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문제

다솜이는 눈이 오늘 날 아침이면 어김없이 테라스에 나가 발자국을 남기는 것이 취미다. 하지만 늦게 일어나는 조교는 단지 발자국만을 보며 추측할 수 밖에 없다. 그러던 어느 날 아침에 조교는 다솜이의 발자취가 궁금해졌다. 하지만 이미 너무 많은 학생들이 왔다 갔다 했기에 발자취를 찾기란 쉬운 일이 아니었다.
  한 가지 확실한 점은 학생들은 언제나 일직선으로 이동하며 눈을 밟으며 걸음폭도 일정하다는 것이다.


  위의 그림과 같이 다솜이를 포함한 학생들의 발자취는 뛰는 간격이나 방향이 서로 제각각일 수 있다.

위의 그림에서와 같이 학생들은 밖에서부터 일직선으로 들어와 정확한 보폭으로 눈을 밟고 지나간다.

그림 3과 같이 아침에 여러 학생들은 제각기 다른 방향에서 들어와 서로 다른 방향으로 밟고 지나간다. 물론 여러 명이 한 곳을 밟을 수도 있다. 그리고 늦잠을 잔 조교는 그림 4와 같이 밟고 지나간 최종 상태만 볼 수 있다.
  그림 4를 보면, 학생들이 몇 명 들어 와서 어느 방향으로 어느 간격으로 눈 위에 발자국을 내고 갔는지 그 경로를 재현할 수 있다. 단, 세 번 이상 눈 위에 발자국을 낸 학생의 경로만 생각하기로 한다. 그렇다면, 그림 4의 상황이라면 그림 3과 같은 세 개의 학생 경로가 있다고 말할 수 있다. (물론 다른 가짓수도 있을 수 있다.)

한편, 첫째 칸에서 수직으로 내려가는 경로는 보폭이 4인 학생의 경로일 수도 있지만, 이 경로는 총 발자국이 두 개이기 때문에 고려하지 않는다. 그리고 둘째 줄 셋째 칸(2, 3), 셋째 줄 넷째 칸(3, 4), 여섯째 줄 일곱째(6, 7) 칸을 지나는 경로는 일직선이고 세 개의 발자국을 남겼지만, 간격이 일정하지 않기 때문에 한 학생이 밟고 간 경로로 인정하지 않는다.

어떤 학생이 밟고 간 경로에 속하는 선상에 다른 학생이 밟고 간 곳이 있을 수 있다. 예를 들어 위 그림에서 (2, 1), (2, 3), (2, 5), (2, 7)은 한 학생이 밟고 간 경로인데, 이 선 안에는 (2, 6)처럼 다른 학생이 밟고 간 곳이 있을 수 있다. 그리고 사실은, 발자국이 있지만 이번에 학생이 남긴게 아닌 이 전에 눈이 왔을 때 남은 발자국일 수도 있다

세상에서 제일 착한 다솜이는 가장 많은 발자국을 남기고 지나가기에 이를 찾는 것이 문제이다. 그림 4의 경우, 정답은 7이다. 여섯째 줄부터 가로로 일곱 발자국을 모두 밟고 지나간 학생을 다솜이로 추정할 수 있기 때문이다.

입력

첫째 줄에는 테라스의 세로, 가로 크기를 나타내는 정수 R과 C(1≤R, C≤5000)가 있다. 그리고 둘째 줄에는 학생들의 발자국 수 N(3≤N≤5000)이 있다.  그리고 다음 N 줄에는 발자국이 있는 좌표를 나타내는 정수 두 개가 뒤따른다. 처음 것이 줄(세로) 좌표이고 나중 것이 칸(가로) 좌표이다. (1≤줄 좌표≤R, 1≤칸 좌표≤C) 두 정수는 공백 한 칸으로 구분된다. 여러 학생에게 밟혔더라도 좌표는 한 번만 나온다.

출력

다솜이로 추정되는 가장 많은 발자국을 남기고 간 학생의 발자국 수를 출력한다. 만약 학생의 경로를 하나도 발견하지 못했다면 그냥 0을 출력한다.

