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문제

재현이는 얼마 전, 그래프 탐색에 대해서 배웠다. 강의를 맡은 지학이는 그래프 탐색을 통해서 풀 수 있는 대표적인 문제인 Flood Fill에 대해서 알려주었다.

  • N * N 격자에 M개의 점이 있을 때, 상하좌우에 붙어있는 인접한 셀을 연결되어 있다고 하자, 연결되어 있는 점들의 집합을 “섬” 이라고 부르면, 섬의 개수와, 가장 큰 섬의 크기는 얼마인가?

이 문제를 푸는 법은 여러 방법이 있지만, 재현이는 깊이 우선 탐색으로 문제를 풀었다. 재현이는 점들을 정점이라 생각하고, 연결되어 있는 점들 사이에 간선을 이으면, 그래프의 컴포넌트의 개수와, 가장 큰 컴포넌트의 크기를 묻는 문제로 변환됨을 알아냈다. 재현이는 이를 깊이 우선 탐색으로 구한 후 지학이에게 자랑하였다.

지학이는 재현이가 이 문제를 푼 것을 본 후 감탄하여, 문제를 조금 더 어렵게 해서 주었다.

  • N은 무조건 1,000,000,000이다. 고로 격자의 크기는 1,000,000,000 * 1,000,000,000 크기이다.
  • (x1, y1) 점과 (x2, y2) 점의 “택시 거리” 는 |x2 - x1| + |y2 - y1| 으로 정의된다. 이 정의대로라면 상하좌우에 인접한 셀은 택시 거리가 1 이하인 셀의 쌍이었다는 것을 알 수 있다. 지학이는 “붙어있다” 의 정의를 바꿨다
  • 택시 거리가 1 이하가 아니라, 택시 거리가 D 이하이면 붙어 있는 것이다.

이 문제는 갓 그래프 탐색을 배운 재현이에게 너무 어려운 문제였고, 재현이는 여러분들에게 도움을 요청했다. 재현이를 도와주자!

입력

첫 번째 줄에 M과 D가 주어진다. (1 <= M <= 100,000, 1 <= D <= 1,000,000,000)

이후 M개의 줄에 점의 좌표 Xi, Yi가 주어진다. (1 <= Xi, Yi <= 1,000,000,000)

출력

지학이가 어렵게 바꾼 문제의 정의대로, 섬의 개수와 가장 큰 섬의 크기를 출력하라.

예제 입력 1

4 2
1 1
3 3
2 2
10 10

예제 출력 1

2 3

힌트

(1,1) - (3,3) - (2,2) 셀은 연결되었으며, (10, 10) 셀은 혼자이다. 섬의 개수는 2개이고, 가장 큰 섬은 크기가 3이다.

