<p>You are given an array of integers <code>arr</code> and an integer <code>target</code>.</p>

<p>You have to find <strong>two non-overlapping sub-arrays</strong> of <code>arr</code> each with a sum equal <code>target</code>. There can be multiple answers so you have to find an answer where the sum of the lengths of the two sub-arrays is <strong>minimum</strong>.</p>

<p>Return <em>the minimum sum of the lengths</em> of the two required sub-arrays, or return <code>-1</code> if you cannot find such two sub-arrays.</p>

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

<pre>
<strong>Input:</strong> arr = [3,2,2,4,3], target = 3
<strong>Output:</strong> 2
<strong>Explanation:</strong> Only two sub-arrays have sum = 3 ([3] and [3]). The sum of their lengths is 2.
</pre>

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

<pre>
<strong>Input:</strong> arr = [7,3,4,7], target = 7
<strong>Output:</strong> 2
<strong>Explanation:</strong> Although we have three non-overlapping sub-arrays of sum = 7 ([7], [3,4] and [7]), but we will choose the first and third sub-arrays as the sum of their lengths is 2.
</pre>

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

<pre>
<strong>Input:</strong> arr = [4,3,2,6,2,3,4], target = 6
<strong>Output:</strong> -1
<strong>Explanation:</strong> We have only one sub-array of sum = 6.
</pre>

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

<ul>
	<li><code>1 &lt;= arr.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= arr[i] &lt;= 1000</code></li>
	<li><code>1 &lt;= target &lt;= 10<sup>8</sup></code></li>
</ul>

class Solution:
    def minSumOfLengths(self, arr: List[int], target: int) -> int:
        # prefix sum + L to R & R to L
        n, l, mn = len(arr), [], float("inf")
        ls, d = list(accumulate(arr, initial=0)), {0: 0}
        for i in range(1, n+1):
            li, ps = mn, ls[i] - target
            if ps in d: li = min(li, i - d[ps])
            d[ls[i]] = max(d.get(ls[i], float("-inf")), i)
            mn = min(mn, li)
            l.append(li)
        rs, d = list(accumulate(arr[::-1], initial=0))[::-1], {0: n}
        mn, ans = float("inf"), float("inf")
        for i in range(n-1, -1, -1):
            ans = min(ans, l[i] + mn)
            ri, ps = mn, rs[i] - target
            if ps in d: ri = min(ri, d[ps] - i)
            d[rs[i]] = min(d.get(rs[i], float("inf")), i)
            mn = min(mn, ri)
        return -1 if ans == float("inf") else ans
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