2 Main techniques and algorithm for MWPSM

ICorollary 4. There exists an algorithm which finds an8/3-approximation to MPSM on strings of lengthn over constant-sized alphabets inO(n)time.

ITheorem 5. There exists a single-pass streaming algorithm which finds a 4-approximation to the size of an MPSM on strings of lengthn over alphabets of sizeαusing onlyO(α²lgn) space.

1.4 Preliminaries

Let A = a1a2. . . an and B =b1b2. . . bn be two strings of lengthn with ai and bj being the i^th and j^th characters of their respective strings. A duo D^A_i = (a_i, a_i+1) contains a pair of consecutive characters ai and ai+1. We use D^A = (D₁^A, . . . , D^A_n−1) and D^B = (D^B₁, . . . , D^B_n−1) to denote the sets ofduosforAandB, respectively. We similarly define a triplet T_i^A= (a_i, a_i+1, a_i+2) as a set of three consecutive charactersa_i,a_i+1, anda_i+2 in the string and sets of tripletsT^A= (T₁^A, . . . , T_n−2^A ) andT^B= (T₁^B, . . . , T_n−2^B ) for stringsAand B, respectively. Observe that the duos D_i^AandD^A_i+1 correspond to the first two and last two characters, respectively, of the tripletT_i^A. We refer to duos D^A_i andD_i+1^A as subsetsof the triplet T_i^A.

A proper mappingπfromAto B is a one-to-one mapping from the letters ofAto the letters ofB withai=b_π(i) for alli= 1, . . . , n. Recall that a duo (ai, ai+1) is preserved if and only if a_i is mapped to some b_j anda_i+1 is mapped to b_j+1. We call a pair of duos (D^A_i , D^B_j)preservable if and only ifai=bj andai+1=bj+1. For MWPSM, letw(D_i^A, D^B_j)

be the weight gained by mappingD_i^Ato D_j^B.

For consistency, we define the concept of conflicting pairs of duos using the terminology of [6]. Two preservable pairs of duos (D^A_i , D^B_j) and (D_h^A, D^B<sub>`</sub>) are said to beconflicting if no proper mapping can preserve both of them. These conflicts can be of two types type 1 and type 2. Intype 1 conflicts, eitheri=h∧j6=` ori6=h∧j =`. Intype 2 conflicts, either i=h+ 1∧j6=`+ 1 ori6=h+ 1∧j=`+ 1.

The algorithms here only show how to map the characters of the preserved duos. In all cases, note that any unmapped characters can be mapped arbitrarily to identical characters in the other string in linear time.

(1)

A: . . . A A A C A G T C T. . .

B: . . . A A A G T C A T C. . .

3 5 4 3 1 3 2 1 5

(2)

T^A0:

T^B0:

AAA ACA AGT TCT

AAA AGT TCA ATC

6 4 1 2 5

(3)

T^A0:

T^B00:

AAA ACA AGT TCT

AAG GTC CAT

4 1 3 2 1

Figure 2 Illustration of how to generate an ATM instance from an MWPSM instance. (1) Substrings of the original two strings,AandB, starting at some odd index and featuring weighted edges representing the weight of preserving a pair of duos. (2) The graphG⁰ with thicker edges representing an exact match between two triplets. In the case of multiple edges between a pair of triplets (e.g. the five edges between the “AAA” triplets), we only show the heaviest weight edge. (3) The graphG⁰⁰. Note that that the weight of a mapping which maps the two “AGTC” strings to each other is 6, which can be achieved by a matching inG⁰, but not inG⁰⁰.

D^A_h andD<sub>`</sub><sup>B</sup>withh∈ {i, i+ 1},`∈ {j, j+ 1}, andD_h^A=D^B_{`</sub>, we add an edgee= (T<sub>i</sub><sup>A0</sup>, T<sub>j</sub><sup>B0</sup>) with weightw(e) =w(D<sup>A</sup><sub>h</sub>, D<sup>B</sup><sub>`} ). Additionally, ifT_i^A0 =T_j^B0, we add an edgee= (T_i^A0, T_j^B0) between them with weightw(e) =w(D_i^A, D^B_j) +w(D_i+1^A , D_j+1^B ). In other words, the edge gets the combined weight of the duo pairs preserved by mapping the substringT_i^A0 to the substringT_j^B0. The graphG⁰⁰ is defined similarly. There could be up to five edges total if the triplets contain one letter repeated (e.g. “AAA”). In the case of multiple edges between a pair of triplets, we only need to consider the heaviest edge among them since each triplet can be matched at most once. However, we keep all edges for the sake of simplifying some of the proofs. Figure 2 illustrates the procedure of generating an ATM instance.

2.2 MWPSM algorithm and analysis

Let OP TG⁰ and OP TG⁰⁰ be the weights of maximum weight matchings in G⁰ and G⁰⁰, respectively. Note that we can find these matchings in the time it takes to compute maximum weight bipartite matching. Since our graphs haveO(n) vertices and could haveO(n²) edges,

this takes O(n²lgn+n·n²) = O(n³) time [19]. Lemma 6 states that either OP TG⁰ or OP TG⁰⁰ will be a (3/4)-approximation to the weight of an optimal solution to MWPSM, OP T_{M W P SM}. LetOP T_{AT M} = max(OP T_G⁰, OP T_G⁰⁰).

