Danny Hucke

The Smallest Grammar Problem Revisited

In a seminal paper of Charikar et al. on the smallest grammar problem, the authors derive upper and lower bounds on the approximation ratios for several grammar-based compressors, but in all cases there is a gap between the lower and upper bound. Here we close the gaps for LZ78 and BISECTION by showing that the approximation ratio of LZ78 is \(\varTheta ( (n/\log n)^{2/3})\), whereas the approximation ratio of BISECTION is \(\varTheta ( (n/\log n)^{1/2})\). We also derive a lower bound for a smallest grammar for a word in terms of its number of LZ77-factors, which refines existing bounds of Rytter. Finally, we improve results of Arpe and Reischuk relating grammar-based compression for arbitrary alphabets and binary alphabets.

Constructing small tree grammars and small circuits for formulas

It is shown that every tree of size n over a fixed set of different ranked symbols can be produced by a so called straight-line linear context-free tree grammar of size O ( n log n ) , which can be used as a compressed representation of the input tree. Two algorithms for computing such tree grammar are presented: One working in logarithmic space and the other working in linear time. As a consequence, it is shown that every arithmetical formula of size n, in which only m n different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size O ( n log m log n ) and depth O ( log n ) . This refines a classical result of Brent from 1974, according to which an arithmetical formula of size n can be transformed into a logarithmic depth circuit of size O ( n ) . A grammar-based tree compressor based on BISECTION is developed.The output grammar has size O ( n / log n ) , where n is the size of the input tree.A logspace as well as a linear time implementation is presented.

Querying Regular Languages over Sliding Windows.

We study the space complexity of querying regular languages over data streams in the sliding window model. The algorithm has to answer at any point of time whether the content of the sliding window belongs to a fixed regular language. A trichotomy is shown: For every regular language the optimal space requirement is either in Theta(n), Theta(log(n)), or constant, where

Tree Compression Using String Grammars

n

Universal tree source coding using grammar-based compression

is the size of the sliding window.

Circuits and Expressions over Finite Semirings

We study the compressed representation of a ranked tree by a straight-line program (SLP) for its preorder traversal string, and compare it with the previously studied representation by straight-line context-free tree grammars (also known as tree straight-line programs or TSLPs). Although SLPs may be exponentially more succinct than TSLPs, we show that many simple tree queries can still be performed efficiently on SLPs, such as computing the height of a tree, tree navigation, or evaluation of Boolean expressions. Other problems like pattern matching and evaluation of tree automata become intractable.

Circuit Evaluation for Finite Semirings

We apply so-called tree straight-line programs to the problem of universal source coding for binary trees. We derive an upper bound on the maximal pointwise redundancy (or worst-case redundancy) that improve previous bounds on the average case redundancy obtained by Zhang, Yang, and Kieffer using directed acyclic graphs. Using this, we obtain universal codes for new classes of tree sources.

Constructing Small Tree Grammars and Small Circuits for Formulas.

The computational complexity of the circuit and expression evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or a multiplicative identity. The following dichotomy is shown: If a finite semiring is such that (i) the multiplicative semigroup is solvable and (ii) it does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 ≠ 0, then the circuit evaluation problem is in DET ⊆ NC2, and the expression evaluation problem for the semiring is in TC0. For all other finite semirings, the circuit evaluation problem is P-complete and the expression evaluation problem is NC1-complete. As an application, we determine the complexity of intersection non-emptiness problems for given context-free grammars (regular expressions) with a fixed regular language.

Automata Theory on Sliding Windows.

The circuit evaluation problem for finite semirings is considered, where semirings are not assumed to have an additive or multiplicative identity. The following dichotomy is shown: If a finite semiring R (i) has a solvable multiplicative semigroup and (ii) does not contain a subsemiring with an additive identity 0 and a multiplicative identity 1 != 0, then its circuit evaluation problem is in the complexity class DET (which is contained in NC^2). In all other cases, the circuit evaluation problem is P-complete.