Long Way Home

题意翻译

有 $n$ 座城市，城市间有 $m$ 条双向道路，通过第 $i$ 条道路需要花费 $w_i$ 的时间，任意两个城市之间都有航班，乘坐城市 $u$ 和 $v$ 之间的航班需要花费 $(u-v)^2$ 的时间。 现在请对于任意城市 $i(1 \le i \le n)$，求出从城市 $1$ 出发，到达城市 $i$ 所需要的最短时间，注意从城市 $1$ 到 $i$ 的过程中**最多乘坐 $k$ 次航班**。

题目描述

Stanley lives in a country that consists of $ n $ cities (he lives in city $ 1 $ ). There are bidirectional roads between some of the cities, and you know how long it takes to ride through each of them. Additionally, there is a flight between each pair of cities, the flight between cities $ u $ and $ v $ takes $ (u - v)^2 $ time. Stanley is quite afraid of flying because of watching "Sully: Miracle on the Hudson" recently, so he can take at most $ k $ flights. Stanley wants to know the minimum time of a journey to each of the $ n $ cities from the city $ 1 $ .

输入输出格式

输入格式

In the first line of input there are three integers $ n $ , $ m $ , and $ k $ ( $ 2 \leq n \leq 10^{5} $ , $ 1 \leq m \leq 10^{5} $ , $ 1 \leq k \leq 20 $ ) — the number of cities, the number of roads, and the maximal number of flights Stanley can take. The following $ m $ lines describe the roads. Each contains three integers $ u $ , $ v $ , $ w $ ( $ 1 \leq u, v \leq n $ , $ u \neq v $ , $ 1 \leq w \leq 10^{9} $ ) — the cities the road connects and the time it takes to ride through. Note that some pairs of cities may be connected by more than one road.

输出格式

Print $ n $ integers, $ i $ -th of which is equal to the minimum time of traveling to city $ i $ .

输入输出样例

输入样例 #1

3 1 2
1 3 1

输出样例 #1

0 1 1

输入样例 #2

4 3 1
1 2 3
2 4 5
3 4 7

输出样例 #2

0 1 4 6

输入样例 #3

2 1 1
2 1 893746473

输出样例 #3

0 1

输入样例 #4

5 5 2
2 1 33
1 5 93
5 3 48
2 3 21
4 2 1

输出样例 #4

0 1 2 2 3

说明

In the first sample, it takes no time to get to city 1; to get to city 2 it is possible to use a flight between 1 and 2, which will take 1 unit of time; to city 3 you can get via a road from city 1, which will take 1 unit of time. In the second sample, it also takes no time to get to city 1. To get to city 2 Stanley should use a flight between 1 and 2, which will take 1 unit of time. To get to city 3 Stanley can ride between cities 1 and 2, which will take 3 units of time, and then use a flight between 2 and 3. To get to city 4 Stanley should use a flight between 1 and 2, then take a ride from 2 to 4, which will take 5 units of time.

题意翻译

有 $n$ 座城市，城市间有 $m$ 条双向道路，通过第 $i$ 条道路需要花费 $w_i$ 的时间，任意两个城市之间都有航班，乘坐城市 $u$ 和 $v$ 之间的航班需要花费 $(u-v)^2$ 的时间。 现在请对于任意城市 $i(1 \le i \le n)$，求出从城市 $1$ 出发，到达城市 $i$ 所需要的最短时间，注意从城市 $1$ 到 $i$ 的过程中**最多乘坐 $k$ 次航班**。