Line Intersection Searching Amid Unit Balls in 3-Space

Abstract

Let \(\mathscr {B}\) be a set of n unit balls in \({\mathbb {R}}^3\). We present a linear-size data structure for storing \(\mathscr {B}\) that can determine in \(O^*(\sqrt{n})\) time whether a query line intersects any ball of \(\mathscr {B}\) and report all k such balls in additional O(k) time. The data structure can be constructed in \(O(n\log n)\) time. (The \(O^*(\cdot )\) notation hides subpolynomial factors, e.g., of the form \(O(n^{{\varepsilon }})\), for arbitrarily small \({\varepsilon }> 0\), and their coefficients which depend on \({\varepsilon }\).) We also consider the dual problem: Let \(\mathscr {L}\) be a set of n lines in \({\mathbb {R}}^3\). We preprocess \(\mathscr {L}\), in \(O^*(n^2)\) time, into a data structure of size \(O^*(n^2)\) that can determine in \(O(\log {n})\) time whether a query unit ball intersects any line of \(\mathscr {L}\), or report all k such lines in additional O(k) time.