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1
Constructing Splits Graphs
Author:
Andreas W.M. Dress
Daniel H. Huson
Presented by:
Bakhtiyar Uddin
2
Constructing Splits Graphs
Agenda:
1. Objective
2. Definitions, Theorems and Notations
3. Constructing Plane Splits Graphs
4. Constructing Non Planar Splits Graphs
5. Conclusion
3
Constructing Splits Graphs
Objective
4
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).
5
Constructing Splits GraphsExample: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}
6
Constructing Splits GraphsExample: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}
f1
f2f3
f5
f4
v0 u’2
v1
v2
v4
v5
v3
g1
u’1g2u’3 u’4g3
g4 g5
7
Constructing Splits Graphs
This problem has been addressed by earlier publications. But in practice, the proposed approach is only feasible for small split systems.
8
Constructing Splits Graphs
Definitions, Theorems and Notations
9
Constructing Splits GraphsSigma: 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
10
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
Constructing Splits Graphs
11
Constructing Splits GraphsTrivial 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.
12
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}
13
Constructing Splits Graphs
cat
dog
mouse
turtle
parrot
owl
Example of a circular split system
14
Constructing Splits Graphs
Constructing Plane Splits Graphs
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.
16
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.
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
18
Constructing Splits GraphsAlgorithm 1: Add trivial splitsInput: 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: v1
v6v5
v4
v3
v2
v7
f1 f2
f7
f4
f5
f3
f6
19
Constructing Splits GraphsAlgorithm 1: Add trivial splitsInput: 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)
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)
21
Constructing Splits GraphsAlgorithm 2: Find Shortest PathInput: 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)
Example:
Input: G_(t-1) = S_t = {x2, x3, x4}/{x1, x5, x6, x7}
v1
v6
v5
v4
v3 v2
v7
f1
f2
f7
f4
f5
f3
f6
v1
v6
v5
v4
v3 v2
v7
f1
f2
f7
f4
f5
f3
f6
e0
e1
e3e2
Output: Path P = (v2, e0, u1, e1, u2, e2, u3, e3, v4)
u3
u2u1
(The algorithm labels edges and vertices)
22
Constructing Splits GraphsAlgorithm 2: Find Shortest PathInput: 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)
v7
23
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}
24
Constructing Splits Graphs
Example:
Input: G_(t-1) =
v1
v6
v5
v4
v3 v2
v7
f1
f2
f7
f4
f5
f3
f6
v1
v6
v5
v4
v3 v2
v7
f1
f2
f7
f4
f5
f3
f6
e0
e1
e2Output:
u’3
u’2u’1
S_t = {x2, x3, x4}/{x1, x5, x6, x7}
P = (v2, e0, u1, e1, u2, e2,
u3, e3, v4) (shortest path between nu(x2)=v2 and nu(x4)= v4)
g1g2
g3
u3
u2u1
u3
u2 u1
25
Constructing Splits Graphs
v1
v6
v5
v4
v3 v2
f1
f2
f7
f4
f5
f3
f6
e0
e1
e3e2
u3
u2u1
S_t = {x2, x3, x4}/{x1, x5, x6, x7}
P = (v2, e0, u1, e1, u2, e2,
u3, e3, v4) (shortest path between nu(x2)=v2 and nu(x4)= v4)
26
Constructing Splits Graphs
v4
v3 v2
f2
f4
f3 e0
e1
e3 e2
u2u1
v1
v6
v5
f1
f7
f5
f6
e1
