Caso 1: Oreo y el Super Bowl

One can show that the proposed algorithm works correctly by verifying that the trivial cases are solved correctly and that every newly computed MCS of two

5.3 Computing a maximum common subgraph of two outerplanar graphs 119 Algorithm 5.3 Computing an MCS of two block-splitting-subgraphs G|∗

oG_[r0,r[

and H|∗ oH_[s0,s[

Require: g1= r0, g2, . . . , gm= r are the vertices of the path [r0, r[ with orienta-

tion oG _{of the block B}

G, h1 = s0, h2, . . . , hm= s are the vertices of the path

[s0_{, s[ with orientation o}H _{of the block B} H

1: function MatchBlockSplittingSubgraph(G|∗_oG_[r0_,r[, H|∗_oH_[s0_,s[)

2: if not(MatchVertex(r0, r) and MatchVertex(s0, s))

3: return G(∅, ∅)

4: if oG[r0, r[ or oH_[s0_{, s[ consists of two vertices in the Hamiltonian path}

along orientation oG _{or o}H 5: if MatchEdge({r0, r_{}, {s}0_{, s})} 6: return σ(Gr_B0_G, Hs0 BH) ∪ G({r}, {{r 0_{, r}})} 7: else 8: return G(∅, ∅) 9: else 10: M CS← G(∅, ∅)

11: for all pairs of edges {r0, r00} ∈ oG[r0, r[ and{s0, s00} ∈ oG[s0, s[ do 12: if MatchEdge({r0, r00_{}, {s}0_{, s}00_}) 13: CS_{← σ(G|}∗ oG_[r00,r[, H|∗oH_[s00,s[) ∪ σ(G r0 BG, H s0 BH) ∪ G(∅, {r 0_{, r}00_}) 14: if |CS| > |MCS| 15: M CS← CS 16: return MCS

relevant subgraphs is a correct increment of the MCS of their earlier computed children.

Matching of two BPSs We first describe the computation of an MCS St _of

two BPSs Gr _{and H}s_{. We have to prove the following invariant:}

σ(Gr, Hs) = max{St| St vϕGG r ∧ St vϕH H s } (5.1) with ϕG(t) = r ∧ ϕH(t) = s.

First, if the roots r and s are different, we have defined that there exists no MCS. In this case, it holds that λG(r) 6= λH(s), so no isomorphism mappings ϕG

and ϕH can exist such that with ϕG(t) = r ∧ ϕH(t) = s. Therefore, the set in

Eq. 5.1 is empty, such that the invariant corresponds to the empty MCS.

Second, when the roots are equal, that is, λG(r) = λH(s), we consider two

cases:

• Base case When either Gr _{or H}s _{consists of a single vertex, no maximal}

B

G

B

H $ $

B

G

B

H 3 3

G_|

∗_![r!_,r[

H|

∗_![s!_,s[ r! r r!! r!!! Gr! BG GrB!!!G s! s s!! s!!! HBs!H HBs!!!H

Figure 5.8: An example matching between two BSSs G|∗

oG_[r0,r[and H|∗oH_[s0,s[.

case, St _{is a graph consisting of one single vertex labeled λ}

G(r). Since r

and s are the roots of Gr _{and H}s_{, it is trivial to show that S}t_{is subgraph}

isomorphic to Gr _{and H}s_{, that is, the following holds:}

StvϕGG

r

∧ St

vϕH H

s_.

Now we just need to show that

ϕG(t) = r ∧ ϕH(t) = s.

However, since λG(r) = λH(s) = λG(t), we can show that such a mapping

exists. Moreover, since there is only one possible St _{consisting of one single}

vertex labeled λG(r), it is automatically the largest one.

• Induction step In order to prove the correctness in the general case, we assume a matching between the children of Gr _{and H}s_{and assume without}

loss of generality that only one pair of BPS children (Gx_{and H}y_{) has been}

selected in the matching. Because of the induction hypothesis, we can also assume to have a solution for an MCS Tu _{between G}x _{and H}y_{, that is, we}

know that: Tu vϕGG x ∧ Tu vϕH H y ∧ ϕG(u) = x ∧ ϕH(u) = y.

