Packed Acyclic Deterministic Finite Automata: A Fast and Space-Efficient Data Structure for Pattern Searching

核心概念

This paper introduces PADFA, a novel data structure for pattern searching that outperforms traditional tries and minimal ADFA's in both speed and memory efficiency by leveraging packed strings and heavy path decomposition.

摘要

Bibliographic Information:

Shibata, H., Ishihata, M., & Inenaga, S. (2024). Packed Acyclic Deterministic Finite Automata. arXiv preprint arXiv:2410.07602.

Research Objective:

This paper introduces a new data structure called Packed Acyclic Deterministic Finite Automata (PADFA) designed to improve the efficiency of pattern searching in large dictionaries. The authors aim to demonstrate the superiority of PADFA over traditional data structures like tries and minimal ADFA's in terms of both speed and memory usage.

Methodology:

The authors propose a method for constructing PADFA from existing ADFA structures by employing techniques like Symmetric Centroid Path Decomposition (SymCPD) to extract heavy paths, which are then stored as packed strings. The remaining light edges are organized using Biased Search Trees (BST) for efficient access. The authors theoretically analyze the time and space complexity of PADFA and compare it with existing approaches. They also conduct experiments on real-world datasets to evaluate the practical performance of PADFA.

Key Findings:

• PADFA achieves near time-optimal pattern searching with a time complexity of O(m/α + log k), where m is the pattern length, α is the number of characters packed into a word, and k is the dictionary size.
• For sufficiently long patterns, PADFA achieves fully time-optimal pattern searching with a complexity of O(m/α).
• PADFA constructed from a minimal ADFA consumes fewer bits than a trie when the dictionary size is relatively smaller than the number of states in the minADFA.
• Empirical results on real-world datasets demonstrate that PADFA improves both space and time efficiency compared to traditional tries and minimal ADFA's.

Main Conclusions:

The authors conclude that PADFA offers a significant advancement in pattern searching by effectively leveraging packed strings and heavy path decomposition. Its superior performance in terms of both speed and memory efficiency makes it a promising alternative to traditional data structures for various applications involving pattern searching.

Significance:

This research contributes to the field of string algorithms and data structures by introducing a novel and efficient approach for pattern searching. The proposed PADFA structure has the potential to improve the performance of various applications that rely on efficient pattern matching, such as information retrieval, natural language processing, and bioinformatics.

Limitations and Future Research:

The paper primarily focuses on the theoretical analysis and empirical evaluation of PADFA for pattern searching. Future research could explore the application of PADFA in other domains like pattern matching and regular expression matching. Additionally, investigating the dynamic update capabilities of PADFA for evolving dictionaries could further enhance its practical applicability.

统计

The alphabet size (σ) for the url dataset is 93.
The url dataset contains 862,665 strings with a total length of 72,540,387 characters and an average length of 84.089 characters.
The alphabet size (σ) for the city dataset is 78.
The city dataset contains 177,030 strings with a total length of 1,970,082 characters and an average length of 11.183 characters.
The alphabet size (σ) for the prot dataset is 25.
The prot dataset contains 157,237 strings with a total length of 46,687,247 characters and an average length of 295.046 characters.

引用

"In this paper, we introduce a packed ADFA (PADFA), the first approach to apply the packing technique to ADFA."
"We theoretically show that a PDFA for any ADFA achieves the time-optimal pattern searching, i.e., O(m/α), if m is sufficiently long compared to k."
"Additionally, we demonstrate that a PDFA for any minADFA consumes fewer bit of memory than a trie if it has a sufficiently large number of states compared to k."

从中提取的关键见解

Packed Acyclic Deterministic Finite Automata

by Hiroki Shiba... 在 arxiv.org 10-11-2024

https://arxiv.org/pdf/2410.07602.pdf

更深入的查询

How does the performance of PADFA compare to other recent advancements in compressed data structures for pattern searching, particularly in scenarios with highly repetitive patterns?

PADFA shines in scenarios with highly repetitive patterns, especially when compared to other compressed data structures. Here's why:

Exploiting Repetitiveness: PADFA leverages the inherent repetitiveness within the data through its use of heavy path decomposition. This technique identifies frequently traversed paths in the underlying ADFA and encodes them as single packed strings. This compression becomes particularly effective when dealing with datasets containing many similar or repeating patterns, leading to:

Reduced Space: Fewer packed strings are needed to represent the dictionary, resulting in significant space savings compared to structures like traditional tries or even compressed tries that don't optimize for repetitive sequences as effectively.
Faster Search: Traversing a single packed string representing a heavy path is significantly faster than navigating multiple individual edges. This translates to substantial speedups in pattern searching, especially for longer patterns frequently occurring within the data.

Comparison with other structures:

Compact Tries/Patricia Tries: While these structures also compress unary paths, they might not achieve the same level of compression as PADFA for datasets with complex repetitive patterns that extend beyond simple unary paths.
CDAWGs (Compact Directed Acyclic Word Graphs): CDAWGs are known for their space efficiency, especially for highly repetitive text data. However, they can be more complex to implement and query compared to PADFAs. The choice between PADFA and CDAWG would depend on the specific application requirements and the trade-off between space efficiency and query time.

