Let us now define a one-to-one mapping u from the set of signed
permutations of order n into the set of unsigned permutations of
order 2n. Let
be a signed permutation. To obtain
replace each positive element x in
by 2x-1 and 2x, and
each negative element -x by 2x and 2x-1. For any signed
permutation
,
let
.
This description of
is equivalent to the following description: given a
permutation
,
obtain a graph of 2n+2 vertices by replacing each positive element x in
by 2x-1and 2x, each negative element -x by 2x and 2x-1, and
augment with begin and end vertices, 0 and 2n+1. Black edges
connect vertices
and gray edges connect
vertices 2i and 2i+1.
We may now limit our discussion to signed permutations given that
the reversals we perform on our unsigned permutation have a
one-to-one correspondance to reversals in signed permutations.
Thus, sorting
by reversals is equivalent to sorting the
unsigned permutation
by even reversals. From now on we
will consider the latter problem and by reversals we will always
mean an even reversal.
Note that in
every vertex has exactly one black
edge and one gray edge incident on it. Therefore, there is a unique decomposition
of
into cycles. The edges of each cycle are alternating
gray and black. Our goal is to sort our given graph
into
n+1 trivial cycles.
Notice that in this formulation all reversals are one of three
types:
Let
and let
be the number of cycles in
.
Figure 10.10(a) shows the breakpoint graph of the
permutation
.
It has eight breakpoints
and decomposes into two alternating cycles, i.e.
,
and
.
The two cycles are shown in figure
10.10(b).
An alternative construction of the breakpoint graph constructs
,
with vertices
,
black edges
,
and grey edges (2i,2i+1) for all
.
All vertices of
are in disjoint cycles,
but we ignore trivial cycles, i.e. cycles of length 2. Figure 10.11(a) shows the
breakpoint graph of
,
that has seven
breakpoints and decomposes into two cycles (the trivial cycle that is composed of the vertices {2,3} is ignored). Let
denote the number of cycles in
.
For an arbitrary reversal
on a permutation
,
define
and
.
When the reversal
and the
permutation
will be clear from the context we will
abbreviate
to
and
to
.
As Bafna and
Pevzner [2] observed, the following values are taken
by
and
depending on the types of the gaps
acts on (see figure 10.9):
We call a reversal proper if
,
i.e. it is either
of type 4a, 4b, or 4d. We say that a reversal
acts
on a gray edge e if it acts on the breakpoints which
correspond to the black edges incident on e. A gray edge is
oriented if a reversal acting on it is proper, otherwise it
is unoriented. Notice that a gray edge
is
oriented if and only if k+l is even. For example, the gray edge
(0,1) in the graph of figure 10.10(a) is
unoriented, while the gray edge (7,6) is oriented.
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