<p>Given a <strong>1-indexed</strong>&nbsp;<code>m x n</code> integer matrix <code>mat</code>, you can select any cell in the matrix as your <strong>starting cell</strong>.</p>

<p>From the starting cell, you can move to any other cell <strong>in the</strong> <strong>same row or column</strong>, but only if the value of the destination cell is <strong>strictly greater</strong> than the value of the current cell. You can repeat this process as many times as possible, moving from cell to cell until you can no longer make any moves.</p>

<p>Your task is to find the <strong>maximum number of cells</strong> that you can visit in the matrix by starting from some cell.</p>

<p>Return <em>an integer denoting the maximum number of cells that can be visited.</em></p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/23/diag1drawio.png" style="width: 200px; height: 176px;" /></strong></p>

<pre>
<strong>Input:</strong> mat = [[3,1],[3,4]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> The image shows how we can visit 2 cells starting from row 1, column 2. It can be shown that we cannot visit more than 2 cells no matter where we start from, so the answer is 2. 
</pre>

<p><strong class="example">Example 2:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/23/diag3drawio.png" style="width: 200px; height: 176px;" /></strong></p>

<pre>
<strong>Input:</strong> mat = [[1,1],[1,1]]
<strong>Output:</strong> 1
<strong>Explanation:</strong> Since the cells must be strictly increasing, we can only visit one cell in this example. 
</pre>

<p><strong class="example">Example 3:</strong></p>

<p><strong class="example"><img alt="" src="https://assets.leetcode.com/uploads/2023/04/23/diag4drawio.png" style="width: 350px; height: 250px;" /></strong></p>

<pre>
<strong>Input:</strong> mat = [[3,1,6],[-9,5,7]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> The image above shows how we can visit 4 cells starting from row 2, column 1. It can be shown that we cannot visit more than 4 cells no matter where we start from, so the answer is 4. 
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == mat.length&nbsp;</code></li>
	<li><code>n == mat[i].length&nbsp;</code></li>
	<li><code>1 &lt;= m, n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= m * n &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>5</sup>&nbsp;&lt;= mat[i][j] &lt;= 10<sup>5</sup></code></li>
</ul>

from collections import deque
from itertools import chain
from math import inf
from operator import itemgetter


class Solution:
    def maxIncreasingCells(self, mat: List[List[int]]) -> int:
        n, m = len(mat), len(mat[0])
        
        # store the max stricly increasing cells able to achieve per row/cell
        rowMax = [-inf for i in range(n)]
        colMax = [-inf for i in range(m)]

        # use heap or just sort
        vals = sorted(
            [(mat[i][j], (i, j)) for i in range(n) for j in range(m)],
            key=itemgetter(0),
            reverse=True,
        )
        
        maxIncr, prevVal = -inf, vals[0]
        toBeUpdated = deque()
        for val, (i, j) in vals:
            # do not updated rowMax and colMax until value changes
            # since future cells can only jump to a strictly increasing cell
            # we should not enable this jump until the value changes
            if val != prevVal:
                while toBeUpdated:
                    x, (i1, j1) = toBeUpdated.pop()
                    rowMax[i1] = max(rowMax[i1], x)
                    colMax[j1] = max(colMax[j1], x)

            # max of either ending at current cell
            # jumping to visited cell on same row
            # jumping to visited cell on same column
            maxIncrForCell = max(1, rowMax[i]+1, colMax[j]+1)
            maxIncr = max(maxIncr, maxIncrForCell)
            
            toBeUpdated.append((maxIncrForCell, (i, j)))
            prevVal = val
        
        return maxIncr
1486