Render(m(v)) /* Generate Rendering Sequence for Mesh M on Vertex Set V *

An approximate max-ow min-cut theorem for uniform multicommod-ity ow problems with applications to approximation algorithms. Therefore, without loss of generality, we may assume that jS 1 j < p m 1 and proceed recursively to separate V 1 , until a component of the required size is obtained. C(n), the complexity of this procedure, satisses C(n) O(n) + C(2n=3) implying C(n) = O(n). Theorem 6 implies the existence of a O(n) procedure separate(G; k), which, given a planar graph G(V) on n vertices, computes a triple < U; S; W > such that S separates G to U and W, (W = V ? U ? S), k=3 jUj 2k=3 and jSj < 12:5 p k. Given a polygon mesh M, using Theorem 6, we separate a small submesh M 0 from M, use minimum time render to generate a rendering sequence for the polygons deened on M 0 (including the separator), and then discard M 0. The process is continued with the remainder of the mesh (again including the separator). Assume the vertex stack size is k. The following algorithm generates a rendering sequence for a polygon mesh under this constraint. d is a suuciently large constant, whose exact value may be determined later. Theorem 7 Algorithm render generates a rendering sequence requiring a stack of size k, which pushes n(1 + c=k) vertices, for some constant c. The algorithm runs in O(n 2 =k 2) time. Proof: At each iteration of the loop, minimum time render pushes the vertices of S once in order to render M(U S), but these are pushed again in the rendering of M(W S). The number of iterations is O(n=k 2). By Theorem 6, at each iteration jSj = O(k), so the number of vertices pushed at that iteration is jU Sj jUj + jSj = jUj + O(k). The total number of vertices pushed during render is therefore = n + O(n=k 2)O(k) n(1 + c=k), for some constant c. render does not require a stack of size more than k since this holds for minimum time render. Each iteration runs in O(n) time, so the total run time is O(n 2 =k 2). 6 Conclusion We have explored the advantages of extending the architecture of contemporary graphics engines to larger vertex stores, in order to render polygon meshes more rapidly. We have shown that any n-vertex mesh may be rendered in …