Schnyder woods for higher genus triangulated surfaces with ...

More documents

Recommendations

Info

$(PUC - Rio de Janeiro ALOCA\307\303O.xlsx)$

Luca Castelli Aleardi, Eric Fusy and Thomas Lewiner 6 v n−1 v 0 v 1 Figure 4: Some steps of the algorithm that computes a Schnyder wood of a planar triangulation T (rooted at (v 0 ,v 1 )). The traversal is performed as a sequence of conquer operations on free vertices: edges in S out (dark region) are oriented and directed according to the Schnyder rule. v between e and the right (left) neighbour of v, looking toward S out , have color 1 (0, resp.). Using these invariants, it is straightforward to prove the following lemma (see [7] for a detailed presentation): Lemma 6 Given a planar triangulation T with outer face (v 0 ,v 1 ,v n−1 ) the algorithm COMPUTESCHNYDERPLANAR computes in linear time a Schnyder wood of T . (d) A new traversal for genus g surfaces This section presents our main contribution, an algorithm for traversing a triangulated surface S of arbitrary genus g ≥ 0, which extends the algorithm of Brehm in a natural way. As in the planar case, the traversal is a greedy sequence of conquer operations on free vertices, with here the important difference that these operations are interleaved with 2g merge/split operations so as to make the genus of the conquered area increase from 0 to g. We start with C := {C 0 } := {(v 0 ,v 1 ,v n−1 )}. At the beginning S in is a topological disk delimited by the simple cycle C 0 . As in the planar case, we make use of the operation conquer(v) that consists in transferring the (remaining) faces incident to v from S out to S in . Associated with such a conquest is the colorient rule for coloring and orienting the edges that are conquered, as defined in Section 3(c). We also make use of the handle operations split and merge, as defined in Section 3(b). COMPUTESCHNYDERANYGENUS(S) (S a triangulated surface of genus g) while {C} ≠ {v 0 ,v 1 } If there exists a free vertex v on some C i ∈C conquer(v); colorient(v); otherwise, if there exists a split edge e =(u, w) ∈ S out \∂S in ; split(u,w); thereby adding a new cycle C ′ to C; otherwise, find a merge edge e =(u, w) ∈ S out ; merge(u,w); thereby removing a cycle from C; end while Figure 7 shows the traversal algorithm executed on a toroidal triangulation. Notice the subtlety that, for nonzero genus, the multiple vertices, i.e., the vertices incident to merge/split edges, are conquered several times, as illustrated in Figure 6. Precisely, for a vertex v incident to k ≥ 0 merge/split edges, conquer(v) occurs k +1times. Our main result is expressed by the following Theorem 7 Any triangulated surface S of genus g admits a genus g Schnyder wood, which can be computed in linear time. The validity of this property relies on the Lemmas stated in the next section. (e) Termination and complexity of the algorithm Lemma 8 For any triangulated surface S of genus g, COMPUTESCHNYDERANYGENUS(S) terminates. Proof : It suffices to show that a merge or split operation is always possible whenever there remain no free vertices on ∂S in . The main idea is to use an induction argument (on the number of faces of S out , which is decreasing along the conquest) and to proceed by contradiction to show the existence of non-contractible non-separating cycles corresponding to split/merge edges. More precisely, assume by contradiction that we have an example of situation where we are stuck, i.e., no free vertex on the boundary, and only contractible chordal edges or non-contractible separating chordal edges. Assume also that this is the smallest example of such a situation in terms of the number of faces in the non discovered area (notice that this number is at least 2 otherwise we would be at the last step of the algorithm where there is no problem; one just conquer the vertex not incident to the root edge in the last remaining not conquered face). Let us first observe that there cannot be a contractible chordal edge, since it would yield a free vertex on ∂S in , precisely, a free vertex in the sub-path P of ∂S in such that P + e is a contractible cycle (this is due to the well known property that a triangulated dissection of a polygon, with a base edge on the polygon, has at least one vertex not incident to the base edge nor incident to any internal chord, see [7]). Hence, all edges in S out \∂S in have to be non-contractible separating edges. Let e be such Preprint MAT. 18/08, communicated on September 1 st , 2008 to the Department of Mathematics, Pontifícia Universidade Católica — Rio de Janeiro, Brazil.
