<p>You are given <code>m</code> <code>arrays</code>, where each array is sorted in <strong>ascending order</strong>.</p>

<p>You can pick up two integers from two different arrays (each array picks one) and calculate the distance. We define the distance between two integers <code>a</code> and <code>b</code> to be their absolute difference <code>|a - b|</code>.</p>

<p>Return <em>the maximum distance</em>.</p>

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

<pre>
<strong>Input:</strong> arrays = [[1,2,3],[4,5],[1,2,3]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> One way to reach the maximum distance 4 is to pick 1 in the first or third array and pick 5 in the second array.
</pre>

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

<pre>
<strong>Input:</strong> arrays = [[1],[1]]
<strong>Output:</strong> 0
</pre>

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

<ul>
	<li><code>m == arrays.length</code></li>
	<li><code>2 &lt;= m &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= arrays[i].length &lt;= 500</code></li>
	<li><code>-10<sup>4</sup> &lt;= arrays[i][j] &lt;= 10<sup>4</sup></code></li>
	<li><code>arrays[i]</code> is sorted in <strong>ascending order</strong>.</li>
	<li>There will be at most <code>10<sup>5</sup></code> integers in all the arrays.</li>
</ul>

# approach 1:
# just sort all of the numbers and go in order of least to greatest
# when sorting all the numbers keep track of which arrays each number belongs to.

class Solution:
    def maxDistance(self, arrays: List[List[int]]) -> int:

        nums = defaultdict(set)
        for i, arr in enumerate(arrays):
            for el in arr:
                nums[el].add(i)

        list_nums = sorted([(k, list(v)) for k, v in nums.items()], key = lambda x : x[0])
        print(list_nums)
        
        def find_dist(l, r):
            if l > r:
                return 0

            if len(list_nums[l][1]) >= 2 or len(list_nums[r][1]) >= 2:
                return list_nums[r][0] - list_nums[l][0]

            if len(list_nums[l][1]) == 1 or len(list_nums[r][1]) == 1:
                if list_nums[l][1][0] != list_nums[r][1][0]:
                    return list_nums[r][0] - list_nums[l][0]

            return max(find_dist(l, r - 1), find_dist(l + 1, r))

        return find_dist(0, len(list_nums) - 1)


        
44971