Problem:
Single Source Shortest Path

For a given weighted graph G(V, E) and a source r, find the source shortest path to each vertex from the source (SSSP: Single Source Shortest Path).

Input

An edge-weighted graph G (V, E) and the source r.

|V| |E| r
s0 t0 d0
s1 t1 d1
:
s|E|-1 t|E|-1 d|E|-1

|V| is the number of vertices and |E| is the number of edges in G. The graph vertices are named with the numbers 0, 1,..., |V|-1 respectively. r is the source of the graph.

si and ti represent source and target vertices of i-th edge (directed) and di represents the cost of the i-th edge.

Output

Print the costs of SSSP in the following format.

c0
c1
:
c|V|-1

The output consists of |V| lines. Print the cost of the shortest path from the source r to each vertex 0, 1, ... |V|-1 in order. If there is no path from the source to a vertex, print INF.

Constraints

1 ≤ |V| ≤ 100000

0 ≤ di ≤ 10000

0 ≤ |E| ≤ 500000

There are no parallel edges

There are no self-loops

Sample Input 1

4 5 0
0 1 1
0 2 4
1 2 2
2 3 1
1 3 5

Sample Output 1

0
1
3
4

Sample Input 2

4 6 1
0 1 1
0 2 4
2 0 1
1 2 2
3 1 1
3 2 5

Sample Output 2

3
0
2
INF

Solution in Haskell (judge verdict: Time Limit Exceeded)

import Data.IntMap.Strict ((!))
import qualified Data.IntMap.Strict as M
import qualified Data.Set as S
import qualified Data.ByteString.Char8 as B
import Control.Applicative
import Data.Maybe
import Data.List
import Data.Ord

type Weight = Int
type Vertex = Int

type PriorityQueue = S.Set (Weight, Vertex)
nullq = S.null
emp = S.empty
push c v = S.insert (c,v)
pop = S.deleteFindMin

inf = 10^9+7
readEdge = map (fst . fromJust . B.readInt) . B.words

buildGraph edges =
 let g = groupBy (\ (x, _, _) (y, _, _) -> x == y) (sort edges)
 in M.fromList (map (\ x -> (f1 $ head x, map (\ (_, y, z) -> (y, z)) x)) g)

f1 :: (t, t1, t2) -> t
f1 (x, _, _) = x
 
f3 :: (t, t1, t2) -> t2
f3 (_, _, x) = x

main = do
 ([nv,ne,v0]:es) <- map (map (fst . fromJust . B.readInt) . B.words) . B.lines <$> B.getContents
 let g = buildGraph (map (\[s,t,d]->(s,t,d)) es)
 mapM_ (putStrLn . maybe "INF" show . sumOfWeight g nv ne v0 es) [0..nv-1]

sumOfWeight g nv ne v0 es v = M.lookup v (dijkstra g im0 q0)
 where
  im0 = M.singleton v0 0
  q0 = push 0 v0 emp

dijkstra g im q
 | nullq q = im
 | otherwise = dijkstra g im' q''
 where
  ((d,v),q') = pop q
  im' = foldl' (\acc (b,c) -> M.insert b (d+c) acc) im vns
  q'' = foldl' (\acc (b,c) -> push (d+c) b acc) q' vns
  vns = filter (\(b,c) -> M.notMember b im || d+c < im!b) (M.findWithDefault [] v g)