시간 제한메모리 제한제출정답맞힌 사람정답 비율
3 초 1024 MB43272158.333%

문제

정점이 $N$개인 트리가 주어진다. 각 정점은 1번부터 $N$번까지 차례대로 번호가 부여되어 있다. $i$번째 간선은 $A_i$번 정점과 $B_i$번 정점을 연결하며, 가중치는 $C_i$다. $(1 \leq i < N)$

트리에서 두 정점 사이의 거리는 그 둘을 잇는 최단경로 상의 간선의 가중치의 최댓값으로 정의한다. 단, 같은 두 정점 사이의 거리는 0으로 정의한다.

트리에 사는 사람들이 $N$개의 모임을 개최하려 한다. $i$번째 모임에는 1 이상 $i$ 이하의 번호를 가진 정점에 사는 사람들이 참석한다. 올해에는 코로나바이러스 전파 상황을 고려해 모임을 $X$개의 장소에서 각자 모인 후, 인터넷으로 진행하기로 했다. 각 모임은 트리 상의 서로 다른 $X$개의 정점 $v_1,\cdots,v_X$에서 이루어진다. 모임마다 고르는 정점은 독립적이다. 정점들이 정해지면 각 사람은 $v_1, \cdots, v_X$ 중 필요한 이동 거리가 최소인 정점 중 하나를 골라 이동하게 된다.

코로나바이러스 전파 상황에 따라 $X$의 값을 $1$부터 $K$까지의 값 중 하나로 정하기로 하였다. 모임을 미리 준비하기 위해 각 모임에 대해, $X$의 값이 $1$일 때부터 $K$일 때까지 사람들이 이동하는 거리의 최댓값의 최솟값의 합을 구하는 프로그램을 작성하시오.

입력

첫 줄에 트리의 정점 개수를 의미하는 정수 $N$과 정수 $K$가 사이에 공백을 두고 주어진다. $(1 \leq K \leq N \leq 300\,000)$

두번째 줄부터 $(N-1)$개의 줄에 걸쳐, 트리의 간선에 대한 정보가 주어진다. $(i+1)$번째 줄에는 세 개의 정수 $A_i,B_i,C_i$가 사이에 공백을 두고 주어진다. $(1 \leq i <N)$. 이는 $A_i$번 정점과 $B_i$번 정점을 연결하는 가중치 $C_i$의 간선이 존재함을 의미한다. $(1 \leq A_i,B_i,C_i \leq N)$

출력

첫 번째 줄부터 $N$개의 줄에 걸쳐, 답을 차례대로 출력한다. $i$번째 줄에는 $i$번째 모임에 대한 답을 출력한다 $(1 \leq i \leq N)$.

