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1 Constructing Splits Graphs Author: Andreas W.M. Dress Daniel H. Huson Presented by: Bakhtiyar Uddin

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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

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3 Constructing Splits Graphs Objective

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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).

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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}

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6 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} f1 f2 f3 f5 f4 v0u’2 v1 v2 v4 v5 v3 g1 u’1g2 u’3 u’4 g3 g4 g5

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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.

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8 Constructing Splits Graphs Definitions, Theorems and Notations

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9 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

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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

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11 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.

12 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

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13 Constructing Splits Graphs cat dog mouse turtle parrot owl Example of a circular split system

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14 Constructing Splits Graphs Constructing Plane Splits Graphs

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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.

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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.

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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

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18 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: v1 v6 v5 v4 v3 v2 v7 f1f2 f7 f4 f5 f3 f6

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19 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)

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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)

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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) Example: Input : G_(t-1) = S_t = {x2, x3, x4}/{x1, x5, x6, x7} v1 v6 v5 v4 v3v2 v7 f1 f2 f7 f4 f5 f3 f6 v1 v6 v5 v4 v3v2 v7 f1 f2 f7 f4 f5 f3 f6 e0 e1 e3 e2 Output: Path P = (v2, e0, u1, e1, u2, e2, u3, e3, v4) u3 u2 u1 (The algorithm labels edges and vertices)

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22 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) v7

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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}

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24 Constructing Splits Graphs Example: Input : G_(t-1) = v1 v6 v5 v4 v3v2 v7 f1 f2 f7 f4 f5 f3 f6 v1 v6 v5 v4 v3v2 v7 f1 f2 f7 f4 f5 f3 f6 e0 e1 e2 Output: u’3 u’2 u’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) g1 g2 g3 u3 u2 u1 u3 u2u1

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25 Constructing Splits Graphs v1 v6 v5 v4 v3v2 f1 f2 f7 f4 f5 f3 f6 e0 e1 e3 e2 u3 u2 u1 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)

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26 Constructing Splits Graphs v4 v3v2 f2 f4 f3 e0 e1 e3e2 u2 u1 v1 v6 v5 f1 f7 f5 f6 e1 e2 u3 u2 u1 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

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27 Constructing Splits Graphs v4 v3v2 f2 f4 f3 e0 e1 e3e2 u2 u1 v1 v6 v5 f1 f7 f5 f6 e1 e2 u3 u2 u1 g1 g2 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

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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 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*
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"name": "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 representing Sigma_t Note: Sigma_t = Sigma_(t-1) U {S_t} Algorithm: 1 For each i = 1….",
"description": "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
*

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29 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 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*
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"name": "29 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 representing Sigma_t Note: Sigma_t = Sigma_(t-1) U {S_t} Algorithm: 1 For each i = 1….",
"description": "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
*

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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 r_1 r_2 l_yl_1 l_2 r_x e_(i-1)e_i g_i e_(i-1) e_i r_1 r_2 r_x e’_i g_i l_1 l_2 l_y l_1 l_2 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*
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"name": "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 r_1 r_2 l_yl_1 l_2 r_x e_(i-1)e_i g_i e_(i-1) e_i r_1 r_2 r_x e’_i g_i l_1 l_2 l_y l_1 l_2 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
*

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31 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

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32 Constructing Splits Graphs Algorithm 1 creates the star: f1 f2 f3 f5 f4 v0 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

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33 Constructing Splits Graphs f1 f2 f3 f5 f4 v0 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

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34 Constructing Splits Graphs f1 f2 f3 f5 f4 v0 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

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35 Constructing Splits Graphs f1 f2 f3 f5 f4 v0 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

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36 Constructing Splits Graphs f1 f2 f3 f5 f4 v0 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

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37 Constructing Splits Graphs f1 f2 f3 f5 f4 v0 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

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38 Constructing Splits Graphs f1 f2 f3 f5 f4 v0 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

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39 Constructing Splits Graphs f1 f2 f3 f5 f4 v0 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

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40 Constructing Splits Graphs f1 f2 f3 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) 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)

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41 Constructing Splits Graphs f1 f2 f3 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 g2 u’3 u’4 g3 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)

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42 Constructing Splits Graphs f1 f2 f3 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 g2 u’3 u’4 g3 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)

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43 Constructing Splits Graphs Constructing Non planar Splits Graphs

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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.

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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

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46 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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47 Constructing Splits Graphs f1 f2 f3 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 g2 u’3 u’4 g3 g4 g5

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48 Constructing Splits Graphs f1 f2 f3 f5 f4 v0u’2 v1 v2 v4 v5 v3 g1 u’1 g2 u’3 u’4 g3 g4 g5 split S = {mouse, parrot}/{dog, cat, turtle} Convex Hull of the nodes {mouse, parrot} = H_A

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49 Constructing Splits Graphs f1 f2 f3 f5 f4 v0u’2 v1 v2 v4 v5 v3 g1 u’1 g2 u’3 u’4 g3 g4 g5 split S = {mouse, parrot}/{dog, cat, turtle} Convex Hull of the nodes {dog, cat, parrot} = H_A’

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50 Constructing Splits Graphs f1 f2 f3 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 g2 u’3 u’4 g3 g4 g5

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51 Constructing Splits Graphs f1 f2 f3 f5 f4 v0u’2 v1 v2 v4 v5 v3 g1 u’1 g2 u’3 u’4 g3 g4 g5 split S = {mouse, parrot}/{dog, cat, turtle}

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52 Constructing Splits Graphs f1 f2 f3 f5 f4 v0u’2 v1 v2 v4 v5 v3 g1 u’1g2 u’3 u’4 g3 g4 g5 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) u’5 u’6 u’7g6 g7

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53 Constructing Splits Graphs f1 f2 f3 f5 f4 v0u’2 v1 v2 v4 v5 v3 g1 u’1g2 u’3 u’4 g3 g4 g5 split S = {mouse, parrot}/{dog, cat, turtle} S is the partition obtained by removing the brown color edges u’5 u’7g6 u’6 g7

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54 Constructing Splits Graphs Conclusion

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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.

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