문제
$N\times N$ 크기의 행렬 $D$가 있다. 당신은 정점이 $N$개이고 아래 조건들을 만족하는 무방향 연결 그래프를 구성해야 한다. 각 정점은 1ドル$부터 $N$까지 번호가 매겨져 있으며, 각 간선에는 양의 정수 가중치를 원하는 대로 부여할 수 있다.
- 모든 정점 쌍 $(u,v)$에 대해, $u$와 $v$ 사이의 최단 경로의 길이는 $D_{u,v}$이다.
- 모든 간선의 가중치의 합은 가능한 최소여야 한다.
조건을 만족하는 그래프가 존재하는지 판별하고, 있다면 그 중 아무거나 하나를 출력하라.
출력
문제의 조건을 만족하는 그래프가 존재하지 않는다면, $-1$을 출력한다.
조건을 만족하는 그래프가 존재한다면,
- 첫 번째 줄에 간선의 개수를 나타내는 정수 $M$을 출력한다.
- 다음 $M$개 줄 중 $i$번째 줄에 세 정수 $u_i,ドル $v_i,ドル $c_i$를 공백으로 구분하여 출력한다. 이것은 $i$번 간선이 두 정점 $u_i$와 $v_i$를 잇고 가중치가 $c_i$라는 것을 나타낸다.
- 같은 쌍의 정점을 연결하는 간선은 최대 하나여야 하고, 각 간선의 가중치는 10ドル^9$ 이하여야 한다.
제한
- 2ドル\leq N\leq 300$
- $D_{i,i}=0$ (1ドル\le i\le N$)
- 1ドル\leq D_{i,j}=D_{j,i}\leq 10^9$ (1ドル\le i<j\le N$)
- 1ドル\leq u_i,v_i\leq N$ (1ドル\le i\le M$)
- $u_i\neq v_i$ (1ドル\le i\le M$)
- 1ドル\leq c_i\leq 10^9$ (1ドル\le i\le M$)
- 같은 쌍의 정점을 연결하는 간선은 최대 하나여야 한다.
서브태스크
| 번호 | 배점 | 제한 | | 1 | 9 | 이 서브태스크에서는 조건을 만족하는 그래프가 존재하는지만 판별해도 된다. 즉, 채점 프로그램은 출력이 $-1$인지 아닌지의 여부만 확인한다.
|
| 2 | 19 | $N \leq 50$
|
| 3 | 72 | 추가적인 제약 조건이 없다.
|
[{"problem_id":"31815","problem_lang":"0","title":"Construct a Graph","description":"<p>$N\\times N$ \ud06c\uae30\uc758 \ud589\ub82c $D$\uac00 \uc788\ub2e4. \ub2f9\uc2e0\uc740 \uc815\uc810\uc774 $N$\uac1c\uc774\uace0 \uc544\ub798 \uc870\uac74\ub4e4\uc744 \ub9cc\uc871\ud558\ub294 \ubb34\ubc29\ud5a5 \uc5f0\uacb0 \uadf8\ub798\ud504\ub97c \uad6c\uc131\ud574\uc57c \ud55c\ub2e4. \uac01 \uc815\uc810\uc740 $1$\ubd80\ud130 $N$\uae4c\uc9c0 \ubc88\ud638\uac00 \ub9e4\uaca8\uc838 \uc788\uc73c\uba70, \uac01 \uac04\uc120\uc5d0\ub294 \uc591\uc758 \uc815\uc218 \uac00\uc911\uce58\ub97c \uc6d0\ud558\ub294 \ub300\ub85c \ubd80\uc5ec\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\ubaa8\ub4e0 \uc815\uc810 \uc30d $(u,v)$\uc5d0 \ub300\ud574, $u$\uc640 $v$ \uc0ac\uc774\uc758 \ucd5c\ub2e8 \uacbd\ub85c\uc758 \uae38\uc774\ub294 $D_{u,v}$\uc774\ub2e4.<\/li>\r\n\t<li>\ubaa8\ub4e0 \uac04\uc120\uc758 \uac00\uc911\uce58\uc758 \ud569\uc740 \uac00\ub2a5\ud55c \ucd5c\uc18c\uc5ec\uc57c \ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uadf8\ub798\ud504\uac00 \uc874\uc7ac\ud558\ub294\uc9c0 \ud310\ubcc4\ud558\uace0, \uc788\ub2e4\uba74 \uadf8 \uc911 \uc544\ubb34\uac70\ub098 \ud558\ub098\ub97c \ucd9c\ub825\ud558\ub77c.<\/p>\r\n","input":"<p>\uccab \ubc88\uc9f8 \uc904\uc5d0 \uc815\uc810\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218 $N$\uc774 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\ub2e4\uc74c $N$\uac1c \uc904 \uc911 $i$\ubc88\uc9f8 \uc904\uc5d0\ub294 $N$\uac1c\uc758 \uc815\uc218 $D_{i,1},D_{i,2},\\ldots ,D_{i,N}$\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\ubb38\uc81c\uc758 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uadf8\ub798\ud504\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4\uba74, $-1$\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uadf8\ub798\ud504\uac00 \uc874\uc7ac\ud55c\ub2e4\uba74,<\/p>\r\n\r\n<ul>\r\n\t<li>\uccab \ubc88\uc9f8 \uc904\uc5d0 \uac04\uc120\uc758 \uac1c\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uc815\uc218 $M$\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/li>\r\n\t<li>\ub2e4\uc74c $M$\uac1c \uc904 \uc911 $i$\ubc88\uc9f8 \uc904\uc5d0 \uc138 \uc815\uc218 $u_i$, $v_i$, $c_i$\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \ucd9c\ub825\ud55c\ub2e4. \uc774\uac83\uc740 $i$\ubc88 \uac04\uc120\uc774 \ub450 \uc815\uc810 $u_i$\uc640 $v_i$\ub97c \uc787\uace0 \uac00\uc911\uce58\uac00 $c_i$\ub77c\ub294 \uac83\uc744 \ub098\ud0c0\ub0b8\ub2e4.