Charles Paperman

Mateusz Gienieczko, Martín Muñoz, Filip Murlak et al.

Streaming allows executing queries over massive JSON or XML documents whose size makes it infeasible to fully parse them into a tree. Earliest query answering is a radical approach to reducing latency and memory footprint. To minimize latency, a document node must be returned as soon as the node is guaranteed to be an answer regardless of how the document ends. Similarly, to minimize memory footprint, a node must be discarded as soon as it cannot become an answer regardless of how the document ends. For simple queries that select nodes based on the path from the root, the decision for each node can be made on the spot, but practical languages such as XPath or JSONpath support filters, which allow selecting nodes based on information collected from various parts of the document, possibly further down the stream. This makes earliest query answering a challenging task, as candidate nodes must be kept in memory until it becomes clear that they can be safely returned or discarded. We show that this can be done for all unary queries expressible in monadic second order logic (MSO), while ensuring constant update time -- provided that nodes are returned by passing a suitable iterator, rather than one by one.

Antoine Amarilli, Sebastien Labbe, Charles Paperman

We introduce the task of out-of-order membership to a formal language L, where the letters of a word w are revealed one by one in an adversarial order. The length |w| is known in advance, but the content of w is streamed as pairs (i, w[i]), received exactly once for each position i, in arbitrary order. We study efficient algorithms for this task when L is regular, seeking tight complexity bounds as a function of |w| for a fixed target language. Most of our results apply to an algebraically defined variant dubbed out-of-order evaluation: this problem is defined for a fixed finite monoid or semigroup S, and our goal is to compute the ordered product of the streamed elements of w. We show that, for any fixed regular language or finite semigroup, both problems can be solved in constant time per streamed symbol and in linear space. However, the precise space complexity strongly depends on the algebraic structure of the target language or evaluation semigroup. Our main contributions are therefore to show (deterministic) space complexity characterizations, which we do for out-of-order evaluation of monoids and semigroups. For monoids, we establish a trichotomy: the space complexity is either Θ(1), Θ(log n), or Θ(n), where n = |w|. More specifically, the problem admits a constant-space solution for commutative monoids, while all non-commutative monoids require Ω(log n) space. We further identify a class of monoids admitting an O(log n)-space algorithm, and show that all remaining monoids require Ω(n) space. For general semigroups, the situation is more intricate. We characterize a class of semigroups admitting constant-space algorithms for out-of-order evaluation, and show that semigroups outside this class require at least Ω(log n) space.