Non-redundant random generation algorithms for weighted context-free languages

TL;DR

This paper introduces efficient algorithms for non-redundant random generation of words in weighted context-free languages, improving over rejection-based methods with near-linear complexity after precomputation.

Contribution

It presents two novel algorithms based on recursive and unranking methods for non-redundant generation, with complexity significantly better than rejection-based approaches.

Findings

Algorithms achieve non-redundant generation in O(k·n·log n) operations

Precomputation of Θ(n) numbers enables efficient generation

Rejection-based approach has exponential worst-case complexity

Abstract

We address the non-redundant random generation of $k$ words of length $n$ in a context-free language. Additionally, we want to avoid a predefined set of words. We study a rejection-based approach, whose worst-case time complexity is shown to grow exponentially with $k$ for some specifications and in the limit case of a coupon collector. We propose two algorithms respectively based on the recursive method and on an unranking approach. We show how careful implementations of these algorithms allow for a non-redundant generation of $k$ words of length $n$ in $\mathcal{O}(k\cdot n\cdot \log{n})$ arithmetic operations, after a precomputation of $\Theta(n)$ numbers. The overall complexity is therefore dominated by the generation of $k$ words, and the non-redundancy comes at a negligible cost.