Problem Statement

We define the density of a non-empty simple undirected graph as $\displaystyle\frac{(\text{number$\ of\ $edges})}{(\text{number$\ of\ vertices})}.
You are given positive integers N and D. Determine whether there is a simple undirected graph G with N vertices and DN edges that satisfies the following condition. If it exists, find one such graph.
Condition: Let V be the vertex set of G.
For any non-empty proper subset X of V, the density of the induced subgraph of G by X is strictly less than D.

What is an induced subgraph?

For a vertex subset X of graph G, the induced subgraph of G by X is the graph with vertex set X and edge set containing all edges of G that connect two vertices in X.
In the above condition, note that we only consider vertex subsets that are neither empty nor the entire set.

Input

The input is given from Standard Input in the following format:
N D

Output

If there is a simple undirected graph that satisfies the condition, print Yes; otherwise, print No.
Furthermore, in the case of Yes, print a graph that satisfies the condition in the following format in the next DN lines.
If multiple graphs satisfy the condition, any of them will be considered correct.
A_1 B_1
A_2 B_2
$\vdots$
A_{DN} B_{DN}

• $1 \leq A_i, B_i \leq N$ \ \ $(1 \leq i \leq$ DN)
• $A_i \neq B_i$ \ \ $(1 \leq i \leq$ DN)
• {A_i, B_i$} \neq${A_j, B_j} \ $(1 \leq i < j \leq$ DN)
• The vertices of the graph are numbered from 1 to N.
• The i-th line of the output represents that the i-th edge connects vertex A_i and vertex B_i.
• The order of the edges (which edge is printed first) and the order of the vertices (which endpoint of each edge is printed first) may be arbitrary.

Constraints

• $N \geq 1$
• $D \geq 1$
• DN $\leq 5 \times 10^4$

Sample Input 1

3 1

Sample Output 1

Yes
1 2
1 3
2 3

The printed graph has vertex set {1, 2, 3} and edge set {(1, 2), (1, 3), (2, 3)}, and is simple.
The vertex set X has 6 non-empty proper subsets: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}.

• For X = {1}, {2}, {3}, the edge sets of the induced subgraphs by X are empty, and the densities are $\displaystyle\frac{0}{1} = 0$.
• For X = {1, 2}, {1, 3}, {2, 3}, the edge sets of the induced subgraphs by X are {(1, 2)}, {(1, 3)}, {(2, 3)}, respectively, and the densities are all $\displaystyle\frac{1}{2}$.

In all cases, the density of the induced subgraph is strictly less than D = 1, so this graph satisfies the condition.

Sample Input 2

4 2

Sample Output 2

No

A simple undirected graph with 4 vertices and 8 edges does not exist.

见题面 Input 部分

见题面 Output 部分