>>> RECALL

3.1 A tight lower bound for srd(\pi) 

Definition 3.4: The /breakpoint graph/ of a signed permutation \pi is the graph
BG(\pi) = (V, E), whose vertex set V contains, for 1 \leq g \leq n, two vertices
g^t and g^h called the /tail/ and the /head/ of gene g, plus two vertices 0^h
and n+1^t. The edge set E is the union of two perfect matchings R and D of V:

- "reality edges" R contains edge from \pi_i^h if \pi_i is non-negative, and
  from \pi_i^t otherwise, to \pi_i^t if \pi_{i+1} is non-negative, and to
  \pi_{i+1}^h otherwise, for 0 \leq i \leq n.
- "desire edges" D := {{g_h, (g+1)_t} | 0 \leq g \leq n } (adjacencies of the
  identity) --> Question, how would BG(id) look like?

(BG(\pi^2) drawn using two different colors for the two matchings R and D) 

(Question: How many cycles has BG(\pi^2) ? How many cycles has BG(id)?)
<<<

Lemma 3.1: A reversal changes the number of cycles of the BP graph at most by 1

The identity permutation of n characters has n cycles. Together with Lemma 3.1
we can derive a lower bound on srd(\pi):

    srd(\pi) >= n-c, 

where c is the number of cycles in BG(pi).

This bound is usually /tight/, that is, most of the times it is exactly the
reversal distance. It is not tight whenever it is not possible to increase the
number of cycles in BG(.) with a reversal.

3.2 Of hurdles, super-hurdles and fortresses

The permutation

(draw the following in a way so that you can also draw BG(\pi^3)) 
\pi^3 = (0 -2 -3 1 4 6 5 7 8) 

has four cycles, but can not be sorted with less than five reversals. Why?
Because of unoriented components!

Definition 3.5 (component):  A /component/ is an interval, (i^h ... (i+j)^t) for
i, i+j >= 0 or ((i+j)^t ... i^h) for i+j, i <0, whose set of unsigned elements
is {i,...,i+j}, but not the union of smaller such intervals.

Two cycles are /interleaving/ if they have crossing edges. 

Observation 3.1: Components correspond to maximal subsets of interleaving
cycles.

-> \pi^3 has three components: [0 -2 -3 1 (4] 6 5 [7)8]

Components have an orientation, according to the orientation of their
constituting cycles.

Definition 3.6 (orientation of cycles): Two reality edges in the same cycle are
/convergent/ if, traversing the cycle, both edges are entered from the right or
both from the left side, otherwise they are /divergent/. A desire edge is called
/oriented/ if its two incident reality edges are divergent, otherwise the edge
is unoriented.  A non-trivial cycle is called /oriented/ if it contains at least
one oriented (desire) edge.

-> \pi^3 has two unoriented and one oriented cycle

Definition 3.7: A component is /trivial/ if it is of the form (i,i+1) or
(-(i+1),-i) and otherwise /non-trivial/. A component is /unoriented/ if it is
non-trivial and all its elements have the same sign, otherwise it is /oriented/.

-> \pi^3 has 1 oriented, 1 unoriented, and 1 trivial component

A /hurdle/ is an unoriented component that does not separate other two
unoriented components. Removing a hurdle requires an additional reversal. But
some hurdles cause, when removed, the creation of a new hurdle. Such hurdles are
called /super-hurdles/. Super-hurdles can be removed in most cases, except for a
/fortress/, which is a permutation that has has an odd number of hurdles, and
all are super-hurdles.

(show fortress with Java-Program)

Theorem 3.1 (Hannenhalli-Pevzner duality theorem): The reversal distance for a
signed permutation \pi of n symbols is
    srd(\pi) = n+1 - c + h + f,

where c is the number of cycles, h the number of hurdles, and f = 1 if \pi has a
fortress, and f = 0 otherwise.