Date Published: May 2, 2012
Publisher: BioMed Central
Author(s): Susana Vinga, Alexandra M Carvalho, Alexandre P Francisco, Luís MS Russo, Jonas S Almeida.
Chaos 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.
The 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.
The 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.
Biological sequence analysis remains a central problem in bioinformatics, boosted in recent years by the emergence of new generation sequencing high-throughput techniques [1,2]. With the advances of new technologies, the need for efficient algorithms for string representation and processing is increasing continuously.
We start this section by introducing some notation and recalling the definition and relevant properties concerning CGR. Thereafter, we embed CGR maps in the Cantor set in order to elicit longest common suffixes in an efficient and unambiguous way. As we shall see in the Results section this enables the retrieval of longest common extension queries in constant time, which in turn yield efficient algorithms for more complex string matching problems.
In this section we present several exact and approximate string matching problems, along with their CGR solution. These include the longest common extension problem, as well as finding all palindrome and tandem repeats, exact and approximate string matching, and longest common substring. In addition, we show how CGR can be used as a rolling hash for the Rabin-Karp string search algorithm. A study of the time complexity attained by these algorithms is given throughout this section and a summary in provided in Table 1.
Graph-based data structures such as suffix trees are heavily applied in sequence comparison and bioinformatics problems, having attained a high performance level. This optimization makes them the most common data structures for string-related operations.
The authors declare that they have no competing interests.