P17338 [MX-X30-T4] Hypercube.

You are given two integers $n,k$. Now you have a graph with $2^n$ vertices, numbered $0\sim 2^n-1$. For any two vertices $x,y$, there is an edge between them if and only if $\mathrm{popc}(x\oplus y)=1$, that is, these two numbers differ in exactly one bit in binary. You need to choose $k$ simple cycles (cycles with no repeated vertices) in this graph, such that each vertex belongs to exactly one simple cycle. You need to output a construction or report that there is no solution.

This problem contains multiple test cases. The first line contains an integer $T$ indicating the number of test cases. Each test case contains one line with two integers $n,k$.

Output separately for each test case. If you think this test case has no solution, output one line with the string $\texttt{No}$. Otherwise, first output one line with the string $\texttt{Yes}$, followed by $k$ lines. In each line, the first integer $l$ denotes the length of the cycle, then output $v_1,v_2,v_3,\dots,v_l$ in the order along the cycle. You must ensure that these vertices are all distinct, and for $1\le i

| Test Point ID | $n\le $ | Special Property | |:-:|:-:|:-:| | $1\sim 3$ | $3$ | None | | $4,5$ | $13$ | A | | $6\sim 8$ | $13$ | None | | $9,10$ | $17$ | None | Special property A: $k=1$. For all testdata, $1\le T\le 10$, $1\le n\le 17$, $1\le k \le 2^{n}$. Translated by ChatGPT 5