Anil Maheshwari

Ahmad Biniaz, Prosenjit Bose, Chaeyoon Chung et al.

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are between any two successive hyperplanes. Let $A(S,m)$ be the set of different closest pairs in the ${{m+1} \choose 2}$ vertical slabs that are bounded by $H_i$ and $H_j$, over all $1 \leq i < j \leq m+1$. We prove tight bounds for the largest possible size of $A(S,m)$, over all point sets of size $n$, and for all values of $1 \leq m \leq n$. As a result of these bounds, we obtain, for any constant $ε>0$, a data structure of size $O(n)$, such that for any vertical query slab $Q$, the closest pair in the set $Q \cap S$ can be reported in $O(n^{1/2+ε})$ time. Prior to this work, no linear space data structure with sublinear query time was known.

Pegah Kamousi, Sylvain Lazard, Anil Maheshwari et al.

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor $\mathcal{F}_{FPS}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{FPS}$ can be bounded in terms of the minimal value $\mathcal{F}^*$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{FPS}\leq 2 r_e^2 \mathcal{F}^*+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we show that it is NP-complete to find $k$ sources that minimize the stretch factor.