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
represents a different order of genes. We write this
sequence horizontally, using the terms left and right to
denote directions along it.
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This problem has been investigated in the last few years with the following results:
We denote the number of breakpoints in
by
.
When
performing a reversal, transforming
into
,
we denote
by
.
Lemma 10.2 gives rise to the following algorithm:
If there exists a decreasing strip, find and perform a good
reversal (
). Else reverse an increasing strip, thus
creating a decreasing strip (
).
This algorithm leads to performance of at most 4 times the
optimum, since there are at most
reversals.
Lemma 10.3 gives rise to the following algorithm:
For as long as possible, either:
Or, if no such reversal exists:
This algorithm leads to performance of at most twice the optimum, since
on the average.