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We generalize regular subdivisions (polyhedral complexes resulting from the projection of the lower faces of a polyhedron) introducing the class of recursively-regular subdivisions. Informally speaking, a recursively-regular subdivision is a subdivision that can be obtained by splitting some faces of a regular subdivision by other regular subdivisions (and continue recursively). We also define the finest regular coarsening and the regularity tree of a polyhedral complex. We prove that recursively-regular subdivisions are not necessarily connected by flips and that they are acyclic with respect to the in-front relation. We show that the finest regular coarsening of a subdivision can be efficiently computed, and that whether a subdivision is recursively regular can be efficiently decided. As an application, we also extend a theorem known since 1981 on illuminating space by cones and present connections of recursive regularity to tensegrity theory and graph-embedding problems.

We show how to preprocess a polygonal domain with a fixed starting point $s$ in order to answer efficiently the following queries: Given a point $q$, how should one move from $s$ in order to see $q$ as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach $q$, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from $s$ to a query segment in the domain.

Given n red and n blue points in general position in the plane, it is well-known that there is a perfect matching formed by non-crossing line segments. We characterize the bichromatic point sets which admit exactly one non-crossing matching. We give several geometric descriptions of such sets, and find an O(n log n) algorithm that checks whether a given bichromatic set has this property.

Constant-work-space algorithms may use only constantly many cells of storage in addition to their input, which is provided as a read-only array. We show how to construct several geometric structures efficiently in the constant-work-space model. Traditional algorithms process the input into a suitable data structure (like a doubly-connected edge list) that allows efficient traversal of the structure at hand. In the constant-work-space setting, however, we cannot afford to do this. Instead, we provide operations that compute the desired features on the fly by accessing the input with no extra space. The whole geometric structure can be obtained by using these operations to enumerate all the features. Of course, we must pay for the space savings by slower running times. While the standard data structure allows us to implement traversal operations in constant time, our schemes typically take linear time to read the input data in each step.We begin with two simple problems: triangulating a planar point set and finding the trapezoidal decomposition of a simple polygon. In both cases adjacent features can be enumerated in linear time per step, resulting in total quadratic running time to output the whole structure. Actually, we show that the former result carries over to the Delaunay triangulation, and hence the Voronoi diagram. This also means that we can compute the largest empty circle of a planar point set in quadratic time and constant work-space. As another application, we demonstrate how to enumerate the features of an Euclidean minimum spanning tree (EMST) in quadratic time per step, so that the whole EMST can be found in cubic time using constant work-space.Finally, we describe how to compute a shortest geodesic path between two points in a simple polygon. Although the shortest path problem in general graphs is NL-complete (Jakoby and Tantau 2003), this constrained problem can be solved in quadratic time using only constant work-space.

We are pleased to welcome readers to this inaugural issue of the Journal of Computational Geometry.

We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We achieve this by analyzing polyominoes on a different surface, a so-called $\textit{twisted cylinder}$ by the transfer matrix method. A bijective representation of the "states'' of partial solutions is crucial for allowing a compact representation of the successive iteration vectors for the transfer matrix method.