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문제

요즘 고려대학교에서 가장 유행하는 게임은 "구슬을 꿰어라" 이다. 게임 이름에서 알 수 있듯이 이 게임은 구슬과 실을 이용해서 한다. 실은 총 두 가지 색상이 있고, 빨간색과 파란색이다. 구슬은 1번부터 n번가지 번호가 매겨져 있다. 게임은 구슬 1개를 가지고 시작한다. 이제 다음과 같은 과정을 통해 새로운 구슬을 실을 이용해 추가할 수 있다.

  • Append(w, v): 새로운 구슬 w를 이미 가지고 있는 구슬 v에 빨간 실을 이용해 연결한다.
  • Insert(w, u, v): 새로운 구슬 w를 이미 빨간 실로 연결된 구슬 u와 v사이에 파란 실 두 개를 이용해 연결한다. 이 때, u와 v를 연결하는 빨간 실은 제거해야 한다. 즉, 이미 존재하는 빨간 실 u - v가 새로운 파란 실 두 개 u - w, w - v로 교체된다.

모든 실은 특정 길이를 가진다. 게임이 종료되었을 때, 최종 점수는 구슬을 연결하는 파란 실 길이의 합이다.

게임이 모두 종료된 상태가 주어진다. 상태는 실이 연결하고 있는 두 구슬의 번호와 길이로 이루어져 있으며, 색상은 알 수 없다.

문제에서 주어진 연결 상태를 만드는 방법은 여러가지가 있다. 이 때, 모든 방법 중에서 가장 큰 최종 점수를 구하는 프로그램을 작성하시오.

입력

첫재 줄에 구슬의 수 n이 주어진다. (1 ≤ n ≤ 200000) 구슬은 1번부터 n번가지 번호가 매겨져 있다. 다음 n-1개 줄에는 실의 정보 ai, bi, ci가 주어진다. (1 ≤ ai < bi ≤ n, 1 ≤ ci ≤ 10000) ai와 bi는 실이 연결하는 구슬의 번호이고, ci는 길이이다.

출력

가능한 점수 중에서 가장 큰 최종 점수를 출력한다.

예제 입력 1

5
1 2 10
1 3 40
1 4 15
1 5 20

예제 출력 1

60

힌트

예제의 경우에 아래와 같이 연결하면, 60점을 얻을 수 있다. 시작은 3번 구슬로 시작한다.

  • 5와 3을 연결한다. (선의 길이는 상관없다)
  • 1을 3-5 사이에 넣는다. (길이 40, 20 선을 이용)
  • 2와 1을 길이가 10인 선을 이용해 연결한다.
  • 4와 1을 길이가 15인 선으로 연결한다.

연결은 아래와 같아지고, 이 방법보다 더 큰 점수를 얻을 수 있는 방법은 없다.

