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code_segments/segment_294.txt

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+[Ken Arai - COMPLEX](https://soundcloud.com/diatomichail2/complex)
+
+⠀
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+This is the hard version of the problem. In this version, the constraints on $n$ and the time limit are higher. You can make hacks only if both versions of the problem are solved.
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+A set of (closed) segments is complex if it can be partitioned into some subsets such that
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+  * all the subsets have the same size; and    * a pair of segments intersects if and only if the two segments are in the same subset. 
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+You are given $n$ segments $[l_1, r_1], [l_2, r_2], \ldots, [l_n, r_n]$. Find the maximum size of a complex subset of these segments.
+
+Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.
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+The first line of each test case contains a single integer $n$ ($1 \le n \le 3 \cdot 10^5$) — the number of segments.
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+The second line of each test case contains $n$ integers $l_1, l_2, \ldots, l_n$ ($1 \le l_i \le 2n$) — the left endpoints of the segments.
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+The third line of each test case contains $n$ integers $r_1, r_2, \ldots, r_n$ ($l_i \leq r_i \le 2n$) — the right endpoints of the segments.
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+It is guaranteed that the sum of $n$ over all test cases does not exceed $3 \cdot 10^5$.
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+For each test case, output a single integer: the maximum size of a complex subset of the given segments.
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+In the first test case, all pairs of segments intersect, therefore it is optimal to form a single group containing all of the three segments.
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+In the second test case, there is no valid partition for all of the five segments. A valid partition with four segments is the following: $\\{\\{ [1, 5], [2, 4] \\}, \\{ [6, 9], [8, 10] \\}\\}$.
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+In the third test case, it is optimal to make a single group containing all the segments except the second.