Behnam Iranfar; Mohammad Farshi
Abstract
Given a point set $S\subset \mathbb{R}^d$, the $\theta$-graph of $S$ is as follows: for each point $s\in S$, draw cones with apex at $s$ and angle $\theta$ %fix a line through $p$ at each cone and connect $s$ to the point in each cone such that the projection of the point on the bisector of the cone ...
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Given a point set $S\subset \mathbb{R}^d$, the $\theta$-graph of $S$ is as follows: for each point $s\in S$, draw cones with apex at $s$ and angle $\theta$ %fix a line through $p$ at each cone and connect $s$ to the point in each cone such that the projection of the point on the bisector of the cone is the closest to~$s$. One can define the $\theta$- graph on an uncertain point set, i.e. a point set where each point $s_i$ exists with an independent probability $\pi_i \in (0,1]$. In this paper, we propose an algorithm that computes the expected weight of the $\theta$-graph on a given uncertain point set. The proposed algorithm takes $O(n^2\alpha(n^2,n)^{2d})$ time and $O(n^2)$ space, where $n$ is the number of points, $d$ and $\theta$ are constants, and $\alpha$ is the inverse of the Ackermann's function.
