A Dichotomy for Regular Expression Membership Testing (1611.00918v2)

Abstract: We study regular expression membership testing: Given a regular expression of size $m$ and a string of size $n$, decide whether the string is in the language described by the regular expression. Its classic $O(nm)$ algorithm is one of the big success stories of the 70s, which allowed pattern matching to develop into the standard tool that it is today. Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. However, a systematic study of tractable cases was made possible only recently, with the first conditional lower bounds reported by Backurs and Indyk [FOCS'16]. Restricted to any "type" of homogeneous regular expressions of depth 2 or 3, they either presented a near-linear time algorithm or a quadratic conditional lower bound, with one exception known as the Word Break problem. In this paper we complete their work as follows: 1) We present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing. 2) We classify all types, except for the Word Break problem, into almost-linear time or quadratic time assuming the Strong Exponential Time Hypothesis. This extends the classification from depth 2 and 3 to any constant depth. 3) For the Word Break problem we give an improved $\tilde{O}(n m^{1/3} + m)$ algorithm. Surprisingly, we also prove a matching conditional lower bound for combinatorial algorithms. This establishes Word Break as the only intermediate problem. In total, we prove matching upper and lower bounds for any type of bounded-depth homogeneous regular expressions, which yields a full dichotomy for regular expression membership testing.

• The paper establishes a dichotomy for regular expression membership testing, classifying instances into near-linear or quadratic complexity based on type and depth, assuming the Strong Exponential Time Hypothesis (SETH).
• It identifies the Word Break problem as an intermediate complexity case, presenting an improved $O(nm^{1/3} + m)$ algorithm with a matching conditional lower bound.
• The research extends the classification of homogeneous regular expressions by depth and type beyond previous work, providing a structured approach to understanding pattern matching complexity.

Overview of Regular Expression Membership Testing Dichotomy

The paper "A Dichotomy for Regular Expression Membership Testing" by Karl Bringmann, Allan Grønlund, and Kasper Green Larsen presents a comprehensive exploration of regular expression membership testing through the lens of computational complexity. Regular expression membership testing, a fundamental problem in computer science, involves determining whether a given string belongs to the language described by a regular expression. While an $O(nm)$ algorithm for general cases has existed since the 1970s, this paper aims to delineate cases where faster algorithms are possible, establish conditional lower bounds, and propose a dichotomy characterizing tractable and intractable instances based on the Strong Exponential Time Hypothesis (SETH).

Contributions and Results

The paper extends the prior work by Backurs and Indyk, who first proposed conditional lower bounds to special cases of regular expression membership testing. Specifically, it provides a dichotomy based on the type and depth of the regular expression:

1. Algorithms and Bounds: It introduces almost-linear time algorithms and establishes matching conditional lower bounds, effectively creating a dichotomy where each case of homogenous regular expressions of bounded depth is categorized into classes solvable in either near-linear time or requiring time at least quadratic, assuming SETH.
2. Word Break Problem: It highlights the Word Break problem as an intermediate complexity case, presenting both an improved algorithm with a runtime of $O(n m^{1/3} + m)$ and a matching conditional lower bound, showing that it stands uniquely between almost-linear and quadratic complexities.
3. Characterization by Type: The researchers classify homogeneous regular expressions by depth and type, further extending the classification beyond depth three previously established by Backurs and Indyk.

Implications

The results presented bolster our understanding of pattern matching complexity, illustrating that within the landscape of regular expressions, certain types offer the potential for significant computational speedups or face inherent barriers, barring advancements in complexity theory or hypothetical breakthroughs. Practically, this dichotomy aids in identifying cases where optimizations are meaningful versus those where efforts may yield diminishing returns due to theoretical hardness.

Future Prospects in AI and Algorithm Design

The paper's approach provides insights into how we can leverage computational complexity assumptions like SETH to guide algorithmic advancements. This approach encourages future research to identify other intermediate or special cases within broader algorithmic challenges and apply similar fine-grained complexity analyses. Furthermore, it opens avenues for exploring "combinatorial" algorithms that sidestep impractical methodologies such as fast matrix multiplication, potentially impacting fields like AI where discrete pattern matching guides learning models.

This systematic layout of complexity for regular expression membership testing serves as an archetype for analyzing computational problems through stratified complexity lenses. As AI continues to integrate nuanced data processing tasks, such rigorous complexity categorizations will be instrumental in refining underlying algorithmic frameworks.