Line Intersection Searching Amid Unit Balls in 3-Space

Let B be a set of n unit balls in R3. We present a linear-size data structure for storing B that can determine in O∗(n) time whether a query line intersects any ball of B and report all k such balls in additional O(k) time. The data structure can be constructed in O(nlogn) time. (The O∗(·) notation hides subpolynomial factors, e.g., of the form O(nε), for arbitrarily small ε>0, and their coefficients which depend on ε.) We also consider the dual problem: Let L be a set of n lines in R3. We preprocess L, in O∗(n2) time, into a data structure of size O∗(n2) that can determine in O(logn) time whether a query unit ball intersects any line of L, or report all k such lines in additional O(k) time.

Duke Scholars

Author Pankaj K. Agarwal Computer Science