Unsigned Permutations

We will assume that we are able to identify genes along the chromosome, and we will discuss a single chromosome. We will also assume that all the genes are different. The order of the genes, which might be different in different taxa, is a permutation of these genes. Thus we will be discussing sequences of unsigned, different integers, where each permutation $\pi = (\pi_1 \ldots \pi_n)$ represents a different order of genes. We write this sequence horizontally, using the terms left and right to denote directions along it.

 

 

  

Figure: Example of reversals; the underlined segments show where the reversals took place. The reversal distance between $\pi _{1}$ and $\pi _{3}$ is 3.

 

This problem has been investigated in the last few years with the following results:

1.

2-approximation algorithm [16]

2.

1.75-approximation algorithm [2]

3.

NP Completeness proof [6]

4.

1.5-approximation algorithm [7]

5.

1.375-approximation algorithm [11]

 

We denote the number of breakpoints in $\pi$ by $b(\pi)$. When performing a reversal, transforming $\pi$ into $\pi '$, we denote $b(\pi') - b(\pi)$ by $\Delta b$.

 

 

 

  

Figure 10.3: Two possible cases to reduce a breakpoint using a decreasing strip (K=4).

Lemma 10.2 gives rise to the following algorithm:

If there exists a decreasing strip, find and perform a good reversal ( $\Delta b=-1$). Else reverse an increasing strip, thus creating a decreasing strip ( $\Delta b=0$).

This algorithm leads to performance of at most 4 times the optimum, since there are at most $2b(\pi)$ reversals.

 

   

Lemma 10.3 gives rise to the following algorithm:

For as long as possible, either:

1.

Perform a good reversal using a decreasing strip, resulting in a permutation with a decreasing strip ( $\Delta b=-1$).

Or, if no such reversal exists:

2.

Perform a reversal with $\Delta b=-2$, and then reverse any strip.

This algorithm leads to performance of at most twice the optimum, since $\Delta b=-1$ on the average.