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Constructing Splits GraphsPowerPoint Presentation

Constructing Splits Graphs

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Constructing Splits Graphs. Author: Andreas W.M. Dress Daniel H. Huson Presented by: Bakhtiyar Uddin. Constructing Splits Graphs. Agenda: Objective Definitions, Theorems and Notations Constructing Plane Splits Graphs Constructing Non Planar Splits Graphs Conclusion.

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### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

### Constructing Splits Graphs

- Agenda:
- Objective
- Definitions, Theorems and Notations
- Constructing Plane Splits Graphs
- Constructing Non Planar Splits Graphs
- Conclusion

Objective

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).

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}

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

f4

g4

v4

g2

u’1

u’4

u’3

g3

f3

f2

v3

v2

This problem has been addressed by earlier publications. But in practice, the proposed approach is only feasible for small split systems.

Definitions, Theorems and Notations

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

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

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.

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}

Constructing Plane 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.

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.

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

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

f3

v7

f7

f4

f6

f5

v4

v6

v5

- 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:
- Create a new vertex v0
- 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)
- Set E(v_0) = (f_1,f_2,…,f_n)

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 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)

Example:

Input: G_(t-1) = S_t = {x2, x3, x4}/{x1, x5, x6, x7}

v3

v2

f2

f3

v4

f4

f1

v1

v5

f5

f6

f7

v6

v7

v3

v2

f2

f3

e0

e1

Output: Path P = (v2, e0, u1, e1, u2, e2, u3, e3, v4)

u1

u2

v4

f4

e2

e3

f1

v1

u3

v5

(The algorithm labels edges and vertices)

f5

f6

f7

v6

v7

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)

v7

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}

Example:

Input: G_(t-1) =

v3

v2

f2

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)

f3

u2

u1

v4

f4

u3

f1

v1

v5

f5

f6

f7

v6

v7

v3

v2

f2

e0

f3

u1

e1

u2

g2

f4

g1

v4

u3

u’1

Output:

u’2

e2

g3

f1

v1

u’3

v5

f5

f6

f7

v6

v7

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)

v3

v2

f2

f3

e0

e1

u1

u2

v4

f4

e2

e3

f1

v1

u3

v5

f5

f6

f7

v6

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)

v3

v2

f2

f3

e0

u2

e1

u1

v4

e3

f4

e2

e1

u3

u1

u2

e2

f1

v1

u3

v5

f5

f6

f7

v6

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)

v3

v2

f2

f3

e0

u2

e1

u1

g2

v4

g1

e3

f4

e2

e1

u3

u1

u2

e2

g3

f1

v1

u3

v5

f5

f6

f7

v6

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 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)

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 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)

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

r_2

r_x

r_1

e_i

e_(i-1)

u_i

l_1

l_y

l_2

g_i

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)

Then,

If i = 1

If 1<i<k

If i = k

e’_i

u_i

l_y

l_1

r_2

l_2

r_x

r_1

g_i

e_i

e_(i-1)

u_i

e’_(i-1)

e’_i

u_i

l_1

l_y

l_2

g_i

g_i

E(u_i) = (e_(i-1), r_1, r_2, …, r_x, e_i, f_i)

e’_(i-1)

u_i

l_1

l_y

l_2

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

Algorithm 1 creates the star:

v5

v1

E(v0) = (f1,f2,…f5)

E(dog) = (f1)

E(cat) = (f2)

E(parrot) = (f3)

E(turtle) = (f4)

E(mouse) = (f5)

f5

f1

v0

f4

v4

f2

f3

v2

v3

v5

v1

E(v0) = (f1,f2,…f5)

E(dog) = (f1)

E(cat) = (f2)

E(parrot) = (f3)

E(turtle) = (f4)

E(mouse) = (f5)

f5

f1

v0

f4

v4

f2

f3

v2

v3

Iteration 1:

Consider S1 = {dog,cat}/{mouse, turtle, parrot}

v5

v1

E(v0) = (f1,f2,…f5)

E(dog) = (f1)

E(cat) = (f2)

