Problem:
Score : 600 points

Problem Statement

On the xy-plane, Snuke is going to travel from the point (x_s, y_s) to the point (x_t, y_t).
He can move in arbitrary directions with speed 1.
Here, we will consider him as a point without size.

There are N circular barriers deployed on the plane.
The center and the radius of the i-th barrier are (x_i, y_i) and r_i, respectively.
The barriers may overlap or contain each other.

A point on the plane is exposed to cosmic rays if the point is not within any of the barriers.

Snuke wants to avoid exposure to cosmic rays as much as possible during the travel.
Find the minimum possible duration of time he is exposed to cosmic rays during the travel.

Constraints

All input values are integers.

-10^9 ≤ x_s, y_s, x_t, y_t ≤ 10^9

(x_s, y_s) ≠ (x_t, y_t)

1≤N≤1,000

-10^9 ≤ x_i, y_i ≤ 10^9

1 ≤ r_i ≤ 10^9

Input

The input is given from Standard Input in the following format:

x_s y_s x_t y_t
N
x_1 y_1 r_1
x_2 y_2 r_2
:
x_N y_N r_N

Output

Print the minimum possible duration of time Snuke is exposed to cosmic rays during the travel.
The output is considered correct if the absolute or relative error is at most 10^{-9}.

Sample Input 1

-2 -2 2 2
1
0 0 1

Sample Output 1

3.6568542495

An optimal route is as follows:

Sample Input 2

-2 0 2 0
2
-1 0 2
1 0 2

Sample Output 2

0.0000000000

An optimal route is as follows:

Sample Input 3

4 -2 -2 4
3
0 0 2
4 0 1
0 4 1

Sample Output 3

4.0000000000

An optimal route is as follows:

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

{-# LANGUAGE BangPatterns #-}
import Control.Arrow
import Control.Monad.Fix
import Data.Foldable
import qualified Data.Vector as V
import qualified Data.Vector.Mutable as VMM
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VM
import qualified Data.ByteString.Char8 as B
import Data.ByteString.Internal (accursedUnutterablePerformIO)
import qualified Data.Set as S
import System.IO.Unsafe
import Debug.Trace

uread :: VU.Unbox a => Int -> VM.IOVector a -> a
uread !i !v = accursedUnutterablePerformIO $ VM.unsafeRead v i

uwrite :: VU.Unbox a => Int -> a -> VM.IOVector a -> VM.IOVector a
uwrite !i !a !v = accursedUnutterablePerformIO $ VM.unsafeWrite v i a >> return v

umodify :: VU.Unbox a => Int -> (a -> a) -> VM.IOVector a -> VM.IOVector a
umodify !i !f !v = accursedUnutterablePerformIO $ VM.unsafeModify v f i >> return v

uthaw :: VU.Unbox a => VU.Vector a -> VM.IOVector a
uthaw !v = accursedUnutterablePerformIO $ VU.unsafeThaw v

ufreeze :: VU.Unbox a => VM.IOVector a -> VU.Vector a
ufreeze !v = accursedUnutterablePerformIO $ VU.unsafeFreeze v

dijkstra
  :: Int  -- ^ nodes
  -> V.Vector (VU.Vector (Int, Double))  -- ^ edges from i-th node: (end, cost)
  -> Int  -- ^ start node
  -> VU.Vector Double  -- ^ constructed graph
dijkstra !n !edges !s = ufreeze $ fst $ fix
  ( \f (costs, visited) -> case findMinView costs visited of
    (-1) -> (costs,visited)
    n ->
      let visited' = uwrite n True visited
          c = uread n costs
      in if c <= uread n costs
          then f $ flip (,) visited' $ VU.foldl'
            ( \cs (e,ec) ->
              let c' = c + ec in 
                if c' < uread e costs
                  then uwrite e c' cs
                  else cs
            )
            costs
            (edges V.! n)
          else f (costs, visited')
  )
  (costs0, visited0)
 where
  costs0 :: VM.IOVector Double
  costs0 = uwrite s 0 (uthaw $ VU.replicate n (2^60))

  visited0 :: VM.IOVector Bool
  visited0 = uthaw $ VU.replicate n False

  findMinView :: VM.IOVector Double -> VM.IOVector Bool -> Int
  findMinView vec visited = fst $ foldl' (\(j,acc) i -> let a = uread i vec in if a < acc then (i,a) else (j,acc)) (-1,fromIntegral (maxBound :: Int)) $ filter (\i -> uread i visited == False) [0..n-1]

solve :: Int -> Int -> Int -> Int -> Int -> VU.Vector (Int, Int, Int) -> Double
solve xs ys xt yt n !xyr = graph VU.! (n + 1)
 where
  big = fromIntegral (maxBound :: Int)

  dist2C :: (Int, Int, Int) -> (Int, Int, Int) -> Double
  dist2C (x1, y1, r1) (x2, y2, r2) =
    ( sqrt (fromIntegral $ (x1 - x2) ^ 2 + (y1 - y2) ^ 2)
      - fromIntegral (r1 + r2)
      )
      `max` 0

  toIndex (!i,!j)
    | i <= j = i * (n + 2) + j
    | otherwise = toIndex (j,i)

  distMemo = ufreeze $ foldl'
    ( \acc (i, j) ->
      let c1 = if i == n
            then (xs, ys, 0)
            else if i == n + 1 then (xt, yt, 0) else xyr VU.! i
          c2 = if j == n
            then (xs, ys, 0)
            else if j == n + 1 then (xt, yt, 0) else xyr VU.! j
      in  uwrite (toIndex (i, j)) (dist2C c1 c2) acc
    )
    (uthaw $ VU.replicate ((n + 2) * (n + 2)) big)
    [ (i, j) | i <- [0 .. n + 1], j <- [i+1 .. n + 1] ]
  edges = accursedUnutterablePerformIO $ V.generateM
    (n + 2)
    ( \e -> return $ VU.map (\i -> (i, distMemo VU.! toIndex (i, e)))
      $ VU.fromList [0 .. n + 1]
    )
  graph = dijkstra (n + 2) edges n

main = do
  let readInt = fmap (second B.tail) . B.readInt

  (xs, ys, xt, yt) <-
    (\vec -> (vec VU.! 0, vec VU.! 1, vec VU.! 2, vec VU.! 3))
    .   VU.unfoldrN 4 readInt
    <$> B.getLine
  n   <- (\vec -> (vec VU.! 0)) . VU.unfoldrN 1 readInt <$> B.getLine
  xyr <-
    VU.replicateM n
    $   (\vec -> (vec VU.! 0, vec VU.! 1, vec VU.! 2))
    .   VU.unfoldrN 3 readInt
    <$> B.getLine
  print $ solve xs ys xt yt n xyr