King Escape
Contest: Lyft Level 5 Challenge 2018 - Elimination Round
Difficulty rating: 1000
Tags: dfs and similar, graphs, implementation

Alice and Bob are playing chess on a huge chessboard with dimensions $$$n \times n$$$. Alice has a single piece left — a queen, located at $$$(a_x, a_y)$$$, while Bob has only the king standing at $$$(b_x, b_y)$$$. Alice thinks that as her queen is dominating the chessboard, victory is hers.

But Bob has made a devious plan to seize the victory for himself — he needs to march his king to $$$(c_x, c_y)$$$ in order to claim the victory for himself. As Alice is distracted by her sense of superiority, she no longer moves any pieces around, and it is only Bob who makes any turns.

Bob will win if he can move his king from $$$(b_x, b_y)$$$ to $$$(c_x, c_y)$$$ without ever getting in check. Remember that a king can move to any of the $$$8$$$ adjacent squares. A king is in check if it is on the same rank (i.e. row), file (i.e. column), or diagonal as the enemy queen.

Find whether Bob can win or not.

Input format:
The first line contains a single integer $$$n$$$ ($$$3 \leq n \leq 1000$$$) — the dimensions of the chessboard.

The second line contains two integers $$$a_x$$$ and $$$a_y$$$ ($$$1 \leq a_x, a_y \leq n$$$) — the coordinates of Alice's queen.

The third line contains two integers $$$b_x$$$ and $$$b_y$$$ ($$$1 \leq b_x, b_y \leq n$$$) — the coordinates of Bob's king.

The fourth line contains two integers $$$c_x$$$ and $$$c_y$$$ ($$$1 \leq c_x, c_y \leq n$$$) — the coordinates of the location that Bob wants to get to.

It is guaranteed that Bob's king is currently not in check and the target location is not in check either.

Furthermore, the king is not located on the same square as the queen (i.e. $$$a_x \neq b_x$$$ or $$$a_y \neq b_y$$$), and the target does coincide neither with the queen's position (i.e. $$$c_x \neq a_x$$$ or $$$c_y \neq a_y$$$) nor with the king's position (i.e. $$$c_x \neq b_x$$$ or $$$c_y \neq b_y$$$).

Output format:
Print "YES" (without quotes) if Bob can get from $$$(b_x, b_y)$$$ to $$$(c_x, c_y)$$$ without ever getting in check, otherwise print "NO".

You can print each letter in any case (upper or lower).

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the diagrams below, the squares controlled by the black queen are marked red, and the target square is marked blue.

In the first case, the king can move, for instance, via the squares $$$(2, 3)$$$ and $$$(3, 2)$$$. Note that the direct route through $$$(2, 2)$$$ goes through check.

In the second case, the queen watches the fourth rank, and the king has no means of crossing it.

In the third case, the queen watches the third file.

Example input:
8
4 4
1 3
3 1

Example output:
YES

Example input:
8
4 4
2 3
1 6

Example output:
NO

Editorial:
Imagine a two-dimensional plane with origin positioned on the black queen. We notice that the queen partitions the board into up to four connected components, with each quadrant being one of them. The answer is thus "YES" if and only if the source king position and the target position are in the same quadrant.
This condition can be checked in $$$\mathcal O(1)$$$ using few simple ifs.
In case you prefer brute force over case analysis, it is also possible to construct the grid graph and find the connected components or use DFS to find the path. This runs in $$$\mathcal O(n^2)$$$.