예제 입력 1

6 7       
14
2 1
6 6
4 2
2 5
2 6
2 7
3 4
6 1
6 2
2 3
6 3
6 4
6 5
6 7

예제 출력 1

7
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(\ubb3c\ub860 \ub2e4\ub978 \uac00\uc9d3\uc218\ub3c4 \uc788\uc744 \uc218 \uc788\ub2e4.)<\/p>\r\n<p>\ud55c\ud3b8, \uccab\uc9f8 \uce78\uc5d0\uc11c \uc218\uc9c1\uc73c\ub85c \ub0b4\ub824\uac00\ub294 \uacbd\ub85c\ub294 \ubcf4\ud3ed\uc774 4\uc778 \ud559\uc0dd\uc758 \uacbd\ub85c\uc77c \uc218\ub3c4 \uc788\uc9c0\ub9cc, \uc774 \uacbd\ub85c\ub294 \ucd1d \ubc1c\uc790\uad6d\uc774 \ub450 \uac1c\uc774\uae30 \ub54c\ubb38\uc5d0 \uace0\ub824\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub9ac\uace0 \ub458\uc9f8 \uc904 \uc14b\uc9f8 \uce78(2, 3), \uc14b\uc9f8 \uc904 \ub137\uc9f8 \uce78(3, 4), \uc5ec\uc12f\uc9f8 \uc904 \uc77c\uacf1\uc9f8(6, 7) \uce78\uc744 \uc9c0\ub098\ub294 \uacbd\ub85c\ub294 \uc77c\uc9c1\uc120\uc774\uace0 \uc138 \uac1c\uc758 \ubc1c\uc790\uad6d\uc744 \ub0a8\uacbc\uc9c0\ub9cc, \uac04\uaca9\uc774 \uc77c\uc815\ud558\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc5d0 \ud55c \ud559\uc0dd\uc774 \ubc1f\uace0 \uac04 \uacbd\ub85c\ub85c \uc778\uc815\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n<p>\uc5b4\ub5a4 \ud559\uc0dd\uc774 \ubc1f\uace0 \uac04 \uacbd\ub85c\uc5d0 \uc18d\ud558\ub294 \uc120\uc0c1\uc5d0 \ub2e4\ub978 \ud559\uc0dd\uc774 \ubc1f\uace0 \uac04 \uacf3\uc774 \uc788\uc744 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \uc704 \uadf8\ub9bc\uc5d0\uc11c (2, 1), (2, 3), (2, 5), (2, 7)\uc740 \ud55c \ud559\uc0dd\uc774 \ubc1f\uace0 \uac04 \uacbd\ub85c\uc778\ub370, \uc774 \uc120 \uc548\uc5d0\ub294 (2, 6)\ucc98\ub7fc \ub2e4\ub978 \ud559\uc0dd\uc774 \ubc1f\uace0 \uac04 \uacf3\uc774 \uc788\uc744 \uc218 \uc788\ub2e4. \uadf8\ub9ac\uace0 \uc0ac\uc2e4\uc740, \ubc1c\uc790\uad6d\uc774 \uc788\uc9c0\ub9cc \uc774\ubc88\uc5d0 \ud559\uc0dd\uc774 \ub0a8\uae34\uac8c \uc544\ub2cc \uc774 \uc804\uc5d0 \ub208\uc774 \uc654\uc744 \ub54c \ub0a8\uc740 \ubc1c\uc790\uad6d\uc77c \uc218\ub3c4 \uc788\ub2e4<\/p>\r\n<p>\uc138\uc0c1\uc5d0\uc11c \uc81c\uc77c \ucc29\ud55c \ub2e4\uc19c\uc774\ub294 \uac00\uc7a5 \ub9ce\uc740 \ubc1c\uc790\uad6d\uc744 \ub0a8\uae30\uace0 \uc9c0\ub098\uac00\uae30\uc5d0 \uc774\ub97c \ucc3e\ub294 \uac83\uc774 \ubb38\uc81c\uc774\ub2e4. \uadf8\ub9bc 4\uc758 \uacbd\uc6b0, \uc815\ub2f5\uc740 7\uc774\ub2e4. \uc5ec\uc12f\uc9f8 \uc904\ubd80\ud130 \uac00\ub85c\ub85c \uc77c\uacf1 \ubc1c\uc790\uad6d\uc744 \ubaa8\ub450 \ubc1f\uace0 \uc9c0\ub098\uac04 \ud559\uc0dd\uc744 \ub2e4\uc19c\uc774\ub85c \ucd94\uc815\ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>","input":"<p>\uccab\uc9f8 \uc904\uc5d0\ub294 \ud14c\ub77c\uc2a4\uc758 \uc138\ub85c, \uac00\ub85c \ud06c\uae30\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218 R\uacfc C(1&le;R, C&le;5000)\uac00 \uc788\ub2e4. \uadf8\ub9ac\uace0 \ub458\uc9f8 \uc904\uc5d0\ub294 \ud559\uc0dd\ub4e4\uc758 \ubc1c\uc790\uad6d \uc218 N(3&le;N&le;5000)\uc774 \uc788\ub2e4.&nbsp; \uadf8\ub9ac\uace0 \ub2e4\uc74c N \uc904\uc5d0\ub294 \ubc1c\uc790\uad6d\uc774 \uc788\ub294 \uc88c\ud45c\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218 \ub450 \uac1c\uac00 \ub4a4\ub530\ub978\ub2e4. \ucc98\uc74c \uac83\uc774 \uc904(\uc138\ub85c) \uc88c\ud45c\uc774\uace0 \ub098\uc911 \uac83\uc774 \uce78(\uac00\ub85c) \uc88c\ud45c\uc774\ub2e4. (1&le;\uc904 \uc88c\ud45c&le;R, 1&le;\uce78 \uc88c\ud45c&le;C) \ub450 \uc815\uc218\ub294 \uacf5\ubc31 \ud55c \uce78\uc73c\ub85c \uad6c\ubd84\ub41c\ub2e4. \uc5ec\ub7ec \ud559\uc0dd\uc5d0\uac8c \ubc1f\ud614\ub354\ub77c\ub3c4 \uc88c\ud45c\ub294 \ud55c \ubc88\ub9cc \ub098\uc628\ub2e4.