<p>You are given two integer arrays <code>nums1</code> and <code>nums2</code>. We write the integers of <code>nums1</code> and <code>nums2</code> (in the order they are given) on two separate horizontal lines.</p>

<p>We may draw connecting lines: a straight line connecting two numbers <code>nums1[i]</code> and <code>nums2[j]</code> such that:</p>

<ul>
	<li><code>nums1[i] == nums2[j]</code>, and</li>
	<li>the line we draw does not intersect any other connecting (non-horizontal) line.</li>
</ul>

<p>Note that a connecting line cannot intersect even at the endpoints (i.e., each number can only belong to one connecting line).</p>

<p>Return <em>the maximum number of connecting lines we can draw in this way</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2019/04/26/142.png" style="width: 400px; height: 286px;" />
<pre>
<strong>Input:</strong> nums1 = [1,4,2], nums2 = [1,2,4]
<strong>Output:</strong> 2
<strong>Explanation:</strong> We can draw 2 uncrossed lines as in the diagram.
We cannot draw 3 uncrossed lines, because the line from nums1[1] = 4 to nums2[2] = 4 will intersect the line from nums1[2]=2 to nums2[1]=2.
</pre>

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

<pre>
<strong>Input:</strong> nums1 = [2,5,1,2,5], nums2 = [10,5,2,1,5,2]
<strong>Output:</strong> 3
</pre>

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

<pre>
<strong>Input:</strong> nums1 = [1,3,7,1,7,5], nums2 = [1,9,2,5,1]
<strong>Output:</strong> 2
</pre>

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

<ul>
	<li><code>1 &lt;= nums1.length, nums2.length &lt;= 500</code></li>
	<li><code>1 &lt;= nums1[i], nums2[j] &lt;= 2000</code></li>
</ul>

from functools import cache

class Solution:
    def maxUncrossedLines(self, nums1: List[int], nums2: List[int]) -> int:
        """
        If we draw a line from nums1[i] to nums2[j] and j > i 
        then we cant draw a line from
        nums1[i+d] to nums1[j-b] were d, b >= 0


        We can try to solve this problem recursivley, considerring
        prefixes of the lists ending ad i and j, solving for
        the max number of lines for nums1[:i], nums2[:j]

        Function:
            MUL[i, j] = the max number of uncrossed lines
                        where one of nums1[i] or nums2[j]
                        are the endpoint of a line
        
        Base Case:
            MUL[0, j] = 1 if nums1[0] == nums2[j] else 0,
            MUL[i, 0] = 1 if nums1[i] == nums2[0] else 0
        
        Recursive Case:
            MUL[i, j] = 1 + MUL[i-1][j-1] if nums1[i] == nums2[j] else max(
                MUL[i-1][j],
                MUL[i][j-1]
            )
        """

        @cache
        def helper(i: int, j: int) -> int:
            nonlocal nums1, nums2
            if i == 0 and j == 0:
                return int(nums1[i] == nums2[j])
            
            if i == 0:
                if nums1[0] == nums2[j]:
                    return 1
                return helper(0, j - 1)
            
            if j == 0:
                if nums1[i] == nums2[0]:
                    return 1
                return helper(i - 1, 0)
            
            if nums1[i] == nums2[j]:
                return 1 + helper(i - 1, j - 1)
            
            return max(
                helper(i-1, j),
                helper(i, j-1)
            )
        
        return helper(len(nums1) - 1, len(nums2) - 1)
60300