Well Separated Pair Decomposition for Unit–Disk Graph

Keywords and Synonyms

Clustering

Problem Definition

Notations

Given a finite point set A in ℝ^d, its bounding box R(A) is the d‑dimensional hyper‐rectangle $[a_1,b_1] \times [a_2,b_2] \times \dots \linebreak \times [a_d,b_d]$ that contains A and has minimum extension in each dimension.

Two point sets A, B are said to be well‐separated with respect to a separation parameter $s \,{\mathchar"313E}\, 0$ if there exist a real number $r\,{\mathchar"313E}\, 0$ and two d‑dimensional spheres C _A and C _B of radius r each, such that the following properties are fulfilled.

• 1. $C_A \cap C_B = \emptyset$

• 2. C _A contains the bounding box R(A) of A

• 3. C _B contains the bounding box R(B) of B

• 4. $|C_AC_B| \geq s\cdot r$.

Here $|C_AC_B|$ denotes the smallest Euclidean distance between two points of C _A and C _B , respectively. An example is depicted in Fig. 1. Given the bounding boxes R(A), R(B), it takes time only O(d) to test if A and B are well‐separated with respect to s.

Two...