 Research article
 Open Access
 Published:
Efficient parallel and out of core algorithms for constructing large bidirected de Bruijn graphs
BMC Bioinformatics volume 11, Article number: 560 (2010)
Abstract
Background
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 bidirected edges which can overlap with a node and ends up generating Θ(n Σ) messages (Σ being the size of the alphabet).
Results
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 outofcore model and in this case it has an optimal I/O complexity of $\Theta (\frac{n\mathrm{log}(n/B)}{B\mathrm{log}(M/B)})$ (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 outofcore algorithm by comparing it with the algorithm used by VELVET to build the bidirected 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 bidirected chain compaction problem.
Conclusions
The bidirected de Bruijn graph is a fundamental data structure for any sequence assembly program based on Eulerian approach. Our algorithms for constructing Bidirected de Bruijn graphs are efficient in parallel and out of core settings. These algorithms can be used in building large scale bidirected de Bruijn graphs. Furthermore, our algorithms do not employ any alltoall communications in a parallel setting and perform better than the prior algorithms. Finally our outofcore algorithm is extremely memory efficient and can replace the existing graph construction algorithm in VELVET.
Background
Sequencing genomes is one of the most fundamental problems in modern Biology and has immense impact on biomedical research. De novo sequencing is computationally more challenging when compared to sequencing with a reference genome. On the other hand the existing sequencing technology is not mature enough to identify/read the entire sequence of the genome  especially for complex organisms like the mammals. However small fragments of the genome can be read with acceptable accuracy. The shotgun sequencing employed in many sequencing projects breaks the genome randomly at many places and generates a large number of small fragments (reads) of the genome. The problem of reassembling all the fragmented reads into a small sequence close to the original sequence is known as the Sequence Assembly (SA) problem.
Although the SA problem seems similar to the Shortest Common Super string (SCS) problem, there are in fact some fundamental differences. Firstly, the genome sequence might contain several repeating regions. However, in any optimal solution to the SCS problem we will not be able to find repeating regions  because we want to minimize the length of the solution string. In addition to the repeats, there are other issues such as errors in reads and double strandedness of the reads which make the reduction to SCS problem very complex.
Existing algorithms to address the SA problem can be classified into two broad categories. The first class of algorithms model a read as a vertex in a directed graph  known as the overlap graph[1]. The second class of algorithms model every substring of length k (i.e., a kmer) in a read as a vertex in a (subgraph of) a de Bruijn graph [2].
In an overlap graph, for every pair of overlapping reads, directed edges are introduced consistent with the orientation of the overlap. Since the transitive edges in the overlap graph are redundant for the assembly process they are removed and the resultant graph is called the string graph[1]. The edges of the string graph are classified into optional, required and exact. The SA problem is reduced to the identification of a shortest walk in the string graph which includes all the required and exact constraints on the edges. Identifying such a walk  minimum Swalk, on the string graph is known to be NPhard [3].
In the de Bruijn graph model every substring of length k in a read acts as a vertex [2]. A directed edge is introduced between two kmers if there exists some read in which these two kmers overlap by exactly k  1 symbols. Thus every read in the input is mapped to some path in the de Bruijn graph. The SA problem is reduced to a Chinese Postman Problem (CPP) on the de Bruijn graph, subject to the constraint that the resultant CPP tour include all the paths corresponding to the reads. This problem (Shortest Super Walk) is also known to be NPhard. Thus solving the SA problem exactly on both of these graph models is intractable.
Overlap graph based algorithms were found to perform better (see [4–7]) with Sanger based read methods. Sanger methods produce reads typically around 1000 base pairs long and are very reliable. Unfortunately Sanger based methods are very expensive. To overcome the issues with Sanger reads, new read technologies such as sequencing by synthesis (Solexa) and pyrosequencing (454 sequencing) have emerged. These rapidly emerging read technologies are collectively termed as the Next Generation Sequencing (NGS) technologies. These NGS technologies can produce shorter genome fragments (anywhere from 25 to 500 basepairs long). NGS technologies have drastically reduced the sequencing cost per basepair when compared to Sanger technology. The reliability of NGS technology is acceptable, although it is relatively lower than the Sanger technology. In the recent past, the sequencing community has witnessed an exponential growth in the adoption of these NGS technologies.
On the other hand these NGS technologies can increase the number of reads in the SA problem by a large magnitude. The computational cost of building an overlap graph on these short reads is much higher than that of building a de Bruijn graph. De Bruijn graph based algorithms handle short reads very efficiently (see [8]) in practice compared to the overlap graph based algorithms. However the major bottleneck in using de Bruijn graph based assemblers is that they require a large amount of memory to build the de Bruijn graph.
This limits the applicability of these methods to large scale SA problems. In this paper we address this issue and present algorithms to construct large de Bruijn graphs very efficiently. Our algorithm is optimal in the sequential, parallel, and outofcore models. A recent work by Jackson and Aluru [9] yielded parallel algorithms to build these de Bruijn graphs efficiently. They present a parallel algorithm that runs in O(n/p) time using p processors (assuming that n is a constantdegree polynomial in p). The message complexity of their algorithm is Θ(n Σ). By message complexity we mean the total number of messages (i.e., kmers) communicated by all the processors in the entire algorithm. The distributed de Bruijn graph building algorithm in ABySS [10] is similar to the algorithm of [9].
