CF1844F2 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.

A sequence $ x $ is lexicographically smaller than a sequence $ y $ if and only if one of the following holds:

• $ x $ is a prefix of $ y $ , but $ x \\ne y $ ;
• 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 $ .