A16002. Min Cost Permutation (Hard Version)

The only difference between this problem and the easy version is the constraints on $t$ and $n$ .

You are given an array of $n$ positive integers $a_1,\dots,a_n$ , and a (possibly negative) integer $c$ .

Across all permutations $b_1,\dots,b_n$ of the array $a_1,\dots,a_n$ , consider the minimum possible value of $$$$\sum_{i=1}^{n-1} |b_{i+1}-b_i-c|. $$ Find the lexicographically smallest permutation $b$ of the array $a$ that achieves this minimum.</p><p>A sequence $x$ is lexicographically smaller than a sequence $y$ if and only if one of the following holds:</p><ul><li> $x$ is a prefix of $y$ , but $x \\ne y$ ;</li><li> in the first position where $x$ and $y$ differ, the sequence $x$ has a smaller element than the corresponding element in $y$$$.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 10^4$ ). The description of the test cases follows.

The first line of each test case contains two integers $n$ and $c$ ( $1 \le n \le 2 \cdot 10^5$ , $-10^9 \le c \le 10^9$ ).

The second line of each test case contains $n$ integers $a_1,\dots,a_n$ ( $1 \le a_i \le 10^9$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output $n$ integers $b_1,\dots,b_n$ , the lexicographically smallest permutation of $a$ that achieves the minimum $\sum\limits_{i=1}^{n-1} |b_{i+1}-b_i-c|$ .

In the first test case, it can be proven that the minimum possible value of $\sum\limits_{i=1}^{n-1} |b_{i+1}-b_i-c|$ is $27$ , and the permutation $b = [9,3,1,4,5,1]$ is the lexicographically smallest permutation of $a$ that achieves this minimum: $|3-9-(-7)|+|1-3-(-7)|+|4-1-(-7)|+|5-4-(-7)|+|1-5-(-7)| = 1+5+10+8+3 = 27$ .

In the second test case, the minimum possible value of $\sum\limits_{i=1}^{n-1} |b_{i+1}-b_i-c|$ is $0$ , and $b = [1,3,5]$ is the lexicographically smallest permutation of $a$ that achieves this.

In the third test case, there is only one permutation $b$ .