5.3 Computing a maximum common subgraph of two outerplanar graphs 121 Tu_{, or,} St vϕG G r ∧ St vϕH H s ∧ ϕG(t) = r ∧ ϕH(t) = s, (5.2)

and that it is maximal. We start by proving that St

vϕG G

r_{. First of all, the}

BPS Gr _{is exactly the same as its child BPS G}x_{, except for one additional}

vertex (r) and edge ({r, x}). St_{can thus have at most one vertex and edge}

more than Tu_{. Since we have assumed that the root r can be mapped, we}

only need to consider the possible matching of {r, x}. However, if this edge cannot be mapped, then the child Gx_{would not have been considered in the}

maximal matching. We can thus conclude that it can be mapped, and thus St

vϕGG

r_.

The proof for St

vϕH H

s _{is identical.}

Now we only have to prove that ϕG(t) = r ∧ ϕH(t) = s. However, again

because we have assumed equal root vertices, we can choose such an isomor- phism mapping.

We still have to proof that St _{is maximal, that is, there exists no other}

graph that also fulfils the requirements in Eq. 5.2 and that is strictly larger. However, since Gr_{is only one vertex and one edge larger than G}x_{, and since}

we have extended the MCS Tu_{with exactly one vertex and one edge to reach}

St, we know that there cannot exist an Stthat is strictly larger.

A similar proof can be constructed in the case of two BSS children that are being matched. Gr _{and H}s_{then have children G|}∗

oG_[r0,r[ and H|∗oH_[s0,s[,

respectively. An MCS between the children has been computed in a previ- ous step, and only the mapping of the edges {r, r0_{} and {s, s}0_{} needs to be}

checked. If the edges have equal labels, their weight in the maximal match- ing is equal to the size of the MCS together with the edge {r, r0_{}. If not,}

their weight in the maximal matching is 0.

Matching of two BSSs We now turn to the computation of an MCS S|∗ oS_[t0,t[

of two BSSs G|∗

oG_[r0,r[ and H|∗oH_[s0,s[, splitting a block BG and BH, respectively.

We have to prove the following invariant: σ(G|∗ oG_[r0_,r[, H|∗_oH_[s0_,s[) = max S∈ρ(G|∗ oG[r0,r[,H| ∗ oH [s0 ,s[) S. (5.3) with ρ(G|∗

oG_[r0,r[, H|∗oH_[s0,s[) the set of all graphs S|o∗S_[t0,t[ for which S|∗oS_[t0,t[ vϕG

G|∗

oG_[r0,r[ and S|∗oS_[t0,t[ vϕH H|∗oH_[s0,s[ and for which there are two BBP subgraph

isomorphism mappings ϕG : S|∗_oS_[t0,t[ → G|∗oG_[r0,r[ and ϕH : S|∗oS_[t0,t[ → H|∗oH_[s0,s[

such that ϕG(t0) = r0, ϕG(t) = r, ϕH(t0) = s0 and ϕH(t) = s.

First, if the endpoints r and s0 _{or r and s are different, we have defined that}

there exists no MCS. In this case, the set ρ is empty, such that the invariant corresponds to the empty MCS.

Second, if the endpoints are equal, that is λG(r0) = λH(s0) and λG(r) = λH(s),

we consider two cases: • Base case When G|∗

oG_[r0,r[or H|∗oH_[s0,s[consists of two vertices of the Hamil-

tonian path, then it is checked if λG({r0, r}) = λH({s0, s}). If this is false, ρ

is again empty, and there is no MCS. Otherwise, S is the result of combin- ing the two vertices r0 _{and r and their corresponding edge {r}0_{, r}, together}

with the MCS M of the BPSs Gr0

BG and H

s0

BH. Because of the induction

hypothesis we can assume that M is maximal, and since the new MCS has been extended with the maximal amount of vertices and edges, S is again maximal.

• Induction step We assume there is an MCS S0_|∗

oS0_[t00,t[ of BSSs G|∗oG_[r00,r[

and H|∗

oH_[s00,s[. It is straightforward to show that, if the edges {r0, r00} and

{s0_{, s}00_{} can be matched, that is, λ}

G({r0, r00}) = λH({s0, s00}), the MCS S0

can be correctly extended to S by adding the edge {r0_{, r}00_{} together with}

σ(Gr0 BG, H

s0

BH) (see Fig. 5.8).

In document Ahora o nunca: El poder del Newsjacking desde el contexto publicitario. Catalina De La Vega Martínez (página 46-52)