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Constructing Splits Graphs

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

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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: • Objective • Definitions, Theorems and Notations • Constructing Plane Splits Graphs • Constructing Non Planar Splits Graphs • 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 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}

  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} v5 v1 f5 f1 v0 g1 u’2 g5 f4 g4 v4 g2 u’1 u’4 u’3 g3 f3 f2 v3 v2

  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 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

  10. 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

  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. 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 Example of a circular split system mouse dog turtle parrot cat owl

  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 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 f3 v7 f7 f4 f6 f5 v4 v6 v5

  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: • 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)

  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 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} 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

  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

  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) = 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

  25. Constructing Splits Graphs 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

  26. Constructing Splits Graphs 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

  27. Constructing Splits Graphs 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

  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<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 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)

  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 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

  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

  32. Constructing Splits Graphs 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

  33. Constructing Splits Graphs 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}

  34. Constructing Splits Graphs 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)

  35. Constructing Splits Graphs 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)

  36. Constructing Splits Graphs 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)

  37. Constructing Splits Graphs 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)

  38. Constructing Splits Graphs 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)

  39. Constructing Splits Graphs 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)

  40. Constructing Splits Graphs 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)

  41. Constructing Splits Graphs 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)

  42. Constructing Splits Graphs 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)

  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 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}

  47. Constructing Splits Graphs 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)

  48. Constructing Splits Graphs 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

  49. Constructing Splits Graphs 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’

  50. Constructing Splits Graphs 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.

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