Problem:
Score : 500 points

Problem Statement

There are N cities numbered 1 to N, connected by M railroads.

You are now at City 1, with 10^{100} gold coins and S silver coins in your pocket.

The i-th railroad connects City U_i and City V_i bidirectionally, and a one-way trip costs A_i silver coins and takes B_i minutes.
You cannot use gold coins to pay the fare.

There is an exchange counter in each city. At the exchange counter in City i, you can get C_i silver coins for 1 gold coin.
The transaction takes D_i minutes for each gold coin you give.
You can exchange any number of gold coins at each exchange counter.

For each t=2, ..., N, find the minimum time needed to travel from City 1 to City t. You can ignore the time spent waiting for trains.

Constraints

2 \leq N \leq 50

N-1 \leq M \leq 100

0 \leq S \leq 10^9

1 \leq A_i \leq 50

1 \leq B_i,C_i,D_i \leq 10^9

1 \leq U_i < V_i \leq N

There is no pair i, j(i \neq j) such that (U_i,V_i)=(U_j,V_j).

Each city t=2,...,N can be reached from City 1 with some number of railroads.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M S
U_1 V_1 A_1 B_1
:
U_M V_M A_M B_M
C_1 D_1
:
C_N D_N

Output

For each t=2, ..., N in this order, print a line containing the minimum time needed to travel from City 1 to City t.

Sample Input 1

3 2 1
1 2 1 2
1 3 2 4
1 11
1 2
2 5

Sample Output 1

2
14

The railway network in this input is shown in the figure below.

In this figure, each city is labeled as follows:

The first line: the ID number i of the city (i for City i)

The second line: C_i / D_i

Similarly, each railroad is labeled as follows:

The first line: the ID number i of the railroad (i for the i-th railroad in input)

The second line: A_i / B_i

You can travel from City 1 to City 2 in 2 minutes, as follows:

Use the 1-st railroad to move from City 1 to City 2 in 2 minutes.

You can travel from City 1 to City 3 in 14 minutes, as follows:

Use the 1-st railroad to move from City 1 to City 2 in 2 minutes.

At the exchange counter in City 2, exchange 3 gold coins for 3 silver coins in 6 minutes.

Use the 1-st railroad to move from City 2 to City 1 in 2 minutes.

Use the 2-nd railroad to move from City 1 to City 3 in 4 minutes.

Sample Input 2

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

Sample Output 2

5
5
7

The railway network in this input is shown in the figure below:

You can travel from City 1 to City 4 in 7 minutes, as follows:

At the exchange counter in City 1, exchange 2 gold coins for 6 silver coins in 2 minutes.

Use the 2-nd railroad to move from City 1 to City 3 in 4 minutes.

Use the 4-th railroad to move from City 3 to City 4 in 1 minutes.

Sample Input 3

6 5 1
1 2 1 1
1 3 2 1
2 4 5 1
3 5 11 1
1 6 50 1
1 10000
1 3000
1 700
1 100
1 1
100 1

Sample Output 3

1
9003
14606
16510
16576

The railway network in this input is shown in the figure below:

You can travel from City 1 to City 6 in 16576 minutes, as follows:

Use the 1-st railroad to move from City 1 to City 2 in 1 minute.

At the exchange counter in City 2, exchange 3 gold coins for 3 silver coins in 9000 minutes.

Use the 1-st railroad to move from City 2 to City 1 in 1 minute.

Use the 2-nd railroad to move from City 1 to City 3 in 1 minute.

At the exchange counter in City 3, exchange 8 gold coins for 8 silver coins in 5600 minutes.

Use the 2-nd railroad to move from City 3 to City 1 in 1 minute.

Use the 1-st railroad to move from City 1 to City 2 in 1 minute.

Use the 3-rd railroad to move from City 2 to City 4 in 1 minute.

At the exchange counter in City 4, exchange 19 gold coins for 19 silver coins in 1900 minutes.

Use the 3-rd railroad to move from City 4 to City 2 in 1 minute.

Use the 1-st railroad to move from City 2 to City 1 in 1 minute.

Use the 2-nd railroad to move from City 1 to City 3 in 1 minute.

Use the 4-th railroad to move from City 3 to City 5 in 1 minute.

At the exchange counter in City 5, exchange 63 gold coins for 63 silver coins in 63 minutes.

Use the 4-th railroad to move from City 5 to City 3 in 1 minute.

Use the 2-nd railroad to move from City 3 to City 1 in 1 minute.

Use the 5-th railroad to move from City 1 to City 6 in 1 minute.

Sample Input 4

4 6 1000000000
1 2 50 1
1 3 50 5
1 4 50 7
2 3 50 2
2 4 50 4
3 4 50 3
10 2
4 4
5 5
7 7

Sample Output 4

1
3
5

The railway network in this input is shown in the figure below:

Sample Input 5

2 1 0
1 2 1 1
1 1000000000
1 1

Sample Output 5

1000000001

The railway network in this input is shown in the figure below:

You can travel from City 1 to City 2 in 1000000001 minutes, as follows:

At the exchange counter in City 1, exchange 1 gold coin for 1 silver coin in 1000000000 minutes.