[{"problem_id":"6181","problem_lang":"0","title":"\ud50c\ub7ec\ub4dc \ud544 (Flood Fill)","description":"<p>\uc7ac\ud604\uc774\ub294 \uc5bc\ub9c8 \uc804, \uadf8\ub798\ud504 \ud0d0\uc0c9\uc5d0 \ub300\ud574\uc11c \ubc30\uc6e0\ub2e4. \uac15\uc758\ub97c \ub9e1\uc740 \uc9c0\ud559\uc774\ub294 \uadf8\ub798\ud504 \ud0d0\uc0c9\uc744 \ud1b5\ud574\uc11c \ud480 \uc218 \uc788\ub294 \ub300\ud45c\uc801\uc778 \ubb38\uc81c\uc778 Flood Fill\uc5d0 \ub300\ud574\uc11c \uc54c\ub824\uc8fc\uc5c8\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>N * N \uaca9\uc790\uc5d0 M\uac1c\uc758 \uc810\uc774 \uc788\uc744 \ub54c, \uc0c1\ud558\uc88c\uc6b0\uc5d0 \ubd99\uc5b4\uc788\ub294 \uc778\uc811\ud55c \uc140\uc744 \uc5f0\uacb0\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790, \uc5f0\uacb0\ub418\uc5b4 \uc788\ub294 \uc810\ub4e4\uc758 \uc9d1\ud569\uc744 &ldquo;\uc12c&rdquo; \uc774\ub77c\uace0 \ubd80\ub974\uba74, \uc12c\uc758 \uac1c\uc218\uc640, \uac00\uc7a5 \ud070 \uc12c\uc758 \ud06c\uae30\ub294 \uc5bc\ub9c8\uc778\uac00?<\/li>\r\n<\/ul>\r\n\r\n<p>\uc774 \ubb38\uc81c\ub97c \ud478\ub294 \ubc95\uc740 \uc5ec\ub7ec \ubc29\ubc95\uc774 \uc788\uc9c0\ub9cc, \uc7ac\ud604\uc774\ub294 \uae4a\uc774 \uc6b0\uc120 \ud0d0\uc0c9\uc73c\ub85c \ubb38\uc81c\ub97c \ud480\uc5c8\ub2e4. \uc7ac\ud604\uc774\ub294 \uc810\ub4e4\uc744 \uc815\uc810\uc774\ub77c \uc0dd\uac01\ud558\uace0, \uc5f0\uacb0\ub418\uc5b4 \uc788\ub294 \uc810\ub4e4 \uc0ac\uc774\uc5d0 \uac04\uc120\uc744 \uc774\uc73c\uba74, \uadf8\ub798\ud504\uc758 \ucef4\ud3ec\ub10c\ud2b8\uc758 \uac1c\uc218\uc640, \uac00\uc7a5 \ud070 \ucef4\ud3ec\ub10c\ud2b8\uc758 \ud06c\uae30\ub97c \ubb3b\ub294 \ubb38\uc81c\ub85c \ubcc0\ud658\ub428\uc744 \uc54c\uc544\ub0c8\ub2e4. \uc7ac\ud604\uc774\ub294 \uc774\ub97c \uae4a\uc774 \uc6b0\uc120 \ud0d0\uc0c9\uc73c\ub85c \uad6c\ud55c \ud6c4 \uc9c0\ud559\uc774\uc5d0\uac8c \uc790\ub791\ud558\uc600\ub2e4.<\/p>\r\n\r\n<p>\uc9c0\ud559\uc774\ub294 \uc7ac\ud604\uc774\uac00 \uc774 \ubb38\uc81c\ub97c \ud47c \uac83\uc744 \ubcf8 \ud6c4 \uac10\ud0c4\ud558\uc5ec, \ubb38\uc81c\ub97c \uc870\uae08 \ub354 \uc5b4\ub835\uac8c \ud574\uc11c \uc8fc\uc5c8\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>N\uc740 \ubb34\uc870\uac74 1,000,000,000\uc774\ub2e4. \uace0\ub85c \uaca9\uc790\uc758 \ud06c\uae30\ub294 1,000,000,000 * 1,000,000,000 \ud06c\uae30\uc774\ub2e4.<\/li>\r\n\t<li>(x1, y1) \uc810\uacfc (x2, y2) \uc810\uc758 &ldquo;\ud0dd\uc2dc \uac70\ub9ac&rdquo; \ub294 |x2 - x1| + |y2 - y1| \uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \uc774 \uc815\uc758\ub300\ub85c\ub77c\uba74 \uc0c1\ud558\uc88c\uc6b0\uc5d0 \uc778\uc811\ud55c \uc140\uc740 \ud0dd\uc2dc \uac70\ub9ac\uac00 1 \uc774\ud558\uc778 \uc140\uc758 \uc30d\uc774\uc5c8\ub2e4\ub294 \uac83\uc744 \uc54c \uc218 \uc788\ub2e4. \uc9c0\ud559\uc774\ub294 &ldquo;\ubd99\uc5b4\uc788\ub2e4&rdquo; \uc758 \uc815\uc758\ub97c \ubc14\uafe8\ub2e4<\/li>\r\n\t<li>\ud0dd\uc2dc \uac70\ub9ac\uac00 1 \uc774\ud558\uac00 \uc544\ub2c8\ub77c, \ud0dd\uc2dc \uac70\ub9ac\uac00 D \uc774\ud558\uc774\uba74 \ubd99\uc5b4 \uc788\ub294 \uac83\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc774 \ubb38\uc81c\ub294 \uac13 \uadf8\ub798\ud504 \ud0d0\uc0c9\uc744 \ubc30\uc6b4 \uc7ac\ud604\uc774\uc5d0\uac8c \ub108\ubb34 \uc5b4\ub824\uc6b4 \ubb38\uc81c\uc600\uace0, \uc7ac\ud604\uc774\ub294 \uc5ec\ub7ec\ubd84\ub4e4\uc5d0\uac8c \ub3c4\uc6c0\uc744 \uc694\uccad\ud588\ub2e4. \uc7ac\ud604\uc774\ub97c \ub3c4\uc640\uc8fc\uc790!<\/p>\r\n","input":"<p>\uccab \ubc88\uc9f8 \uc904\uc5d0 M\uacfc D\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &lt;= M &lt;= 100,000, 1 &lt;= D &lt;= 1,000,000,000)<\/p>\r\n\r\n<p>\uc774\ud6c4 M\uac1c\uc758 \uc904\uc5d0 \uc810\uc758 \uc88c\ud45c Xi, Yi\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &lt;= Xi, Yi &lt;= 1,000,000,000)<\/p>\r\n","output":"<p>\uc9c0\ud559\uc774\uac00 \uc5b4\ub835\uac8c \ubc14\uafbc \ubb38\uc81c\uc758 \uc815\uc758\ub300\ub85c, \uc12c\uc758 \uac1c\uc218\uc640 \uac00\uc7a5 \ud070 \uc12c\uc758 \ud06c\uae30\ub97c \ucd9c\ub825\ud558\ub77c.