P9698 [GDCPC 2023] Path Planning

There is a grid with $n$ rows and $m$ columns. Each cell of the grid has an integer in it, where $a_{i, j}$ indicates the integer in the cell located at the $i$-th row and the $j$-th column. Each integer from $0$ to $(n \times m - 1)$ (both inclusive) appears exactly once in the grid. Let $(i, j)$ be the cell located at the $i$-th row and the $j$-th column. You now start from $(1, 1)$ and need to reach $(n, m)$. When you are in cell $(i, j)$, you can either move to its right cell $(i, j + 1)$ if $j < m$ or move to its bottom cell $(i + 1, j)$ if $i < n$. Let $\mathbb{S}$ be the set consisting of integers in each cell on your path, including $a_{1, 1}$ and $a_{n, m}$. Let $\text{mex}(\mathbb{S})$ be the smallest non-negative integer which does not belong to $\mathbb{S}$. Find a path to maximize $\text{mex}(\mathbb{S})$ and calculate this maximum possible value.

There are multiple test cases. The first line of the input contains an integer $T$ indicating the number of test cases. For each test case: The first line contains two integers $n$ and $m$ ($1 \le n, m \le 10^6$, $1 \le n \times m \le 10^6$) indicating the number of rows and columns of the grid. For the following $n$ lines, the $i$-th line contains $m$ integers $a_{i, 1}, a_{i, 2}, \cdots, a_{i, m}$ ($0 \le a_{i, j} < n \times m$) where $a_{i, j}$ indicates the integer in cell $(i, j)$. Each integer from $0$ to $(n \times m - 1)$ (both inclusive) appears exactly once in the grid. It's guaranteed that the sum of $n \times m$ of all test cases will not exceed $10^6$.

For each test case output one line containing one integer indicating the maximum possible value of $\text{mex}(\mathbb{S})$.

For the first sample test case there are $3$ possible paths. - The first path is $(1, 1) \to (1, 2) \to (1, 3) \to (2, 3)$. $\mathbb{S} = \{1, 2, 4, 5\}$ so $\text{mex}(\mathbb{S}) = 0$. - The second path is $(1, 1) \to (1, 2) \to (2, 2) \to (2, 3)$. $\mathbb{S} = \{1, 2, 0, 5\}$ so $\text{mex}(\mathbb{S}) = 3$. - The third path is $(1, 1) \to (2, 1) \to (2, 2) \to (2, 3)$. $\mathbb{S} = \{1, 3, 0, 5\}$ so $\text{mex}(\mathbb{S}) = 2$. So the answer is $3$. For the second sample test case there is only $1$ possible path, which is $(1, 1) \to (1, 2) \to (1, 3) \to (1, 4) \to (1, 5)$. $\mathbb{S} = \{1, 3, 0, 4, 2\}$ so $\text{mex}(\mathbb{S}) = 5$.