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문제

철승이는 사물을 좌표계로 생각하기를 좋아한다. 어느 날 집안에 파리가 앉았던 벽의 위치마다 점을 찍고있던 철승이는 엄마에게 등짝 스매싱을 맞는다.

철승이네 집의 벽은 한 변이 1cm인 정사각형의 패턴이 그려진 벽지로 도배되어 있으며, 똑똑한 파리는 이 패턴 정사각형의 모서리에만 앉았다.

< 파리가 앉은 예시 >

새로운 벽지로 모든 점들을 덮으려고하던 철승이는 갑자기 궁금증이 생겼다. 정사각형 모양의 벽지조각 3개만 사용해서 모든 점들을 가린다고 하면 조각의 한 변은 최소 몇 cm이어야 할까?

크기가 똑같고, 벽지와 같은 무늬의 정사각형 종이 3장으로 모든 점들을 가린다고 할 때에 한 종이의 변은 최소 몇 cm인지 계산해보자. 벽지는 깔끔하게 다시 도배해야 하므로 모든 종이는 정확히 정사각형 패턴들과 모서리가 일치해야한다. 종이의 모서리나 면 끝과 점한 점은 가려졌다고 가정한다.

아래의 예제는 정사각형 종이 3장으로 위의 점들을 가리는 경우들을 보여준다.

두 경우에서 오른쪽의 방법이 종이가 더 작으므로 더 좋은 방법이 된다. 철승이의 궁금증을 풀어주기 위해 최소의 벽지 조각의 크기를 계산하자.

모든 종이의 크기는 정수여야하며, 한 점만 가리는 종이(크기가 0인)가 있을 수 있다. 세 종이의 크기는 모두 동일해야 한다.

입력

입력의 첫 줄에는 테스트 케이스의 수 T가 주어진다.

이 후 T개의 테스트 케이스마다, 첫 줄에는 점의 수 n (1 ≤ n ≤ 100,000)이 주어진다.

그 후 n줄에 걸쳐서 점의 위치 xi, yi (-1,000,000,000 ≤ xi, yi ≤ 1,000,000,000) 가 주어진다.

출력

각 테스트 케이스마다 한 줄에 최소의 종이조각의 크기를 정수로 출력하시오. 단, 종이조각의 크기는 0이 될 수 있음에 유의하시오.

예제 입력 1

3
5
1 0
0 1
5 2
2 3
3 2
11
1 8
3 8
5 5
5 10
6 4
7 7
8 8
9 2
11 6
12 3
10 4
4
1 1
2 8
-21 45
1 1

예제 출력 1

1
4
0
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You are given a set P of n points in the plane. Any real number x &ge; 0 is called 3SQ-sufficient for P if there exist three congruent axis-parallel squares of side length x such that the union of the three squares contains all points in the set P. If a square has side length zero, it is degenerated to a point, so you can consider a point itself as a square of side length zero. Your program is to find the smallest 3SQ-sufficient number for a given input set P of points.<\/p>\r\n\r\n<p style=\"text-align:center\"><img alt=\"\" src=\"\/upload\/images2\/3sq1.png\" style=\"height:190px; width:221px\" \/><\/p>\r\n\r\n<p style=\"text-align:center\">Figure 1. 11 points in the plane<\/p>\r\n\r\n<p>For example, let P be the set of 11 points in the plane as shown in Figure 1. Then, one can find three congruent axis-parallel squares of side length 5 whose union contains all these 11 points in the set P, as shown in Figure 2(a). This means that number 5 is 3SQ-sufficient for this set P of 11 points. Notice that points on the boundary of a square are considered to be contained in the square.<\/p>\r\n\r\n<p style=\"text-align:center\"><img alt=\"\" src=\"\/upload\/images2\/3sq2.png\" style=\"height:196px; width:493px\" \/><\/p>\r\n\r\n<p style=\"text-align:center\">Figure 2. Two numbers 5 and 4 are 3SQ-sufficient<\/p>\r\n\r\n<p>You can find an even smaller 3SQ-sufficient number for this case; there exist three axis-parallel squares of side length 4 whose union contains all the 11 points as illustrated in Figure 2(b). More effort to find a further smaller 3SQ-sufficient number will be however worthless because number 4 is in fact the smallest 3SQ-sufficient number for this set P. Therefore, if these 11 points of Figure 1 are given as input of your program, then your program must output 4 as the answer.<\/p>\r\n","input":"<p>Your program is to read from standard input. The input consists of T test cases. The number of test cases T is given in the first line of the input. Each test case starts with a line containing an integer, n (1 &le; n &le; 100,000), where n is the number of points in the data set P. In the following n lines, each of the n points in P is given line by line. Each point is represented by two numbers separated by a single space, which are the x-coordinate and the y-coordinate of the point, respectively. Each coordinate is given as an integer between &minus;1,000,000,000 and 1,000,000,000, inclusively. Note that there may be two or more points having the same coordinates in the input data set P.<\/p>\r\n","output":"<p>Your program is to write to standard output. Print exactly one line for each test case. The line should contain an integer representing the smallest 3SQ-sufficient integer for a given input set P of points.&nbsp;<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]

출처

ICPC > Regionals > Asia Pacific > Korea > Asia Regional - Daejeon 2014 I번