Connected Component on a Chessboard
Contest: Codeforces Round 575 (Div. 3)
Difficulty rating: 1800
Tags: constructive algorithms, implementation

You are given two integers $$$b$$$ and $$$w$$$. You have a chessboard of size $$$10^9 \times 10^9$$$ with the top left cell at $$$(1; 1)$$$, the cell $$$(1; 1)$$$ is painted white.

Your task is to find a connected component on this chessboard that contains exactly $$$b$$$ black cells and exactly $$$w$$$ white cells. Two cells are called connected if they share a side (i.e. for the cell $$$(x, y)$$$ there are at most four connected cells: $$$(x - 1, y), (x + 1, y), (x, y - 1), (x, y + 1)$$$). A set of cells is called a connected component if for every pair of cells $$$C_1$$$ and $$$C_2$$$ from this set, there exists a sequence of cells $$$c_1$$$, $$$c_2$$$, ..., $$$c_k$$$ such that $$$c_1 = C_1$$$, $$$c_k = C_2$$$, all $$$c_i$$$ from $$$1$$$ to $$$k$$$ are belong to this set of cells and for every $$$i \in [1, k - 1]$$$, cells $$$c_i$$$ and $$$c_{i + 1}$$$ are connected.

Obviously, it can be impossible to find such component. In this case print "NO". Otherwise, print "YES" and any suitable connected component.

You have to answer $$$q$$$ independent queries.

Input format:
The first line of the input contains one integer $$$q$$$ ($$$1 \le q \le 10^5$$$) — the number of queries. Then $$$q$$$ queries follow.

The only line of the query contains two integers $$$b$$$ and $$$w$$$ ($$$1 \le b, w \le 10^5$$$) — the number of black cells required and the number of white cells required.

It is guaranteed that the sum of numbers of cells does not exceed $$$2 \cdot 10^5$$$ ($$$\sum w + \sum b \le 2 \cdot 10^5$$$).

Output format:
For each query, print the answer to it.

If it is impossible to find the required component, print "NO" on the first line.

Otherwise, print "YES" on the first line. In the next $$$b + w$$$ lines print coordinates of cells of your component in any order. There should be exactly $$$b$$$ black cells and $$$w$$$ white cells in your answer. The printed component should be connected.

If there are several answers, you can print any. All coordinates in the answer should be in the range $$$[1; 10^9]$$$.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Example input:
3
1 1
1 4
2 5

Example output:
YES
2 2
1 2
YES
2 3
1 3
3 3
2 2
2 4
YES
2 3
2 4
2 5
1 3
1 5
3 3
3 5

Editorial:
I'll consider the case when $$$b \ge w$$$, the other case is symmetric and the answer I construct is the same but shifted by $$$1$$$ to the right.
Consider the given field as a matrix where $$$x$$$ is the number of row and $$$y$$$ is the number of column. Firstly, let's build the line of length $$$2w-1$$$ from the cell $$$(2, 2)$$$ to the cell $$$(2, 2w)$$$. Then $$$b$$$ will decrease by $$$w-1$$$ and $$$w$$$ will (formally) become $$$0$$$.
Then we have two black cells to the left and to the right ($$$(2, 1)$$$ and $$$(2, 2w+1)$$$) and $$$w-1$$$ black cells to the up (all cells ($$$1, 2w+2*i$$$) for all $$$i$$$ from $$$0$$$ to $$$w-1$$$) and $$$w-1$$$ black cells to the down (all cells ($$$3, 2w+2*i$$$) for all $$$i$$$ from $$$0$$$ to $$$w-1$$$). Let's add the required number of cells to the answer.
If even after adding all these cells $$$b$$$ still be greater than $$$0$$$ then the answer is "NO" (maybe there will be a proof why it is so but you can read it already from other participants). Otherwise the answer is "YES" and we constructed the required component.