Abstract
BackgroundChaos Game Representation (CGR) is an iterated function that bijectively maps discrete sequences into a continuous domain. As a result, discrete sequences can be object of statistical and topological analyses otherwise reserved to numerical systems. Characteristically, CGR coordinates of substrings sharing an L-long suffix will be located within 2-L distance of each other. In the two decades since its original proposal, CGR has been generalized beyond its original focus on genomic sequences and has been successfully applied to a wide range of problems in bioinformatics. This report explores the possibility that it can be further extended to approach algorithms that rely on discrete, graph-based representations.ResultsThe exploratory analysis described here consisted of selecting foundational string problems and refactoring them using CGR-based algorithms. We found that CGR can take the role of suffix trees and emulate sophisticated string algorithms, efficiently solving exact and approximate string matching problems such as finding all palindromes and tandem repeats, and matching with mismatches. The common feature of these problems is that they use longest common extension (LCE) queries as subtasks of their procedures, which we show to have a constant time solution with CGR. Additionally, we show that CGR can be used as a rolling hash function within the Rabin-Karp algorithm.ConclusionsThe analysis of biological sequences relies on algorithmic foundations facing mounting challenges, both logistic (performance) and analytical (lack of unifying mathematical framework). CGR is found to provide the latter and to promise the former: graph-based data structures for sequence analysis operations are entailed by numerical-based data structures produced by CGR maps, providing a unifying analytical framework for a diversity of pattern matching problems.
Highlights
Chaos Game Representation (CGR) is an iterated function that bijectively maps discrete sequences into a continuous domain
Graph-based data structures such as suffix trees are heavily applied in sequence comparison and bioinformatics problems, having attained a high performance level
In this work we have shown that CGR can go beyond these applications and demonstrate how typical string operations can be recoded and solved using CGR
Summary
Chaos Game Representation (CGR) is an iterated function that bijectively maps discrete sequences into a continuous domain. These theoretical results allowed for the development of compressed aligners that have significantly reduced the resources necessary for sequence alignment [6,7,8], namely for aligning reads from pyro-sequencing platforms [9] In parallel to these efforts, a particular prolific technique at engendering novel sequence analysis algorithms is the Chaos Game Representation (CGR), based on Iterated Function Systems (IFS), firstly proposed more than two decades ago [10]. This compact, lossless and computationally efficient representation allows the visualization of biological sequences and patterns, with a convenient visual appearance. It was shown that CGR represents a generalization of genomic signatures [13,14], allowing to characterize different species by associating them with distinctive sequence statistics [12]
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