One of the major contributions of our work is to show that we can build a bidirected de Bruijn graph in Θ(n/p) time with a message complexity of Θ(n). An experimental comparison of these two algorithms on an SGI Altix machine shows that our algorithm is considerably faster. In addition, our algorithm works optimally in an outofcore setting. In particular, our algorithm requires only $\Theta (\frac{n\mathrm{log}(n/B)}{B\mathrm{log}(M/B)})$ I/O operations.
Methods
Preliminaries
Let s ∈ Σ ^{n} be a string of length n. Any substring s_{ j } (i.e., s[j]s[j + 1] . . . s[j + k  1], n  k + 1 ≥ j ≥ 1) of length k is called a k mer of s. The set of all k mers of a given string s is called the kspectrum of s and is denoted by $\mathbb{S}$ (s, k). Given a kmer s_{ j } , ${\overline{s}}_{j}$ denotes the reverse complement of s_{ j } (e.g., if s_{ j } = AAGTA then ${\overline{s}}_{j}=TACTT$). Let ≤ be the partial ordering among the strings of equal length such that s_{ i } ≤ s_{ j } indicates that the string s_{ i } is lexicographically smaller than s_{ j } . Given any kmer s_{ i } , let ${\widehat{s}}_{i}$ be the lexicographically smaller string between s_{ i } and ${\overline{s}}_{j}$. We call ${\widehat{s}}_{i}$ the canonical kmer of s_{ i } . In other words, if ${s}_{i}\le {\overline{s}}_{i}$ then ${\widehat{s}}_{i}={s}_{i}$ otherwise ${\widehat{s}}_{i}={\overline{s}}_{i}$. A kmolecule of a given kmer s_{ i } is a tuple $({\widehat{s}}_{i},{\overline{\widehat{s}}}_{i})$ consisting of the canonical kmer of s_{ i } and the reverse complement of the canonical kmer. We refer to the reverse compliment of the canonical kmer as noncanonical kmer.
A bidirected graph is a generalized version of a standard directed graph. In a directed graph every edge has only one arrow head (▷ or ◁). On the other hand in a bidirected graph every edge has two arrow heads attached to it (◁▷,◁◁,▷◁, or ▷▷). Let V be the set of vertices and E = {(v_{ i }, v_{ j } , o_{ 1 }, o_{2})v_{ i }, v_{ j } ∈ V Λ o_{ 1 }, o_{2} ∈ {◁, ▷}} be the set of bidirected edges in a bidirected graph G(V, E). For any edge e = (v_{ i }, v_{ u }, o_{ 1 }, o_{2}) ∈ E, o_{ 1 } = e[o^{+}] and o_{2} = e[o^{} ] refer to the orientations of the arrow heads on the vertices v_{ i } and v_{ j } , respectively. A walk W (v_{ i }, v_{ j } ) between two nodes v_{ i }, v_{ j } ∈ V of a bidirected graph G(V, E) is a sequence ${v}_{i},{e}_{{i}_{1}},{v}_{{i}_{1}},{e}_{{i}_{2}},{v}_{{i}_{2}},\dots ,{v}_{{i}_{m}},{e}_{{i}_{m+1}},{v}_{j}$, such that for every intermediate vertex ${v}_{i}{{l}_{}}_{}$, 1 ≤ l ≤ m the orientation of the arrow head on the incoming edge adjacent on ${v}_{i}{{l}_{}}_{}$ should match the orientation of the arrow head on the outgoing edge. To make this clearer, let ${e}_{i}{{l}_{}}_{},{v}_{i}{{l}_{}}_{},{e}_{i}{{l+1}_{}}_{}$ be a subsequence in the walk W (v_{ i }, v_{ j } ). If ${e}_{i}{{}_{l}}_{l}=({v}_{i}{{l1}_{l}}_{l},{v}_{i}l{{l}_{}}_{},{o}_{1},\text{}{o}_{\text{2}})$ and ${e}_{i}{{l+1}_{}}_{}=({v}_{i}{{l}_{}}_{},{v}_{i}{{l+1}_{}}_{},{{o}^{\text{'}}}_{1},{{o}^{\text{'}}}_{2})$ then for the walk to be valid it should be the case that ${o}_{\text{2}}={{o}^{\text{'}}}_{1}$. Figure 1(a) illustrates an example of a bidirected graph. Figure 1(b) shows a simple bidirected walk between the nodes A and E. A bidirected walk between two nodes may not be simple. Figure 1(c) shows a bidirected walk between A and E which is not simple  because B repeats twice.
