Intersection Searching amid Tetrahedra in Four Dimensions

We develop data structures for intersection queries in four dimensions that involve segments, triangles and tetrahedra. Specifically, we study three main problems: (i) Preprocess a set of $n$ tetrahedra in $\reals^4$ into a data structure for answering segment-intersection queries amid the given tetrahedra (referred to as \emph{segment-tetrahedron intersection queries}). (ii) Preprocess a set of $n$ triangles in $\reals^4$ into a data structure that supports triangle-intersection queries amid the input triangles (referred to as \emph{triangle-triangle intersection queries}). (iii) Preprocess a set of $n$ segments in $\reals^4$ into a data structure that supports tetrahedron-intersection queries amid the input segments (referred to as \emph{tetrahedron-segment intersection queries}). In each problem we want either to detect an intersection, or to count or report all intersections. As far as we can tell, these problems have not been previously studied. For problem (i), we first present a "standard" solution which, for any prespecified value $n \le s \le n^6$ of a so-called storage parameter $s$, yields a data structure with $O^*(s)$ storage and expected preprocessing, which answers an intersection query in $O^*(n/s^{1/6})$ time (here and in what follows, the $O^*(\cdot)$ notation hides subpolynomial factors). For problems (ii) and (iii), using similar arguments, we present a solution that has the same asymptotic performance bounds. We then improve the solution for problem (i), and present a more intricate data structure that uses $O^*(n^{2})$ storage and expected preprocessing, and answers a segment-tetrahedron intersection query in $O^*(n^{1/2})$ time.