Abstract
We present a general framework for balancing expressions (terms) in the form of so-called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations, which insert terms/contexts into contexts. In Ref. [16], it was shown that one can compute for a given term of size n in logspace a tree straight-line program of depth O(log n) and size O(n/ log n). In the present article, it is shown that the conversion can be done in DLOGTIME-uniform TC0. This allows reducing the term evaluation problem over an arbitrary algebra A to the term evaluation problem over a derived two-sorted algebra F (A). Three applications are presented: (i) an alternative proof for a recent result by Krebs et al. [25] on the expression evaluation problem is given; (ii) it is shown that expressions for an arbitrary (possibly non-commutative) semiring can be transformed in DLOGTIME-uniform TC0 into equivalent circuits of logarithmic depth and size O(n/ log n); and, (iii) a corresponding result for regular expressions is shown.
- Karl R. Abrahamson, Norm Dadoun, David G. Kirkpatrick, and Teresa M. Przytycka. 1989. A simple parallel tree contraction algorithm. Journal of Algorithms 10, 2, 287--302 Google Scholar
Digital Library
- Eric Allender, Jia Jiao, Meena Mahajan, and V. Vinay. 1998. Non-commutative arithmetic circuits: Depth reduction and size lower bounds. Theoretical Computer Science 209, 1--2, 47--86. Google Scholar
Digital Library
- David A. M. Barrington, Neil Immerman, and Howard Straubing. 1990. On uniformity within . Journal of Computer and System Sciences 41, 3, 274--306. Google Scholar
Digital Library
- Philip Bille, Inge Li Gørtz, Gad M. Landau, and Oren Weimann. 2015. Tree compression with top trees. Information and Computation 243, 166--177. Google Scholar
Digital Library
- Maria Luisa Bonet and Samuel R. Buss. 1994. Size-depth tradeoffs for Boolean fomulae. Information Processing Letters 49, 3, 151--155. Google Scholar
Digital Library
- Richard P. Brent. 1974. The parallel evaluation of general arithmetic expressions. Journal of the ACM 21, 2, 201--206. Google Scholar
Digital Library
- Nader H. Bshouty, Richard Cleve, and Wayne Eberly. 1995. Size-depth tradeoffs for algebraic formulas. SIAM Journal on Computing 24, 4, 682--705. Google Scholar
Digital Library
- S. Buss, S. Cook, A. Gupta, and V. Ramachandran. 1992. An optimal parallel algorithm for formula evaluation. SIAM Journal on Computing 21, 4, 755--780. Google Scholar
Digital Library
- Samuel R. Buss. 1987. The Boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Annual Symposium on Theory of Computing (STOC’87). ACM Press, 123--131. Google Scholar
Digital Library
- Samuel R. Buss. 1993. Algorithms for Boolean formula evaluation and for tree-contraction. Proof Theory, Complexity, and Arithmetic 95--115.Google Scholar
- M. Charikar, E. Lehman, A. Lehman, D. Liu, R. Panigrahy, M. Prabhakaran, A. Sahai, and A. Shelat. 2005. The smallest grammar problem. IEEE Transactions on Information Theory 51, 7, 2554--2576. Google Scholar
Digital Library
- Stephen A. Cook and Pierre McKenzie. 1987. Problems complete for deterministic logarithmic space. Journal of Algorithms 8, 3, 385--394. Google Scholar
Digital Library
- Nachum Dershowitz and Jean-Pierre Jouannaud. 1990. Rewrite systems. In Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics (B). Elsevier, 243--320. Google Scholar
Digital Library
- Bartlomiej Dudek and Pawel Gawrychowski. 2018. Slowing down top trees for better worst-case compression. In Proceedings of the Annual Symposium on Combinatorial Pattern Matching (CPM’18), volume 105 of LIPIcs. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, 16:1--16:8.Google Scholar
- Michael Elberfeld, Andreas Jakoby, and Till Tantau. 2012. Algorithmic meta theorems for circuit classes of constant and logarithmic depth. In Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science, STACS 2012, volume 14 of LIPIcs. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, 66--77.Google Scholar
- Moses Ganardi, Danny Hucke, Artur Jez, Markus Lohrey, and Eric Noeth. 2017. Constructing small tree grammars and small circuits for formulas. Journal of Computer and System Sciences 86, 136--158. Google Scholar
Digital Library
- Moses Ganardi, Danny Hucke, Daniel König, and Markus Lohrey. 2018. Circuits and expressions over finite semirings. Accepted for publication in ACM Transactions on Computation Theory. Google Scholar
Digital Library
- Adrià Gascón, Markus Lohrey, Sebastian Maneth, Carl Philipp Reh, and Kurt Sieber. 2018. Grammar-based compression of unranked trees. In Proceedings of 13th International Computer Science Symposium in Russia (CSR’18), volume 10846 of Lecture Notes in Computer Science. Springer, 118--131.Google Scholar
