Constructing Splits Graphs
Author:
Andreas W.M. Dress
Daniel H. Huson
Presented by:
Bakhtiyar Uddin
1
Constructing Splits Graphs
Agenda:
1. Objective
2. Definitions, Theorems and Notations
3. Constructing Plane Splits Graphs
4. Constructing Non Planar Splits Graphs
5. Conclusion
2
Constructing Splits Graphs
Objective
3
Constructing Splits Graphs
Objective:
Given a set of splits (not necessarily compatible), generate a splits graph.
The algorithm is designed to handle large split systems.
Note: Splits graph is a graphical representation of an arbitrary splits system (set of splits).
4
Constructing Splits Graphs
Example:
Input:
Set of taxa, X = {dog, cat, mouse, turtle, parrot}
Circular ordering of X = (dog, cat, mouse, turtle, parrot)
Splits System:
S1 = {dog, cat} / {mouse, turtle, parrot}
S2 = {turtle, parrot} / {cat, dog, mouse}
S3 = {dog, mouse} / {cat, turtle, parrot}
S4 = {mouse, parrot} / {dog, cat, turtle}
5
Constructing Splits Graphs
Example:
Input:
Set of taxa, X = {dog, cat, mouse, turtle, parrot}
Circular ordering of X = (dog, cat, mouse, turtle, parrot)
Splits System:
S1 = {dog, cat} / {mouse, turtle, parrot}
S2 = {turtle, parrot} / {cat, dog, mouse}
S3 = {dog, mouse} / {cat, turtle, parrot}
S4 = {mouse, parrot} / {dog, cat, turtle}
v5
v1
f5
f1
v0 g1 u’2
g5
g4
f2
u’3
g3 u’4
f4
g2
v4
u’1
f3
v3
v2
6
Constructing Splits Graphs
This problem has been addressed by earlier publications. But in practice,
the proposed approach is only feasible for small split systems.
7
Constructing Splits Graphs
Definitions, Theorems and
Notations
8
Constructing Splits Graphs
Sigma: Set of splits
C: Set of colors
X: set of taxa
X-split: Partitioning of X into two non empty and complementary sets A and A’
EtoC: E -> C
Assigns a color to each edge
nu: X -> V
Mapping from set of taxa X to a node v in a graph.
Properly colored:
A path is properly colored if each edge in P has a different color.
Isometric coloring:
Coloring of the edges such that every shortest paths between any two vertices are properly colored and utilize
the same set of colors
9
Constructing Splits Graphs
Splits Graph:
A graph G = (V,E) is called a splits graph if it is:
1) Finite, simple, connected, bipartite
2) And there exists an isometric and surjective(onto C) edge coloring.
Theorem:
Assume G = (V,E) is a splits graph and EtoC is an appropriate edge coloring. For any color c in C, the graph
G_c, obtained by deleting all edges of color c, consists of precisely two separate connected components.
Thus, given a splits Graph G(V,E), there exists a set of color C such that it has one-one mapping with Sigma
(set of splits on G). We can use the set C as the range for EtoC.
Also, let StoC be the mapping from split to color.
StoC: Sigma -> C
10
Constructing Splits Graphs
Trivial Split:
A partition with a single element in one of the splits.
I represent the set of trivial splits as Sigma_O.
I represent the set of non trivial splits as Sigma_I
Frontier of G:
Frontier of G consists of the set of vertices and edges of G that are incident to the unbounded face of G
Outer-labeled graph:
G is outer-labeled if al labeled vertices of G are of degree one and contained in the frontier of G.
Convex sub graph:
G’ is a convex sub graph of a graph G is an induced subgraph of G such that for any pair of v and w belonging
to G’, all shortest paths between v and w that belongs to G also belongs to G’.
Convex Hull:
Convex Hull H_A is the smallest convex sub graph containing all the elements in A.
