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Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line segment is minimized. In the continuous median line segment problem, a real number $\ell>0$ is given, and the goal is to locate a line segment of length $\ell$ in $\mathbb{R}^d$ such that the sum of the Euclidean distances between $P$ and the line segment is minimized. We show how to compute $(1+\epsilon\Delta)$- and $(1+\epsilon)$-approximations to a discrete median line segment in time $O(n\epsilon^{-2d}\log n)$ and $O(n^2\epsilon^{-d})$, respectively, where $\Delta$ is the spread of line segments spanned by pairs of points. While developing our algorithms, by using the principle of pair decomposition, we derive new data structures that allow us to quickly approximate the sum of the distances from a set of points to a given line segment or point. To our knowledge, our utilization of pair decompositions for solving minsum facility location problems is the first of its kind; it is versatile and easily implementable. We prove that it is impossible to construct a continuous median line segment for $n\geq3$ non-collinear points in the plane by using only ruler and compass. In view of this, we present an $O(n^d\epsilon^{-d})$-time algorithm for approximating a continuous median line segment in $\mathbb{R}^d$ within a factor of $1+\epsilon$. The algorithm is based upon generalizing the point-segment pair decomposition from the discrete to the continuous domain. Last but not least, we give an $(1+\epsilon)$-approximation algorithm, whose time complexity is sub-quadratic in $n$, for solving the constrained median line segment problem in $\mathbb{R}^2$ where an endpoint or the slope of the median line segment is given at input., Comment: There is an error in the packing argument used to determine the number of well separated pairs/subsets. The error first appears in the proof in Appendix C -- on page 17, in the first mathematical expression, the term $||ab||$ should be $||ab||/E[x]$

Consider an input consisting of a set of $n$ disjoint triangular obstacles in $\mathbb{R}^3$ and a target point $t$ in the free space, all enclosed by a large sphere $S$ of radius $R$ centered at $t$. An articulated probe is modeled as two line segments $ab$ and $bc$ connected at point $b$. The length of $ab$ can be equal to or greater than $R$, while $bc$ is of a given length $r \leq R$. The probe is initially located outside $S$, assuming an unarticulated configuration, in which $ab$ and $bc$ are collinear and $b \in ac$. The goal is to find a feasible (obstacle-avoiding) probe trajectory to reach $t$, with the condition that the probe is constrained by the following sequence of moves -- a straight-line insertion of the unarticulated probe into $S$, possibly followed by a rotation of $bc$ at $b$ for at most $\pi/2$ radians, so that $c$ coincides with $t$. We prove that if there exists a feasible probe trajectory, then a set of extremal feasible trajectories must be present. Through careful case analysis, we show that these extremal trajectories can be represented by $O(n^4)$ combinatorial events. We present a solution approach that enumerates and verifies these combinatorial events for feasibility in overall $O(n^{4+\epsilon})$ time using $O(n^{4+\epsilon})$ space, for any constant $\epsilon > 0$. The enumeration algorithm is highly parallel, considering that each combinatorial event can be generated and verified for feasibility independently of the others. In the process of deriving our solution, we design the first data structure for addressing a special instance of circular sector emptiness queries among polyhedral obstacles in three dimensional space, and provide a simplified data structure for the corresponding emptiness query problem in two dimensions., Comment: 38 pages, 14 figures. A preliminary version of this work was presented at the 31st Annual Canadian Conference on Computational Geometry

An articulated probe is modeled in the plane as two line segments, $ab$ and $bc$, joined at $b$, with $ab$ being very long, and $bc$ of some small length $r$. We investigate a trajectory planning problem involving the articulated two-segment probe where the length $r$ of $bc$ can be customized. Consider a set $P$ of simple polygonal obstacles with a total of $n$ vertices, a target point $t$ located in the free space such that $t$ cannot see to infinity, and a circle $S$ centered at $t$ enclosing $P$. The probe initially resides outside $S$, with $ab$ and $bc$ being collinear, and is restricted to the following sequence of moves: a straight line insertion of $abc$ into $S$ followed by a rotation of $bc$ around $b$. The goal is to compute a feasible obstacle-avoiding trajectory for the probe so that, after the sequence of moves, $c$ coincides with $t$. We prove that, for $n$ line segment obstacles, the smallest length $r$ for which there exists a feasible probe trajectory can be found in $O(n^{2+\epsilon})$ time using $O(n^{2+\epsilon})$ space, for any constant $\epsilon > 0$. Furthermore, we prove that all values $r$ for which a feasible probe trajectory exists form $O(n^2)$ intervals, and can be computed in $O(n^{5/2})$ time using $O(n^{2+\epsilon})$ space. We also show that, for a given $r$, the feasible trajectory space of the articulated probe can be characterized by a simple arrangement of complexity $O(n^2)$, which can be constructed in $O(n^2)$ time. To obtain our solutions, we design efficient data structures for a number of interesting variants of geometric intersection and emptiness query problems., Comment: 42 pages, 19 figures. A preliminary version of this work was presented at the 32nd Annual Canadian Conference on Computational Geometry