CF1311E.Construct the Binary Tree

You are given two integers $n$ and $d$. You need to construct a rooted binary tree consisting of $n$ vertices with a root at the vertex $1$ and the sum of depths of all vertices equals to $d$.

A tree is a connected graph without cycles. A rooted tree has a special vertex called the root. A parent of a vertex $v$ is the last different from $v$ vertex on the path from the root to the vertex $v$. The depth of the vertex $v$ is the length of the path from the root to the vertex $v$. Children of vertex $v$ are all vertices for which $v$ is the parent. The binary tree is such a tree that no vertex has more than $2$ children.

You have to answer $t$ independent test cases.

给你两个整数 $n$ 和 $d$。你需要构造一棵包含 $n$ 个顶点的有根二叉树，其中顶点 $1$ 为根，且所有顶点的深度之和恰好等于 $d$。

树是一种无环的连通图。有根树中有一个特殊的顶点称为根。顶点 $v$ 的父节点是指从根到 $v$ 的路径上最后一个不同于 $v$ 的顶点。顶点 $v$ 的深度定义为从根到 $v$ 的路径长度。顶点 $v$ 的子节点是指所有以 $v$ 为父节点的顶点。二叉树是一种每个顶点至多有两个子节点的树。

你需要回答 $t$ 个相互独立的测试用例。

The first line of the input contains one integer $t$ ($1 \le t \le 1000$) — the number of test cases.

The only line of each test case contains two integers $n$ and $d$ ($2 \le n, d \le 5000$) — the number of vertices in the tree and the required sum of depths of all vertices.

It is guaranteed that the sum of $n$ and the sum of $d$ both does not exceed $5000$ ($\sum n \le 5000, \sum d \le 5000$).

输入的第一行包含一个整数 $t$（$1 \le t \le 1000$），表示测试用例的数量。

每个测试用例仅一行，包含两个整数 $n$ 和 $d$（$2 \le n, d \le 5000$），分别表示树中顶点的数量以及所有顶点深度之和的要求值。

保证所有测试用例的 $n$ 之和与 $d$ 之和均不超过 $5000$（即 $\sum n \le 5000,\ \sum d \le 5000$）。

For each test case, print the answer.

If it is impossible to construct such a tree, print "NO" (without quotes) in the first line. Otherwise, print "{YES}" in the first line. Then print $n-1$ integers $p_2, p_3, \dots, p_n$ in the second line, where $p_i$ is the parent of the vertex $i$. Note that the sequence of parents you print should describe some binary tree.

对于每个测试用例，请输出答案。

如果无法构造出满足条件的树，则在第一行输出 "NO"（不带引号）。否则，在第一行输出 "{YES}"。然后在第二行输出 $n-1$ 个整数 $p_2, p_3, \dots, p_n$，其中 $p_i$ 表示顶点 $i$ 的父节点。注意：你所输出的父节点序列必须描述一棵二叉树。

Pictures corresponding to the first and the second test cases of the example:

对应示例中第一个和第二个测试用例的图片：