F. Messenger
Misha needs to send packages to his friend Nadia. Both of them often travel across Russia, which is very large. So they decide to hire a messenger. Since the cost of the messenger service depends on the time it takes to deliver the package, they need your h...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
Misha needs to send packages to his friend Nadia. Both of them often travel across Russia, which is very large. So they decide to hire a messenger. Since the cost of the messenger service depends on the time it takes to deliver the package, they need your help to optimize a little bit. Assume Misha and Nadia move on a two-dimensional plane, each visiting a sequence of places and moving along straight line segments from place to place. Your task is to find the shortest possible delivery time given their two paths. Misha hands the package to the messenger at some point along his path. The messenger moves without delay along a straight line from the pick-up to intercept Nadia, who is traveling along her path. Misha, Nadia and the messenger move with a constant speed of 1 distance unit per time unit. The delivery time is the time between Misha handing over the package and Nadia receiving it.
Input
The input consists of a single test case. The test case contains two path descriptions, the first for Misha and the second for Nadia. Each path description starts with a line containing an integer n, the number of places visited (2 ≤ n ≤ 50 000). This is followed by n lines, each with two integers xi and yi specifying the coordinates of a place (0 ≤ xi , yi ≤ 30 000). Coordinates of the places are listed in the order in which they are to be visited, and successive places do not have the same coordinates. Misha and Nadia start their journeys at the same time, visiting the places along their paths without stopping. The length of each path is at most 106 . The package must be picked up at the latest when Misha reaches his final place and it must be delivered at the latest when Nadia reaches her final place.
Output
Display the minimal time needed for delivery. Give the answer with an absolute error of at most 10−3 or a relative error of at most 10−5 . If the package cannot be delivered, display impossible instead.
Sample Tests
2
0 0
0 10
2
4 10
4 04.000002
0 0
1 0
3
2 0
3 0
3 105.00000Editorial
The solution write-up is rendered from the LaTeX source, with equations kept live through MathJax.
Key Observations
Write the structural observations that make the problem tractable.
State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
If the constraints matter, explain exactly which part of the solution they enable.
Algorithm
Describe the data structures and the state maintained by the algorithm.
Explain the processing order and why it is sufficient.
Mention corner cases explicitly if they affect the implementation.
Correctness Proof
We prove that the algorithm returns the correct answer.
Lemma 1.
State the first key claim.
Proof.
Provide a concise proof.
Lemma 2.
State the next claim if needed.
Proof.
Provide a concise proof.
Theorem.
The algorithm outputs the correct answer for every valid input.
Proof.
Combine the lemmas and finish the argument.
Complexity Analysis
State the running time and memory usage in terms of the input size.
Implementation Notes
Mention any non-obvious implementation detail that is easy to get wrong.
Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
Code
C++ solution used for this page.
#include <bits/stdc++.h>
using namespace std;
namespace {
void solve() {
// Fill in the full solution logic for the problem here.
}
} // namespace
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
solve();
return 0;
}
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