CF1715E Long Way Home

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 $ .

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.

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

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.