The Center Star Method for (SP) Alignment

In this section, we present an approximation algorithm for calculating the optimal multiple alignment under the SP metric (see e.g. [4] [pp 348-350]). The algorithm achieves an approximation ratio of two. Definitions: Assume 25#25 is our scoring function i.e., the price of aligning the character x with the character y. For each x,y and z,

• 30#30.
• 29#29.
• 28#28

We denote by D(S,Y) the score of the optimal alignment between sequences S and Y.
31#31

  

Figure 4.2: A generic center star for six strings, where the center string (S_c) is S₃.

32#32

The Center Star Algorithm:

1.

Find 33#33 minimizing 34#34 and let 35#35.

2.

Add the sequences in 36#36 to 37#37 one by one so that the alignment of every newly added sequence with S_t is optimal. Add spaces, when needed, to all pre-aligned sequences.

Running time analysis:

1.

38#38 O(n²) for step 1.

2.

39#39 for step 2. (Since the worst-case length of S'_c after the addition of i strings is 40#40)

Approximation analysis:

• Let 37#37 denote the multiple alignment of the algorithm.
• Let d(i,j) be the score of the pairwise alignment it induces on S_i, S_j. ( Note that 41#41 ).
• Let 42#42
• Let 43#43 denote the optimal alignment of 44#44.
• Let d^*(i,j) denote the value of alignment between S_i and S_j induced by 43#43.

We note that 45#45 the SP score of M. We will assume w.l.o.g that S₁ is the center found by the algorithm, so for each 46#46. We also note that 47#47
 48#48

49#49
Theorem 4.1 implies that calculating the multiple alignment of the center star produces a multiple alignment with a value which is at most 50#50 times the value of the optimal alignment. For example 51#51, 52#52.