7 Schnyder woods for higher genus triangulated surfaces (planar case) w C C S out invariants S out conquer(w)+ colororient(w) Figure 5: These pictures show the result of a colorient operation in the planar case. an edge. Cutting S out along e consists in splitting into two smaller surfaces S 1 and S 2 , each of which is partitioned into a conquered area and a not conquered area. Assume w.l.o.g. that S 1 is the area whose boundary (between conquered and not conquered) does not contain the base-edge; and now define e as the base-edge of that boundary. Any free vertex v for S 1 is also a free vertex for S (indeed by definition v is not incident to any chord for S 1 nor is it an extremity of e; hence v is not incident to any chord of S). Therefore, as S has no free vertex, S 1 has neither a free vertex. Now observe that a non-contractible non-separating cycle for S 1 is also non-contractible and non-separating for S. Hence there can not be any candidate for a merge or split in S 1 (since there is no such candidate in S). We have thus reached a contradiction, as S 1 has fewer faces than S in its not conquered area (because of the minimality of S). Lemma 9 For any triangulated surface S of genus g having n vertices, the procedure COMPUTESCHNYDERANYGENUS(S) can be implemented to run in O(n) time. Proof : Implementing the algorithm to run in linear time requires some bookkeeping. For this purpose, we maintain the list of chordal edges for ∂S in , and, which is more difficult, we maintain a classification of these chordal edges as contractible, non-contractible separating, split edges, and merge edge. Such a classification is done by maintaining a spanning cut-graph G c for the dual of S out , such that G c contains the dual of all chordal edges; and by maintaining a depth-firstsearch spanning tree of G c so as to test which edges of G c disconnect G c . All this bookkeeping can be done in amortized linear cost, as detailed below. Firstly, consider a special vertex spanning tree T p of S out , including the boundary ∂S in in the following way (blue edges in Fig. 8). For each boundary component C i of ∂S in choose a boundary edge (x i 0,x i 1) (where for the first component we take the root edge (v 0 ,v 1 )) and add all the edges in ∂S in \ C i to the set T p (which initially consists of a set of disjoints paths). Then perform a depth-first traversal S out starting from v 0 , that computes a spanning tree of the primal graph of S out : add its edges to T p . Note that by construction chordal edges do not appear in T p , as each extremity already lies to the paths added in the first step. Now consider the dual map G c (red edges in Fig. 8) of T p , which is a sub-map of the dual graph of S out , defined in the following manner: its nodes correspond to the faces of S out , while its arcs are exactly the dual of that edges not appearing in T p . If we denote by g ′ the genus of S out , then G c is a map of genus g ′ , having only one face. We firstly simplify this structure, by recursively eliminating all degree one nodes (vertices which are leaves at some step of this decimating procedure). Note that this transformation does not affect the topology of the resulting map obtained by G c , which has still genus g ′ and one face, thus containing 2g ′ non trivial cycles. Observe that each chordal edge has a corresponding dual arc in G c . We now compute a vertex spanning tree of such map, which will contain ‖G c ‖−2g ′ arcs (as it cannot contain any cycle, by construction). Note that the cycle contained in map G c must be non-trivial (non-contractible and non-separating). For example, the existence of a contractible cycle would imply the existence of a vertex v which would not appear in T p (which by construction is a spanning tree of all the vertices of S out ). On the other hand, the existence of a non-contractible but Preprint MAT. 18/08, communicated on September 1 st , 2008 to the Department of Mathematics, Pontifícia Universidade Católica — Rio de Janeiro, Brazil.
Page 1 and 2: Schnyder woods for higher genus tri
Page 3 and 4: Schnyder woods for higher genus tri
Page 5 and 6: 3 Schnyder woods for higher genus t
Page 7: 5 Schnyder woods for higher genus t
Page 11 and 12: 9 Schnyder woods for higher genus t
Page 13 and 14: 11 Schnyder woods for higher genus
Page 15 and 16: 13 Schnyder woods for higher genus

Schnyder woods for higher genus triangulated surfaces with ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?