[{"problem_id":"10068","problem_lang":"0","title":"\uad6c\uc2ac\uc774 \uc11c\ub9d0\uc774\ub77c\ub3c4 \uaff0\uc5b4\uc57c \ubcf4\ubc30","description":"<p>\uc694\uc998 \uace0\ub824\ub300\ud559\uad50\uc5d0\uc11c \uac00\uc7a5 \uc720\ud589\ud558\ub294 \uac8c\uc784\uc740 &quot;\uad6c\uc2ac\uc744 \uaff0\uc5b4\ub77c&quot; \uc774\ub2e4. \uac8c\uc784 \uc774\ub984\uc5d0\uc11c \uc54c \uc218 \uc788\ub4ef\uc774 \uc774 \uac8c\uc784\uc740 \uad6c\uc2ac\uacfc \uc2e4\uc744 \uc774\uc6a9\ud574\uc11c \ud55c\ub2e4. \uc2e4\uc740 \ucd1d \ub450 \uac00\uc9c0 \uc0c9\uc0c1\uc774 \uc788\uace0, \ube68\uac04\uc0c9\uacfc \ud30c\ub780\uc0c9\uc774\ub2e4. \uad6c\uc2ac\uc740 1\ubc88\ubd80\ud130 n\ubc88\uac00\uc9c0 \ubc88\ud638\uac00 \ub9e4\uaca8\uc838 \uc788\ub2e4. \uac8c\uc784\uc740 \uad6c\uc2ac 1\uac1c\ub97c \uac00\uc9c0\uace0 \uc2dc\uc791\ud55c\ub2e4. \uc774\uc81c \ub2e4\uc74c\uacfc \uac19\uc740 \uacfc\uc815\uc744 \ud1b5\ud574 \uc0c8\ub85c\uc6b4 \uad6c\uc2ac\uc744 \uc2e4\uc744 \uc774\uc6a9\ud574 \ucd94\uac00\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>Append(w, v): \uc0c8\ub85c\uc6b4 \uad6c\uc2ac w\ub97c \uc774\ubbf8 \uac00\uc9c0\uace0 \uc788\ub294 \uad6c\uc2ac v\uc5d0 \ube68\uac04 \uc2e4\uc744 \uc774\uc6a9\ud574 \uc5f0\uacb0\ud55c\ub2e4.<\/li>\r\n\t<li>Insert(w, u, v): \uc0c8\ub85c\uc6b4 \uad6c\uc2ac w\ub97c \uc774\ubbf8 \ube68\uac04 \uc2e4\ub85c \uc5f0\uacb0\ub41c \uad6c\uc2ac u\uc640 v\uc0ac\uc774\uc5d0 \ud30c\ub780 \uc2e4 \ub450 \uac1c\ub97c \uc774\uc6a9\ud574 \uc5f0\uacb0\ud55c\ub2e4. \uc774 \ub54c, u\uc640 v\ub97c \uc5f0\uacb0\ud558\ub294 \ube68\uac04 \uc2e4\uc740 \uc81c\uac70\ud574\uc57c \ud55c\ub2e4. \uc989, \uc774\ubbf8 \uc874\uc7ac\ud558\ub294 \ube68\uac04 \uc2e4 u - v\uac00 \uc0c8\ub85c\uc6b4 \ud30c\ub780 \uc2e4 \ub450 \uac1c u - w, w - v\ub85c \uad50\uccb4\ub41c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\ubaa8\ub4e0 \uc2e4\uc740 \ud2b9\uc815 \uae38\uc774\ub97c \uac00\uc9c4\ub2e4. \uac8c\uc784\uc774 \uc885\ub8cc\ub418\uc5c8\uc744 \ub54c, \ucd5c\uc885 \uc810\uc218\ub294 \uad6c\uc2ac\uc744 \uc5f0\uacb0\ud558\ub294 \ud30c\ub780 \uc2e4 \uae38\uc774\uc758 \ud569\uc774\ub2e4.<\/p>\r\n\r\n<p>\uac8c\uc784\uc774 \ubaa8\ub450 \uc885\ub8cc\ub41c \uc0c1\ud0dc\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uc0c1\ud0dc\ub294 \uc2e4\uc774 \uc5f0\uacb0\ud558\uace0 \uc788\ub294 \ub450 \uad6c\uc2ac\uc758 \ubc88\ud638\uc640 \uae38\uc774\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\uc73c\uba70, \uc0c9\uc0c1\uc740 \uc54c \uc218 \uc5c6\ub2e4.<\/p>\r\n\r\n<p>\ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \uc5f0\uacb0 \uc0c1\ud0dc\ub97c \ub9cc\ub4dc\ub294 \ubc29\ubc95\uc740 \uc5ec\ub7ec\uac00\uc9c0\uac00 \uc788\ub2e4. \uc774 \ub54c, \ubaa8\ub4e0 \ubc29\ubc95 \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \ucd5c\uc885 \uc810\uc218\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc7ac \uc904\uc5d0 \uad6c\uc2ac\uc758 \uc218 n\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; n &le; 200000)&nbsp;\uad6c\uc2ac\uc740 1\ubc88\ubd80\ud130 n\ubc88\uac00\uc9c0 \ubc88\ud638\uac00 \ub9e4\uaca8\uc838 \uc788\ub2e4. \ub2e4\uc74c n-1\uac1c \uc904\uc5d0\ub294 \uc2e4\uc758 \uc815\ubcf4 a<sub>i<\/sub>, b<sub>i<\/sub>, c<sub>i<\/sub>\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; a<sub>i<\/sub> &lt; b<sub>i<\/sub> &le; n, 1 &le; c<sub>i<\/sub> &le; 10000) a<sub>i<\/sub>\uc640 b<sub>i<\/sub>\ub294 \uc2e4\uc774 \uc5f0\uacb0\ud558\ub294 \uad6c\uc2ac\uc758 \ubc88\ud638\uc774\uace0, c<sub>i<\/sub>\ub294 \uae38\uc774\uc774\ub2e4.