A de Bruijn graph D^{k} (s) of order k on a given string s is defined as follows. The vertex set V of D^{k} (s) is defined as the kspectrum of s (i.e., $V=\mathbb{S}\left(s,\text{}k\right)$). We use the notation suf(v_{ i }, l) (pre(v_{ i }, l), respectively) to denote the suffix (prefix, respectively) of length l in the string v_{ i } . Let the symbol ◦ denote the concatenation operation between two strings. The set of directed edges E of D^{k} (s) is defined as follows:
E = {(v_{ i }, v_{ j } ) suf(v_{ i }, k  1) = pre(v_{ j } , k  1) Λ v_{ i }[1] ◦ suf(v_{ i }, k  1) ◦ v_{ j } [k] ∈ $\mathbb{S}$(s, k + 1)}. We can also define de Bruijn graphs for sets of strings as follows. If S = {s_{ 1 }, s_{2} . . . s_{ n } } is any set of strings, a de Bruijn graph B^{k} (S) of order k on S has $V={\cup}_{i=1}^{n}\mathbb{S}({s}_{i},k)$ and E = {(v_{ i }, v_{ j } ) suf(v_{ i }, k  1) = pre(v_{ j } , k  1) Λ ∃ l : v_{ i }[1] ◦ suf(v_{ i }, k  1) ◦ v_{ j } [k] ∈ $\mathbb{S}$(s_{ l }, k + 1)}. To model the double strandedness of the DNA molecules we should also consider the reverse complements ($\overline{S}=\{{\overline{s}}_{1},{\overline{s}}_{2},\dots ,{\overline{s}}_{n}\}$) while we build the de Bruijn graph.
To address this a bidirected de Bruijn graph $B{D}^{k}(S\cup \overline{S})$ has been suggested in [3]. The set of vertices V of $B{D}^{k}(S\cup \overline{S})$ consists of all possible kmolecules from S ∪ $\overline{S}$. The set of bi directed edges for $B{D}^{k}(S\cup \overline{S})$ is defined as follows. Let x, y be two kmers which are next to each other in some input string $z\in (S\cup \overline{S})$. Then an edge is introduced between the k molecules v_{ i } and v_{ j } corresponding to x and y, respectively. Please note that two consecutive kmers in some input string always overlap by k  1 symbols. The converse need not be true. The orientations of the arrow heads on the edges are chosen as follows. If the canonical kmers of nodes v_{ i } and v_{ j } overlap then an edge (v_{ i }, v_{ j } ▷, ▷) is introduced. If the canonical kmer of v_{ i } overlaps with the noncanonical kmer of v_{ j } then an edge (v_{ i }, v_{ j }, ▷, ◁) is introduced. Finally if the noncanonical kmer of v_{ i } overlaps with canonical kmer of v_{ j } then an edge (v_{ i }, v_{ j }◁, ▷) is introduced.
Figure 2 illustrates a simple example of the bidirected de Bruijn graph of order k = 3 from a set of reads ATGG, CCAT, GGAC, GTTC, TGGA and TGGT observed from a DNA sequence ATGGACCAT and its reverse complement ATGGTCCAT. Consider two 3molecules v_{ 1 } = (GGA, TCC) and v_{ 2 } = (GAC, GTC). Because the canonical kmer x = GGA in v_{ 1 } overlaps with the canonical kmer y = GAC in v_{ 2 } by string GA, an edge (v_{ 1 }, v_{ 2 } , ▷, ▷) is introduced. Note that the noncanonical kmer GTC in v_{ 2 } also overlaps with the noncanonical kmer T CC in v_{ 2 } by string T C, so the two overlapping strings GA and TC are drawn above the edge (v_{ 1 }, v_{ 2 } , ▷, ▷) in Figure 2. A bidirected walk on the example bidirected de Bruijn graph as illustrated by the dashed line corresponds to the original DNA sequence with the first letter omitted TGGACCAT. We would like to remark that the parameter k is always chosen to be odd to ensure that the forward and reverse complements of a kmer are not the same.
Results
Our algorithm to construct bidirected de Bruijn graphs
In this section we describe our algorithm BiConstruct to construct a bidirected de Bruijn graph on a given set of reads. The following are the main steps in our algorithm to build the bidirected de Bruijn graph. Let R_{ f } = {r_{1}, r_{2} . . . r_{ n } }, r_{ i } ∈ Σ ^{r} be the input set of reads. $\overline{{R}_{f}}=\{\overline{{r}_{1}},\overline{{r}_{2}}\mathrm{...}\overline{{r}_{n}}\}$ is a set of reverse complements. Let $R*\text{}={R}_{f}\cup {\overline{R}}_{f}$ and ${R}^{k}{+\text{1}}^{}={\cup}_{r\in {R}^{*}}\mathbb{S}\left(r,k+1\right)$. R^{k+1}is the set of all (k + 1)mers from the input reads and their reverse complements.

