Sliding Window Algorithms for Regular Languages

Abstract

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 [14,15] 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, [16] extended this result 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 onesided 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.

Exploring the Spatial Distribution Characteristics and Correlation Factors of Wayfinding Performance on City-Scale Road Networks Based on Massive Trajectory Data

“…e global WPL is calculated based on the entire driving route to evaluate the overall wayfinding performance on this route. A sliding window [40][41][42] with a width of 3 km and a step length of one road segment is designed to calculate the local WPL. As shown in Figure 4, the sliding window slides along the actual driving route (black solid line) progressively-e.g., from W1, W2 to W3 in the figure.…”

Derandomization for Sliding Window Algorithms with Strict Correctness∗

“…In the following discussion, let us fix the error probability λ = 1/3 (using probability amplification, one can reduce λ to any constant > 0). In our recent paper [15] we studied the space complexity of the membership problem for regular languages with respect to λ-correct randomized sliding window algorithms. It turned out that in this setting, one can gain from randomization.…”

“…From our results in [13,14], it follows that the optimal space complexity of a deterministic sliding window algorithm for the membership problem for ab * is Θ(log n). On the other hand, it is shown in [15] that there is an λ-correct randomized sliding window algorithm for ab * with (worst-case) space complexity O(log log n) (this is also optimal). In fact, we proved in [15] that for every regular language L, the space optimal λ-correct randomized sliding window algorithm for L has either constant, doubly logarithmic, logarithmic, or linear space complexity, and the corresponding four space classes can be characterized in terms of simple syntactic properties.…”

“…Other algorithmic problems that were addressed in the extensive literature on sliding window streams include the computation of statistical data (e.g. computation of the variance and k-median [5], and quantiles [4]), optimal sampling from sliding windows [9], membership problems for formal languages [13][14][15][16], computation of edit distances [10], database querying (e.g. processing of join queries over sliding windows [18]) and graph problems (e.g.…”

“…A second ingredient of many sliding window algorithms is randomization. Following our recent work [13][14][15] we model a randomized streaming algorithm for a given approximation problem as a probabilistic automaton over a finite alphabet. Probabilistic automata were introduced by Rabin [23] and can be seen as a common generalization of deterministic finite automata and Markov chains.…”