Further Advantages in Repetitive Scenarios:

Entropy Compression: As mentioned in the paper, the FID (Fully Indexable Dictionary) used in PADFA can be further compressed using entropy encoding techniques. This additional compression becomes even more effective in scenarios with high repetitiveness, as the distribution of characters and patterns becomes more skewed.
In conclusion, PADFA's ability to effectively exploit repetitiveness through heavy path decomposition and packed strings makes it a strong contender, particularly in scenarios with highly repetitive patterns, potentially outperforming other compressed data structures in terms of both space and time efficiency.

While PADFA demonstrates advantages in speed and space efficiency, could its complex structure pose challenges in terms of implementation complexity and maintainability compared to simpler data structures like tries?

You are right to point out that the complexity of PADFA, while advantageous for performance, could potentially lead to challenges in implementation and maintainability compared to simpler structures like tries.

Implementation Complexity:

Heavy Path Decomposition: Implementing the two-stage HPD algorithm correctly and efficiently requires a good understanding of graph algorithms and data structures.
Packed Strings: Managing packed strings and their operations, while often abstracted by libraries, adds another layer of complexity compared to handling individual characters in tries.
Intertwined Components: The interplay between packed strings, BSTs, and the FID in PADFA necessitates careful implementation to ensure correctness and efficiency.

Maintainability:

Code Readability: The intricate structure of PADFA can make the code harder to understand and debug compared to the straightforward structure of tries.
Modification Challenges: Adding new features or modifying existing functionalities might require a deeper understanding of the underlying algorithms and data structures compared to simpler structures.

Mitigating Factors:

Abstraction and Libraries: Utilizing well-tested libraries for handling packed strings, BSTs, and FIDs can significantly reduce the implementation burden.
Modular Design: Adopting a modular design approach can improve code readability and maintainability by separating the implementation of different components within PADFA.
Thorough Documentation: Comprehensive documentation is crucial for explaining the design choices, algorithms, and data structures used in PADFA, making it easier for others (or even the original developers at a later time) to understand and maintain the code.

Trade-off Considerations:

Performance vs. Complexity: The decision to use PADFA over a simpler trie involves a trade-off between performance gains and increased implementation and maintenance complexity.
Project Requirements: For applications where space and time efficiency are paramount and the development resources allow for managing the complexity, PADFA offers significant advantages. However, for projects with limited resources or where simplicity is prioritized, a trie might be a more practical choice.
In summary, while PADFA's complexity can pose implementation and maintainability challenges, these can be mitigated through careful design, utilization of libraries, and thorough documentation. The decision to employ PADFA ultimately depends on the specific application requirements and the trade-off between performance gains and development overhead.

Considering the increasing prevalence of approximate string matching tasks, could the principles of packed strings and heavy path decomposition used in PADFA be extended to develop efficient data structures for fuzzy pattern searching?

This is an insightful question! While PADFA, as described, focuses on exact string matching, the principles it utilizes – packed strings and heavy path decomposition – hold promising potential for adaptation to fuzzy pattern searching scenarios. Here's how these principles could be extended:

Packed Strings for Approximate Matching:

Word-Level Comparisons: Instead of character-by-character comparisons, packed strings could facilitate word-level or even q-gram comparisons, which are commonly used in approximate string matching algorithms. This could lead to significant speedups, especially for larger edit distances or when searching for patterns with minor variations.
Bit-Parallelism: The inherent bit-level representation of packed strings opens up possibilities for leveraging bit-parallel algorithms, which are known for their efficiency in approximate string matching. Techniques like Shift-Or could be adapted to work with packed strings, enabling faster computation of edit distances or similarity scores.

Heavy Path Decomposition for Fuzzy Search:

Approximate Heavy Paths: The concept of heavy paths could be modified to identify "approximately heavy paths," which represent sequences of nodes frequently traversed even with minor variations allowed in the patterns. This could involve considering edit distances or similarity scores when determining the "heaviness" of a path.
Clustering Similar Paths: Heavy path decomposition could be used to cluster similar paths together, even if they don't share the exact same sequence of characters. This clustering could then be exploited by approximate matching algorithms to reduce the search space and improve efficiency.

Challenges and Considerations:

Increased Complexity: Adapting these principles to fuzzy searching would undoubtedly introduce additional complexity, both in terms of data structure design and query algorithms.
Balancing Act: Finding the right balance between exploiting the efficiency of packed strings and the flexibility required for approximate matching would be crucial.
Theoretical Analysis: Rigorous theoretical analysis would be needed to prove the efficiency and correctness of such adaptations.
In conclusion, while not directly applicable to fuzzy searching in its current form, PADFA's underlying principles of packed strings and heavy path decomposition offer intriguing possibilities for developing efficient data structures for approximate string matching. Further research in this direction could lead to novel and efficient solutions for this increasingly important task.