| # 1334_D. Minimum Euler Cycle |
| |
| ## Problem Description |
| You are given a complete directed graph K_n with n vertices: each pair of vertices u ≠ v in K_n have both directed edges (u, v) and (v, u); there are no self-loops. |
| |
| You should find such a cycle in K_n that visits every directed edge exactly once (allowing for revisiting vertices). |
|
|
| We can write such cycle as a list of n(n - 1) + 1 vertices v_1, v_2, v_3, ..., v_{n(n - 1) - 1}, v_{n(n - 1)}, v_{n(n - 1) + 1} = v_1 — a visiting order, where each (v_i, v_{i + 1}) occurs exactly once. |
| |
| Find the lexicographically smallest such cycle. It's not hard to prove that the cycle always exists. |
| |
| Since the answer can be too large print its [l, r] segment, in other words, v_l, v_{l + 1}, ..., v_r. |
|
|
| Input |
|
|
| The first line contains the single integer T (1 ≤ T ≤ 100) — the number of test cases. |
|
|
| Next T lines contain test cases — one per line. The first and only line of each test case contains three integers n, l and r (2 ≤ n ≤ 10^5, 1 ≤ l ≤ r ≤ n(n - 1) + 1, r - l + 1 ≤ 10^5) — the number of vertices in K_n, and segment of the cycle to print. |
| |
| It's guaranteed that the total sum of n doesn't exceed 10^5 and the total sum of r - l + 1 doesn't exceed 10^5. |
| |
| Output |
| |
| For each test case print the segment v_l, v_{l + 1}, ..., v_r of the lexicographically smallest cycle that visits every edge exactly once. |
|
|
| Example |
|
|
| Input |
|
|
|
|
| 3 |
| 2 1 3 |
| 3 3 6 |
| 99995 9998900031 9998900031 |
|
|
|
|
| Output |
|
|
|
|
| 1 2 1 |
| 1 3 2 3 |
| 1 |
|
|
| Note |
|
|
| In the second test case, the lexicographically minimum cycle looks like: 1, 2, 1, 3, 2, 3, 1. |
|
|
| In the third test case, it's quite obvious that the cycle should start and end in vertex 1. |
|
|
| ## Contest Information |
| - **Contest ID**: 1334 |
| - **Problem Index**: D |
| - **Points**: 0.0 |
| - **Rating**: 1800 |
| - **Tags**: constructive algorithms, graphs, greedy, implementation |
| - **Time Limit**: {'seconds': 2, 'nanos': 0} seconds |
| - **Memory Limit**: 256000000 bytes |
|
|
| ## Task |
| Solve this competitive programming problem. Provide a complete solution that handles all the given constraints and edge cases. |