11
Constructing Splits Graphs
Circular Split System:
Split system Sigma for a set of taxa X is circular if there exists an ordered list (x_1,x_2,….,x_n) of elements of X
and every split in S belonging to Sigma is interval realizable, ie there exists p,q with 1<p<q<=n such that
S = {x_p, x_(p+1),…,x_q}/(X-{x_p, x_(p+1),…,x_q})
Example:
Given ordering (x1,x2,x3,x4) of X = {x1,x2,x3,x4}
Sigma = { {x1,x2}/{x3,x4}, {x2,x3}/{x1,x4} } is a circular split system
Theorem:
A set of X-splits Sigma is circular iff there exists an outer-labeled plane splits graph G that represents
Sigma U Sigma_O, where Sigma_O = { {x}/(X-{x}) | x belongs to X}
12
Constructing Splits Graphs
Example of a circular split system
mouse
dog
turtle
parrot
cat
owl
13
Constructing Splits Graphs
Constructing Plane Splits
Graphs
14
Constructing Splits Graphs
Input: A set of taxa X = {x_1,x_2,….,x_n}
A set of nontrivial X-splits, Sigma_I, such
that Sigma_I is circular with respect to the
ordering (x_1,x_2,….,x_n)
A set of trivial X-splits, Sigma_O
Output: Outer-labeled plane splits graph G
representing Sigma_I and Sigma_O.
15
Constructing Splits Graphs
Input: A set of taxa X = {x_1,x_2,….,x_n}
A set of nontrivial X-splits, Sigma_I, such that Sigma_I is circular with respect to the ordering (x_1,x_2,….,x_n)
A set of trivial X-splits, Sigma_O
Output: Outer-labeled plane splits graph G representing Sigma_I and Sigma_O.
Algorithm:
Apply Algorithm 1 to obtain a star graph (G_0, nu) representing Sigma_O.
Order the set Sigma_I by increasing the size of the split part containing x1
For each split S_t in Sigma_I, do:
Determine p,q such that S_t = {x_p, …, x_q}/( X - {x_p,…,x_q} )
Apply Algorithm 2 to find the shortest path P from nu(x_p) to
nu(x_q)
Apply Algorithm 3 to G_(t-1), S_t and P to obtain G_t.
16
Constructing Splits Graphs
Algorithm 1: Add trivial splits
Input: An ordering (x_1,x_2,…, x_n) of X and the set of
all trivial X-splits Sigma_O = {S1_O, S2_O,…,Sn_O}
Output: Outer-labeled plane splits graph G_0 = (V,E)
representing Sigma_O
17
Constructing Splits Graphs
Algorithm 1: Add trivial splits
Input: An ordering (x_1,x_2,…, x_n) of X and the set of all trivial X-splits Sigma_O = {S1_O, S2_O,…,Sn_O}
Output: Outer-labeled plane splits graph G_0 = (V,E) representing Sigma_O
Example:
Input: Ordering (x1,x2,x3,x4,x5,x6,x7)
Sigma_O = { {x1}/{x2, …, x7}, {x2}/{x1, x3, …, x7},
{x3}/{x1, x2, x4, …, x7}, {x4}/{x1, …, x3, x5, x6, x7},
{x5}/{x1, …, x4, x6, x7}, {x6}/{x1, …, x5, x7},
{x7}/{x1, …, x6} }
Output:
v2
v1
f1
f2
v3
v7
f3
f7
f4
f6
f5
v4
v6
v5
18
Constructing Splits Graphs
Algorithm 1: Add trivial splits
Input: An ordering (x1,x2,…, xn) of X and the set of all trivial X-splits Sigma_O = {S1_O, S2_O,…,Sn_O}
Output: Outer-labeled plane splits graph G_0 = (V,E) representing Sigma_O
Algorithm:
1.
Create a new vertex v0
2.
For each new taxon xi in {x_1,x_2,…,x_n}
2.1 Create a new vertex v_i and set nu(x_i) = v_i
2.2 Create a new edge f_i and set set c(f_i) = {x_i}/(X-{x_i})
2.3 Set E(v_i) = (f_i)
3.
