Falling Portals
Falling Portals
There are n worlds, each with a portal. Initially, world i (for 1 ≤ i ≤ n) is at x-coordinate i, and y-coordinate Ai
(1 ≤ Ai
≤ 109
). There is also a cow on each world. At time 0, all y-coordinates are distinct and the worlds start falling: world i moves continuously in the negative y direction at a speed of i units per second.
At any time when two worlds are at the same y-coordinate (possibly a fractional time), the portals "align", meaning that a cow on one of the worlds can choose to travel instantaneously to the other world.
For each i, the cow on world i wants to travel to world Qi
(Qi
≠ i). Help each cow determine how long her journey will take, if she travels optimally.
Each query output should be a fraction a / b where a and b are positive and relatively prime integers, or −1 if it the journey is impossible.
Input
The first line contains a single integer n (2 ≤ n ≤ 2 * 105
). The next line contains n space-separated integers A1
, A2
, ..., An
. The next line contains n integers Q1
, Q2
, ..., Qn
.
Output
Print n lines, the i-th of which contains the journey length for cow i.
4 3 5 10 2 3 3 2 1
7/2 7/2 5/1 -1