<\/p>\r\n","output":"<p>\uac00\ub2a5\ud55c \uc810\uc218 \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \ucd5c\uc885 \uc810\uc218\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"<p>\uc608\uc81c\uc758 \uacbd\uc6b0\uc5d0 \uc544\ub798\uc640 \uac19\uc774 \uc5f0\uacb0\ud558\uba74, 60\uc810\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4. \uc2dc\uc791\uc740 3\ubc88 \uad6c\uc2ac\ub85c \uc2dc\uc791\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>5\uc640 3\uc744 \uc5f0\uacb0\ud55c\ub2e4. (\uc120\uc758 \uae38\uc774\ub294 \uc0c1\uad00\uc5c6\ub2e4)<\/li>\r\n\t<li>1\uc744 3-5 \uc0ac\uc774\uc5d0 \ub123\ub294\ub2e4. (\uae38\uc774 40, 20 \uc120\uc744 \uc774\uc6a9)<\/li>\r\n\t<li>2\uc640 1\uc744 \uae38\uc774\uac00 10\uc778 \uc120\uc744 \uc774\uc6a9\ud574 \uc5f0\uacb0\ud55c\ub2e4.<\/li>\r\n\t<li>4\uc640 1\uc744 \uae38\uc774\uac00 15\uc778 \uc120\uc73c\ub85c \uc5f0\uacb0\ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc5f0\uacb0\uc740 \uc544\ub798\uc640 \uac19\uc544\uc9c0\uace0, \uc774 \ubc29\ubc95\ubcf4\ub2e4 \ub354 \ud070 \uc810\uc218\ub97c \uc5bb\uc744 \uc218 \uc788\ub294 \ubc29\ubc95\uc740 \uc5c6\ub2e4.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/bead.png\" style=\"height:227px; width:240px\" \/><\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"10068","problem_lang":"1","title":"Beads and wires","description":"<p>A popular child game at the time of Leonardo was called Beads-and-wires. Not surprisingly, it was played with beads and wires. The wires are red or blue. The beads are numbered from 1 to n. The game starts with a single bead. From this moment on, new beads can be added using wires as follows.<\/p>\r\n\r\n<ul>\r\n\t<li>Append(w, v): a new bead w is attached to an existing bead v through a piece of red wire.<\/li>\r\n\t<li>Insert(w, u, v): a new bead w is inserted by substituting an already existing red wire connecting existing beads u and v, with two new blue wires by putting w between u and v; in other words, the existing red wire u &minus; v is replaced by two new blue wires u &minus; w and w &minus; v.<\/li>\r\n<\/ul>\r\n\r\n<p>Every piece of wire (both blue or red) has a certain length. At the end of the game, what counts is the sum of lengths of blue wires only (the length of red wires doesn&rsquo;t count): this is called the final score. You are given the final configuration reached by a beads-and-wire game, specifying how beads are connected to one another and providing also the length of each piece of wire, but omitting the color of the wires.<\/p>\r\n\r\n<p>You have to write a program that finds the maximal possible final score for that configuration. More precisely, among all the feasible beads-and-wire games that end with that configuration, you have to find one that has the maximum final score (sum of blue wire length) and you have to output that value.<\/p>\r\n","input":"<p>First line contains positive integer n &mdash; number of beads, which are numbered from 1 to n. Next n &minus; 1 lines contains 3 numbers each: a<sub>i<\/sub>, b<sub>i<\/sub> (1 &le; a<sub>i<\/sub> &lt; b<sub>i<\/sub> &le; n) and 1 &le; c<sub>i<\/sub> &le; 10000 &mdash; two beads connected by a piece of wire, and the length of that piece.<\/p>\r\n","output":"<p>Output one integer &ndash; the maximum final score.<\/p>\r\n","hint":"<p>For the first sample testcase, the final score of 60 can be obtained as follows: start with the single bead 3.<\/p>\r\n\r\n<ul>\r\n\t<li>append 5 to 3 (with a wire of arbitrary length).<\/li>\r\n\t<li>insert 1 on the wire 3 &minus; 5 (using wires of length 40 and 20).<\/li>\r\n\t<li>append 2 to 1 with a wire of length 10.<\/li>\r\n\t<li>append 4 to 1 with a wire of length 15.<\/li>\r\n<\/ul>\r\n\r\n<p>The configuration obtained this way is shown in the first figure. It is possible to see that there is no way to obtain the same configuration (apart for the colors) with a larger final score.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/bead.png\" style=\"height:227px; width:240px\" \/><\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]

출처

Olympiad > Asia-Pacific Informatics Olympiad > APIO 2014 3번