e2
u3
u2u1
S_t = {x2, x3, x4}/{x1, x5, x6, x7}
P = (v2, e0, u1, e1, u2, e2,
u3, e3, v4) (shortest path between nu(x2)=v2 and nu(x4)= v4)
u3
27
Constructing Splits Graphs
v4
v3 v2
f2
f4
f3 e0
e1
e3 e2
u2u1
v1
v6
v5
f1
f7
f5
f6
e1
e2
u3
u2u1
g1g2
g3
S_t = {x2, x3, x4}/{x1, x5, x6, x7}
P = (v2, e0, u1, e1, u2, e2,
u3, e3, v4) (shortest path between nu(x2)=v2 and nu(x4)= v4)
u3
28
Constructing Splits GraphsAlgorithm 3: Add non-trivial circular splitInput: 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}
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)
29
Constructing Splits GraphsAlgorithm 3: Add non-trivial circular splitInput: 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}
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)
Complexity:
O(k2 + nk)
30
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)
Else
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
u_i
u_i
u_i
u_i
u_i
r_1r_2
l_yl_1l_2
r_x
e_(i-1)e_i
g_i
e_(i-1)e_ir_1
r_2r_x
e’_i
e’_i
g_i
g_i
g_i
l_1l_2
l_y
l_y
l_y
l_1
l_1
l_2
l_2
e’_(i-1)
e’_(i-1)
Then,
E(u_i) = (e_(i-1), r_1, r_2, …, r_x, e_i, f_i)
If i = 1
If 1<i<k
If i = k
And, E(u’_i) =
(g_i, e’_i, l_1, l_2, .., l_y)
(e’_(i-1), g_i, e’_i, l_1, l_2, …, l_y)
(e’_(i-1), g_i, l_1, l_2, …, l_y)
31
Constructing Splits GraphsExample:
Input: Set of taxa X such that X is circular with respect to ordering. X = (dog, cat, mouse, turtle, parrot)
Set of non-trivial splitsSigma_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
32
Constructing Splits GraphsAlgorithm 1 creates the star:
f1
f2
f3
f5
f4v0
E(v0) = (f1,f2,…f5)
E(dog) = (f1)
E(cat) = (f2)
E(parrot) = (f3)
E(turtle) = (f4)
E(mouse) = (f5)
v1
v2
v4
v5
v3
33
Constructing Splits Graphs
f1
f2
f3
f5
f4v0
E(v0) = (f1,f2,…f5)
E(dog) = (f1)
E(cat) = (f2)
E(parrot) = (f3)
E(turtle) = (f4)
E(mouse) = (f5)
Iteration 1:
Consider S1 = {dog,cat}/{mouse, turtle, parrot}
v1
v2
v4
v5
v3
34
Constructing Splits Graphs
f1
f2
f3
f5
f4v0
E(v0) = (f1,f2,…f5)
E(dog) = (f1)
E(cat) = (f2)
E(parrot) = (f3)
E(turtle) = (f4)
E(mouse) = (f5)
Iteration 1:
Consider S1 = {dog,cat}/{mouse, turtle, parrot}
Algorithm 2 will generate the path P = ( v1, f1, v0, f2, v2)
v1
v2
v4
v5
v3
35
Constructing Splits Graphs
f1
f2
f3
f5
f4v0
E(v0) = (f1,f2,…f5)
E(dog) = (f1)
E(cat) = (f2)
E(parrot) = (f3)
E(turtle) = (f4)
E(mouse) = (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)
u’1
v1
v2
v4
v5
v3
g1
36
Constructing Splits Graphs
f1
f2
f3
f5
f4v0
E(v0) = (f1,f2,g1)
E(v1) = (f1)
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
E(v5) = (f5)
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)
u’1
v1
v2
v4
v5
v3
g1
37
Constructing Splits Graphs
f1
f2
f3
f5
f4v0
E(v0) = (f1,f2,g1)
E(v1) = (f1)
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
E(v5) = (f5)
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)
u’1
v1
v2
v4
v5
v3
g1
38
Constructing Splits Graphs
f1
f2f3
f5
f4v0
E(v0) = (f1,f2,g1)
E(v1) = (f1)
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
E(v5) = (f5)
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)
u’1
v1
v2
v4
v5
v3
g1
u’2
39
Constructing Splits Graphs
