Question

Given radius of expansion $R$, the size of the merging cluster is $w(R)$, which is the largest integer that is a power of 2 and is less than or equal to $R+1$, i.e., $w(R)=2^r \leq R+1<2^{r+1}$ for $r \geq 0$. Let $v s_{d_1}$ and $v s_{d_2}$ be two adjacent vertex subsets such that vertex $d_1$ is adjacent to $d_2$ in the clockwise or counterclockwise directions. Prove that the size of the union of any two adjacent vertex subsets is less than or equal to $w(R)+1$ and that this bound is attainable.

   Given radius of expansion $R$, the size of the merging cluster is $w(R)$, which is the largest integer that is a power of 2 and is less than or equal to $R+1$, i.e., $w(R)=2^r \leq R+1<2^{r+1}$ for $r \geq 0$. Let $v s_{d_1}$ and $v s_{d_2}$ be two adjacent vertex subsets such that vertex $d_1$ is adjacent to $d_2$ in the clockwise or counterclockwise directions. Prove that the size of the union of any two adjacent vertex subsets is less than or equal to $w(R)+1$ and that this bound is attainable.

Applications of spatial data structures to computer graphics

Hanan Samet 1st Edition

Chapter 6, Problem 92

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Step 1

In this case, $w(R)=2^0=1$. We have two adjacent vertex subsets, $v s_{d_1}$ and $v s_{d_2}$, with $d_1$ adjacent to $d_2$. The size of each vertex subset is at most $w(R)+1=2$, so the size of their union is at most $2+2=4$. Now, let's consider the case where Show more…

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Given radius of expansion $R$, the size of the merging cluster is $w(R)$, which is the largest integer that is a power of 2 and is less than or equal to $R+1$, i.e., $w(R)=2^r \leq R+1<2^{r+1}$ for $r \geq 0$. Let $v s_{d_1}$ and $v s_{d_2}$ be two adjacent vertex subsets such that vertex $d_1$ is adjacent to $d_2$ in the clockwise or counterclockwise directions. Prove that the size of the union of any two adjacent vertex subsets is less than or equal to $w(R)+1$ and that this bound is attainable.

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