Schnyder woods for higher genus triangulated surfaces with ...

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Luca Castelli Aleardi, Eric Fusy and Thomas Lewiner 10 S in S in S in S in spanning tree T p dual map G c G c after simplification S in Figure 8: Special edges (dual of green edges) can efficiently be found by computing a spanning tree of the map G c . 2, and the only sink for ingoing edges of color 2 is v n−1 . Hence to prove that T 2 is a tree rooted at v n−1 it suffices to prove that T 2 is connected, precisely that each vertex is the origin of a path in T 2 going to the root v n−1 . This property is clearly maintained for the vertices of the conquered area all along the algorithm. Indeed each time a vertex w appears in S in , it is due to the conquest of a vertex v previously on the boundary of S in . In addition by definition of a conquest, there is an edge of color 2 from w to v. Hence the path of color 2 from v to v n−1 is extended to a path from w to v n−1 . Let us now prove that the graph C formed by T 2 and the special edges is a unicellular map on the surface. Just before the last step of the algorithm (at that time the entire cutgraph is discovered), the unique face t not in S in is the face incident to the base edge on the other side of the root face. Hence ∂S in has a unique component (cycle) and this cycle is contractible. According to Lemma 11, S in \C is planar and C has a unique face. The interior of C is the union of a planar annulus (S in ) and of the triangular face t. Hence the interior of C is a topological disk, i.e., C is a unicellular map. 4 The maps in color 1 and 2 are also cellular In this section we show that g-Schnyder woods provide a characterization in terms of a special class of genus g spanning maps, which can be viewed as a good generalization of plane trees. Proposition 13 Consider a genus g Schnyder wood computed by the algorithm COMPUTESCHNYDERANYGENUS. Then the embedded graph T 0 (T 1 ) formed by the edges in color 0 (1, resp.) is cellular and has at most 1+2g faces. Proof : When doing all the steps of COMPUTESCHNYDER- ANYGENUS, in reverse chronological order, one grows the surface S out until it occupies all the surface S except the root face. During that process, let T 1 be the (growing) part of T 1 already discovered (i.e., the part of T 1 in S out ). To prove the result, it suffices show the following invariant when growing S out : S out \T 1 remains planar, each boundary of S out is enclosed inside a unique facial walk of T 1 , and each facial walk of T 1 contains at most one boundary of ∂S in . Let us first describe the effect of conquest, split, and merge on S out when doing these operations in reverse chronological order. One checks that a (reverse) conquest adds a vertex on the boundary of S out , and does not change the topology (number of boundary components and genus) of S out . In addition it is well known in the study of topological surfaces [31] that a (reverse) split is also a split for S out and that a reverse merge is also a merge for S out . The only difference is that the two bases of the bridge-band for S in are the sides of the bridgeband for S out and the two sides of the bridge-band for S in are the bases of the bridge-band for S out . In each case, the conquests performed at the vertices on each side of the thin bridge-band create a bridge of edges of color 1 traversing the band transversally. By an argument similar to the one used in Lemma 11 (see also Figure 10) this ensures that S out \T 1 remains planar and, more generally, that the invariants stated above still hold. Finally, the bound 1 + 2g on the number of faces is due to the fact that each special edge contributes at most by one to the number of new faces of T i , for i =0, 1 (see Fig. 12). Remark 1 The properties of T 0 ,T 1 ,T 2 stated in Proposition 13 can be considered as extensions of the fundamental property of planar Schnyder woods, which states that T 0 , T 1 , and T 2 are plane trees. Figure 11 shows an example in genus 1. 5 Application to encoding In the planar case, Schnyder woods yield a simple encoding procedure for triangulations, as described in [19] and more recently in [1]. Precisely, a Schnyder wood with n inner vertices is encoded by two parenthesis words W, W ′ each of size 2n, where W is the classical Dyck encoding for the tree T 2 ; and W ′ is obtained from a clockwise walk around T 2 starting at v n−1 , writing an opening (closing) parenthesis each time an outgoing edge in color 1 (ingoing edge in color 0, resp.) is crossed, and completing the end of the word with closing parenthesis. Hence, for a triangulation with n +3vertices, the coding word has length 4n. This code is both simple and quite compact, as the length 4n is not far from the information-theory lower bound of ( log2 ( 4 4 /3 3) · n ) ≈ 3.245 · n, which is attained in the planar case by a bijective construction due to Poulalhon and Schaeffer [29]. Such a bijective construction is still to be found for higher genus, but we can here at least extend to higher genus the simple encoding procedure based on Schnyder woods. To Preprint MAT. 18/08, communicated on September 1 st , 2008 to the Department of Mathematics, Pontifícia Universidade Católica — Rio de Janeiro, Brazil.