Cross Ref
- Hillel Gazit, Gary L. Miller, and Shang-Hua Teng. 1988. Optimal tree contraction in an EREW model. In S. K. Tewksbury, B. W. Dickinson, and S. C. Schwartz (Eds.). Concurrent Computations: Algorithms, Architecture and Technology. New York, Plenum Press, 139--156.Google Scholar
- Hermann Gruber and Markus Holzer. 2008. Finite automata, digraph connectivity, and regular expression size. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming, ICALP 2008, Part II, volume 5126 of Lecture Notes in Computer Science. Springer, 39--50. Google Scholar
Digital Library
- William Hesse, Eric Allender, and David A. Mix Barrington. 2002. Uniform constant-depth threshold circuits for division and iterated multiplication. Journal of Computer and System Sciences 65, 4, 695--716. Google Scholar
Digital Library
- Danny Hucke and Markus Lohrey. 2017. Universal tree source coding using grammar-based compression. In Proceedings of the IEEE International Symposium on Information Theory, ISIT 2017. IEEE, 1753--1757.Google Scholar
Cross Ref
- Neil Immerman. 1999. Descriptive Complexity. Graduate texts in computer science. Springer.Google Scholar
- Artur Jez and Markus Lohrey. 2016. Approximation of smallest linear tree grammar. Information and Computation 251, 215--251. Google Scholar
Digital Library
- Andreas Krebs, Nutan Limaye, and Michael Ludwig. 2018. 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. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, 36:36--36:15.Google Scholar
- Markus Lohrey. 2001. On the parallel complexity of tree automata. In Proceedings of the 12th International Conference on Rewrite Techniques and Applications, RTA 2001, volume 2051 of Lecture Notes in Computer Science. Springer, 201--215. Google Scholar
Digital Library
- Markus Lohrey. 2015. Grammar-based tree compression. In Proceedings of the 19th International Conference on Developments in Language Theory, DLT 2015, volume 9168 of Lecture Notes in Computer Science. Springer, 46--57.Google Scholar
Cross Ref
- Markus Lohrey, Sebastian Maneth, and Roy Mennicke. 2013. XML tree structure compression using RePair. Information Systems 38, 8, 1150--1167. Google Scholar
Digital Library
- Gary L. Miller and Shang-Hua Teng. 1987. Dynamic Parallel Complexity of Computational Circuits. In Proceedings of the 19th Annual Symposium on Theory of Computing, STOC 1987. ACM Press, 254--263. Google Scholar
Digital Library
- Gary L. Miller and Shang-Hua Teng. 1997. Tree-based parallel algorithm design. Algorithmica 19, 4, 369--389.Google Scholar
Cross Ref
- Mike Paterson and Leslie G. Valiant. 1976. Circuit size is nonlinear in depth. Theoretical Computer Science 2, 3, 397--400.Google Scholar
- Walter L. Ruzzo. 1981. On uniform circuit complexity. Journal of Computer and System Sciences 22, 3, 365--383.Google Scholar
- Philip M. Spira. 1971. On time-hardware complexity tradeoffs for Boolean functions. In Proceedings of the 4th Hawaii International Symposium on System Sciences. 525--527.Google Scholar
- Leslie G. Valiant, Sven Skyum, S. Berkowitz, and Charles Rackoff. 1983. Fast parallel computation of polynomials using few processors. SIAM Journal on Computing 12, 4, 641--644.Google Scholar
Cross Ref
- Heribert Vollmer. 1999. Introduction to Circuit Complexity. Springer. Google Scholar
Digital Library
- Wolfgang Wechler. 1992. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer.Google Scholar
Index Terms
A Universal Tree Balancing Theorem
Recommendations
Arithmetic circuits: The chasm at depth four gets wider
In their paper on the ''chasm at depth four'', Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o^(^m^) also admit arithmetic circuits of depth four and size 2^o^(^m^). This theorem ...
Finding a k-Tree Core and a k-Tree Center of a Tree Network in Parallel
A k-tree core of a tree network is a subtree with exactly k leaves that minimizes the total distance from vertices to the subtree. A k-tree center of a tree network is a subtree with exactly k leaves that minimizes the distance from the farthest vertex ...
Bottom-Up tree evaluation in tree-based genetic programming
ICSI'10: Proceedings of the First international conference on Advances in Swarm Intelligence - Volume Part IIn tree-based genetic programming (GP) performance optimization, the primary optimization target is the process of fitness evaluation This is because fitness evaluation takes most of execution time in GP Standard fitness evaluation uses the top-down ...






Comments