<\/p>\r\n","hint":"<p>(1,1) - (3,3) - (2,2) \uc140\uc740 \uc5f0\uacb0\ub418\uc5c8\uc73c\uba70, (10, 10) \uc140\uc740 \ud63c\uc790\uc774\ub2e4. \uc12c\uc758 \uac1c\uc218\ub294 2\uac1c\uc774\uace0, \uac00\uc7a5 \ud070 \uc12c\uc740 \ud06c\uae30\uac00 3\uc774\ub2e4.<\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"6181","problem_lang":"1","title":"Cow Neighborhoods","description":"<p>Those Who Know About Cows are aware of the way cows group into &quot;Cow Neighborhoods&quot;. They have observed Farmer John&#39;s N (1 &lt;= N &lt;= 100,000) cows (conveniently numbered 1..N) as they graze, each at her own unique integer rectilinear coordinate, on a pasture whose X and Y coordinates are in the range 1..1,000,000,000.<\/p>\r\n\r\n<p>Two cows are neighbors if at least one of two criteria is met:<\/p>\r\n\r\n<ol>\r\n\t<li>If the cows are no further than some integer Manhattan distance C (1 &lt;= C &lt;= 1,000,000,000) apart, they are neighbors. [Manhattan distance is calculated as d = |x1-x2| + |y1-y2|.]<\/li>\r\n\t<li>If cow A is a neighbor of cow Z and cow B is a neighbor of cow Z, then cow A is a neighbor of cow B (the &quot;transitive closure of neighbors&quot;).<\/li>\r\n<\/ol>\r\n\r\n<p>Given the locations of the cows and the distance C, determine the the number of neighborhoods and the number of cows in the largest neighborhood.<\/p>\r\n\r\n<p>By way of example, consider the pasture below. When C = 4, this pasture has four neighborhoods: a big one on the left, two neighborhoods of size 1 (the lonesome cows), and a huge neighborhood on the right with 60 different cows.<\/p>\r\n\r\n<pre>\r\n.....................................*.................\r\n....*...*..*.......................***.................\r\n......*...........................****.................\r\n..*....*..*.......................*...*.******.*.*.....\r\n........................*.............***...***...*....\r\n*..*..*...*..........................*..*...*..*...*...\r\n.....................................*..*...*..*.......\r\n.....................................*..*...*..*.......\r\n...*................*..................................\r\n.*..*............................*.*.*.*.*.*.*.*.*.*.*.\r\n.*.....*..........................*.*.*.*.*.*.*.*.*.*.*\r\n....*..................................................<\/pre>\r\n\r\n<p>The input file describes cow locations by integer X,Y coordinates, where the lower left corner is (1,1) and cows close to that corner appear at both (2,2) and (5,1) in the example above.<\/p>\r\n\r\n<p>For a given pasture, determine both the number of cow neighborhoods and the number of cows resident in the largest cow neighborhood.<\/p>\r\n\r\n<p>The above picture is sample test case 2, which will be evaluated for you upon submission.<\/p>\r\n","input":"<ul>\r\n\t<li>Line 1: Two space-separated integers: N and C<\/li>\r\n\t<li>Lines 2..N+1: Line i+1 describes cow i&#39;s location with two space-separated integers: X_i and Y_i<\/li>\r\n<\/ul>\r\n\r\n<p>&nbsp;<\/p>\r\n","output":"<ul>\r\n\t<li>Line 1: A single line with a two space-separated integers: the number of cow neighborhoods and the size of the largest cow neighborhood.<\/li>\r\n<\/ul>\r\n","hint":"<p>There are 2 neighborhoods, one formed by the first three cows and the other being the last cow. The largest neighborhood therefore has size 3.<\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]