11 Schnyder woods for higher genus triangulated surfaces S in S out w r w w z S in S in u z u conquer(w r )+ colorient(w r ) w l w z u conquer(u)+ colorient(u) w z u conquer(w l )+ colorient(w l ) w z u conquer(w)+ colorient(w) w r conquer(w l )+ Figure 9: The correctness of our construction relies on a new invariant describing the edge orientation around multiple vertices. encode the Schnyder wood we proceed in a similar way as in the planar case except that we have to deal with the special edges. First, the one-face map C formed by T 2 and the special edges (the cut-graph of the Schnyder wood) is encoded by the Dyck word W for T 2 , augmented by O(g) pointers, each of size O(log(n)) bits, so as to locate the two extremities of each special edge. We obtain the second encoding word W ′ from a clockwise walk along the unique face of C starting at v n−1 , writing an opening (closing) parenthesis each time an outgoing edge in color 1 (ingoing edge in color 0, resp.) is crossed, and completing the end of the word with closing parenthesis. Lemma 14 Let S be a triangulated surface endowed with a g-Schnyder wood. Then the information given by edges of color 0 is redundant, i.e., the Schnyder wood can be recovered even after all edges of color 0 have been deleted. Proof : When all edges of color 0 are deleted, we obtain a map M on the surface with only edges of color 1 and 2 (and the special edges). In each face f of this map, there is one corner that received all ends of the edges of color 0 (before they were deleted) that were inside f. This corner can be recovered since it is the unique corner around the face such that the right edge of the angle (looking toward the interior of f) is outgoing of color 1 and the left edge of the angle is outgoing of color 2. Lemma 15 Consider a genus g Schnyder wood computed by the algorithm COMPUTESCHNYDERANYGENUS, whose cut-graph is unfolded into a flat polygon. Let e = (u, v) be an edge of color 1 (oriented from u to v). Then u is encountered before v during a cw walk along the polygon (i.e., with the interior of the polygon on its right) starting at v n−1 . Proof : Among the two walks along the polygon that join u and v, consider the one, called W e , that does not pass by v n−1 . Proving the lemma is equivalent to proving that the region enclosed by e and W e is on the left of e. Now consider the step of the algorithm where u is conquered. Before that step, e is on a certain component (cycle) C of the boundary ∂S in . By definition of a conquest, e has the conquered area on its left when it is treated. According to Lemma 11, C is enclosed inside a face f of the current cut-graph C. Let W e be the walk around f from u to v. Clearly the area enclosed by e and W e is on the left of e, and W e does not pass by v n−1 (to be proved). In addition, this walk along C is now completely comprised inside the conquered region after the conquest involving e, hence it will not change until the end of the algorithm, i.e., W e will still be the unique walk along the final cut-graph C joining u and v without passing by v n−1 . This concludes the proof. Lemma 15 is crucial to encode the edges of color 1 (red edges). When walking around the cut-graph (with the face on the right), write an opening parenthesis when crossing an outgoing edge of color 1 and a closing parenthesis when crossing an ingoing edge of color 1. According to the lemma above, the word we obtain is a parenthesis word (each prefix has at least as many opening as closing parentheses) and each matching corresponds to an edge of color 1. In addition, from the parenthesis word, it is easy to place the ingoing half-edge and outgoing half-edges of color 1 (before doing the matchings so as to form complete edges of color 1). Indeed, one can recognize around the cut graph (using the local conditions of Schnyder woods) which corners can receive ingoing edges of color 1 and which corners can receive ingoing edges of color 1. To conclude, the edges of color 0 are redundant, the cutgraph can be encoded by a parenthesis word of length 2n (for the tree T 2 ) plus O(g) pointers each of length O(log(n)) (for the special edges), and the edges of color 1 can be encoded by a parenthesis word of length 2(n +4g) (since there are n +4g edges of color 1, each of which corresponds to a matching of the parenthesis word). All in all, we obtain the following result: Proposition 16 A triangulated surface of genus g with n+3 vertices can be encoded — via a genus g Schnyder wood — by a binary word of length 4n + O(g log(n)). Coding and decoding can be done in linear time. We mention that one could also design a more sophisticated code that supports queries: this time we should need to encode the three maps T i using multiple parenthesis systems (as done in [12, 2]). Preprint MAT. 18/08, communicated on September 1 st , 2008 to the Department of Mathematics, Pontifícia Universidade Católica — Rio de Janeiro, Brazil.
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