On the total correctness of lawson's oriented walk algorithm

Lawson's oriented walk is a simple algorithm for point location without preprocessing. Originally formulated for planar Delaunay triangulations 2], it generalizes immediately to tesselations of a convex, polyhedral domain into convex, polyhedral cells in d-space. Given a query point q, the algorithm works as follows. 1. Pick some cell C of the tesselation. 2. For each facet F of C, determine the position of q relative to the hyperplane that contains F. 3. If q and C lie on opposite sides of the hyper-plane, step through F: If F is an interior facet of the tesselation, iterate Steps 2{4 for the cell on the other side of F. If F is a boundary facet, report that q lies outside the domain. 4. If q lies on the same side as C (or on the hyper-plane) for all facets of C, report that q 2 C. Partial correctness of the oriented walk is easily seen. If it terminates, it locates q correctly in some cell or outside the domain. Although the algorithm is widely used, it seems that its total correctness has not been investigated. The importance of this question is illustrated by a recent publication. M ucke et al. 3] use an oriented walk in 3-dimensional Delaunay triangu-lations. Innnite loops can occur in this setting. Figure 1 shows an innnitely looping oriented walk in a planar triangulation. In some triangles of the loop, there are two alternative edges through which the walk could step. The (hypothetical) implementation at hand tests the unfavorable edge rst, and therefore loops. A diierent implementation might test the favorable edge rst and terminate. If a walk happens to follow the line segment from some starting point to q, it will terminate. Thus, a given tesselation can only contain a potential innnite loop. Whether or not an oriented walk gets caught in this loop is implementation-dependent. Proposition 1. The oriented walk algorithm is only partially correct in planar triangulations. Supported by Deutsche Forschungsgemeinschaft (DFG), grant MU 744/3-2. q Figure 1. An innnitely looping oriented walk. The example in Figure 1 can bèinnated' to higher dimensions. Let T be a (d ? 1)-dimensional triangu-lation. Let S be a suuciently large sphere in d-space, and let H be the equatorial hyperplane of S, as in Figure 2. Embed T into H such that it lies inside S. Project the vertices of T into the northern hemisphere of S, …