[STEP1] Generate canonical edges: Let (x, y) = (z[1 . . . k], z[2 . . . k + 1]) be the kmers corresponding to a (k + 1)mer z[1 . . . k + 1] ∈ R^{k+1}. Recall that $\widehat{x}$ and ŷ are the canonical k mers of x and y, respectively. Create a canonical bidirected edge $({\widehat{v}}_{i},{\widehat{v}}_{j},{o}_{1},{o}_{2})$ for each (k + 1)mer as follows.
$$(\widehat{{v}_{i}},\widehat{{v}_{j}},{o}_{1},{o}_{2})=\{\begin{array}{l}\begin{array}{cc}(\widehat{x},\widehat{y},\u22b3,\u22b3)& x=\widehat{x},y=\widehat{y}\\ (\widehat{y},\widehat{x},\u22b2,\u22b2)& \begin{array}{l}\text{IF}\widehat{x}\le \widehat{y},\\ \text{ELSE}\end{array}\end{array}\\ \begin{array}{c}x\ne \widehat{x}\wedge y=\widehat{y}\\ (\widehat{x},\widehat{y},\u22b2,\u22b3)& \text{IF}\widehat{x}\le \widehat{y}\end{array}\\ \begin{array}{cc}(\widehat{y},\widehat{x},\u22b2,\u22b3)& \text{ELSE}\end{array}\\ \begin{array}{c}x=\widehat{x}\wedge y\ne \widehat{y}\\ (\widehat{x},\widehat{y},\u22b3,\u22b2)& \text{IF}\widehat{x}\le \widehat{y}\end{array}\\ \begin{array}{cc}(\widehat{y},\widehat{x},\u22b3,\u22b2)& \text{ELSE}\end{array}\\ \begin{array}{c}x\ne \widehat{x}\wedge y\ne \widehat{y}\\ (\widehat{x},\widehat{y},\u22b2,\u22b2)& \text{IF}\widehat{x}\le \widehat{y},\end{array}\\ \begin{array}{cc}(\widehat{y},\widehat{x},\u22b3,\u22b3)& \text{ELSE}\end{array}\end{array}$$ 
[STEP2] Reduce multiplicity: Sort all the bidirected edges in [STEP1], using radix sort. Since the parameter k is always odd this guarantees that a pair of canonical kmers has exactly one orientation. Remove the duplicates and record the multiplicities of each canonical edge. Gather all the unique canonical edges into an edge list ℰ.

[STEP3] Collect bidirected vertices: For each canonical bidirected edge $({\widehat{v}}_{i},{\widehat{v}}_{j},{o}_{1},{o}_{2})\in \mathcal{E}$, collect the canonical kmers ${\widehat{v}}_{i}$, ${\widehat{v}}_{j}$ into list $\mathcal{V}$. Sort the list $\mathcal{V}$ and remove duplicates, such that $\mathcal{V}$ contains only the unique canonical kmers.

[STEP4] Adjacency list representation: The list ℰ is the collection of all the edges in the bidirected graph and the list $\mathcal{V}$ is the collection of all the nodes in the bidirected graph. It is now easy to use ℰ and generate the adjacency lists representation for the bidirected graph. This may require one extra radix sorting step.
Analysis of the algorithm BiConstruct
Theorem 1. Algorithm BiConstruct builds a bidirected de Bruijn graph of order k in Θ(n) time. Here n is number of characters/symbols in the input.
Proof. Without loss of generality assume that all the reads are of the same size r. Let N be the number of reads in the input. This generates a total of (r  k)N (k + 1)mers in [STEP1]. The radix sort needs to be applied in at most 2k log(Σ) passes, resulting in 2k log(Σ)(r  k)N operations. Since n = Nr is the total number of characters/symbols in the input, the radix sort takes Θ(kn log(Σ)) operations assuming that in each pass of sorting only a constant number of symbols are used. If k log(Σ) = O(log N ), the sorting takes only O(n) time. In practice since N is very large in relation to k and Σ, the above condition readily holds. Since the time for this step dominates that of all the other steps, the runtime of the algorithm BiConstruct is Θ(n).
A parallel algorithm for building bidirected de Bruijn graphs
In this section we illustrate a parallel implementation of our algorithm. Let p be the number of processors available. We first distribute N/p reads for each processor. All the processors can execute [STEP1] in parallel. In [STEP2] we need to perform parallel sorting on the list ℰ. Parallel radix/bucket sort  which does not use any alltoall communications can be employed to accomplish this. For example, the integer sorting algorithm of Kruskal, Rudolph and Snir takes $O\left(\frac{n}{p}\frac{\mathrm{log}n}{\mathrm{log}(n/p)}\right)$ time. This will be O(n/p) if n is a constant degree polynomial in p. In other words, for coarsegrain parallelism the run time is asymptotically optimal  which means optimality within a constant. In practice coarsegrain parallelism is what we have. Here n = N (r  k + 1). We call this algorithm ParBiConstruct.
Theorem 2. Algorithm ParBiConstruct builds a bidirected de Bruijn graph in time O(n/p). The message complexity is O(n).
The algorithm of Jackson and Aluru [9] first identifies the vertices of the bidirected graph  which they call representative nodes. Then for every representative node Σ manytomany messages are generated. These messages correspond to potential bidirected edges which can be adjacent on that representative node. A bidirected edge is successfully created if both the representatives of the generated message exist in some processor, otherwise the edge is dropped. This results in generating a total of Θ(nΣ) manytomany messages. The authors in the same paper demonstrate that communicating manytomany messages is a major bottleneck and does not scale well. We remark that the algorithm BiConstruct does not involve any manytomany communications and does not have any scaling bottlenecks.
The algorithm presented in [9] can potentially generate spurious bidirected edges. According to the definition [3] of the bidirected de Bruijn graph in the context of SA problem, a bidirected edge between two kmers/vertices exists if and only if there exists some read in which these two kmers are adjacent. We illustrate this by a simple example. Consider a read r_{ i } = AATGCATC. If we wish to build a bidirected graph of order 3, then AAT, ATG, TGC, GCA, CAT , and ATC form a subset of the vertices of the bidirected graph. In this example we see that the kmers AAT and ATC overlap by exactly 2 symbols. However there cannot be any bidirected edge between them according to the definition  because they are not adjacent. On the other hand the algorithm presented in [9] generates the following edges with respect to the kmer AAT : {(AAT, ATA), (AAT, ATG), (AAT, ATT ), (AAT, ATC)}. The edges (AAT, ATA) and (AAT, ATC) are purged since the kmers ATA and ATC are missing. However bidirected edges with corresponding orientations are established between ATG and ATC. Unfortunately (AAT, ATC) is a spurious edge and can potentially generate wrong assembly solutions. In contrast to their algorithm [9] our algorithm does not use alltoall communications  although we use pointtopoint communications.
Out of core algorithms for building bidirected de Bruijn graphs
Theorem 3. There exists an outofcore algorithm to construct a bidirected de Bruijn graph using an optimal number of I/O ' s.
Proof. Replace the radix sorting with an external Rway merge sort which takes only $\Theta (\frac{n\mathrm{log}(n/B)}{B\mathrm{log}(M/B)})$ I/O' s. Here M is the main memory size, n is the sum of the lengths of all reads, and B is the block size of the disk.
Simplified bidirected de Bruijn graphs
The bidirected de Bruijn graph constructed in the previous section may contain several linear chains. These chains have to be compacted to save space as well as time. The graph that results after this compaction step is referred to as the simplified bidirected graph. A linear chain of bidirected edges between nodes u and v can be compacted only if we can find a valid bidirected walk connecting u and v. All the kmers/vertices in a compactable chain can be merged into a single node, and relabelled with the corresponding forward and reverse complementary strings. In Figure 3 we can see that the nodes X_{1} and X_{3} can be connected with a valid bidirected walk and hence these nodes are merged into a single node. In practice the compaction of chains plays a very crucial role. It has been reported that merging the linear chains can reduce the number of nodes in the graph by up to 30% [8].
Although the bidirected chain compaction problem seems like a list ranking problem there are some fundamental differences. Firstly, a bidirected edge can be traversed in both the directions. As a result, applying pointer jumping directly on a bidirected graph can lead to cycles and cannot compact the bidirected chains correctly. Figure 4 illustrates the first phase of pointer jumping. Pointer jumping is an operation on a directed chain/linked list which changes the neighbour of a list node to its neighbour's neighbour. As we can see, the green arcs indicate valid pointer jumps from the starting nodes. However since the orientation of the node Y_{4} is reverse relative to the direction of pointer jumping a cycle results. In contrast, a valid bidirected chain compaction would merge all the nodes between Y_{1} and Y_{5} since there is a valid bidirected walk between Y_{1} and Y_{5}. On the other hand, bidirected chain compaction may result in changing the orientation of some bidirected edges and these edges should be recognised and updated accordingly. Consider a bidirected chain in Figure 3(a). This chain contains two possible bidirected walks  Y_{2} to Y_{3} and X_{1} to X_{3}, see Figure 3(b). The walk from Y_{3} to Y_{2} (Y_{2} to Y_{3}) spells out a label TAGG(CCTA) after compaction. Once we perform this compaction (see Figure 3(c)) the orientation of the edge between Y_{3} and Y_{4} in the original graph is no longer valid, because the label CCTA on the newly compacted node cannot overlap with the label GGT on the node Y_{4}. However the label TAGG on the newly compacted node overlaps with label GGT on the Y_{4} and hence its orientation should be updated. On the other hand after the edge between X_{1} and Y_{1} does not need any update after compacting the nodes X_{1}, X_{2} and X_{3}, see Figure 3(b) and Figure 3(c).
Since bidirected chain compaction has a lot of practical importance, efficient and correct algorithms are essential. We now provide algorithms for the bidirected chain compaction problem. Our key idea here is to transform a bidirected graph into a directed graph and then apply list ranking. We define the ListRankingTransform as an operation which replaces every bidirected edge with a pair of directed edges with some orientation  see Figure 5 for these orientations.
Given a list of candidate canonical bidirected edges, we apply a ListRankingTransform (see Figure 5) which introduces two new nodes v^{+}, v^{} for every node v in the original graph. Directed edges corresponding to the orientation are introduced. See Figure 5.
Lemma 1. Let BG(V, E) be a bidirected graph; let BG^{t} (V^{t}, E^{t} ) be the directed graph after applying ListRankingTransform. Two nodes u, v ∈ V are connected by a bidirected path iff u^{+} ∈ V^{t} (u^{} ∈ V^{t}) is connected to one of v^{+}(v^{}) or v^{}(v^{+}) by a directed path.
Proof. We first prove the forward direction by induction on the number of nodes in the bi directed graph. Consider the basis of induction when V  = 2, let v_{0}, v_{1} ∈ V. Clearly we are only interested when v_{0} and v_{1} are connected by a bidirected edge. By the definition of ListRankingTransform the Lemma in this case is trivially true. Now consider a bidirected graph with V  = n + 1 nodes, if the path between v_{ i }, i < n and v_{ j } , j < n does not involve node v_{ n } the lemma still holds by induction on the sub bidirected graph BG(V  {v_{n}}, E). Now assume that v_{ i } . . . v_{ p }, v_{ n }, v_{ q } . . . v_{ j } is the bidirected path between v_{ i } and v_{ j } involving the node v_{ n } . (See Figure 6(a)). Figure 6(a) shows what the transformed directed graph looks like. Observe the colors of bidirected edges and the corresponding directed edges. By the induction hypothesis on the sub bidirected paths v_{ i } . . . v_{ p }, v_{ n } and v_{ n }, v_{ q } . . . v_{ j } we have the following. ${v}_{i}^{+}$ is connected to ${v}_{n}^{+}$ or ${v}_{n}^{}$ by some directed path P_{1} (See Figure 6(b)); ${v}_{n}^{+}$ is connected to ${v}_{j}^{+}$ or ${v}_{j}^{}$ by some directed path P_{2}. We examine three possible cases depending on how the directed edge from P_{1} and P_{2} is incident on ${v}_{n}^{+}$ In CASE1 we have both P_{1} and P_{2} pointing into node ${v}_{n}^{+}$. This implies that the orientation of the bidirected edges in the original graph is according to Figure 6(b). In this case we cannot have a bidirected walk involving the node v_{ n } , which contradicts our original assumption. Similarly CASE2 (Figure 6(c)) would also lead to a similar contradiction. Only CASE3 would let node v_{ n } be involved in a bidirected walk. In this case ${v}_{i}^{+}$ will be connected to either ${v}_{j}^{+}$ or ${v}_{j}^{}$ by concatenation of the paths P_{1} and P_{2}. We can make a similar argument to prove the reverse direction.
Algorithm for bidirected chain compaction
Given a bidirected graph we now give an outline of the algorithm which compacts all the valid bidirected chains.

STEP1: Apply the ListRankingTransform for each bidirected edge. Let the resultant directed graph be G(V, E).

STEP2: Identify a subset of nodes V' = {v : v ∈ V , (d_{i n}(v) > 1 or d_{ou t}(v) > 1)} and a subset of edges E' = {(u, v) : (u, v) ∈ E, (u ∈ V' or v ∈ V')}.

STEP3: Apply pointer jumping on the directed graph G(V  V', E  E').

STEP4: Now let $({v}_{i}^{+},\dots ,{v}_{j}^{})$ be a maximal chain obtained after pointer jumping. Due to the symmetry in the graph there exists a corresponding complementary chain $({v}_{j}^{+},\dots ,{v}_{i}^{})$. Each chain is replaced with a single node and its label is the concatenation of all the labels in the chain. We should stick to some convention while we give a new number to the newly created node. For example, we can choose min{v_{ i }, v_{ j }} as the new number for the newly created node. In our example if v_{ i }= min{v_{ i }, v_{ j }} then we replace the chain $({v}_{i}^{+},\dots ,{v}_{j}^{})$ with node ${v}_{i}^{+}$ and relabel it with the concatenated label. Similarly, the chain $({v}_{j}^{+},\dots ,{v}_{i}^{})$ will be replaced with the node ${v}_{i}^{}$ and relabeled accordingly.

STEP5: Finally, to maintain the connectivity we need to update the edges in E' to reflect the changes during compaction. Coming back to our example, we have replaced the chain $({v}_{i}^{+},\dots ,{v}_{j}^{})$ with the node ${v}_{i}^{+}$. As a result we need to replace any edge $(x,{v}_{j}^{})\in E\text{'}$ with the edge $(x,{v}_{i}^{+})$. Similarly we also need to update any edges adjacent on ${v}_{j}^{+}$.
Note that all of the above steps can be accomplished with some constant number of radix sorting operations. Figure 3 illustrates the compaction algorithm on a bidirected graph. The red nodes in Figure 3 indicate the nodes in the set V'. Red colored edges indicate the edges in E'. After list ranking we will have four maximal chains as follows: $({Y}_{2}^{},{Y}_{3}^{+}),({Y}_{3}^{},{Y}_{2}^{+}),({X}_{1}^{},{X}_{2}^{+},{X}_{3}^{})({X}_{3}^{+},{X}_{2}^{},{X}_{1}^{+})$. Now if we stick to the convention described in STEP5 we renumber the new node corresponding to the chains $({Y}_{2}^{},{Y}_{3}^{+}),({Y}_{3}^{},{Y}_{2}^{+})$ as Y_{2}. As a result the edges $({Y}_{3}^{+},{Y}_{4}^{})$ and $({Y}_{4}^{+},{Y}_{3}^{})$ are updated (shown in green in Figure 3(c)) as $({Y}_{2}^{},{Y}_{4}^{})$ and $({Y}_{4}^{+},{Y}_{2}^{+})$. Finally the directed edges are replaced with bidirected edges in Figure 3(d).
Analysis of bidirected compaction on parallel and outofcore models
Let ℰ _{ l } be the list of candidate edges for compaction. To do compaction in parallel, we can use a Segmented Parallel Prefix[11] on p processors to accomplish this in time $O(2{\mathcal{E}}_{l}/p+\mathrm{log}(p))$. List ranking can also be done outofcore as follows. Without loss of generality we can treat the input for the list ranking problem as a set S of ordered tuples of the form (x, y). Given S we create a copy and call it S'. We now perform an external sort of S and S' with respect to y (i.e., using the y value of tuple (x, y) as the key) and x, respectively. The two sorted lists are scanned linearly to identify tuples (x, y) ∈ S, (x',y') ∈ S' such that y =x'. These two tuples are merged into a single tuple (x, y') and is added to a list ${\mathcal{E}}_{l}{\text{'}}^{}$. This process is now repeated on ${\mathcal{E}}_{l}{\text{'}}^{}$. Note that if the underlying graph induced by ℰ _{ l } does not have any cycles then ${\mathcal{E}}_{l}^{\text{'}}\le {\mathcal{E}}_{l}/2$; which means that the size of ${\mathcal{E}}_{l}{\text{'}}^{}$ geometrically decreases after every iteration. The I/O complexity of each iteration is dominated by the external sorting and hence bidirected compaction can be accomplished outofcore with $\Theta ({\mathrm{log}}_{2}(C)\frac{{\mathcal{E}}_{l}}{B}{\mathrm{log}}_{\frac{M}{B}}(\frac{{\mathcal{E}}_{l}}{B}))$ I/O operations, where C is the length of the longest chain. Care should be taken to deal with cycles. The sortmerge algorithm mentioned above will run forever in the presence of cycles. To address this we can maintain what is known as join count in every sortmerge phase. Given any S the join count of a tuple (x, y) ∈ S is an indicator variable J(x, y) = 1, if ∃(y, z) ∈ S; else 0. Finally J(S) = Σ_{(x, y )∈S}J(x, y). Notice that the function J(S) strictly decreases in every sort and merge phase and finally becomes zero when there are no cycles. However in the presence of cycles the function J(S) decreases and then remains constant. Thus if the function J(S) remains constant in any two consecutive sortmerge phases we can stop iterating and report that there are some cycles. Once we stop the list ranking we can easily detect the edges in the cycles. Our implementation of this outofcore list ranking based on this idea is available at http://trinity.engr.uconn.edu/~vamsik/exlistrank.
Improving the construction of the bidirected de Bruijn graph in some practical assemblers
In this section we briefly describe how our algorithms can be used to speedup some of the existing SA programs. As an example, we consider VELVET [8]. VELVET is a suite of programs  velveth and velvetg, which has recently gained acclamation in assembling short reads. The VELVET program builds a simplified bidirected graph from a set of reads. We now briefly describe the algorithm used in VELVET to build this graph. The VELVET program puts all the kmers from the input into a hash table and then identifies the kmers which are present in at least 2 reads  this information is called the roadmap in VELVET' s terminology. The program then builds a de Bruijn graph using these kmers. Finally it takes every read and threads it on these kmers. The worst case time complexity is O(n log(n))  assuming that the implementation of hash table is based on a balanced search tree. However VELVET uses a Splay tree so this would be the amortized runtime rather than the worst case. Since VELVET builds this graph entirely inmemory, this has some serious scalability problems especially on large scale assembly projects. However VELVET has some very good assembly heuristics to remove errors and identify redundant assembly paths, etc. Our outofcore algorithm can act as a replacement to the code in VELVET that performs inmemory graph construction. The internal de Bruijn graph of VELVET is slightly different from the bidirected graph our algorithm builds. It is easy to see the equivalence between these two representations. (See Figure 7). We have implemented an outofcore algorithm that takes in a file with reads and the value of k and generates the graph for VELVET program. To be more precise, VELVET program creates a file with the name Graph in the directory when we run the velvetg program. We have modified the code in the VELVET program by adding an option which quits after it builds the Graph file without any simplification. This gives us a chance to compare the VELVET' s algorithm which can build the Graph file with our algorithm. The results and more details about the program are in the results section.
Discussion
Before we go into the discussion we briefly describe our experimental setting. We used a SGIAltix 64bit, 64 node supercomputer with a RAM of 2 GB per node for our parallel algorithm experiments. For our sequential outofcore experiments we used a 32bit x86 machine with 1 GB of RAM. All our algorithms are implemented in C under Linux environment.
We have compared the performance of our algorithm and that of Jackson and Aluru [9]. We refer to the later algorithm as JA. To make this comparison fair, we have implemented the JA algorithm (because their implementation is unavailable) also on the same platform that our algorithm runs on. We have used the SGI/Altix IA64 machine for all of our experiments. Our implementation uses MPI for communication between the processors. We used a test set of 8 million unpaired reads obtained from sequencing a plant genome at CSHL. The performance of both the algorithms is measured with various values of k (the de Bruijn graph parameter) on multiple processors. Both the algorithms (JA and ParBiConstruct) use the same underlying parallel sorting routines.
Table 1 shows the user and system times for both our algorithm and the JA algorithm. We can clearly see that the system time (communication time) is consistently higher for the JA algorithm. Also notice that as we increase the value of k keeping the number of processors fixed both the algorithms become faster. On the other hand ParBiConstruct is consistently superior than the JA algorithm for all the parameters. Also notice the speedup (time taken by the JA algorithm divided by the time taken by ParBiConstruct) of ParBiConstruct in Table 1. We obtain a maximum speedup of 24.9X on the real time when k = 21 and p = 64. This is clearly because the communication cost is very high and the JA algorithm enumerates all the possible overlaps. Finally, over all the settings for k and p ParBiConstruct is around 8X faster than the JA algorithm with a maximum speedup of 25X when k = 21, p = 64.
The user time of our algorithm is also consistently superior compared to the user time of JA. This clearly is because we do much less local computations. In contrast, JA needs to do a lot of local processing, which arises from processing all the received edges, removing redundant ones, and collecting the necessary edges to perform manytomany communications.
We also compare the memory used by both the algorithms. We briefly describe how we obtained the memory reports in our experiments. Since the memory used by each processor is different during execution at any given instance we addup the memory used by each processor and divide by the number of processors and we report this number in our experiments. We obtained the resident memory and shared memory from the top command and averaged it over the number of memory probe samples obtained by top. Table 2 gives details of the memory usage of both the algorithms. From these results it is clear that our approach is also efficient compared to the JA algorithm. ParBiConstruct uses upto 4X less resident memory compared to the JA algorithm.
Since JA takes a significant amount of time for inputs larger than 8 million, we have compared these algorithms only for input sizes up to 8 million. The experimental results reported in [9] start with at least 64 processors. We however show the scalablity of our algorithm for up to 128 million (randomly generated) reads in Table 3. Table 3 clearly demonstrates the scalability of our algorithm. We make our implementations and all the details of test cases used available at http://trinity.engr.uconn.edu/~vamsik/ParBiDirected.
Outofcore algorithm versus VELVET graph building algorithm
Our aim is to study the computational efficiency of the current VELVET' s algorithm to build the de Bruijn graph and our algorithm. To accomplish this we have modified the code of VELVET to stop once it completes building the graph from the reads. This is done as follows. Firstly we run the velveth program to complete the building of RoadMaps. The code of velvetg is modified such that the program dumps out the Graph file after threading of the reads. Our outofcore algorithm generates Graph file directly by taking the reads file and the value of k. We have used a low end desktop 32bit machine with 1 GB RAM to demonstrate the scalability of our outofcore algorithm. Our results indicate that the VELVET algorithm starts virtual memory trashing[12] for around 4 million reads with k = 21. This trashing leads to massive increase in the pagefaults and stalls the program from progressing further. Thus the VELVET algorithm cannot build large bidirected graphs. In contrast to VELVET our algorithm works with a constant (user specified) amount of memory and scales well for building large amounts of reads  which we demonstrate in Table 4.
The program is available at http://trinity.engr.uconn.edu/~vamsik/exbuildvgraph/.
Conclusions
In this paper we have presented an efficient algorithm to build a bidirected de Bruijn graph, which is a fundamental data structure for any sequence assembly program  based on an Eulerian approach. Our algorithm is also efficient in parallel and out of core settings. These algorithms can be used in building large scale bidirected de Bruijn graphs. Also, our algorithm does not employ any alltoall communications in a parallel setting and performs better than that of [9]. Finally, our outofcore algorithm can build these graphs with a constant amount of RAM, and hence can act as a replacement for the graph construction algorithm employed by VELVET [8].
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Acknowledgements
This work has been supported in part by the following grants: NSF 0326155, NSF 0829916 and NIH 1R01LM01010101A1.
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VK designed and implemented the algorithms, and drafted the manuscript. SR participated in designing the algorithms, revised the manuscript, and coordinated all phases of the project. HD also participated in the design of algorithms and contributed towards the figures in the manuscript. MV and VT introduced the sequence assembly problem to the team and provided short read data to validate our method. All the authors read and approved the final manuscript.
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Kundeti, V.K., Rajasekaran, S., Dinh, H. et al. Efficient parallel and out of core algorithms for constructing large bidirected de Bruijn graphs. BMC Bioinformatics 11, 560 (2010). https://doi.org/10.1186/1471210511560
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Keywords
 Next Generation Sequencing Technology
 List Ranking
 Message Complexity
 Graph File
 Radix Sort