Matrix Game
Contest: Codeforces Round 648 (Div. 2)
Difficulty rating: 1100
Tags: games, greedy, implementation

Ashish and Vivek play a game on a matrix consisting of $$$n$$$ rows and $$$m$$$ columns, where they take turns claiming cells. Unclaimed cells are represented by $$$0$$$, while claimed cells are represented by $$$1$$$. The initial state of the matrix is given. There can be some claimed cells in the initial state.

In each turn, a player must claim a cell. A cell may be claimed if it is unclaimed and does not share a row or column with any other already claimed cells. When a player is unable to make a move, he loses and the game ends.

If Ashish and Vivek take turns to move and Ashish goes first, determine the winner of the game if both of them are playing optimally.

Optimal play between two players means that both players choose the best possible strategy to achieve the best possible outcome for themselves.

Input format:
The first line consists of a single integer $$$t$$$ $$$(1 \le t \le 50)$$$ — the number of test cases. The description of the test cases follows.

The first line of each test case consists of two space-separated integers $$$n$$$, $$$m$$$ $$$(1 \le n, m \le 50)$$$ — the number of rows and columns in the matrix.

The following $$$n$$$ lines consist of $$$m$$$ integers each, the $$$j$$$-th integer on the $$$i$$$-th line denoting $$$a_{i,j}$$$ $$$(a_{i,j} \in \{0, 1\})$$$.

Output format:
For each test case if Ashish wins the game print "Ashish" otherwise print "Vivek" (without quotes).

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

Note:
For the first case: One possible scenario could be: Ashish claims cell $$$(1, 1)$$$, Vivek then claims cell $$$(2, 2)$$$. Ashish can neither claim cell $$$(1, 2)$$$, nor cell $$$(2, 1)$$$ as cells $$$(1, 1)$$$ and $$$(2, 2)$$$ are already claimed. Thus Ashish loses. It can be shown that no matter what Ashish plays in this case, Vivek will win.

For the second case: Ashish claims cell $$$(1, 1)$$$, the only cell that can be claimed in the first move. After that Vivek has no moves left.

For the third case: Ashish cannot make a move, so Vivek wins.

For the fourth case: If Ashish claims cell $$$(2, 3)$$$, Vivek will have no moves left.

Example input:
4
2 2
0 0
0 0
2 2
0 0
0 1
2 3
1 0 1
1 1 0
3 3
1 0 0
0 0 0
1 0 0

Example output:
Vivek
Ashish
Vivek
Ashish

Editorial:
Key Idea:
Vivek and Ashish can never claim cells in rows and columns which already have at least one cell claimed. So we need to look at the parity of minimum of the number of rows and columns which have no cells claimed initially.
Solution:
Let $$$a$$$ be the number of rows which do not have any cell claimed in them initially and similarly $$$b$$$ be the number of columns which do not have any cell claimed initially. Each time a player makes a move both $$$a$$$ and $$$b$$$ decrease by $$$1$$$, since they only claim cells in rows and columns with no claimed cells.
If either one of $$$a$$$ or $$$b$$$ becomes $$$0$$$, the player whose turn comes next loses the game. Since both $$$a$$$ and $$$b$$$ decrease by $$$1$$$ after each move, $$$\min(a, b)$$$ becomes $$$0$$$ first. So, if $$$\min(a, b)$$$ is odd, Ashish wins the game otherwise Vivek wins.
Time complexity: $$$O(n \cdot m)$$$