AN EFFICIENT ALGORITHM FOR CHINESE POSTMAN WALK ON BI-DIRECTED DE BRUIJN GRAPHS.

Sequence assembly from short reads is an important problem in biology. It is known that solving the sequence assembly problem exactly on a bi-directed de Bruijn graph or a string graph is intractable. However, finding a shortest double stranded DNA string (SDDNA) containing all the -long words in the reads seems to be a good heuristic to get close to the original genome. This problem is equivalent to finding a cyclic Chinese Postman (CP) walk on the underlying unweighted bi-directed de Bruijn graph built from the reads. The Chinese Postman walk Problem (CPP) is solved by reducing it to a general bi-directed flow on this graph which runs in (|| log(||)) time. In this paper we show that the cyclic CPP on bi-directed graphs can be solved without reducing it to bi-directed flow. We present a Θ((|| + ||) log(||) + ()) time algorithm to solve the cyclic CPP on a weighted bi-directed de Bruijn graph, where = max{|{∣() - () > 0}|, |{∣() - () < 0}|} and = max{∣() - ()}. Our algorithm performs asymptotically better than the bi-directed flow algorithm when the number of nodes is much less than the nodes in the bi-directed graph. From our experimental results on various datasets, we have noticed that the value of /|| lies between 0.08% and 0.13% with 95% probability. Many practical bi-directed de Bruijn graphs do not have cyclic CP walks. In such cases it is not clear how the bi-directed flow can be useful in identifying contigs. Our algorithm can handle such situations and identify maximal bi-directed sub-graphs that have CP walks. A Θ((|| + ||)) time heuristic algorithm based on these ideas has been implemented for the SDDNA problem. This algorithm was tested on short reads from a plant genome and achieves an approximation ratio of at most 1.0134. We also present a Θ((|| + ||) log()) time algorithm for the single source shortest path problem on bi-directed de Bruijn graphs, which may be of independent interest.

Kundeti V ，Rajasekaran S ，Dinh H 《-》

Efficient parallel and out of core algorithms for constructing large bi-directed de Bruijn graphs.

Assembling genomic sequences from a set of overlapping reads is one of the most fundamental problems in computational biology. Algorithms addressing the assembly problem fall into two broad categories - based on the data structures which they employ. The first class uses an overlap/string graph and the second type uses a de Bruijn graph. However with the recent advances in short read sequencing technology, de Bruijn graph based algorithms seem to play a vital role in practice. Efficient algorithms for building these massive de Bruijn graphs are very essential in large sequencing projects based on short reads. In an earlier work, an O(n/p) time parallel algorithm has been given for this problem. Here n is the size of the input and p is the number of processors. This algorithm enumerates all possible bi-directed edges which can overlap with a node and ends up generating Θ(nΣ) messages (Σ being the size of the alphabet). In this paper we present a Θ(n/p) time parallel algorithm with a communication complexity that is equal to that of parallel sorting and is not sensitive to Σ. The generality of our algorithm makes it very easy to extend it even to the out-of-core model and in this case it has an optimal I/O complexity of Θ(nlog(n/B)Blog(M/B)) (M being the main memory size and B being the size of the disk block). We demonstrate the scalability of our parallel algorithm on a SGI/Altix computer. A comparison of our algorithm with the previous approaches reveals that our algorithm is faster--both asymptotically and practically. We demonstrate the scalability of our sequential out-of-core algorithm by comparing it with the algorithm used by VELVET to build the bi-directed de Bruijn graph. Our experiments reveal that our algorithm can build the graph with a constant amount of memory, which clearly outperforms VELVET. We also provide efficient algorithms for the bi-directed chain compaction problem. The bi-directed de Bruijn graph is a fundamental data structure for any sequence assembly program based on Eulerian approach. Our algorithms for constructing Bi-directed de Bruijn graphs are efficient in parallel and out of core settings. These algorithms can be used in building large scale bi-directed de Bruijn graphs. Furthermore, our algorithms do not employ any all-to-all communications in a parallel setting and perform better than the prior algorithms. Finally our out-of-core algorithm is extremely memory efficient and can replace the existing graph construction algorithm in VELVET.

Kundeti VK ，Rajasekaran S ，Dinh H ，Vaughn M ，Thapar V ... -  《BMC BIOINFORMATICS》

On the Hardness of Sequence Alignment on De Bruijn Graphs.

The problem of aligning a sequence to a walk in a labeled graph is of fundamental importance to Computational Biology. For an arbitrary graph and a pattern of length , a lower bound based on the Strong Exponential Time Hypothesis implies that an algorithm for finding a walk in exactly matching significantly faster than time is unlikely. However, for many special graphs, such as de Bruijn graphs, the problem can be solved in linear time. For approximate matching, the picture is more complex. When edits (substitutions, insertions, and deletions) are only allowed to the pattern, or when the graph is acyclic, the problem is solvable in time. When edits are allowed to arbitrary cyclic graphs, the problem becomes NP-complete, even on binary alphabets. Moreover, NP-completeness continues to hold even when edits are restricted to only substitutions. Despite the popularity of the de Bruijn graphs in Computational Biology, the complexity of approximate pattern matching on the de Bruijn graphs remained unknown. We investigate this problem and show that the properties that make the de Bruijn graphs amenable to efficient exact pattern matching do not extend to approximate matching, even when restricted to the substitutions only case with alphabet size four. Specifically, we prove that determining the existence of a matching walk in a de Bruijn graph is NP-complete when substitutions are allowed to the graph. We also demonstrate that an algorithm significantly faster than is unlikely for the de Bruijn graphs in the case where substitutions are only allowed to the pattern. This stands in contrast to pattern-to-text matching where exact matching is solvable in linear time, such as on the de Bruijn graphs, but approximate matching under substitutions is solvable in subquadratic time, where is the text's length.

Gibney D ，Thankachan SV ，Aluru S 《-》

Read mapping on de Bruijn graphs.

Next Generation Sequencing (NGS) has dramatically enhanced our ability to sequence genomes, but not to assemble them. In practice, many published genome sequences remain in the state of a large set of contigs. Each contig describes the sequence found along some path of the assembly graph, however, the set of contigs does not record all the sequence information contained in that graph. Although many subsequent analyses can be performed with the set of contigs, one may ask whether mapping reads on the contigs is as informative as mapping them on the paths of the assembly graph. Currently, one lacks practical tools to perform mapping on such graphs. Here, we propose a formal definition of mapping on a de Bruijn graph, analyse the problem complexity which turns out to be NP-complete, and provide a practical solution. We propose a pipeline called GGMAP (Greedy Graph MAPping). Its novelty is a procedure to map reads on branching paths of the graph, for which we designed a heuristic algorithm called BGREAT (de Bruijn Graph REAd mapping Tool). For the sake of efficiency, BGREAT rewrites a read sequence as a succession of unitigs sequences. GGMAP can map millions of reads per CPU hour on a de Bruijn graph built from a large set of human genomic reads. Surprisingly, results show that up to 22 % more reads can be mapped on the graph but not on the contig set. Although mapping reads on a de Bruijn graph is complex task, our proposal offers a practical solution combining efficiency with an improved mapping capacity compared to assembly-based mapping even for complex eukaryotic data.

Limasset A ，Cazaux B ，Rivals E ，Peterlongo P ... -  《BMC BIOINFORMATICS》

Bit-parallel sequence-to-graph alignment.

Graphs are commonly used to represent sets of sequences. Either edges or nodes can be labeled by sequences, so that each path in the graph spells a concatenated sequence. Examples include graphs to represent genome assemblies, such as string graphs and de Bruijn graphs, and graphs to represent a pan-genome and hence the genetic variation present in a population. Being able to align sequencing reads to such graphs is a key step for many analyses and its applications include genome assembly, read error correction and variant calling with respect to a variation graph. We generalize two linear sequence-to-sequence algorithms to graphs: the Shift-And algorithm for exact matching and Myers' bitvector algorithm for semi-global alignment. These linear algorithms are both based on processing w sequence characters with a constant number of operations, where w is the word size of the machine (commonly 64), and achieve a speedup of up to w over naive algorithms. For a graph with |V| nodes and |E| edges and a sequence of length m, our bitvector-based graph alignment algorithm reaches a worst case runtime of O(|V|+⌈mw⌉|E| log w) for acyclic graphs and O(|V|+m|E| log w) for arbitrary cyclic graphs. We apply it to five different types of graphs and observe a speedup between 3-fold and 20-fold compared with a previous (asymptotically optimal) alignment algorithm. https://github.com/maickrau/GraphAligner. Supplementary data are available at Bioinformatics online.

Rautiainen M ，Mäkinen V ，Marschall T 《-》