0.19}(n)) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size n, in which only m <= n different variables occur, can be transformed (in time O(n * log(n)) into an arithmetical circuit of size O((n * log(m))/log(n)) and depth O(log(n)). This refines a classical result of Brent, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O(n).
  • grammar-based compression
  • tree compression
  • arithmetical circuits


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


  1. T. Akutsu. A bisection algorithm for grammar-based compression of ordered trees. Inf. Process. Lett., 110(18-19):815-820, 2010. Google Scholar
  2. V. Arvind, S. Raja, and A. V. Sreejith. On lower bounds for multiplicative circuits and linear circuits in noncommutative domains. In Proc. CSR 2014, volume 8476 of LNCS, pages 65-76. Springer, 2014. Google Scholar
  3. P. Bille, I. L. Gørtz, G. M. Landau, and O. Weimann. Tree compression with top trees. In Proc. ICALP (1) 2013, volume 7965 of LNCS, pages 160-171. Springer, 2013. Google Scholar
  4. R. P. Brent. The parallel evaluation of general arithmetic expressions. J. ACM, 21(2):201-206, 1974. Google Scholar
  5. N. H. Bshouty, R. Cleve, and W. Eberly. Size-depth tradeoffs for algebraic formulas. SIAM J. Comput., 24(4):682-705, 1995. Google Scholar
  6. G. Busatto, M. Lohrey, and S. Maneth. Efficient memory representation of XML document trees. Inf. Syst., 33(4-5):456-474, 2008. Google Scholar
  7. M. Charikar, E. Lehman, A. Lehman, D. Liu, R. Panigrahy, M. Prabhakaran, A. Sahai, and A. Shelat. The smallest grammar problem. IEEE Trans. Inf. Theory, 51(7):2554-2576, 2005. Google Scholar
  8. H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, D. Lugiez, C. Löding, S. Tison, and M. Tommasi. Tree automata techniques and applications. Google Scholar
  9. N. de Bruijn. A combinatorial problem. Nederl. Akad. Wet., Proc., 49:758-764, 1946. Google Scholar
  10. P. J. Downey, R. Sethi, and R. E. Tarjan. Variations on the common subexpression problem. J. ACM, 27(4):758-771, 1980. Google Scholar
  11. P. Flajolet, P. Sipala, and J.-M. Steyaert. Analytic variations on the common subexpression problem. In Proc. ICALP 1990, volume 443 of LNCS, pages 220-234. Springer, 1990. Google Scholar
  12. L. Gasieniec, R. M. Kolpakov, I. Potapov, and P. Sant. Real-time traversal in grammar-based compressed files. In Proc. DCC 2005, page 458. IEEE Computer Society, 2005. Google Scholar
  13. M. A. Heap and M. R. Mercer. Least upper bounds on OBDD sizes. IEEE Trans. Computers, 43(6):764-767, 1994. Google Scholar
  14. D. Hucke, M. Lohrey, and E. Noeth. Constructing small tree grammars and small circuits for formulas.,, 2014. Google Scholar
  15. A. Jėz and M. Lohrey. Approximation of smallest linear tree grammars. In Proc. STACS 2014, volume 25 of LIPIcs, pages 445-457. Leibniz-Zentrum für Informatik, 2014. Google Scholar
  16. J. C. Kieffer and E.-H. Yang. Grammar-based codes: A new class of universal lossless source codes. IEEE Trans. Inf. Theory, 46(3):737-754, 2000. Google Scholar
  17. J. C. Kieffer, E.-H. Yang, G. J. Nelson, and P. C. Cosman. Universal lossless compression via multilevel pattern matching. IEEE Trans.Inf. Theory, 46(4):1227-1245, 2000. Google Scholar
  18. D. E. Knuth. The Art of Computer Programming, Vol. I: Fundamental Algorithms. Addison-Wesley, 1968. Google Scholar
  19. P. M. Lewis II, R. E. Stearns, and J. Hartmanis. Memory bounds for recognition of context-free and context-sensitive languages. In Proc. 6th Annual IEEE Symp. Switching Circuit Theory and Logic Design, pages 191-202, 1965. Google Scholar
  20. H.-T. Liaw and C.-S. Lin. On the OBDD-representation of general boolean functions. IEEE Trans. Computers, 41(6):661-664, 1992. Google Scholar
  21. M. Lohrey. Traversing grammar-compressed trees with constant delay. Google Scholar
  22. M. Lohrey. Algorithmics on SLP-compressed strings: A survey. Groups Complexity Cryptology, 4(2):241-299, 2012. Google Scholar
  23. M. Lohrey, S. Maneth, and R. Mennicke. XML tree structure compression using RePair. Inf. Syst., 38(8):1150-1167, 2013. Google Scholar
  24. M. Lohrey, S. Maneth, and M. Schmidt-Schauß. Parameter reduction and automata evaluation for grammar-compressed trees. J. Comput. Syst. Sci., 78(5):1651-1669, 2012. Google Scholar
  25. W. L. Ruzzo. Tree-size bounded alternation. J. Comput. Syst. Sci., 21:218-235, 1980. Google Scholar
  26. P. M. Spira. On time-hardware complexity tradeoffs for boolean functions. In Proc. 4th Hawaii Symp. on System Sciences, pages 525-527, 1971. Google Scholar
  27. J. Zhang, E.-H. Yang, and J. C. Kieffer. A universal grammar-based code for lossless compression of binary trees. IEEE Trans. Inf. Theory, 60(3):1373-1386, 2014. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail