Abstract
Computing the edit distance between two genomes under certain operations is a basic problem in the study of genome evolution. The doublecutandjoin (DCJ) model has formed the basis for most algorithmic research on rearrangements over the last few years. The edit distance under the DCJ model can be easily computed for genomes without duplicate genes. In this paper, we study the edit distance for genomes with duplicate genes under a model that includes DCJ operations, insertions and deletions. We prove that computing the edit distance is equivalent to finding the optimal cycle decomposition of the corresponding adjacency graph, and give an approximation algorithm with an approximation ratio of 1.5 + ∈.
Introduction
The combinatorics and algorithmics of genomic rearrangements have been the subject of much research since the problem was formulated in the 1990s [1]. The advent of wholegenome sequencing has provided us with masses of data on which to study genomic rearrangements and has motivated further work. Genomic rearrangements include inversions, transpositions, block exchanges, circularizations, and linearizations, all of which act on a single chromosome, and translocations, fusions, and fissions, which act on two chromosomes. These operations are all implemented in terms of the single doublecutandjoin (DCJ) operation [2, 3], which has formed the basis for much algorithmic research on rearrangements over the last few years [4–7]. A DCJ operation makes two cuts in the genome, either in the same chromosome or in two different chromosomes, producing four cut ends, then rejoins the four cut ends.
A basic problem in genome rearrangements is to compute the edit distance, i.e., the minimum number of operations needed to transform one genome into another. For unichromosomal genomes, Hannenhalli and Pevzner gave the first polynomialtime algorithm to compute the edit distance under signed inversions [8], which was later improved to linear time [9]. For multichromosomal genomes, the edit distance under the HannenhalliPevzner model (signed inversions and translocations) has been studied through a series of papers [8, 10–12], culminating in a fairly complex lineartime algorithm [4]; under DCJ operations, the edit distance can be computed in linear time in a simple and elegant way [2].
All of the above algorithms for computing edit distances assume equal gene content and no duplicate genes. ElMabrouk [13] first extended the results of Hannenhalli and Pevzner to compute the edit distance for inversions and deletions. Chen et al. [14] studied the problem of computing the inversion distance for genomes with equal gene content in the presence of duplicate genesa problem that comes up in determining orthologies, where greedy heuristics were used. Yancopoulos et al. [7] proposed some rules on how to incorporate insertions and deletions into the DCJ model, but no specific algorithms are given. Braga et al. [15] presented a lineartime algorithm to compute the edit distance for DCJ operations, insertions and deletions, but still without duplications. Sébastien Angibaud et al. [16, 17] studied several modelfree measures between genomes with duplicate genes; they first established a onetoone correspondence between genes of both genomes, and then computed the measure between two genomes without duplicate genes.
In this paper, we focus on the problem of computing the edit distance between two genomes in the presence of duplications. We define the edit distance at the adjacency set level on a unitcost model including DCJ operations, insertions and deletions (duplications are a special case of insertions). We reduce the problem of computing such an edit distance to finding the maximum number of certain cycles in the adjacency graph, Finally we give a (1.5 + ∈)approximation algorithm.
Edit distance
We represent the genomes using the notations introduced by Bergeron et al. [2]. Denote each gene g with its two extremities, the head as g_{ h } and the tail as g_{ t }. Two consecutive genes a and b can be connected by one adjacency, which is represented by a pair of extremities; thus adjacencies come in four types: a_{ t }b_{ t }, a_{ h }b_{ t }, a_{ t }b_{ h }, and a_{ h }b_{ h } (there is no order for these two extremities, i.e., a_{ h }b_{ t } = b_{ t }a_{ h }). If gene g lies at one end of a linear chromosome, then this end can be represented by a single extremity, g_{ t } or g_{ h }, called a telomere. The adjacencies and telomeres of a genome form a multiset, called the adjacency set.
We define three operations on an adjacency set. The corresponding operations on the structure of the genome (relative positions and orientations of genes on chromosomes) are illustrated on Figure 1.

1.
DCJ (doublecutandjoin) [2], which acts on one or two elements (adjacencies or telomeres) in one of the following ways: {pq, rs} → {pr, qs} or {ps, qr}(see Figure 1(a)); {pq, r} → {pr, q} or {p, qr}(see Figure 1(b)); {p, q} → {pq}or {pq} → {p, q}(see Figure 1(c)).

2.
Insertion, which inserts a single gene (a pair of extremities) g_{ h }g_{ t } in one of the following ways: {pq} → {pg_{ t }, g_{ h }q} or {pg_{ h }, g_{ t }q} (see the upper arrow in Figure 1(d)); {p} → {pg_{ t }, g_{ h }} or {pg_{ h }, g_{ t }} (see the upper arrow in Figure 1(e)); ∅ → {g_{ t }g_{ h }} (see the upper arrow in Figure 1(f)); ∅ → {g_{ t }, g_{ h }} (see the upper arrow in Figure 1(g)).

3.
Deletion, which deletes a single gene g_{ h }g_{ t } in one of the following ways: {pg_{ t }, g_{ h }q} → {pq} (see the lower arrow in Figure 1(d)); {pg_{ t }, g_{ h }} → {p} (see the lower arrow in Figure 1(e)); {g_{ t }g_{ h }} → ∅ (see the lower arrow in Figure 1(f)); {g_{ t }, g_{ h }} → ∅ (see the lower arrow in Figure 1(g)).
The edit distance between two adjacency sets S_{1} and S_{2}, denoted as d(S_{1}, S_{2}), is the minimum number of operations (including DCJ operations, insertions and deletions) that transform S_{1} into S_{2}. Here we use a unitcost model, in which all operations have the same cost.
Note that the edit distance is defined at the adjacency set level. For genomes without duplicate genes, an adjacency set denotes a unique genomic structure. However, for genomes with duplicate genes, two genomes with different structures may share the same adjacency set as illustrated in Figure 2. Thus, d(S_{1}, S_{2}) defined above is a lower bound for the edit distance between the two genomic structures. Given two adjacency sets S_{1} and S_{2} from two genomes, let E_{ i } be the multiset of extremities collected from all elements in S_{ i }, i = 1, 2. We pair extremities in E_{1}\E_{2} into ghost adjacencies (named for the similar ghost genes of [7]) to yield the adjacency set T_{1}; similarly, we produce T_{2} from E_{2}\E_{1}. Clearly, to transform S_{1} into S_{2}, atleast T_{1} deletions and T_{2} insertions are needed. The following theorem shows that these insertions and deletions are both necessary and sufficient.
Theorem 1. Given two adjacency sets S_{1} and S_{2}, there exists an optimal series of operations with exactly T_{1} deletions, exactly T_{2} insertions and some DCJ operations that transforms S_{1} into S_{2}.
Proof. We prove this theorem by contradiction. Suppose that every optimal series of operations contains more than T_{1} deletions and more than T_{2} insertions. Assume that O_{1}O_{2} ... O_{ m } is an optimal series of operations that contains a minimum number of insertions and deletions. Let S^{0}S^{1}S^{2} ... S^{m} be the trace of S_{1} in the process of transformation, where S^{0} = S_{1} and S^{m} = S_{2}. Note that for any insertion (or deletion) beyond the T_{1} deletions and T_{2} insertions, there must be a matching deletion (or insertion) to preserve gene content. Thus every optimal series of operations has at least a pair of insertion and deletion on the same gene. Without loss of generality, assume O_{ i } inserts a pair of extremities g_{ h }g_{ t } and O_{ j } deletes g_{ h }g_{ t } (i <j), and operations between O_{ i } and O_{ j } do not contain insertion or deletion on g_{ h }g_{ t }. Now we will build a new series of operations {O}_{i}^{\prime}{O}_{i+1}^{\prime}\dots {O}_{j}^{\prime} without the pair of insertion and deletion on g_{ h }g_{ t } to replace O_{ i } ... O_{ j }, which produce the trace {S}^{{i}^{\prime}}{S}^{i+{1}^{\prime}}\cdots {S}^{{j}^{\prime}} and satisfy {S}^{{j}^{\prime}}={S}^{j}. This process is shown in Figure 3. Denote the two extremities inserted in O_{ i } as {g}_{h}^{*} and {g}_{t}^{*} to distinguish them from other g_{ h } and g_{ t }. For k = i, ..., j 1, we will keep the invariant {S}^{k{1}^{\prime}}=\left({S}^{k}\backslash \left\{{p}^{k}{g}_{h}^{*},{q}^{k}{g}_{t}^{*}\right\}\right)\cup \left\{{p}^{k}{q}^{k}\right\}, where p^{k} (q^{k}) is the extremity that shares an adjacency with {g}_{h}^{*}\left({g}_{t}^{*}\right)in S^{k}. Note that p^{k} or q^{k} might be empty if {g}_{h}^{*}or {g}_{t}^{*} forms a telomere, or {g}_{h}^{*}{g}_{t}^{*}forms an adjacency in S^{k}. Clearly this holds for k = i, since we have both {S}^{i{1}^{\prime}}={S}^{i1} and {S}^{i}=\left({S}^{i1}\backslash \left\{{p}^{i}{q}^{i}\right\}\right)\cup \left\{{p}^{i}{g}_{h}^{*},{q}^{i}{g}_{t}^{*}\right\}. To make this invariant hold for k = i + 1, ..., j  1, our new operation {O}_{k1}^{\prime} will mimic operation O_{ k } as follows: if O_{ k } does not affect the adjacencies or telomeres containing {g}_{h}^{*} or {g}_{t}^{*}, then set {O}_{k1}^{\prime}={O}_{k}, and the invariant holds; if O_{ k } acts on at least one of {g}_{h}^{*} or{g}_{t}^{*}, we will build {O}_{k1}^{\prime} from O_{ k } by replacing {g}_{h}^{*}\left({g}_{t}^{*}\right)with p^{k} (q^{k}) in O_{ k }. For example, if O_{ k } is the DCJ operation given by \left\{{p}^{k1}{g}_{h}^{*},cd\right\}\to \left\{{p}^{k1}c,{g}_{h}^{*}d\right\}, then {O}_{k1}^{\prime} would be {p^{k}^{1}q^{k}^{1}, cd} → {p ^{k}^{1}c, q^{k}^{1}d}.
Since O_{ k } does not affect, {g}_{t}^{*} we have q^{k} = q^{k}^{1}. Besides, we have p^{k} = d. Thus we have {S}^{k}\backslash \left\{{p}^{k}{g}_{h}^{*},{q}^{k}{g}_{t}^{*}\right\}\cup \left\{{p}^{k}{q}^{k}\right\}={S}^{k{1}^{\prime}}. Other types of operations can be expressed similarly.
Recall that O_{ j } is a deletion, i.e., {ag_{ h }, bg_{ t }} → {ab}. If g_{ h } and g_{ t } are the same as {g}_{h}^{*} and, {g}_{t}^{*} then we have {S}^{j{2}^{\prime}}={S}^{j}, and we can skip {O}_{j1}^{\prime} and {O}_{j}^{\prime} in our constructed series. If g_{ h } and g_{ t } are different from {g}_{h}^{*} and, {g}_{t}^{*} then we have \left\{a{g}_{h,}b{g}_{t},{p}^{j1}{g}_{h}^{*},{q}^{j1}{g}_{t}^{*}\right\}\subset {S}^{j1}. We can set {O}_{j1}^{\prime} to be {ag_{ h }, bg_{ t }} → {ab, g_{ h }g_{ t }}, and set {O}_{j}^{\prime} to be {p^{j}^{1}q^{j}^{1}, g_{ h }g_{ t }} → {p^{j}^{1}g_{ h }, q^{j}^{1}g_{ t }}. We can verify {S}^{{j}^{\prime}}={S}^{j}, and our constructed series contradicts the optimality of {O}_{1}{O}_{2}\cdots {O}_{m}.
Adjacency graph decomposition
Given two adjacency sets S_{1} and S_{2} from two genomes, their corresponding adjacency graph is defined as a bipartite multigraph, A = {S_{1} ∪ T_{2}, S_{2} ∪ T_{1}, E},in which u ∈ S_{1} ∪ T_{2} and v ∈ S_{2} ∪ T_{1} are linked by one edge if u and v share one extremity, by two edges if they share two extremities. Note that S_{1} ∪ T_{2} and S_{2} ∪ T_{1} have the same set of extremities; we use n to denote half of the number of extremities. In the case of genomes with the same gene content and without duplicate genes, T_{1} = T_{2} = ∅, and each vertex in the adjacency graph has degree 2, which means that the adjacency graph consists of vertexdisjoint cycles and paths. We define the length of a cycle or a path to be the number of edges it contains. Based on Theorem 1, T_{1} = T_{2} = ∅ implies there exists an optimal solution without insertion and deletion, thus d(S_{1}, S_{2}) is just the minimum number of DCJ operations needed to transform S_{1} into S_{2}. When S_{1} has been transformed into S_{2}, the corresponding adjacency graph only consists of cycles of length 2 and paths of length 1. Since each DCJ operation can increase the number of cycles at most by 1, or increase the number of oddlength paths at most by 2, and we can always find out this kind of operation when S_{1} and S_{2} are different, we have d(S_{1}, S_{2})= n  c o/2, where c is the number of cycles and o is the number of oddlength paths in the adjacency graph [2].
In the presence of duplicate genes, the adjacency graph may contain vertices with degree larger than 2, so that there may be multiple ways of choosing vertexdisjoint cycles and paths that cover all vertices as illustrated in Figure 4. We say that a cycle (or path) in the adjacency graph is alternating if no two adjacent edges in this cycle (or path) share the same extremity. A valid decomposition of the adjacency graph is a set of vertexdisjoint alternating cycles and paths that cover all vertices. We say that a cycle of length ℓ is helpful if at most ℓ/2  1 vertices are ghost adjacencies, unhelpful if at least ℓ/2 vertices are ghost adjacencies. In fact, an unhelpful cycle has exactly ℓ/2 ghost adjacencies (all in T_{1} or all in T_{2}), since adjacencies in T_{1} and adjacencies T_{2} do not have common extremities and thus cannot be linked in the adjacency graph. Now we show how to perform DCJ operations, insertions and deletions to transform S_{1} into S_{2} based on a decomposition of the corresponding adjacency graph.
Lemma 1. Given two adjacency sets S_{1} and S_{2}, and a decomposition D of the adjacency graph A = {S_{1} ∪ T_{2}, S_{2} ∪ T_{1}, E} with c helpful cycles and o oddlength paths, we can perform n  c  o/2 operations to transform S_{1} into S_{2}, among which there are T_{1} deletions, T_{2} insertions and n  c  o/2  T_{1}T_{2} DCJ operations.
Proof. We prove this lemma in a constructive way. We will perform operations under the guidance of the graph decomposition. The goal is to transform the adjacency graph into a collection of cycles of length 2 and paths of length 1 without ghost adjacencies, indicating that S_{1} has been transformed into S_{2}. In the following, we will prove that an unhelpful cycle of length ℓ costs ℓ/2 operations, a path of even length ℓ costs ℓ/2 operations, a helpful cycle of length ℓ costs ℓ/2 1 operations, and a path of odd length ℓ costs (ℓ  1)/2 operations. In other words, a helpful cycle requires one less operation than an unhelpful cycle or an evenlength path of the same length.
For a helpful cycle of length ℓ with d adjacencies in T_{1} and i adjacencies in T_{2}, we first perform d deletions guided by this cycle to reduce the size of the cycle to ℓ  2d. Then for each adjacency in T_{2}, we choose one of its nonghost neighbors in S_{1} and perform an insertion to create one more helpful cycle of length 2. After all adjacencies in T_{2} are handled, we transform the cycle of length ℓ into one of length ℓ  2d  2i without ghost adjacencies, on which finally we can perform ℓ/2  d  i  1 DCJ operations to create ℓ/2  d  i cycles of length 2. An example is shown in Figure 5(a).
For a unhelpful cycle of length ℓ with ℓ/2 adjacencies in T_{1}, we can perform ℓ/2 deletions to remove the adjacencies in S_{1}. For a unhelpful cycle of length ℓ with ℓ/2 adjacencies in T_{2}, we can first insert a gene as initial operand, then perform ℓ/2  1 insertions to create ℓ/2 cycles of length 2see Figure 5(b)(d).
For a path with odd length ℓ, we need (ℓ  1)/2 operations, and for a path with even length ℓ, we need ℓ/2 operationssee Figure 5(c)(e).
In sum, there are T_{1} deletions, T_{2} insertions and n  c  o/2  T_{1}  T_{2} DCJ operations.
Lemma 1 states that any decomposition of the adjacency graph gives an upper bound on the edit distance. The following lemma shows that an optimal decomposition also provides a lower bound.
Lemma 2. d\left({S}_{1},{S}_{2}\right)\ge n{\text{max}}_{D\in \mathcal{D}}\left({c}_{D}+{o}_{D}/2\right), where \mathcal{D}is the space of all decompositions of A = {S_{1} ∪ T_{2}, S_{2} ∪ T_{1}, E}, c_{ D } and o_{D} is the number of helpful cycles and oddlength paths in D, respectively.
Proof. Let {\text{\Delta}}_{P}={\text{max}}_{D\in {\mathcal{D}}^{\u2033}}\left({c}_{D}+{o}_{D}/2\right){\text{max}}_{D\in {\mathcal{D}}^{\prime}}\left({c}_{D}+{o}_{D}/2\right), where {\mathcal{D}}^{\prime} and {\mathcal{D}}^{\u2033} are the space of the decomposition before and after performing operation P, and P ∈ {DCJ, INS, DEL}. By Theorem 1, there exists an optimal series of operations with exactly T_{1} deletions and T_{2} insertions. Summing over all Δ_{ P } for these operations in this optimal solution yields {\sum}_{i=1}^{d\left({S}_{1,}{S}_{2}\right)}{{\text{\Delta}}_{P}}_{{}_{i}}=\left(n\left{T}_{1}\right\right){\text{max}}_{D\in \mathcal{D}}\left({c}_{D}+{o}_{D}/2\right), where (n  T_{1}) is the sum of the number of helpful cycles and half of the number of oddlength paths in the optimal decomposition of the adjacency graph when S_{1} has been transformed into S_{2}. Define δ_{ DCJ } = 1, δ_{ INS } = 1 and δ_{ DEL } = 0. In the following, we will prove Δ_{ P } ≤ δ_{ P }, P ∈ {DCJ, INS, DEL}, which implies that {\sum}_{i=1}^{d\left({S}_{1,}{S}_{2}\right)}{{\text{\Delta}}_{P}}_{{}_{i}}\le d\left({S}_{1},{S}_{2}\right)\left{T}_{1}\right. The combination of these two formulas proves this lemma.
We prove Δ_{ P } ≤ δ_{ P } by contradiction. Let A' and A" be the adjacency graphs before and after performing the operation P. Let σ(A') and σ(A") be the optimal decomposition of A' and A", respectively. Suppose Δ_{ P } >δ_{ P }, namely, (c_{ σ }(_{ A" })+ o_{ σ }(_{ A" })/2)  (c_{ σ }(_{ A′ };)+ o_{ σ }(_{ A' })) >δ_{ P }. Note that P is reversible; we denote the reversed operation as \widehat{P}, and \widehat{P} simultaneously transforms σ(A") into a decomposition of A', denoted γ(A'). Since σ(A') is optimal, we have c_{ σ }(_{ A' })+ o_{ σ }(_{ A' })/2 ≥ c_{ γ }(_{ A' })+ o_{ γ }(_{ A' })/2. Thus, to get the contradiction, we only need to prove (c_{ σ }(_{ A" })+ o_{ σ }(_{ A" })/2)  (c_{ γ }(_{ A' })+ o_{ γ }(_{ A' })/2) ≤ δ_{ P }. Recall that γ(A') is obtained from σ(A") by performing operation \widehat{P}, and both σ(A") and γ(A') are decompositions, which includes only vertexdisjoint cycles and paths.
If P is a DCJ operation, then \widehat{P} is still a DCJ operation. A DCJ operation may merge two cycles into one cycle, split one cycle into two cycles, merge two paths into one path, split one path into two paths, merge one path and one cycle into one path, split one path into one cycle and one path, rearrange two odd(even)length paths into two even(odd) paths or make no change in the number of cycles and oddlength paths. Among those possible operations, the following four cases can reduce the number of helpful cycles or oddlength paths: (i) merge two helpful cycles into one helpful cycle; (ii) merge two oddlength paths into one evenlength path; (iii) rearrange two oddlength paths into two evenlength paths; (iv) merge one helpful cycle and one oddlength path into one oddlength path. For any of these four cases, we have (c_{ σ }(_{ A" })+ o_{ σ }(_{ A" })/2)  (c_{ γ }(_{ A' })+ o_{ γ }(_{ A' })/2) = 1. For other possible DCJ operations, we have (c_{ σ }(_{ A" })+ o_{ σ }(_{ A" })/2)  (c_{ γ }(_{ A' })+ o_{ γ }(_{ A' })/2) ≤ 0.
If P is an insertion, then \widehat{P} is a deletion. Similarly, among all possible deletions, the following five cases can reduce the number of helpful cycles or oddlength paths: (i) merge two helpful cycles into one helpful cycle; (ii) merge two oddlength paths into one evenlength path; (iii) rearrange two oddlength paths into two evenlength paths; (iv) merge one helpful cycle and one oddlength path into one oddlength path; (v) change a helpful cycle into an unhelpful one. For any of these five cases, we have (c_{ σ }(_{ A" })+ o_{ σ }(_{ A" })/2)  (c_{ γ }(_{ A' })+ o_{ γ }(_{ A' })/2) = 1. For other possible deletions, we have (c_{ σ }(_{ A" })+ o_{ σ }(_{ A" })/2)  (c_{ γ }(_{ A' })+ o_{ γ }(_{ A' })/2) ≤ 0.
If P is a deletion, then \widehat{P} is an insertion. A insertion may split one cycle into two cycles, split one path into two paths, or split one path into one cycle and one path. All these possible insertions will not reduce the number of helpful cycles or oddlength paths. Thus, any deletion will not increase the number of helpful cycles or the number of oddlength paths, and we have c_{ σ }(_{ A" })+ o_{ σ }(_{ A" })/2 ≤ c_{ γ }(_{ A' })+ o_{ γ }(_{ A' })/2. □
Combining Lemma 1 and Lemma 2, we have the following theorem.
Theorem 2. d\left({S}_{1},{S}_{2}\right)=n{\displaystyle {\text{max}}_{D\in \mathcal{D}}}\left({c}_{D}+{o}_{D}/2\right), where \mathcal{D} is the space of all decompositions of A= {S_{1} ∪ T_{2}, S_{2} ∪ T_{1}, E}, c_{ D } and o_{ D } are the numbers of helpful cycles and oddlength paths in D, respectively.
Approximation algorithm
We design an approximation algorithm by using techniques employed on the problem of BREAKPOINT GRAPH DECOMPOSITION[5, 6, 18–20]. The basic idea is to find the maximum number of vertexdisjoint helpful cycles of length 4 in the adjacency graph. This problem can be reduced to the problem of KSET PACKING problem with k = 4, for which the besttodate algorithm has an approximation ratio of 2 + ∈ [21, 22].
To make use of such algorithm, we must remove telomeres and keep only cycles in the adjacency graph. This can be done by introducing null extremities τ and null adjacencies ττ, which are different from other extremities and adjacencies (the same definition is introduced in [7]). Given two adjacency sets S_{1} and S_{2} with 2k_{1} and 2k_{2} telomeres respectively, we replace each telomere x by the adjacency xτ. If we additionally have k_{1} <k_{2}, we must add (k_{2}  k_{1}) null adjacencies ττ to S_{1} in order to balance the degrees. The corresponding adjacency graph is constructed in the same way as the case without null extremities: two adjacencies are linked by one edge if they share one extremity, by two edges if they share two extremities. Now we prove that this "telomereremoval" operation does not change d(S_{1}, S_{2}).
Theorem 3. Let S_{1} and S_{2} be two adjacency sets and denote by {S}_{1}^{\prime} and {S}_{2}^{\prime}the adjacency sets obtained from S_{1} and S_{2} by removing telomeres. Then we can write d\left({S}_{1},{S}_{2}\right)=d\left({S}_{1}^{\prime},{S}_{2}^{\prime}\right).
Proof. We first prove d\left({S}_{1},{S}_{2}\right)\ge d\left({S}_{1}^{\prime},{S}_{2}^{\prime}\right). Let A = {S_{1} ∪ T_{2}, S_{2} ∪ T_{1}, E} be the adjacency graph with respect to S_{1} and S_{2} and σ(A) be the optimal decomposition of A. Let A\prime =\left\{{S}_{\mathsf{\text{1}}}^{\prime}\cup {T}_{\mathsf{\text{2}}},{S}_{2}^{\prime}\cup {T}_{\mathsf{\text{1}}},E\right\} be the adjacency graph with respect to {S}_{1}^{\prime} and {S}_{2}^{\prime} and σ(A') be the optimal decomposition of A'. Suppose σ(A) contains c helpful cycles, o oddlength paths and e evenlength paths, and among these e evenlength paths, e_{1} of them contain two telomeres in S_{1} and e_{2} of them contain two telomeres in S_{2}. Suppose S_{1} and S_{2} contains 2k_{1} and 2k_{2} telomeres respectively (w.l.o.g., assume k_{1} ≤ k_{2}). Since an oddlength path contains one telomere in each adjacency set while an evenlength path contains two telomeres in one adjacency set, we have o + 2e_{1} = 2k_{1} and o + 2e_{2} = 2k_{2}. We can perform the following modifications on σ(A) to transform it into a decomposition of A' (see Figure 6). Nothing needs to be done for cycles. For oddlength paths, link their two telomeres to form a helpful cycle; for each evenlength path with both telomeres in S_{1}, arbitrarily choose one evenlength path with both telomeres in S_{2} and link these two paths to form a helpful cycle; for the remaining e_{2}  e_{1} evenlength paths, use e_{2}  e_{1} = k_{2}  k_{1} null adjacencies ττ to transform each such path into a helpful cycle. Thus, there are c + e_{2} helpful cycles in this decomposition of A', so that the upper bound on d\left({{S}^{\prime}}_{1},{{S}^{\prime}}_{2}\right) is (n + k_{2})  c e_{2} = n  c  o/2 = d(S_{1}, S_{2}). Now we prove d\left({S}_{\mathsf{\text{1}}},{S}_{\mathsf{\text{2}}}\right)\le d\left({S}_{\mathsf{\text{1}}}^{\prime},{S}_{\mathsf{\text{2}}}^{\prime}\right). Note that σ(A') only consists of vertexdisjoint cycles, and unhelpful cycles cannot contain any null extremity. We claim that, for each helpful cycle in σ(A'), there must be no more than two null extremities τ on each side. Otherwise, we can always choose two nonadjacent edges that are linked through τ, exchange four ends of them, and divide this cycle into two cycles (see Figure 7), contradicting the optimality of σ(A'). Now we transform σ(A') into a decomposition of A by recovering all removed telomeres (see Figure 6). Each cycle falls into one of three cases: (a) it contains one xτ adjacency on each side, then the recovery will yield one oddlength path; (b) it contains one ττ adjacency on one side, then the recovery will yield one evenlength path; (c) it contains two xτlike adjacencies on each side, then the recovery will yield two evenlength paths. In all three cases the value n  c  o/2 remains unchanged, and after the recovery we obtain a decomposition of A. Thus we have d\left({S}_{\mathsf{\text{1}}},{S}_{\mathsf{\text{2}}}\right)\le d\left({S}_{\mathsf{\text{1}}}^{\prime},{S}_{\mathsf{\text{2}}}^{\prime}\right). □
In summary, based on Theorems 2 and 3, we have stated the equivalence of the problem of computing the edit distance and that of finding a valid decomposition with a maximum number of helpful cycles in an adjacency graph without telomeres. The latter problem is NPhard by a reduction from the NPhard problemBREAKPOINT GRAPH DECOMPOSITION[23], since any instance of the BREAKPOINT GRAPH DECOMPOSITION is indeed an adjacency graph without ghost adjacencies. Thus, the problem of computing the edit distance is also NPhard.
Now we give the approximation algorithm and prove that its approximation ratio is 1.5 + ∈.
Approximation Algorithm
Input: Two adjacency sets S_{1} and S_{2} from two genomes
Output: A series of operations to transform S_{1} into S_{2}.
Step 1 Add null adjacencies to S_{1} and S_{2} to obtain {S}_{\mathsf{\text{1}}}^{\prime} and {S}_{\mathsf{\text{2}}}^{\prime} without telomeres. Build the adjacency graph A\prime =\left\{{S}_{\mathsf{\text{1}}}^{\prime}\cup {T}_{\mathsf{\text{2}}},{S}_{2}^{\prime}\cup {T}_{\mathsf{\text{1}}},E\right\}.
Step 2 Collect all helpful cycles of length 4 in A' as \mathcal{C}. Find a subset \mathcal{S} of \mathcal{C} in which no two cycles share one adjacency using the (2 + ε)approximation algorithm for the KSET PACKING problem with k = 4.
Step 3 Remove the adjacencies covered by cycles in \mathcal{S}. Arbitrarily decompose the remaining part of A' into cycles, denoting this set as {\mathcal{S}}^{\prime}.
Step 4 Remove the null adjacencies of cycles in \mathcal{S}\cup {\mathcal{S}}^{\prime} to obtain a decomposition of A. Transform S_{1} into S_{2} according to Lemma 1 guided by these cycles and paths.
The running time of the above algorithm is dominated by the time complexity of the (2 + ε)approximation algorithm for the KSET PACKING problem with k = 4, which is O\left(\mathcal{C}{}^{{\text{log}}_{4}1/\epsilon}\right) and \left\mathcal{C}\right=O\left({n}^{\mathsf{\text{4}}}\right)[21, 22].
Theorem 4. The approximation ratio of the above algorithm is 1.5 + ε.
Proof. Suppose the optimal decomposition of A' contain p helpful cycles of length 4 and q longer helpful cycles. Clearly, we have n ≥ 2p +3q. Based on Theorem 2 and Theorem 3, we know that d(S_{1}, S_{2}) = n  p  q. In the algorithm, we find at least \left\mathcal{S}\righthelpful cycles, which implies that the number of operations that our algorithm outputs is at most n\left\mathcal{S}\right. Since \mathcal{S} is a (2 + ∈)approximation solution, we have \left(2+\epsilon \right)\left\mathcal{S}\right\phantom{\rule{2.77695pt}{0ex}}\ge OPT\ge p, where OPT is the maximum number of independent helpful cycles of length 4 in \mathcal{C}. The approximation ratio is thus
Conclusion
We studied the edit distance problem for two genomes under a unitcost model including DCJ operations, insertions (including duplications) and deletions. We proved that this problem is equivalent to finding maximum number of helpful cycles in the adjacency graph and gave a (1.5 + ∈)approximation algorithm. We made two main assumptions in this work: singlegene insertions and deletions; and unit cost for DCJ operations, insertions and deletions. Both are clearly unrealistic. For example, large segmental duplications are common in many mammalian genomes [24], paracentric rearrangements are more common than pericentric ones, at least in two Drosophila species [25], and short inversions are more common than long ones, in some prokaryotes and in the aforementioned Drosophila [26]. These constraints should be incorporated into our distance computation. Any additional constraint naturally creates complications, but we expect that at least a few natural constraints can be handled within the framework described here.
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Acknowledgements
We thank Bernard Moret for helpful discussions.
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 19, 2012: Proceedings of the Tenth Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/13/S19
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MS and YL conceived the idea, performed the analysis, and wrote the manuscript. All authors read and approved the final manuscript.
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Shao, M., Lin, Y. Approximating the edit distance for genomes with duplicate genes under DCJ, insertion and deletion. BMC Bioinformatics 13 (Suppl 19), S13 (2012). https://doi.org/10.1186/1471210513S19S13
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DOI: https://doi.org/10.1186/1471210513S19S13