(Analysis by Andi Qu)

Although the problem does not state what happens when two particles of the same type meet, this event will never occur in Bessie's experiment setup. This is due to three reasons:

  1. Particles meet in pairs.
  2. And two particles cannot meet each other until all particles between them are gone.
  3. But particles of the same type start with an odd number of particles between them.
So meetings will only ever occur between mootrinos and antimootrinos.

Subtask 1: $O(1)$ for a single pair of particles

The two particles meet only when they are moving toward each other. Because the particles start off moving toward each other, Bessie must observe their disappearance on an odd observation.

On the $k$-th odd observation, the mootrino will be $k \cdot s_1$ units to the right of its initial position. Likewise, the antimootrino will be $k \cdot s_2$ units to the left of its initial position. Therefore, Bessie will observe their disappearance on the $\left\lceil\frac{p_2 - p_1}{s_1 + s_2}\right\rceil$-th odd observation.

We can generalize this solution somewhat by considering what happens if the leftmost particle is an antimootrino instead. In this case, Bessie will observe the particles' disappearance on the $\left\lceil\frac{p_2 - p_1}{s_1 + s_2}\right\rceil$-th even observation.

Putting it together, we get the following function:

long long compute_meeting_time(int a, int b) {
    long long dist = p[b] - p[a];
    long long speed_sum = s[a] + s[b];
    return 2 * ((dist + speed_sum - 1) / speed_sum) - (a % 2 == 0);

which we will use in the subsequent subtasks.

Subtask 2: $O(N \cdot \max\{p_i\})$

Simulating the movement of the particles is sufficient for this subtask.

Subtask 3: $O(N^2)$

The problem with brute-force simulation is that there may be long stretches of observations where nothing happens. (For example, if the particles are all spaced very far apart but all have very low speeds.)

A more efficient algorithm works as follows:

This algorithm runs in $O(N^2)$ because we repeat the steps $\frac{N}{2}$ times, and each step takes $O(N)$ time to execute.

Full credit: $O(N \log N)$

We can improve this algorithm further by noticing that the list of particle pairs to consider only changes by at most $4$ pairs each iteration. Suppose particles $o_1$, $o_2$, $o_3$, and $o_4$ are next to each other (in that order). When we remove the pair $(o_2, o_3)$ from the line:

With this in mind, we can use a priority queue to store the list of adjacent pairs, sorted by meeting time.

We can also use a linked list to help determine which new pairs to add to the priority queue. For each still-existing particle, this linked list would store the indices of the still-existing particle(s) directly adjacent to it.

These improvements bring the time complexity down to $O(N \log N)$ because each step now modifies the priority queue $O(1)$ times, and each modification takes $O(\log N)$ time.

C++ code implementing this solution:

#include <iostream>
#include <queue>
#include <utility>
typedef long long ll;
using namespace std;
ll p[200000], s[200000], ans[200000];
pair<int, int> adj[200000];
ll compute_meeting_time(int a, int b) {
    ll dist = p[b] - p[a], speed_sum = s[a] + s[b];
    return 2 * ((dist + speed_sum - 1) / speed_sum) - (a % 2 == 0);
int main() {
    int t;
    cin >> t;
    while (t--) {
        int n;
        cin >> n;
        for (int i = 0; i < n; i++)
            cin >> p[i];
        for (int i = 0; i < n; i++)
            cin >> s[i];
        for (int i = 0; i < n; i++) {
            adj[i] = {i - 1, i + 1};
            ans[i] = 0;
        priority_queue<pair<ll, pair<int, int>>> pq;
        for (int i = 1; i < n; i++)
            pq.push({-compute_meeting_time(i - 1, i), {i - 1, i}});
        while (pq.size()) {
            pair<ll, pair<int, int>> next_meeting = pq.top();
            ll meeting_time = next_meeting.first;
            pair<int, int> cows = next_meeting.second;
            if (ans[cows.first] || ans[cows.second])
            ans[cows.first] = ans[cows.second] = -meeting_time;
            if (cows.first != 0 && cows.second != n - 1) {
                int prv = adj[cows.first].first, nxt = adj[cows.second].second;
                adj[prv].second = nxt;
                adj[nxt].first = prv;
                pq.push({-compute_meeting_time(prv, nxt), {prv, nxt}});
        for (int i = 0; i < n; i++)
            cout << ans[i] << " \n"[i == n - 1];
    return 0;