Problem C
Svemir
Fourth Great and Bountiful Human Empire is developing a transconduit tunnel network connecting all it’s planets. The Empire consists of $N$ planets, represented as points in the 3D space. The cost of forming a transconduit tunnel between planets $A$ and $B$ is:
\[ \operatorname {TunnelCost}(A,B) = \min \{ |x_ A - x_ B |, |y_ A -y_ B |, |z_ A -z_ B | \} \]where $(x_ A, y_ A, z_ A)$ are the 3D coordinates of planet $A$, and $(x_ B, y_ B, z_ B)$ are coordinates of planet $B$. The Empire needs to build exactly $N - 1$ tunnels in order to fully connect all planets, either by direct links or by chain of links. You need to come up with the lowest possible cost of successfully completing this project.
Input
The first line of input contains one integer $N$ ($1 \le N \le 100\, 000$), the number of planets.
The next $N$ lines contain exactly $3$ integers each. All integers are between $-10^9$ and $10^9$ inclusive. Each line contains the $x$, $y$, and $z$ coordinate of one planet (in order).
No two planets will occupy the exact same point in space.
Output
The first and only line of output should contain the minimal cost of forming the network of tunnels.
| Sample Input 1 | Sample Output 1 |
|---|---|
2 1 5 10 7 8 2 |
3 |
| Sample Input 2 | Sample Output 2 |
|---|---|
3 -1 -1 -1 5 5 5 10 10 10 |
11 |
| Sample Input 3 | Sample Output 3 |
|---|---|
5 11 -15 -15 14 -5 -15 -1 -1 -5 10 -4 -1 19 -4 19 |
4 |