Set E(v_0) = (f_1,f_2,…,f_n)
19
Constructing Splits Graphs
Algorithm 2: Find Shortest Path
Input: Graph, G_(t-1)
Split S_t = {xp, …, xq}/(X - {xp, …, xq})
Output: Shortest path P = (u0, e0, u1, e1, …, uk) from
u0 = nu(xp) and uk = nu(xq)
20
Constructing Splits Graphs
Algorithm 2: Find Shortest Path
Input: Graph, G_(t-1)
Split S_t = {xp, …, xq}/(X - {xp, …, xq})
Output: Shortest path P = (u0, e0, u1, e1, …, uk) from u0 = nu(xp) and uk = nu(xq)
v3
Example:
v2
f2
f3
Input: G_(t-1) =
v4
S_t = {x2, x3, x4}/{x1, x5, x6, x7}
f4
f1
v5
v1
f5
f6
v6
f7
v7
v3
v2
f2
f3
Output: Path P = (v2, e0, u1, e1, u2, e2, u3, e3, v4)
e1
v4
f4
e3
e0
u1
e2 u2
f1
u3
(The algorithm labels edges and vertices)
v1
v5
f5
f6
v6
f7
v7
21
Constructing Splits Graphs
Algorithm 2: Find Shortest Path
Input: Graph, G_(t-1)
Split S_t = {xp, …, xq}/(X - {xp, …, xq})
Output: Shortest path P = (u0, e0, u1, e1, …, uk) from u0 = nu(xp) and uk = nu(xq)
Algorithm:
1. Set u_0 = nu(x_p), e_0=f_p
2. Set i = 0
3. Repeat
3.1 Define u_i to be the vertex opposite to u_(i-1) across e_(i-1)
3.2 Define e_i to be the first successor of e_(i-1) in E(u_i) such that e_i not in
({f_1…f_n}-{f_q})
4. Until e_i = f_q
[have reached nu(x_q)]
5. Set u_i = nu(x_q)
22
v7
Constructing Splits Graphs
Algorithm 2: Add non-trivial circular split
Input: Graph, G_(t-1) representing Sigma_(t-1)
Split S_t = {x_p, …, x_q}/(X - {x_p, …, x_q})
Shortest path P = (u_0, e_0, u_1, e_1, …, u_k)
from u_0 = nu(x_p) and u_k = nu(x_q)
Output: Outer-labeled plane splits graph G_t
representing Sigma_t
Note: Sigma_t = Sigma_(t-1) U {S_t}
23
Constructing Splits Graphs
Example:
v3
Input: G_(t-1) =
v2
f2
f3
u2
v4
S_t = {x2, x3, x4}/{x1, x5,
x6, x7}
u1
f4
u3
v5
f1
v1
P = (v2, e0, u1, e1, u2, e2,
u3, e3, v4)
f5
f6
(shortest path between nu(x2)=v2 and
nu(x4)= v4)
f7
v6
v7
v3
v2
f2 e0
f3
e1
u2
g2
f4
v4
u3
Output:
e2
u’2
u1
g1
u’1
g3
f1
u’3
v1
v5
f5
f6
v6
f7
v7
24
Constructing Splits Graphs
S_t = {x2, x3, x4}/{x1, x5,
x6, x7}
v3
v2
f2
f3
e1
v4
f4
e3
e0
P = (v2, e0, u1, e1, u2, e2,
u3, e3, v4)
u1
e2 u2
f1
u3
v1
(shortest path between nu(x2)=v2 and
nu(x4)= v4)
v5
f5
f6
f7
v6
25
Constructing Splits Graphs
v3
f2
f3
u2
v4
e3 f4
u3
S_t = {x2, x3, x4}/{x1, x5,
x6, x7}
v2
e2
e1
e1
e0
u1
P = (v2, e0, u1, e1, u2, e2,
u3, e3, v4)
u1
e2 u2
f1
u3
v1
(shortest path between nu(x2)=v2 and
nu(x4)= v4)
v5
f5
f6
f7
v6
26
Constructing Splits Graphs
v3
f2
f3
u2
v4
e3 f4
u3
S_t = {x2, x3, x4}/{x1, x5,
x6, x7}
v2
e2
g2
e1
e1
e0
u1
g1
P = (v2, e0, u1, e1, u2, e2,
u3, e3, v4)
u1
e2 u2
g3
f1
u3
v1
(shortest path between nu(x2)=v2 and
