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), S.Set Vertex)
nullq = S.null . fst
emp = (S.empty, S.empty)
push c v (q1,q2)
 | S.member v q2 = (q1,q2)
 | otherwise = (S.insert (c,v) q1, S.insert v q2)
pop (q1,q2) = let ((d,v),q) = S.deleteFindMin q1 in ((d,v), (q, S.delete v q2))

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

main = do
 (ps:es) <- map (map (fst . fromJust . B.readInt) . B.words) . B.lines <$> B.getContents
 let [nv,ne,v0] = ps
 let ges = groupBy (\e1 e2 -> head e1 == head e2) (sort es)
 let g = M.fromDistinctAscList $ map (\x -> ((head (head x)), map (\[_,t,d]->(t,d)) x)) ges
 mapM_ (putStrLn . maybe "INF" show . sumOfWeight g nv ne v0 es) [0..nv-1]

sumOfWeight g nv ne v0 es v = M.lookup v imr
 where
  (imr,_) = until (nullq . snd) (dijkstra g) (im0,q0)
  im0 = M.singleton v0 0
  q0 = push 0 v0 emp

dijkstra g (im,q) = (im',q'')
 where
  ((_,v),q') = pop q
  d = im!v
  im' = foldl' (\acc (b,c) -> M.insertWith min 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)