Core Concepts

We present efficient streaming and read-only algorithms for computing the minimal distance from prefixes of a given string to the languages of palindromes and squares, under both Hamming and edit distances.

Abstract

The paper studies the online variant of the language distance problem for two classical formal languages: the language of palindromes (PAL) and the language of squares (SQ). The task is to compute the minimal distance to these languages from every prefix of a given input string T of length n, focusing on the low-distance regime where only distances smaller than a given threshold k need to be computed.
The authors make the following contributions:
Streaming algorithms:
For PAL and SQ, under Hamming distance, the algorithms use O(k polylog n) space and time per character.
For PAL and SQ, under edit distance, the algorithms use O(k^2 polylog n) space and time per character.
The algorithms are randomized and may err with inverse-polynomial probability in n.
Deterministic read-only online algorithms:
For PAL and SQ, under Hamming distance, the algorithms use O(k polylog n) space and time per character.
For PAL and SQ, under edit distance, the algorithms use O(k^4 polylog n) space and amortized time per character.
The key techniques used include Hamming distance sketches, the structure of k-mismatch occurrences, and efficient offline algorithms for pattern matching with k mismatches.

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Key Insights Distilled From

by Gabriel Bath... at **arxiv.org** 05-01-2024

Deeper Inquiries

The efficient online language distance algorithms presented in this work have various potential applications in the field of computational linguistics and bioinformatics.
Spell Checking and Correction: These algorithms can be used to identify and correct spelling errors in text by computing the distance between a misspelled word and a dictionary of correct words. This can improve the accuracy of spell-checking tools.
Plagiarism Detection: By comparing the language distance between a given text and a database of existing documents, these algorithms can help in detecting instances of plagiarism or unauthorized copying.
Genomic Sequence Analysis: In bioinformatics, these algorithms can be applied to compare DNA or protein sequences to identify similarities or differences, aiding in genetic research and evolutionary studies.
Information Retrieval: The algorithms can be used in search engines to improve the relevance of search results by considering the language distance between search queries and indexed documents.
Machine Translation: By measuring the language distance between source and target languages, these algorithms can enhance the accuracy and quality of machine translation systems.

The techniques used in this paper can be extended to handle other formal languages beyond palindromes and squares by adapting the algorithms to suit the specific properties and structures of those languages. Here are some ways to extend the techniques:
Regular Languages: For regular languages, deterministic finite automata (DFA) can be used to represent the language, and the algorithms can be modified to work with DFA transitions and states.
Context-Free Languages: For context-free languages, techniques from parsing algorithms like CYK or Earley's algorithm can be incorporated to handle the language distance problem efficiently.
Regular Expressions: The algorithms can be adapted to work with regular expressions, allowing for pattern matching and approximate matching with complex patterns.
Natural Language Processing: Extending the techniques to handle natural language processing tasks like sentiment analysis, named entity recognition, or part-of-speech tagging can provide valuable insights into text analysis and understanding.
Graph Languages: For languages represented as graphs, graph algorithms and traversal techniques can be utilized to compute language distances and similarities.

The streaming and read-only algorithms presented in the paper exhibit trade-offs between space, time, and error probability. Here are some fundamental limitations and trade-offs:
Space Complexity: The streaming algorithms require less space compared to the read-only algorithms but may sacrifice accuracy due to the probabilistic nature of the computations. The trade-off lies in balancing space efficiency with error probability.
Time Complexity: The read-only algorithms may have higher time complexity per character compared to streaming algorithms, especially in the worst-case scenarios. This trade-off between time and space efficiency needs to be considered based on the specific requirements of the application.
Error Probability: The streaming algorithms have an error probability that is inverse-polynomial in the input size, which can impact the accuracy of the results. The trade-off here is between achieving faster computations with a certain level of error tolerance.
Scalability: As the input size increases, the algorithms may face scalability challenges in terms of memory usage and processing time. Balancing scalability with efficiency is crucial in real-world applications.
Complexity vs. Accuracy: There is a trade-off between the complexity of the algorithms and the accuracy of the results. More complex algorithms may provide more accurate distance computations but at the cost of increased computational resources.

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