nu(x4)= v4)
v5
f5
f6
f7
v6
27
Constructing Splits Graphs
Algorithm 3: Add non-trivial circular split
Input: Graph, G_(t-1) representing Sigma_(t-1)
Split S_t = {x_p, …, x_q}/(X - {x_p, …, x_q})
Shortest path P = (u_0, e_0, u_1, e_1, …, u_k) from u_0 = nu(x_p) and u_k = nu(x_q)
Output: Outer-labeled plane splits graph G_t
Note: Sigma_t = Sigma_(t-1) U {S_t}
representing Sigma_t
Algorithm:
1 For each i = 1…. k
1.1 Create a new vertex u’_i
1.2 Create a new edge g_i(u’_i, u_i) and EtoS(u’_i) = S_t
1.3 if (i < k) create a new edge e’_i with CtoS(e’_i) = CtoS(e_i)
2 For each I = 1,2,… k
2.1 Assume E(u_i) = (e_(i-1), r_1, r_2, …, r_x, e_i, l_1, l_2, …, l_y)
2.2 Set E(u_i) = (e_(i-1), r_1, r_2, …, r_x, e_i, g_i)
2.3 if (i = 1)
2.3.1 E(u’_i) = (g_i, e’_i, l_1, l_2, .., l_y)
2.4 if (1<i<k)
2.4.1 E(u’_i) = (e’_(i-1), g_i, e’_i, l_1, l_2, …, l_y)
2.5 if (i = k)
2.5.1 E(u’_i) = (e’_(i-1), g_i, l_1, l_2, …, l_y)
28
Constructing Splits Graphs
Algorithm 3: Add non-trivial circular split
Input: Graph, G_(t-1) representing Sigma_(t-1)
Split S_t = {x_p, …, x_q}/(X - {x_p, …, x_q})
Shortest path P = (u_0, e_0, u_1, e_1, …, u_k) from u_0 = nu(x_p) and u_k = nu(x_q)
Output: Outer-labeled plane splits graph G_t
Note: Sigma_t = Sigma_(t-1) U {S_t}
Complexity:
representing Sigma_t
Algorithm:
1 For each i = 1…. k
1.1 Create a new vertex u’_i
O(k2 + nk)
1.2 Create a new edge g_i(u’_i, u_i) and EtoS(u’_i) = S_t
1.3 if (i < k) create a new edge e’_i with CtoS(e’_i) = CtoS(e_i)
2 For each I = 1,2,… k
2.1 Assume E(u_i) = (e_(i-1), r_1, r_2, …, r_x, e_i, l_1, l_2, …, l_y)
2.2 Set E(u_i) = (e_(i-1), r_1, r_2, …, r_x, e_i, g_i)
2.3 if (i = 1)
2.3.1 E(u’_i) = (g_i, e’_i, l_1, l_2, .., l_y)
2.4 if (1<i<k)
2.4.1 E(u’_i) = (e’_(i-1), g_i, e’_i, l_1, l_2, …, l_y)
2.5 if (i = k)
2.5.1 E(u’_i) = (e’_(i-1), g_i, l_1, l_2, …, l_y)
29
Constructing Splits Graphs
Finding ordered list of incident edges recursively (Step 2 of algorithm 3):
For a star graph:
E(v_0) = (f_1,f_2,….,f_n)
E(v_i) = (f_i)
r_2
r_x
Else
e_i
If at the i_th iteration E(u_i) = (e_(i-1), r_1, r_2, …, r_x, e_i, l_1, l_2, …, l_y) for the node u_i
r_1
e_(i-1)
u_i
l_1
l_y
l_2
g_i
And, E(u’_i) =
Then,
If i = 1
e’_i
(g_i, e’_i, l_1, l_2, ..,
l_y)
r_2
r_x
e_i
u_i
l_y
l_1
l_2
r_1
u_i
g_i
e_(i-1)
(e’_(i-1), g_i, e’_i, l_1,
l_2, …, l_y)
e’_i
l_1
E(u_i) = (e_(i-1), r_1, r_2, …, r_x, e_i,
f_i)
If 1<i<k
l_y
l_2
g_i
e’_(i-1)
u_i
g_i
(e’_(i-1), g_i, l_1, l_2,
…, l_y)
e’_(i-1)
u_i
l_1
l_2
l_y
If i = k
30
Constructing Splits Graphs
Example:
Input: Set of taxa X such that X is circular with respect to
