Julio Xochitemol | Searcharxiv

arXiv2025

Streaming algorithms for products of probabilities

We consider streaming algorithms for approximating a product of input probabilities up to multiplicative error of $1-ε$. It is shown that every randomized streaming algorithm for this problem needs space $Ω(\log n + \log b - \log ε) - \mathcal{O}(1)$, where $n$ is length of the input stream and $b$ is the bit length of the input numbers. This matches an upper bound from Alur et al.~up to a constant multiplicative factor. Moreover, we consider the threshold problem, where it is asked whether the product of the input probabilities is below a given threshold. It is shown that every randomized streaming algorithm for this problem needs space $Ω(n \cdot b)$.

cs.DS↗

arXiv2022

Streaming algorithms for groups and semigroups

We investigate deterministic and randomized streaming algorithms for word problems in finitely generated groups and semigroups. For this we introduce the notion of a distinguisher: a randomized streaming algorithm that processes two input words in parallel and, with high probability, reaches identical memory states if the words represent the same element, and distinct states otherwise. We construct such distinguishers with low error probability using logarithmic, and in some cases doubly logarithmic, space. For example, our results apply to linear semigroups and to semigroups obtained (under suitable restrictions) via standard constructions such as graph products, wreath products, and semilattice decompositions. In case of commutative semigroups and cancellative nilpotent semigroups, we achieve space complexity $\mathcal{O}(\log \log n)$. We complement these upper bounds with lower bounds demonstrating that certain well-known semigroups do not admit sublinear-space distinguishers. This includes, for example, free inverse monoids of rank at least two and Thompson's group $F$. Finally, we study randomized streaming algorithms for subgroup membership problems in free groups and their direct products.

math.GR↗