Abstract In the sliding window streaming model the goal is to compute an output value that only depends on the lastnsymbols from the data stream. Thereby, only space sublinear in the window sizenshould be used. Quite often randomization is used in order to achieve this goal. In the literature, one finds two different correctness criteria for randomized sliding window algorithms: (i) one can require that for every data stream and every time instantt, the algorithm computes a correct output value with high probability, or (ii) one can require that for every data stream the probability that the algorithm computes at every time instant a correct output value is high. Condition (ii) is stronger than (i) and is called “strict correctness” in this paper. The main result of this paper states that every strictly correct randomized sliding window algorithm can be derandomized without increasing the worst-case space consumption.

In a seminal paper, Charikar et al. 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 the gaps for LZ78 and BISECTION are closed by showing that the approximation ratio of LZ78 is Θ((n/log n)2/3), whereas the approximation ratio of BISECTION is Θ(√(n/log n)). In addition, the lower bound for RePair is improved from Ω(√(log n)) to Ω(log n/log log n). Finally, results of Arpe and Reischuk relating grammar-based compression for arbitrary alphabets and binary alphabets are improved.

We study the problem of recognizing regular languages in a variant of the streaming model of computation, called the sliding window model. In this model, we are given a size of the sliding window $n$ and a stream of symbols. At each time instant, we must decide whether the suffix of length $n$ of the current stream ("the active window") belongs to a given regular language. Recent works showed that the space complexity of an optimal deterministic sliding window algorithm for this problem is either constant, logarithmic or linear in the window size $n$ and provided natural language theoretic characterizations of the space complexity classes. Subsequently, those results were extended to randomized algorithms to show that any such algorithm admits either constant, double logarithmic, logarithmic or linear space complexity. In this work, we make an important step forward and combine the sliding window model with the property testing setting, which results in ultra-efficient algorithms for all regular languages. Informally, a sliding window property tester must accept the active window if it belongs to the language and reject it if it is far from the language. We consider deterministic and randomized sliding window property testers with one-sided and two-sided errors. In particular, we show that for any regular language, there is a deterministic sliding window property tester that uses logarithmic space and a randomized sliding window property tester with two-sided error that uses constant space.

The definition of kth-order empirical entropy of strings is extended to node-labeled binary trees. A suitable binary encoding of tree straight-line programs (that have been used for grammar-based tree compression before) is shown to yield binary tree encodings of size bounded by the kth-order empirical entropy plus some lower order terms. This generalizes recent results for grammar-based string compression to grammar-based tree compression. A long version of this paper can be found in [11].

The problem of universal source coding for binary trees is considered. Zhang, Yang, and Kieffer derived upper bounds on the average-case redundancy of codes based on directed acyclic graph (DAG) compression for binary tree sources with certain properties. In this paper, a natural class of binary tree sources is presented such that the demanded properties are fulfilled. Moreover, for both subclasses considered in the paper of Zhang, Yang, and Kieffer, their result is improved by deriving bounds on the maximal pointwise redundancy (or worst-case redundancy) instead of the average-case redundancy. Finally, using context-free tree grammars instead of DAGs, upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived. This yields universal codes for new classes of binary tree sources.

A sliding window algorithm receives a stream of symbols and has to output at each time instant a certain value which only depends on the last $n$ symbols. If the algorithm is randomized, then at each time instant it produces an incorrect output with probability at most $ε$, which is a constant error bound. This work proposes a more relaxed definition of correctness which is parameterized by the error bound $ε$ and the failure ratio $ϕ$: A randomized sliding window algorithm is required to err with probability at most $ε$ at a portion of $1-ϕ$ of all time instants of an input stream. This work continues the investigation of sliding window algorithms for regular languages. In previous works a trichotomy theorem was shown for deterministic algorithms: the optimal space complexity is either constant, logarithmic or linear in the window size. The main results of this paper concerns three natural settings (randomized algorithms with failure ratio zero and randomized/deterministic algorithms with bounded failure ratio) and provide natural language theoretic characterizations of the space complexity classes.

In a recent paper we analyzed the space complexity of streaming algorithms whose goal is to decide membership of a sliding window to a fixed language. For the class of regular languages we proved a space trichotomy theorem: for every regular language the optimal space bound is either constant, logarithmic or linear. In this paper we continue this line of research: We present natural characterizations for the constant and logarithmic space classes and establish tight relationships to the concept of language growth. We also analyze the space complexity with respect to automata size and prove almost matching lower and upper bounds. Finally, we consider the decision problem whether a language given by a DFA/NFA admits a sliding window algorithm using logarithmic/constant space.

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.

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.

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 $n$ is the size of the sliding window.

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).