ordering. X = (dog, cat, mouse, turtle, parrot)
Set of non-trivial splits
Sigma_I = { {dog, cat | mouse, turtle, parrot},
{turtle, parrot|cat, dog, mouse},
{dog, mouse | cat, turtle, parrot} }
Set of trivial splits Sigma_O
Output: Outer labeled plane splits graph G representing
Sigma_I and Sigma_O
31
Constructing Splits Graphs
Algorithm 1 creates the star:
v5
v1
E(v0) = (f1,f2,…f5)
f5
f1
E(dog) = (f1)
v0
f4
v4
E(cat) = (f2)
E(parrot) = (f3)
f2
E(turtle) = (f4)
f3
E(mouse) = (f5)
v2
v3
32
Constructing Splits Graphs
v5
v1
E(v0) = (f1,f2,…f5)
f5
f1
E(dog) = (f1)
v0
f4
v4
E(cat) = (f2)
E(parrot) = (f3)
f2
E(turtle) = (f4)
f3
E(mouse) = (f5)
v2
v3
Iteration 1:
Consider S1 = {dog,cat}/{mouse, turtle, parrot}
33
Constructing Splits Graphs
v5
v1
E(v0) = (f1,f2,…f5)
f5
f1
E(dog) = (f1)
v0
f4
v4
E(cat) = (f2)
E(parrot) = (f3)
f2
E(turtle) = (f4)
f3
E(mouse) = (f5)
v2
v3
Iteration 1:
Consider S1 = {dog,cat}/{mouse, turtle, parrot}
Algorithm 2 will generate the path P = ( v1, f1, v0, f2, v2)
34
Constructing Splits Graphs
v5
v1
E(v0) = (f1,f2,…f5)
f5
f1
E(dog) = (f1)
u’1
v0
f2
f4
v4
g1
E(cat) = (f2)
E(parrot) = (f3)
E(turtle) = (f4)
f3
E(mouse) = (f5)
v2
v3
Iteration 1:
Consider S1 = {dog,cat}/{mouse, turtle, parrot}
Algorithm 2 will generate the path P = ( v1, f1, v0, f2, v2)
Algorithm 3 will create a new node u’1 and a new edge g1(v0, u’1)
35
Constructing Splits Graphs
v5
v1
E(v0) = (f1,f2,g1)
f5
f1
E(v1) = (f1)
u’1
v0
f2
f4
v4
g1
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
f3
E(v5) = (f5)
v2
v3
E(u’1) = (g1,f3,f4,f5)
Iteration 1:
Consider S1 = {dog,cat}/{mouse, turtle, parrot}
Algorithm 2 will generate the path P = ( v1, f1, v0, f2, v2)
Algorithm 3 will create a new node u’1 and a new edge g1(v0, u’1)
Algorithm 3 will also modify E(v0) = (f1, f2, g1)
E(u’1) = (g1, f3, f4, f5)
36
Constructing Splits Graphs
v5
v1
E(v0) = (f1,f2,g1)
f5
f1
E(v1) = (f1)
u’1
v0
f2
f4
v4
g1
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
f3
E(v5) = (f5)
v2
v3
E(u’1) = (g1,f3,f4,f5)
Iteration 2:
Consider S2 = {turtle, parrot}/{cat, dog, mouse}
Algorithm 2 will generate the path P = ( v3, f3, u’1, f4, v4)
37
Constructing Splits Graphs
v5
v1
E(v0) = (f1,f2,g1)
f5
f1
E(v1) = (f1)
u’2
u’1
v0
f2
f4
v4
g1
E(v2) = (f2)
E(v3) = (f3)
f3
E(v4) = (f4)
E(v5) = (f5)
v2
v3
E(u’1) = (g1,f3,f4,f5)
Iteration 2:
Consider S2 = {turtle, parrot}/{cat, dog, mouse}
Algorithm 2 will generate the path P = ( v3, f3, u’1, f4, v4)
Algorithm 3 will create a new node u’2 and the new edge g2(u’1, u’2)
38