f1
f2f3
f5
f4v0
E(v0) = (f1,f2,g1)
E(v1) = (f1)
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
E(v5) = (f5)
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)
u’2
v1
v2
v4
v5
v3
g1
u’1
g2
40
Constructing Splits Graphs
f1
f2f3
f5
f4v0
Iteration 3:
Consider S3 = {mouse, dog}/{cat, parrot, turtle}
Algorithm 2 will generate the path P = ( v5, f5, u’2, g1, v0, f1, v1)
u’2
v1
v2
v4
v5
v3
g1
u’1
g2
E(v0) = (f1,f2,g1)
E(v1) = (f1)
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
E(v5) = (f5)
E(u’1) = (g1,f3,f4,f5)
E(u’2) = (g2, f5, g1)
41
Constructing Splits Graphs
f1
f2f3
f5
f4
v0
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)
and two new edges g4(u’3, v0) and g5(u’4, u’2) with EtoC(g4) = EtoC(g5) = StoC(S3)
u’2
v1
v2
v4
v5
v3
g1
u’1
g2u’3 u’4g3
g4 g5
E(v0) = (f1,f2,g1)
E(v1) = (f1)
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
E(v5) = (f5)
E(u’1) = (g1,f3,f4,f5)
E(u’2) = (g2, f5, g1)
42
Constructing Splits Graphs
f1
f2f3
f5
f4
v0
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)
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)
u’2
v1
v2
v4
v5
v3
g1
u’1
g2u’3 u’4g3
g4 g5
E(v0) = (g1, f1, g4)
E(v1) = (f1)
E(v2) = (f2)
E(v3) = (f3)
E(v4) = (f4)
E(v5) = (f5)
E(u’1) = (g1,f3,f4,f5)
E(u’2) = (f5, g1, g5)
E(u’3) = (g3, g4, f2)
E(u’4) = (g5, g3, g2)
43
Constructing Splits Graphs
Constructing Non planar Splits Graphs
44
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.
45
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
46
Constructing Splits GraphsInput: 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}
47
Constructing Splits Graphs
f1
f2f3
f5
f4
v0
Consider the split S = {mouse, parrot}/{dog, cat, turtle} = A/A’(not circular)
u’2
v1
v2
v4
v5
v3
g1
u’1
g2u’3 u’4g3
g4 g5
48
Constructing Splits Graphs
f1
f2f3
f5
f4
v0 u’2
v1
v2
v4
v5
v3
g1
u’1
g2u’3 u’4g3
g4 g5
split S = {mouse, parrot}/{dog, cat, turtle}Convex Hull of the nodes {mouse, parrot} = H_A
49
Constructing Splits Graphs
f1
f2f3
f5
f4
v0 u’2
v1
v2
v4
v5
v3
g1
u’1
g2u’3 u’4g3
g4 g5
split S = {mouse, parrot}/{dog, cat, turtle}Convex Hull of the nodes {dog, cat, parrot} = H_A’
50
Constructing Splits Graphs
f1
f2f3
f5
f4
v0
split S = {mouse, parrot}/{dog, cat, turtle}The intersection of the two convex hulls have edges g5 and g2.
u’2
v1
v2
v4
v5
v3
g1
u’1
g2u’3 u’4g3
g4 g5
51
Constructing Splits Graphs
f1
f2f3
f5
f4
v0 u’2
v1
v2
v4
v5
v3
g1
u’1
g2u’3 u’4g3
g4 g5
split S = {mouse, parrot}/{dog, cat, turtle}
52
Constructing Splits Graphs
f1
f2f3
f5
f4
v0 u’2
v1
v2
v4
v5
v3
g1
u’1g2u’3 u’4g3
g4 g5
split S = {mouse, parrot}/{dog, cat, turtle}
For each edge e in the intersection, create a new edge fEtoC(f) = EtoC(e)
For each node u in the intersection, create a new node vcreate an edge f(u,v)EtoC(f) = StoC(S)
u’5
u’6
u’7g6g7
53
Constructing Splits Graphs
f1
f2f3
f5
f4
v0 u’2
v1
v2
v4
v5
v3
g1
u’1g2u’3 u’4g3
g4 g5
split S = {mouse, parrot}/{dog, cat, turtle}S is the partition obtained by removing the brown color edges
u’5u’7g6
u’6g7
54
Constructing Splits Graphs
Conclusion
55
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.