<\/li>\r\n\t<li>\uac19\uc740 \uc30d\uc758 \uc815\uc810\uc744 \uc5f0\uacb0\ud558\ub294 \uac04\uc120\uc740 \ucd5c\ub300 \ud558\ub098\uc5ec\uc57c \ud558\uace0, \uac01 \uac04\uc120\uc758 \uac00\uc911\uce58\ub294 $10^9$ \uc774\ud558\uc5ec\uc57c \ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","limit":"<ul>\r\n\t<li>$2\\leq N\\leq 300$<\/li>\r\n\t<li>$D_{i,i}=0$ ($1\\le i\\le N$)<\/li>\r\n\t<li>$1\\leq D_{i,j}=D_{j,i}\\leq 10^9$ ($1\\le i&lt;j\\le N$)<\/li>\r\n\t<li>$1\\leq u_i,v_i\\leq N$ ($1\\le i\\le M$)<\/li>\r\n\t<li>$u_i\\neq v_i$ ($1\\le i\\le M$)<\/li>\r\n\t<li>$1\\leq c_i\\leq 10^9$ ($1\\le i\\le M$)<\/li>\r\n\t<li>\uac19\uc740 \uc30d\uc758 \uc815\uc810\uc744 \uc5f0\uacb0\ud558\ub294 \uac04\uc120\uc740 \ucd5c\ub300 \ud558\ub098\uc5ec\uc57c \ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n","subtask1":"<p>\uc774 \uc11c\ube0c\ud0dc\uc2a4\ud06c\uc5d0\uc11c\ub294 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \uadf8\ub798\ud504\uac00 \uc874\uc7ac\ud558\ub294\uc9c0\ub9cc \ud310\ubcc4\ud574\ub3c4 \ub41c\ub2e4. \uc989, \ucc44\uc810 \ud504\ub85c\uadf8\ub7a8\uc740 \ucd9c\ub825\uc774 $-1$\uc778\uc9c0 \uc544\ub2cc\uc9c0\uc758 \uc5ec\ubd80\ub9cc \ud655\uc778\ud55c\ub2e4.<\/p>\r\n","subtask2":"<p>$N \\leq 50$<\/p>\r\n","subtask3":"<p>\ucd94\uac00\uc801\uc778 \uc81c\uc57d \uc870\uac74\uc774 \uc5c6\ub2e4.<\/p>\r\n"},{"problem_id":"31815","problem_lang":"1","title":"Construct a Graph","description":"<p>You are given an $N\\times N$ square matrix $D$. Your task is to construct an undirected connected graph with $N$ vertices that fulfills the following conditions. Each vertex is numbered from $1$ to $N$, and you may assign positive integer weights to each edge as desired.<\/p>\r\n\r\n<ul>\r\n\t<li>For all pairs of vertices $(u,v)$, the length of the shortest path between $u$ and $v$ must be equal to $D_{u,v}$.<\/li>\r\n\t<li>The total sum of edge weights must be minimized.<\/li>\r\n<\/ul>\r\n\r\n<p>Determine whether such a graph exists, and if so, print any graph that satisfies these conditions.<\/p>\r\n","input":"<p>The first line contains an integer $N$ &mdash; the number of vertices.<\/p>\r\n\r\n<p>The $i$-th of the next $N$ lines contains $N$ integers $D_{i,1},D_{i,2},\\ldots ,D_{i,N}$ separated by a space.<\/p>\r\n","output":"<p>If no graph satisfies the conditions, print $-1$.<\/p>\r\n\r\n<p>If there exists a graph that satisfies the conditions:<\/p>\r\n\r\n<ul>\r\n\t<li>On the first line, print an integer $M$ &mdash; the number of edges.<\/li>\r\n\t<li>On the $i$-th of the next $M$ lines, print three integers $u_i$, $v_i$, $c_i$, separated by a space. These integers mean that the $i$-th edge connects two vertices $u_i$ and $v_i$ and its weight is $c_i$.<\/li>\r\n\t<li>There should be at most one edge for each pair of vertices, and the weight of each edge should not exceed $10^9$.<\/li>\r\n<\/ul>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","limit":"<ul>\r\n\t<li>$2\\leq N\\leq 300$<\/li>\r\n\t<li>$D_{i,i}=0$ ($1\\le i\\le N$)<\/li>\r\n\t<li>$1\\leq D_{i,j}=D_{j,i}\\leq 10^9$ ($1\\le i&lt;j\\le N$)<\/li>\r\n\t<li>$1\\leq u_i,v_i\\leq N$ ($1\\le i\\le M$)<\/li>\r\n\t<li>$u_i\\neq v_i$ ($1\\le i\\le M$)<\/li>\r\n\t<li>$1\\leq c_i\\leq 10^9$ ($1\\le i\\le M$)<\/li>\r\n\t<li>There should be at most one edge connecting a pair of vertices.<\/li>\r\n<\/ul>\r\n","subtask1":"<p>In this subtask, you only need to determine whether there exists a graph satisfying the conditions. In other words, the checker only checks whether the output is $-1$ or not.<\/p>\r\n","subtask2":"<p>$N \\leq 50$<\/p>\r\n","subtask3":"<p>No additional constraints.<\/p>\r\n"}]