Constructing Splits Graphs
v5
v1
E(v0) = (f1,f2,g1)
f5
f1
E(v1) = (f1)
u’1
u’2
v0
f2
f4
v4
g2
g1
E(v2) = (f2)
E(v3) = (f3)
f3
E(v4) = (f4)
E(v5) = (f5)
v2
v3
E(u’1) = (g1,f3,f4,f5)
E(u’2) = (g2, f5, g1)
Iteration 2:
Consider S2 = {parrot, turtle}/{cat, dog, mouse}
Algorithm 2 will generate the path P = ( v3, f3, u’1, f4, v4)
Algorithm 3 will create a new node u’2 and the new edge g2(u’1, u’2)
Algorithm 3 will modify E(u’1) = (f3, f4, g2)
E(u’2) = (g2, f5, g1)
39
Constructing Splits Graphs
v5
v1
E(v0) = (f1,f2,g1)
f5
f1
E(v1) = (f1)
u’1
u’2
v0
f2
f4
g2
g1
v4
E(v2) = (f2)
E(v3) = (f3)
f3
E(v4) = (f4)
E(v5) = (f5)
v2
v3
E(u’1) = (g1,f3,f4,f5)
E(u’2) = (g2, f5, g1)
Iteration 3:
Consider S3 = {mouse, dog}/{cat, parrot, turtle}
Algorithm 2 will generate the path P = ( v5, f5, u’2, g1, v0, f1, v1)
40
Constructing Splits Graphs
v5
v1
f5
f1
E(v0) = (f1,f2,g1)
v0 g1 u’2
g4
f2
u’3
g3 u’4
E(v1) = (f1)
u’1
g5
f4
v4
g2
E(v2) = (f2)
E(v3) = (f3)
f3
E(v4) = (f4)
E(v5) = (f5)
v3
v2
E(u’1) = (g1,f3,f4,f5)
E(u’2) = (g2, f5, g1)
Iteration 3:
Consider S3 = {mouse, dog}/{cat, parrot, turtle}
Algorithm 2 will generate the path P = ( v5, f5, u’2, g1, v0, f1, v1)
Algorithm 3 will create:
two new nodes u’3, u’4
a new edge g3(u’3, u’4) with EtoC(g2) = EtoC(g3)
41
and two new edges g4(u’3, v0) and g5(u’4, u’2) with EtoC(g4) = EtoC(g5) = StoC(S3)
Constructing Splits Graphs
v5
v1
f5
f1
E(v0) = (g1, f1, g4)
v0 g1 u’2
g4
f2
u’3
g3 u’4
E(v1) = (f1)
u’1
g5
f4
g2
v4
E(v2) = (f2)
E(v3) = (f3)
f3
E(v4) = (f4)
E(v5) = (f5)
v3
v2
E(u’1) = (g1,f3,f4,f5)
E(u’2) = (f5, g1, g5)
Iteration 3:
Consider S3 = {mouse, dog}/{cat, parrot, turtle}
Algorithm 2 will generate the path P = ( v5, f5, u’2, g1, v0, f1, v1)
E(u’3) = (g3, g4, f2)
E(u’4) = (g5, g3, g2)
Algorithm 3 will create:
two new nodes u’3, u’4
a new edge g3(u’3, u’4) with EtoC(g2) = EtoC(g3)
and two new edges g4(u’3, v0) and g5(u’4, u’2) with EtoC(g4) = EtoC(g5) = StoC(S3)
It will modify E(v0), E(u’2) and create E(u’3) and E(u’4)
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Constructing Splits Graphs
Constructing Non planar
Splits Graphs
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Constructing Splits Graphs
Non circular splits system leads to non-planar splits graphs.
Reminder:
Convex sub graph:
G’ is a convex sub graph of a graph G is an induced subgraph of G
such that for any pair of v and w belonging to G’, all shortest paths
between v and w that belongs to G also belongs to G’.
Convex Hull:
Convex Hull H_A is the smallest convex sub graph containing all the
elements in A.
