Cheat and Win

3000ms

262144K

Description:

Let's consider a $$$(10^9+1) \times (10^9+1)$$$ field. The rows are numbered with integers from $$$0$$$ to $$$10^9$$$ and the columns are numbered with integers from $$$0$$$ to $$$10^9$$$. Let's define as $$$(x, y)$$$ the cell located in the $$$x$$$-th row and $$$y$$$-th column.

Let's call a cell $$$(x, y)$$$ good if $$$x \& y = 0$$$, there $$$\&$$$ is the bitwise and operation.

Let's build a graph where vertices will be all good cells of the field and we will make an edge between all pairs of adjacent by side good cells. It can be proved that this graph will be a tree — connected graph without cycles. Let's hang this tree on vertex $$$(0, 0)$$$, so we will have a rooted tree with root $$$(0, 0)$$$.

Two players will play the game. Initially, some good cells are black and others are white. Each player on his turn chooses a black good cell and a subset of its ancestors (possibly empty) and inverts their colors (from white to black and vice versa). The player who can't move (because all good cells are white) loses. It can be proved that the game is always finite.

Initially, all cells are white. You are given $$$m$$$ pairs of cells. For each pair color all cells in a simple path between them as black. Note that we do not invert their colors, we paint them black.

Sohrab and Mashtali are going to play this game. Sohrab is the first player and Mashtali is the second.

Mashtali wants to win and decided to cheat. He can make the following operation multiple times before the game starts: choose a cell and invert colors of all vertices on the path between it and the root of the tree.

Mammad who was watching them wondered: "what is the minimum number of operations Mashtali should do to have a winning strategy?".

Find the answer to this question for the initial painting of the tree. It can be proved that at least one possible way to cheat always exists.

Input:

The first line contains one integer $$$m$$$ ($$$1 \leq m \leq 10^5$$$).

Each of the next $$$m$$$ lines contains four integers $$$x_{1}$$$, $$$y_{1}$$$, $$$x_{2}$$$, $$$y_{2}$$$ ($$$0 \leq x_i, y_i \leq 10^9$$$, $$$x_i \& y_i = 0$$$). You should color all cells on the path between vertices $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ as black.

Output:

Print a single integer — the minimum number of cheating operations the second player can do.

Sample Input:

1
7 0 0 7

Sample Output:

Sample Input:

Sample Output:

Sample Input:

2
0 1 0 8
1 0 8 0

Sample Output:

Note:

In the first test, you can make one cheating operation with the root of the tree. After that, the second player can win because he can use a symmetric strategy.

In the second test, you can make cheating operations with cells $$$(0, 2), (0, 0), (3, 4)$$$.

In the third test, the second player already has the winning strategy and doesn't need to make any cheating operations.

Informação

Provedor Codeforces

Código CF1439E