Problem:
Score : 500 points

Problem Statement

We have a two-dimensional grid with H \times W squares. There are M targets to destroy in this grid - the position of the i-th target is \left(h_i, w_i \right).

Takahashi will choose one square in this grid, place a bomb there, and ignite it. The bomb will destroy all targets that are in the row or the column where the bomb is placed. It is possible to place the bomb at a square with a target.

Takahashi is trying to maximize the number of targets to destroy. Find the maximum number of targets that can be destroyed.

Constraints

All values in input are integers.

1 \leq H, W \leq 3 \times 10^5

1 \leq M \leq \min\left(H\times W, 3 \times 10^5\right)

1 \leq h_i \leq H

1 \leq w_i \leq W

\left(h_i, w_i\right) \neq \left(h_j, w_j\right) \left(i \neq j\right)

Input

Input is given from Standard Input in the following format:

H W M
h_1 w_1
\vdots
h_M w_M

Output

Print the answer.

Sample Input 1

2 3 3
2 2
1 1
1 3

Sample Output 1

3

We can destroy all the targets by placing the bomb at \left(1, 2\right).

Sample Input 2

3 3 4
3 3
3 1
1 1
1 2

Sample Output 2

3

Sample Input 3

5 5 10
2 5
4 3
2 3
5 5
2 2
5 4
5 3
5 1
3 5
1 4

Sample Output 3

6

Solution in Haskell (GHC 8.8.3) (judge verdict: Accepted)

{-# LANGUAGE BangPatterns #-}
import           Control.Exception           (assert)
import           Control.Monad
import           Control.Monad.Primitive
import qualified Control.Monad.ST            as ST
import qualified Data.Array.IO               as IO
import qualified Data.Array.ST               as ST
import qualified Data.Array.Unboxed          as A
import           Data.Bits
import qualified Data.ByteString.Char8       as BS
import           Data.Char
import           Data.Foldable
import           Data.List
import qualified Data.Map.Strict             as Map
import           Data.Maybe
import qualified Data.Sequence               as Seq
import qualified Data.Set                    as Set
import qualified Data.Vector                 as VB
import qualified Data.Vector.Mutable         as VBM
import qualified Data.Vector.Unboxed         as V
import qualified Data.Vector.Unboxed.Mutable as VM
import           Debug.Trace

readInt = fst . fromJust . BS.readInt
readIntList = map readInt . BS.words
getInt = readInt <$> BS.getLine
getIntList = readIntList <$> BS.getLine
getIntVec n = V.unfoldrN n (BS.readInt . BS.dropWhile isSpace) <$> BS.getLine

readInteger = fst . fromJust . BS.readInteger
readIntegerList = map readInteger . BS.words
getInteger = readInteger <$> BS.getLine
getIntegerList = readIntegerList <$> BS.getLine

inf :: Int
inf = 10^18

main = do
  [h, w, m] <- getIntList
  dh <- VM.new h
  VM.set dh (0::Int)
  dw <- VM.new w
  VM.set dw (0::Int)

  let readInputs :: Int -> Set.Set (Int, Int) -> Int -> Int -> IO (Set.Set (Int, Int), Int, Int)
      readInputs i s my mx
        | i >= m = return (s, my, mx)
        | otherwise = do
            [h', w'] <- getIntList
            let h = h' - 1
                w = w' - 1
                s' = (h, w) `Set.insert` s
            ty <- VM.read dh h
            VM.write dh h (ty + 1)
            tx <- VM.read dw w
            VM.write dw w (tx + 1)
            let my' = max (ty + 1) my
                mx' = max (tx + 1) mx
            readInputs (i+1) s' my' mx'

  (s, my, mx) <- readInputs 0 (Set.empty) 0 0

  let getIs i j d mx is | i >= j = return is
                        | otherwise = do
                            t <- VM.read d i
                            let is' | t == mx = (i:is)
                                    | otherwise = is
                            getIs (i+1) j d mx is'

  ys <- getIs 0 h dh my []
  xs <- getIs 0 w dw mx []

  let found = or [not $ (y, x) `Set.member` s | y <- ys, x <- xs]
      d | found = 0
        | otherwise = 1

  print $ my + mx - d