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Constructing Splits Graphs
Input: Splits Graph G_(t-1) representing
Sigma_(t-1) = Sigma_O U Sigma_I_(t-1)
Split S_t
Output: Splits Graph G_t representing
Sigma_t = Sigma_(t-1) U S_t
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Constructing Splits Graphs
Input: Splits Graph G_(t-1) representing Sigma_(t-1) = Sigma_O U Sigma_I_(t-1)
Split S_t
Output: Splits Graph G_t representing Sigma_t = Sigma_(t-1) U S_t
Algorithm:
Assume S_t = A/A’
1. Compute convex hulls H_A and H_A’
2. Define H_n = intersection of H_A and H_A’
3. F = f_1, f_2, …, f_s denote the set of all edges whose both ends lie in H_n
4. For each i = 1, 2, …, r
4.1 Create a new vertex u’_i
4.2 Create a new edge e_i
4.3 Set EtoC(e_i) = StoC(S_t)
5. For each i = 1,2,…, s
5.1 Create a new edge f’_i
5.2 set EtoC(f’_i) = EtoC(f_i)
6. For each i = 1, 2, …, r
6.1 E_A = set of edges in E(u_i) whose opposite vertices lie in H_A
6.2 E_A’ = set of edges in E(u_i) whose opposite vertices lie in H_A’
6.3 E_n = {g_1, g_2, …, g_q} = set of edges in E(u_i) whose opposite vertices lie in H_n
6.4 E’_n = {g’_1, g’_2, …, g’_q}
6.5 E(u_i) = E_A U E_n U {e_i}
6.6 E(u_i) = E_A’ U E’_n U {e_i}
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Constructing Splits Graphs
v5
v1
f5
f1
v0 g1 u’2
f2
u’1
g5
g4
u’3
g3 u’4
f4
v4
g2
f3
v2
v3
Consider the split S = {mouse, parrot}/{dog, cat, turtle} = A/A’
(not circular)
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Constructing Splits Graphs
v5
v1
f5
f1
v0 g1 u’2
f2
u’1
g5
g4
u’3
g3 u’4
f4
v4
g2
f3
v2
v3
split S = {mouse, parrot}/{dog, cat, turtle}
Convex Hull of the nodes {mouse, parrot} = H_A
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Constructing Splits Graphs
v5
v1
f5
f1
v0 g1 u’2
f2
u’1
g5
g4
u’3
g3 u’4
f4
v4
g2
f3
v2
v3
split S = {mouse, parrot}/{dog, cat, turtle}
Convex Hull of the nodes {dog, cat, parrot} = H_A’
49
Constructing Splits Graphs
v5
v1
f5
f1
v0 g1 u’2
f2
u’1
g5
g4
u’3
g3 u’4
f4
v4
g2
f3
v2
v3
split S = {mouse, parrot}/{dog, cat, turtle}
The intersection of the two convex hulls have edges g5 and g2.
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Constructing Splits Graphs
v5
v1
f5
f1
v0 g1 u’2
f2
u’1
g5
g4
u’3
g3 u’4
f4
v4
g2
f3
v2
v3
split S = {mouse, parrot}/{dog, cat, turtle}
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Constructing Splits Graphs
v5
v1
f5
f1
v0 g1 u’2
g5
g4
f2
u’5
g6
u’3
g3 u’4
u’6
g2
g7
u’7
f4
v4
u’1
f3
v2
v3
split S = {mouse, parrot}/{dog, cat, turtle}
For each edge e in the intersection,
create a new edge f
EtoC(f) = EtoC(e)
For each node u in the intersection,
create a new node v
create an edge f(u,v)
EtoC(f) = StoC(S)
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Constructing Splits Graphs
v5
v1
f5
f1
v0 g1 u’2
g5
g4
f2
u’5
g6
u’3
g3 u’4
u’6
g2
g7
u’7
f4
v4
u’1
f3
v3
v2
split S = {mouse, parrot}/{dog, cat, turtle}
S is the partition obtained by removing the brown color edges
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Constructing Splits Graphs
Conclusion
54
Constructing Splits Graphs
•
The paper include other algorithms
1. Algorithm to compute coordinates.
2. Algorithm to obtain a circular ordering that maximizes the number of
splits in Sigma that are interval-realizable with respect to the given
ordering.
•
To process a large set of splits:
1. First use Algorithm 4 to process the subset of circular splits
2. Use Algorithm 6 to process the remaining splits
•
All the presented algorithms are implemented in a new program called
SplitsTree4.
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