ILemma 6. OP TAT M ≥(3/4)OP TM W P SM.

Proof. We divide the edges of OP TM W P SM into two partitions. The first partition,P^same, includes mappings, in which both letters occur at odd indices or both letters occur at even indices. The second partition,P^{dif f}, includes the remaining mappings wherein one letter is at an odd index and the other is at an even index (this could be odd fromA, even fromB or even fromA, odd fromB).

Note that the mapping of each preserved pair of duos (D^A_i , D_j^B) will be contained in one of these two partitions. Without loss of generality, let the weight ofP^same be at least the weight ofP^{dif f}. We show how to transformOP TM W P SM into a feasible bipartite matching inG⁰ while retaining the full weight ofP^same and at least half of the weight of P^{dif f}. Thus, we retain at least 3/4 of the weight ofOP TM W P SM.

For each triplet in the vertex set of G⁰ that contains one or two preserved duos from P^same, we can add an edge to our matching with weight equal to the weight of the preserved duos. This works because consecutive pairs of preserved duos (D^A_i , D_j^B) and (D^A_i+1, D^B_j+1) withiandj both being odd will correspond to a “double” edge in the ATM instance with weight equal tow(D^A_i , D^B_j ) +w(D_i+1^A , D_j+1^B ). On the other hand, ifiandj are both even, then the duos of (D_i^A, D_j^B) and (D_i+1^A , D_j+1^B ) are contained in four different triplets and will be added separately. Thus, we can maintain all of the weight ofP^same in a matching inG⁰.

A slightly trickier case arises with P^{dif f}. Any consecutive pairs of preserved duos (D^A_i , D^B_j) and (D^A_i+1, D^B_j+1) in P^{dif f} will have iand j of different parity. This results in the duos being contained in three triplets, two from one partition and one from the other.

That means the edges in the ATM instance capturing the weights of the two pairs will be conflicting. Thus we can only preserve the weight of one of the two pairs in our ATM solution.

To guarantee that we add at least half of the weight ofP^{dif f} to our solution, we further partition it into pairs (D^A_i , D^B_j) with i being odd and those with i being even. Then we simply choose the heavier of those two partitions to add to our ATM solution.

For the case where P^{dif f} is heavier thanP^same, we can do a similar construction forG⁰⁰. Thus, our ATM solution in eitherG⁰ orG⁰⁰ could have at least 3/4 the weight of an optimal

solution to MWPSM. J

We can now show how to transform an optimal solution to ATM (the heavier of the two matchings) into a feasible string mapping which preserves at least half of the weight of the ATM solution. LetG= (D^A, D^B, E) be a bipartite graph on the duos ofAandB with edge weights equal to the weight of preserving each pair of duos. We first show how to convert an ATM solution into a matchingM inG. Then, we show how to resolve conflicts of type 2 (conflicts of type 1 will not arise sinceM is a matching).

The transformation is simply a reversal of how we constructed the ATM graphs. For each edge between triplets in our ATM solution (the heavier of the two matchings inG⁰ andG⁰⁰), we add an edge or edges toM corresponding to the duos that “created” that triplet edge.

To resolve conflicts, we consider the conflict graph Cwherein we have a node for each edge inM and an arc between nodes if their corresponding edges are in conflict. We can prove that C has maximum degree 2, meaning it will be a collection of paths and cycles.

Further, we note that each cycle will have even length due to Lemma 7 and the fact that the underlying graph is bipartite. Thus, for each path or cycle, we choose the heavier of

the two maximal independent sets in that path or cycle to add to our final MPSM solution.

Lemma 7 establishes thatC has maximum degree 2.

ILemma 7. Each edge in M conflicts with at most one other edge at each endpoint.

Proof. First, we note that each duo is contained in at most one triplet edge from the ATM solution and therefore can only be matched once inM. In other words, M is a classical matching in the bipartite graph of duos. This follows from the fact that consecutive triplets in a string starting at only odd (or only even) indices will overlap at exactly one letter.

This ensures that no conflicts of type 1 can arise since that would require a duo to be matched twice. We can also show that at most one conflict of type 2 arises at each endpoint.

Without loss of generality, consider the endpointD^A_i . Consider the duosD_i−1^A andD^A_i+1 where such a conflict might arise. Notice that one of these duos must have come from the same triplet asD^A_i , while the other comes from a different triplet. The duo from the same triplet will either be unmatched or matched as a non-conflicting parallel edge. Thus no conflict arises from that duo. The duo from a different triplet could contribute at most one conflicting edge by the above claim that each duo is matched at most once. Applying this argument to both endpoints of a given edge completes the proof. J ILemma 8. M can be converted into M⁰, a feasible solution to MWPSM, such that the weight ofM⁰ is at least(1/2)OP T_{AT M} ≥(3/8)OP T_{M W P SM}.

Proof. The conflict graph on the edges ofM must be a collection of paths and even length cycles since it has maximum degree 2 andGis bipartite. We can simply decompose each path or cycle into two independent sets and choose the heavier of the two. This operation discards at most half of the weight ofM while removing all conflicts and leaving us with a

feasible solution to MWPSM. J

The proofs of Theorem 1 and Corollary 2 follow from the preceding lemmas.