Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices. The i-th edge connects Vertex A_i and B_i bidirectionally.

Takahashi is standing at Vertex u, and Aoki is standing at Vertex v.

Now, they will play a game of tag as follows:

1. If Takahashi and Aoki are standing at the same vertex, the game ends. Otherwise, Takahashi moves to a vertex of his choice that is adjacent to his current vertex.

2. If Takahashi and Aoki are standing at the same vertex, the game ends. Otherwise, Aoki moves to a vertex of his choice that is adjacent to his current vertex.

3. Go back to step 1.

Takahashi performs his moves so that the game ends as late as possible, while Aoki performs his moves so that the game ends as early as possible.

Find the number of moves Aoki will perform before the end of the game if both Takahashi and Aoki know each other's position and strategy.

It can be proved that the game is bound to end.

Constraints

2 \leq N \leq 10^5

1 \leq u,v \leq N

u \neq v

1 \leq A_i,B_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N u v
A_1 B_1
:
A_{N-1} B_{N-1}

Output

Print the number of moves Aoki will perform before the end of the game.

Sample Input 1

5 4 1
1 2
2 3
3 4
3 5

Sample Output 1

2

If both players play optimally, the game will progress as follows:

Takahashi moves to Vertex 3.

Aoki moves to Vertex 2.

Takahashi moves to Vertex 5.

Aoki moves to Vertex 3.

Takahashi moves to Vertex 3.

Here, Aoki performs two moves.

Note that, in each move, it is prohibited to stay at the current vertex.

Sample Input 2

5 4 5
1 2
1 3
1 4
1 5

Sample Output 2

1

Sample Input 3

2 1 2
1 2

Sample Output 3

0

Sample Input 4

9 6 1
1 2
2 3
3 4
4 5
5 6
4 7
7 8
8 9

Sample Output 4

5

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

{-# LANGUAGE BangPatterns #-}

module Main
  ( main
  )
where
 
import           Control.Applicative
import           Control.Monad
import           Data.Bits
import qualified Data.ByteString.Char8         as BS
import           Data.Char
import           Data.List
import           Data.Maybe
import qualified Data.Vector                   as V
 
parse = fst . fromJust
parseInt = parse . BS.readInt
parseInteger = parse . BS.readInteger
readInt = parseInt <$> BS.getLine
readInteger = parseInteger <$> BS.getLine
readIntList = map parseInt . BS.words <$> BS.getLine
 
main = do
  [n, u, v] <- readIntList
  arr       <- concat <$> replicateM
    (n - 1)
    ((\[a, b] -> [(a - 1, b - 1), (b - 1, a - 1)]) <$> readIntList)
  let
    graph    = createUnweightedGraphFromList arr
    dist1    = treeDepthFirstSearch (u - 1) graph
    dist2    = treeDepthFirstSearch (v - 1) graph
    farthest = maximum . map snd . filter (\(a, b) -> a < b) $ zip dist1 dist2
  print $ farthest - 1
 
type Vertex = Int
type Cost = Int
type Capacity = Int
 
data Edge = Edge { from :: !Vertex, to :: !Vertex, cost :: !Cost, cap :: !Capacity } deriving (Eq, Ord, Show)
newUnweightedEdge (!from, !to) = Edge { from = from, to = to, cost = 1, cap = 0 }
createUnweightedEdgeVectorFromList :: [(Vertex, Vertex)] -> V.Vector Edge
createUnweightedEdgeVectorFromList = V.fromList . map newUnweightedEdge
 
type Graph = V.Vector (V.Vector Edge)
createUnweightedGraphFromList :: [(Vertex, Vertex)] -> Graph
createUnweightedGraphFromList =
  V.fromList
    . map createUnweightedEdgeVectorFromList
    . groupBy (\a b -> fst a == fst b)
    . sort
 
treeDepthFirstSearch :: Vertex -> Graph -> [Cost]
treeDepthFirstSearch src =
  map snd . sort . updateTreeDepthFirstSearch [(src, 0, src, 0)] []
 
updateTreeDepthFirstSearch
  :: [(Vertex, Int, Vertex, Cost)]
  -> [(Vertex, Cost)]
  -> Graph
  -> [(Vertex, Cost)]
updateTreeDepthFirstSearch (!(v, i, p, c) : xs) !visited !g =
  let edge  = (g V.! v) V.!? i
      stack = case edge of
        Nothing -> xs
        Just e | to e == p -> (v, i + 1, p, c) : xs
               | otherwise -> (to e, 0, v, c + cost e) : (v, i + 1, p, c) : xs
      newVisited = if i == 0 then (v, c) : visited else visited
  in  updateTreeDepthFirstSearch stack newVisited g
updateTreeDepthFirstSearch [] visited _ = visited