G. The Islands
Problem ID: islands Wen Chen is the captain of a rescue boat. One of his important tasks is to visit a group of islands once a day to check if everything is all right. Captain Wen starts from the west-most island, makes a pass to the east-most island visiti...
Problem Statement
Formatted from the contest statement text, with sample tests broken out into copyable blocks.
Wen Chen is the captain of a rescue boat. One of his important tasks is to visit a group of islands once a day to check if everything is all right. Captain Wen starts from the west-most island, makes a pass to the east-most island visiting some of the islands, then makes a second pass from the east-most island back to the first one visiting the remaining islands. In each pass Captain Wen moves steadily east (in the first pass) or west (in the second pass), but moves as far north or south as he needs to reach the islands. The only complication is that there are two special islands where Wen gets fuel for his boat, so he must visit them in separate passes. Figure 7 shows the two special islands in pink (1 and 3) and one possible path Captain Wen could take.
Calculate the length of the shortest path to visit all
Figure 7
the islands in two passes when each island’s
location and the identification of the two special islands are given.Input
The input consists of multiple test cases. The data for each case begins with a line containing 3 integers n (4 ≤ n ≤ 100), b1, and b2 (0 < b1, b2 < n-1 and b1 ≠ b2), where n is the number of islands (numbered 0 to n-1) and b1 and b2 are the two special islands. Following this, there are n lines containing the integer x- and y-coordinates of each island (0 ≤ x, y ≤ 2000), starting with island 0. No two islands have the same x-coordinate and they are in order from west-most to east-most (that is, minimum x-coordinate to maximum x-coordinate).
Input for the last case is followed by a line containing 3 zeroes.
Output
For each case, display two lines. The first line contains the case number and the length of the shortest tour Captain Wen can take to visit all the islands, rounded and displayed to the nearest hundredth. The second line contains a space-separated list of the islands in the order that they should be visited, starting with island 0 and 1, and ending with island 0. Each test case will have a unique solution. Follow the format in the sample output.
Sample Tests
5 1 3
1 3
3 4
4 1
7 5
8 3
5 3 2
0 10
3 14
4 7
7 10
8 12
0 0 0
This page intentionally left blank.Case 1: 18.18
0 1 4 3 2 0
Case 2: 24.30
0 1 3 4 2 0Editorial
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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