<\/p>","output":"<p>\ub2e4\uc19c\uc774\ub85c \ucd94\uc815\ub418\ub294 \uac00\uc7a5 \ub9ce\uc740 \ubc1c\uc790\uad6d\uc744 \ub0a8\uae30\uace0 \uac04 \ud559\uc0dd\uc758 \ubc1c\uc790\uad6d \uc218\ub97c \ucd9c\ub825\ud55c\ub2e4. \ub9cc\uc57d \ud559\uc0dd\uc758 \uacbd\ub85c\ub97c \ud558\ub098\ub3c4 \ubc1c\uacac\ud558\uc9c0 \ubabb\ud588\ub2e4\uba74 \uadf8\ub0e5 0\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"1752","problem_lang":"1","title":"The Troublesome Frog","description":"<p>In Korea, the naughtiness of the cheonggaeguri, a small frog, is legendary. &nbsp;This is a well-deserved reputation, because the frogs jump through your rice paddy at night, flattening rice plants. &nbsp;In the morning, after noting which plants have been flattened, you want to identify the path of the frog which did the most damage. &nbsp;A frog always jumps through the paddy in a straight line, with every hop the same length:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/f1(1).png\" style=\"height:51px; width:731px\" \/><\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/f2(1).png\" style=\"height:59px; width:360px\" \/><\/p>\r\n\r\n<p>Different frogs can jump with different hop lengths<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/f3(1).png\" style=\"height:223px; width:389px\" \/><\/p>\r\n\r\n<p>And in different directions<\/p>\r\n\r\n<p>Your rice paddy has plants arranged on the intersection points of a grid as shown in Figure-1, and the troublesome frogs hop completely through your paddy, starting outside the paddy on one side and ending outside the paddy on the other side as shown in Figure-2:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/f4.png\" style=\"height:300px; width:866px\" \/><\/p>\r\n\r\n<p>Many frogs can jump through the paddy, hopping from rice plant to rice plant. &nbsp;Every hop lands on a plant and flattens it, as in Figure-3. &nbsp;Note that some plants may be landed on by more than one frog during the night. &nbsp;Of course, you can not see the lines showing the paths of the frogs or any of their hops outside of your paddy &ndash; for the situation in Figure-3, what you can see is shown in Figure-4:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/f5.png\" style=\"height:300px; width:808px\" \/><\/p>\r\n\r\n<p>From Figure-4, you can reconstruct all the possible paths which the frogs may have followed across your paddy. &nbsp;You are only interested in frogs which have landed on at least 3 of your rice plants in their voyage through the paddy. &nbsp;Such a path is said to be a frog path. &nbsp;In this case, that means that the three paths shown in Figure-3 are frog paths (there are also other possible frog paths). The vertical path down column 1 might have been a frog path with hop length 4 except there are only 2 plants flattened so we are not interested; and the diagonal path including the plants on row 2 col. 3, row 3 col. 4, and row 6 col. 7 has three flat plants but there is no regular hop length which could have spaced the hops in this way while still landing on at least 3 plants, and hence it is not a frog path. &nbsp;Note also that along the line a frog path follows there may be additional flattened plants which do not need to be landed on by that path (see the plant at (2, 6) on the horizontal path across row 2 in Figure-4), and in fact some flattened plants may not be explained by any frog path at all.&nbsp;<\/p>\r\n\r\n<p>Your task is to write a program to determine the maximum number of landings in any single frog path (where the maximum is taken over all possible frog paths). &nbsp;In Figure-4 the answer is 7, obtained from the frog path across row 6.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","input":"<p>Your program is to read from standard input. The first line contains two integers R and C, respectively the number of rows and columns in your rice paddy, 1 &le; R,C &le; 5000. &nbsp;The second line contains the single integer N, the number of flattened rice plants, 3 &le; N &le; 5000. &nbsp;Each of the remaining N lines contains two integers, the row number (1&le; row number &le; R) and the column number (1&le; column number &le; C) of a flattened rice plant, separated by one blank. Each flattened plant is only listed once.<\/p>\r\n","output":"<p>Your program is to write to standard output. The output contains one line with a single integer, the number of plants flattened along a frog path which did the most damage if there exists at least one frog path, otherwise, 0.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]

출처

Olympiad > International Olympiad in Informatics > IOI 2002 1번

  • 문제를 번역한 사람: author6