Problem:
Score : 400 points

Problem Statement

There are N pieces of sushi. Each piece has two parameters: "kind of topping" t_i and "deliciousness" d_i.
You are choosing K among these N pieces to eat.
Your "satisfaction" here will be calculated as follows:

The satisfaction is the sum of the "base total deliciousness" and the "variety bonus".

The base total deliciousness is the sum of the deliciousness of the pieces you eat.

The variety bonus is x*x, where x is the number of different kinds of toppings of the pieces you eat.

You want to have as much satisfaction as possible.
Find this maximum satisfaction.

Constraints

1 \leq K \leq N \leq 10^5

1 \leq t_i \leq N

1 \leq d_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
t_1 d_1
t_2 d_2
.
.
.
t_N d_N

Output

Print the maximum satisfaction that you can obtain.

Sample Input 1

5 3
1 9
1 7
2 6
2 5
3 1

Sample Output 1

26

If you eat Sushi 1,2 and 3:

The base total deliciousness is 9+7+6=22.

The variety bonus is 2*2=4.

Thus, your satisfaction will be 26, which is optimal.

Sample Input 2

7 4
1 1
2 1
3 1
4 6
4 5
4 5
4 5

Sample Output 2

25

It is optimal to eat Sushi 1,2,3 and 4.

Sample Input 3

6 5
5 1000000000
2 990000000
3 980000000
6 970000000
6 960000000
4 950000000

Sample Output 3

4900000016

Note that the output may not fit into a 32-bit integer type.

Solution in OCaml (4.02.3) (judge verdict: Time Limit Exceeded)

let rv f p q = f q p
module MultiSet_test = struct
  module Make (X:Map.OrderedType) = struct
    include Map.Make(X)
    let count v set =
      try find v set with Not_found -> 0
    let remove_each v set =
      let k = count v set in
      if k<=0 then raise Not_found
      else if k=1 then remove v set
      else add v (k-1) set
    let add_each v set =
      let k = count v set in
      if k=0 then add v 1 set
      else add v (k+1) set
    let iter_each f set =
      let rec f0 k = function
      | 0 -> ()
      | i -> let () = f k in f0 k (i-1)
      in iter f0 set
    let fold_each f set ac =
      let rec f0 k i ac = match i with
      | 0 -> ac
      | i -> f0 k (i-1) (f k ac)
      in fold f0 set ac
    let bindings_each set =
      fold_each (fun v ls -> v::ls) set []
      |> List.rev
    let min_binding_each set = fst @@ min_binding set
    let max_binding_each set = fst @@ max_binding set
    let exists_each f = exists (fun k _ -> f k)
    let cardinal_each set = (* slow *)
      fold (fun _ v sum -> sum + v) set 0
  end
end
module IRSet = MultiSet_test.Make (struct type t = int let compare = rv compare end);;
Scanf.(Array.(scanf " %d %d" @@ fun n k ->
  let t,d = make n 0,make n 0 in
  let repr = make n 0 in
  init n (fun i -> scanf " %d %d" @@ fun u v ->
    t.(i) <- u-1; d.(i) <- v;
    repr.(t.(i)) <- max v repr.(t.(i))
  ) |> ignore;
  let rest = init n (fun _ -> IRSet.empty) in
  init n (fun i ->
    (* if repr.(t.(i)) <> d.(i) then *)
      rest.(t.(i)) <- IRSet.add_each d.(i) rest.(t.(i))
  ) |> ignore;
  iteri (fun i v ->
    if v>0 then rest.(i) <- IRSet.remove_each v rest.(i)
  ) repr;
  (* iteri (fun i set ->
    Printf.printf "%d: " @@ i+1; IRSet.iter_each (Printf.printf "%2d ") set;
    print_newline ()
  ) rest;
  iteri (fun i v ->
    Printf.printf "%d:%d\n" (i+1) v;
  ) repr; *)
  let repr = to_list repr
  |> List.mapi (fun i v -> v,i)
  |> List.filter (fun (v,i) -> v <> 0)
  |> List.sort (rv compare)
  in
  (* List.iter (fun (v,i) -> Printf.printf "%d:%d; " v @@ i+1) repr; print_newline (); *)
  let _,_,_,res,_ = init n (fun w -> w+1)
  |> fold_left (fun ((best,sum_best,repr,res,car_best) as u) w -> match repr with
    | [] -> u
    | (v,i)::t -> (
      (* Printf.printf "rest.(%d) :: " @@ i+1;
      IRSet.iter_each (Printf.printf "%d ") rest.(i); print_newline (); *)
      let best = IRSet.merge (fun i p q -> match p,q with
      | Some p,Some q -> Some (p+q)
      | Some p,_ | _,Some p -> Some p
      | None,None -> None
      ) best rest.(i) in
      let car_best = car_best + IRSet.cardinal_each rest.(i) in
      (* Printf.printf "car_best=%2d; i=%d\n" car_best @@ i+1; *)
      rest.(i) <- IRSet.empty;
      (* Printf.printf "%d :: " (k-w);
      IRSet.iter_each (Printf.printf "%d ") best; print_newline (); *)
      if k - w < 0 || car_best < k - w then (
        (best,sum_best+v,t,res,car_best)
      ) else
      let u = ref @@ IRSet.empty in
      (* Printf.printf "take: %d\n" (k-w); *)
      let best,sum = init (k-w) (fun i -> i)
      |> fold_left (fun (set,sum) _ ->
        let v = IRSet.min_binding_each set in
        u := IRSet.add_each v !u;
        (IRSet.remove_each v set,sum+v)
        (* (set,sum+v) *)
      ) (best,0)
      in
      let best = IRSet.merge (fun i p q -> match p,q with
      | Some p,Some q -> Some (p+q)
      | Some p,_ | _,Some p -> Some p
      | None,None -> None
      ) best !u in
      (* Printf.printf "  %6d v. %6d :: %2d + %2d + %2d\n"
        res
        (w*w + sum + sum_best + v)
        (w*w)
        sum
        (sum_best + v); *)
      (best,sum_best+v,t,max res (w*w + sum + sum_best + v),car_best)
    )
  ) (IRSet.empty,0,repr,0,0) in
  print_int res
));;