E(parrot) = (f3)

E(turtle) = (f4)

E(mouse) = (f5)

f5

f1

v0

f4

v4

f2

f3

v2

v3

Iteration 1:

Consider S1 = {dog,cat}/{mouse, turtle, parrot}

Algorithm 2 will generate the path P = ( v1, f1, v0, f2, v2)

v5

v1

E(v0) = (f1,f2,…f5)

E(dog) = (f1)

E(cat) = (f2)

E(parrot) = (f3)

E(turtle) = (f4)

E(mouse) = (f5)

f5

f1

v0

u’1

f4

v4

g1

f2

f3

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)

v5

v1

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)

f5

f1

v0

u’1

f4

v4

g1

f2

f3

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)

Algorithm 3 will also modify E(v0) = (f1, f2, g1) E(u’1) = (g1, f3, f4, f5)

v5

v1

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)

f5

f1

v0

u’1

f4

v4

g1

f2

f3

v2

v3

Iteration 2:

Consider S2 = {turtle, parrot}/{cat, dog, mouse}

Algorithm 2 will generate the path P = ( v3, f3, u’1, f4, v4)

v5

v1

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)

f5

f1

u’2

f4

v4

v0

u’1

g1

f2

f3

v3

v2

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)

v5

v1

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)

f5

f1

u’1

f4

v4

v0

u’2

g2

g1

f2

f3

v3

v2

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)

v5

v1

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)

f5

f1

u’1

f4

v4

v0

u’2

g2

g1

f2

f3

v3

v2

Iteration 3:

Consider S3 = {mouse, dog}/{cat, parrot, turtle}

Algorithm 2 will generate the path P = ( v5, f5, u’2, g1, v0, f1, v1)

v5

v1

f5

f1

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)

v0

g1

u’2

u’1

g5

f4

g4

v4

g2

u’4

u’3

g3

f2

f3

v3

v2

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)

v5

v1

f5

f1

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)

v0

g1

u’2

u’1

g5

f4

g4

v4

g2

u’4

u’3

g3

f2

f3

v3

v2

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)

Constructing Non planar 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.

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

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}

v5

v1

f5

f1

v0

g1

u’2

u’1

g5

f4

g4

v4

g2

u’4

u’3

g3

f2

f3

v3

v2

Consider the split S = {mouse, parrot}/{dog, cat, turtle} = A/A’

(not circular)

v5

v1

f5

f1

v0

g1

u’2

u’1

g5

f4

g4

v4

g2

u’4

u’3

g3

f2

f3

v3

v2

split S = {mouse, parrot}/{dog, cat, turtle}

Convex Hull of the nodes {mouse, parrot} = H_A

v5

v1

f5

f1

v0

g1

u’2

u’1

g5

f4

g4

v4

g2

u’4

u’3

g3

f2

f3

v3

v2

split S = {mouse, parrot}/{dog, cat, turtle}

Convex Hull of the nodes {dog, cat, parrot} = H_A’

v5

v1

f5

f1

v0

g1

u’2

u’1

g5

f4

g4

v4

g2

u’4

u’3

g3

f2

f3

v3

v2

split S = {mouse, parrot}/{dog, cat, turtle}

The intersection of the two convex hulls have edges g5 and g2.

v5

v1

f5

f1

v0

g1

u’2

u’1

g5

f4

g4

v4

g2

u’4

u’3

g3

f2

f3

v3

v2

split S = {mouse, parrot}/{dog, cat, turtle}

v5

v1

f5

f1

u’5

v0

g1

u’2

g6

u’7

g7

u’6

g5

f4

g4

v4

g2

u’1

u’4

u’3

g3

f2

f3

v3

v2

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)

v5

v1

f5

f1

u’5

v0

g1

u’2

g6

u’7

g7

u’6

g5

f4

g4

v4

g2

u’1

u’4

u’3

g3

f3

f2

v3

v2

split S = {mouse, parrot}/{dog, cat, turtle}

S is the partition obtained by removing the brown color edges

Conclusion

- 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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