예제 입력 1

10 4
5 1 2
1 6 4
6 2 1
2 8 9
8 3 5
3 4 8
4 10 9
10 9 8
9 7 7

예제 출력 1

0
4
13
21
23
23
30
31
33
34

예제 입력 2

8 3
7 3 4
4 5 2
3 6 1
6 8 6
8 5 1
2 5 8
1 5 2

예제 출력 2

0
8
14
16
16
16
18
18
[{"problem_id":"20559","problem_lang":"0","title":"\uc815\uae30 \ubaa8\uc784 2","description":"<p>\uc815\uc810\uc774 $N$\uac1c\uc778 \ud2b8\ub9ac\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uac01 \uc815\uc810\uc740 1\ubc88\ubd80\ud130 $N$\ubc88\uae4c\uc9c0 \ucc28\ub840\ub300\ub85c \ubc88\ud638\uac00 \ubd80\uc5ec\ub418\uc5b4 \uc788\ub2e4. $i$\ubc88\uc9f8 \uac04\uc120\uc740 $A_i$\ubc88 \uc815\uc810\uacfc $B_i$\ubc88 \uc815\uc810\uc744 \uc5f0\uacb0\ud558\uba70, \uac00\uc911\uce58\ub294 $C_i$\ub2e4. $(1 \\leq i &lt; N)$<\/p>\r\n\r\n<p>\ud2b8\ub9ac\uc5d0\uc11c \ub450 \uc815\uc810 \uc0ac\uc774\uc758 \uac70\ub9ac\ub294 \uadf8 \ub458\uc744 \uc787\ub294 \ucd5c\ub2e8\uacbd\ub85c \uc0c1\uc758 \uac04\uc120\uc758 \uac00\uc911\uce58\uc758 \ucd5c\ub313\uac12\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \ub2e8, \uac19\uc740 \ub450 \uc815\uc810 \uc0ac\uc774\uc758 \uac70\ub9ac\ub294 0\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ud2b8\ub9ac\uc5d0 \uc0ac\ub294 \uc0ac\ub78c\ub4e4\uc774 $N$\uac1c\uc758 \ubaa8\uc784\uc744 \uac1c\ucd5c\ud558\ub824 \ud55c\ub2e4. $i$\ubc88\uc9f8 \ubaa8\uc784\uc5d0\ub294 1 \uc774\uc0c1 $i$ \uc774\ud558\uc758 \ubc88\ud638\ub97c \uac00\uc9c4 \uc815\uc810\uc5d0 \uc0ac\ub294 \uc0ac\ub78c\ub4e4\uc774 \ucc38\uc11d\ud55c\ub2e4. \uc62c\ud574\uc5d0\ub294 \ucf54\ub85c\ub098\ubc14\uc774\ub7ec\uc2a4 \uc804\ud30c \uc0c1\ud669\uc744 \uace0\ub824\ud574 \ubaa8\uc784\uc744 $X$\uac1c\uc758 \uc7a5\uc18c\uc5d0\uc11c \uac01\uc790 \ubaa8\uc778 \ud6c4, \uc778\ud130\ub137\uc73c\ub85c \uc9c4\ud589\ud558\uae30\ub85c \ud588\ub2e4. \uac01 \ubaa8\uc784\uc740 \ud2b8\ub9ac \uc0c1\uc758 \uc11c\ub85c \ub2e4\ub978 $X$\uac1c\uc758 \uc815\uc810 $v_1,\\cdots,v_X$\uc5d0\uc11c \uc774\ub8e8\uc5b4\uc9c4\ub2e4. \ubaa8\uc784\ub9c8\ub2e4 \uace0\ub974\ub294 \uc815\uc810\uc740 \ub3c5\ub9bd\uc801\uc774\ub2e4. \uc815\uc810\ub4e4\uc774 \uc815\ud574\uc9c0\uba74 \uac01 \uc0ac\ub78c\uc740 $v_1, \\cdots, v_X$ \uc911 \ud544\uc694\ud55c \uc774\ub3d9 \uac70\ub9ac\uac00 \ucd5c\uc18c\uc778 \uc815\uc810 \uc911 \ud558\ub098\ub97c \uace8\ub77c \uc774\ub3d9\ud558\uac8c \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ucf54\ub85c\ub098\ubc14\uc774\ub7ec\uc2a4 \uc804\ud30c \uc0c1\ud669\uc5d0 \ub530\ub77c $X$\uc758 \uac12\uc744 $1$\ubd80\ud130 $K$\uae4c\uc9c0\uc758 \uac12 \uc911 \ud558\ub098\ub85c \uc815\ud558\uae30\ub85c \ud558\uc600\ub2e4. \ubaa8\uc784\uc744 \ubbf8\ub9ac \uc900\ube44\ud558\uae30 \uc704\ud574 \uac01 \ubaa8\uc784\uc5d0 \ub300\ud574, $X$\uc758 \uac12\uc774 $1$\uc77c \ub54c\ubd80\ud130 $K$\uc77c \ub54c\uae4c\uc9c0 \uc0ac\ub78c\ub4e4\uc774 \uc774\ub3d9\ud558\ub294 \uac70\ub9ac\uc758 \ucd5c\ub313\uac12\uc758 \ucd5c\uc19f\uac12\uc758 \ud569\uc744 \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0 \ud2b8\ub9ac\uc758 \uc815\uc810 \uac1c\uc218\ub97c \uc758\ubbf8\ud558\ub294 \uc815\uc218 $N$\uacfc \uc815\uc218 $K$\uac00 \uc0ac\uc774\uc5d0 \uacf5\ubc31\uc744 \ub450\uace0 \uc8fc\uc5b4\uc9c4\ub2e4. $(1 \\leq K \\leq N \\leq 300\\,000)$<\/p>\r\n\r\n<p>\ub450\ubc88\uc9f8 \uc904\ubd80\ud130 $(N-1)$\uac1c\uc758 \uc904\uc5d0 \uac78\uccd0, \ud2b8\ub9ac\uc758 \uac04\uc120\uc5d0 \ub300\ud55c \uc815\ubcf4\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. $(i+1)$\ubc88\uc9f8 \uc904\uc5d0\ub294 \uc138 \uac1c\uc758 \uc815\uc218 $A_i,B_i,C_i$\uac00 \uc0ac\uc774\uc5d0 \uacf5\ubc31\uc744 \ub450\uace0 \uc8fc\uc5b4\uc9c4\ub2e4. $(1 \\leq i &lt;N)$. \uc774\ub294 $A_i$\ubc88 \uc815\uc810\uacfc $B_i$\ubc88 \uc815\uc810\uc744 \uc5f0\uacb0\ud558\ub294 \uac00\uc911\uce58 $C_i$\uc758 \uac04\uc120\uc774 \uc874\uc7ac\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4. $(1 \\leq A_i,B_i,C_i \\leq N)$<\/p>\r\n","output":"<p>\uccab \ubc88\uc9f8 \uc904\ubd80\ud130 $N$\uac1c\uc758 \uc904\uc5d0 \uac78\uccd0, \ub2f5\uc744 \ucc28\ub840\ub300\ub85c \ucd9c\ub825\ud55c\ub2e4. $i$\ubc88\uc9f8 \uc904\uc5d0\ub294 $i$\ubc88\uc9f8 \ubaa8\uc784\uc5d0 \ub300\ud55c \ub2f5\uc744 \ucd9c\ub825\ud55c\ub2e4 $(1 \\leq i \\leq N)$.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"20559","problem_lang":"1","title":"Best Meeting Places","description":"<p>A tree with $N$ vertices is given. Vertices are numbered sequentially from $1$ to $N$. The $i$-th edge connects vertices $A_i$ and $B_i$, and has weight $C_i$, for $1 \\leq i \\leq N - 1$.<\/p>\r\n\r\n<p>The <em>teleport distance<\/em> between two vertices of the tree is the maximum weight of the edge on the shortest path connecting them. The teleport distance between a vertex and itself is defined as $0$.<\/p>\r\n\r\n<p>People living on the tree want to hold $N$ meetings. The $i$-th meeting is attended by people living in the vertices numbered from $1$ to $i$. This year, because of the spread of coronavirus, the meeting participants will arrive at $X$ selected locations, and then connect via Internet from these locations.<\/p>\r\n\r\n<p>More formally, for each meeting, we will choose $X$ pairwise distinct vertices $v_1$, $v_2$, $\\ldots$, $v_X$. Once the vertices are determined, each person will move to one of the vertices $v_1$, $\\ldots$, $v_X$ with the minimum teleport distance to it. Let us define the <em>meeting cost<\/em> for the given $X$ and $i$ as the maximum of teleport distances for meeting participants. We will select the vertices $v_1$, $\\ldots$, $v_X$ in such a way that the meeting cost is minimal possible.<\/p>\r\n\r\n<p>The value of $X$ depends on the coronavirus situation, and may vary from $1$ to $K$. To prepare for the meeting in advance, write a program that, for each of the $N$ meetings, finds the sum of the meeting costs for all possible values of $X$ from $1$ to $K$, inclusive.<\/p>\r\n","input":"<p>The first line of input contains two integers $N$ and $K$: the number of vertices and the upper limit for $X$, respectively ($1 \\leq K \\leq N \\leq 3 \\cdot 10^5$).<\/p>\r\n\r\n<p>The following $N - 1$ lines describe the tree. Each of these lines contains three integers, $A_i$, $B_i$, and $C_i$, telling that there is an edge between vertices $A_i$ and $B_i$ with weight $C_i$ ($1 \\leq A_i, B_i, C_i \\leq N$). It is guaranteed that the resulting graph is a tree.<\/p>\r\n","output":"<p>Print $N$ lines. On line $i$, print the sum of meeting costs of $i$-th meeting for all $X$ from $1$ to $K$, inclusive.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English"}]