Problem:
Score : 500 points

Problem Statement

You are given a sequence (P_1,P_2,...,P_N) which is a permutation of the integers from 1 through N.
You would like to sort this sequence in ascending order by repeating the following operation:

Choose an element in the sequence and move it to the beginning or the end of the sequence.

Find the minimum number of operations required.
It can be proved that it is actually possible to sort the sequence using this operation.

Constraints

1 \leq N \leq 2\times 10^5

(P_1,P_2,...,P_N) is a permutation of (1,2,...,N).

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
P_1
:
P_N

Output

Print the minimum number of operations required.

Sample Input 1

4
1
3
2
4

Sample Output 1

2

For example, the sequence can be sorted in ascending order as follows:

Move 2 to the beginning. The sequence is now (2,1,3,4).

Move 1 to the beginning. The sequence is now (1,2,3,4).

Sample Input 2

6
3
2
5
1
4
6

Sample Output 2

4

Sample Input 3

8
6
3
1
2
7
4
8
5

Sample Output 3

5

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

{-# LANGUAGE FlexibleContexts #-}

import Control.Monad
import Control.Applicative
import Data.List
import Data.Array
import Data.Array.IO
import Data.Maybe
import qualified Data.ByteString.Char8 as BS


readIntBS = fst . fromJust . BS.readInt

main = do
  n <- readLn :: IO Int
  ps <- map readIntBS . BS.lines <$> BS.getContents :: IO [Int]

  arr <- newArray (1,n) 0 :: IO (IOUArray Int Int)
  asc <- newArray (1,n) 0 :: IO (IOUArray Int Int)
  dsc <- newArray (1,n) 0 :: IO (IOUArray Int Int)

  forM_ (zip [1..] ps) $ \(i,p) -> do
    writeArray arr p i

  let
    aux1 i a = do
      when (i <= n) $ do
        x <- readArray arr i
        y <- readArray arr (i-1)
        if y < x then
          writeArray asc i a >> aux1 (i+1) a
          else
          writeArray asc i i >> aux1 (i+1) i

    aux2 i a = do
      when (i >= 1) $ do
        x <- readArray arr i
        y <- readArray arr (i+1)
        if y > x then
          writeArray dsc i a >> aux2 (i-1) a
          else
          writeArray dsc i i >> aux2 (i-1) i
          

  writeArray asc 1 1
  aux1 2 1

  writeArray dsc n n
  aux2 (n-1) n

  print =<< (n -) . maximum <$> (zipWith (\x y -> y-x+1) <$> getElems asc <*> getElems dsc)