Use the 1-st railroad to move from City 1 to City 2 in 1 minute.

Solution in OCaml (4.10.0) (judge verdict: Accepted)

module WeightedDirectedGraph
  (* 辺の重み *)
  (Weight : sig
    type t
    val zero : t
    val compare : t -> t -> int
  end)
  (* 経路長を優先度としたヒープの実装 *)
  (Heap : sig
    type t
    type elt (* 頂点に相当 *)
    type key = Weight.t (* 経路長に相当 *)
    (* ヒープが空ならNoneを，
       そうでなければ経路長が最小となるbindingを一つ以上返す
       返したbindingは全てヒープから削除される *)
    val take_min_bindings : t -> (key * elt list) option
    (* ヒープにbindingを追加する
       既に同じ頂点についてのbindingが追加されていたら，
       経路長の短い方だけを残しても良いし，何も考えずに追加してもよい．
       前者の実装ならダイクストラ法の実装が時間計算量O((V + E) log V)，
       空間計算量O(V)に改善する *)
    val add : t -> key -> elt -> unit
  end)
  (* 頂点を添字，経路長を要素とした配列の実装 *)
  (Array : sig
    type t
    type key = Heap.elt
    type elt = Heap.key
    val get : t -> key -> elt
    val set : t -> key -> elt -> unit
  end)
: sig
  type weight = Array.elt
  type vertex = Array.key

  (* ダイクストラ法で最短経路を求める関数 *)
  val shortest_path :
    (* 空なヒープ *)
    Heap.t ->
    (* 全ての頂点についての経路長が無限大で初期化された配列
       Array.getした時に極端に大きな経路長が返ってくる実装でも良いし，
       Not_foundが投げられる実装でも良い *)
    Array.t ->
    (* 最短経路を求めたいグラフの，ある頂点から伸びる辺に対してのイテレータ *)
    (vertex -> (vertex -> (weight -> weight) (* 辺を通った際のコストを加算する関数 *) -> unit) -> unit) ->
    (* 始点 *)
    vertex ->
    (* 終点を受け取って，始点からの最短距離を返す関数
       始点から辿り着けない場合，無限大を返すかNot_foundを投げる（Arrayの実装依存）
       この関数を覚えておけば，呼び出しごとの途中までの計算結果がシェアされる *)
    (vertex -> weight)
end
= struct
  type weight = Array.elt
  type vertex = Array.key

  let shortest_path q d es s =
    (* 始点への経路長を0にする *)
    Array.set d s Weight.zero;
    (* 既に最短距離が確定した辺へのクエリを高速化するため，
       ヒープの最小要素をメモしておく *)
    let min_bindings = ref (Some (Weight.zero, [s])) in
    let rec dijkstra t =
      match !min_bindings with
      (* もう既に全ての頂点までの距離が分かっている *)
      | None -> Array.get d t
      | Some (w, us) ->
          match Array.get d t with
          (* 既に終点までの距離が分かっているので返す *)
          | x when 0 <= Weight.compare w x -> x
          (* 終点までの距離が分かっていないので，ダイクストラ法を続行 *)
          | _ | exception Not_found ->
              List.iter (fun u ->
                (* 未だ頂点uを訪れていない *)
                if 0 <= Weight.compare (Array.get d u) w then
                  es u @@ fun v f ->
                    (* uからvに伸びる辺を通った際の経路長 *)
                    let c = f w in
                    if 0 < Weight.compare (Array.get d v) c then
                      (Heap.add q c v; Array.set d v c)) us;
              min_bindings := Heap.take_min_bindings q;
              dijkstra t in
    dijkstra
end

module PrioritySearchQueue = struct
module type OrderedType = sig
  type t
  val compare : t -> t -> int
end

module Make (Key : OrderedType) (Priority : OrderedType)
: sig
  type key = Key.t
  type priority = Priority.t

  type t
  val is_empty : t -> bool
  val cardinal : t -> int
  val find_opt : key -> t -> priority option
  val mem : key -> t -> bool
  val min_binding_opt : t -> (key * priority) option
  val empty : t
  val singleton : key -> priority -> t
  val add : key -> priority -> t -> t
  val update : key -> (priority option -> priority option) -> t -> t
  val remove : key -> t -> t
  val remove_min_binding : t -> t
  val of_list : (key * priority) list -> t
  val fold : (key -> priority -> 'a -> 'a) -> t -> 'a -> 'a
  val bindings : t -> (key * priority) list
end = struct
  type key = Key.t
  type priority = Priority.t

  type loser_tree =
    | Start
    | LLoser of int * Key.t * Priority.t * loser_tree * Key.t * loser_tree
    | RLoser of int * Key.t * Priority.t * loser_tree * Key.t * loser_tree

  type t =
    | Void
    | Winner of Key.t * Priority.t * loser_tree * Key.t

  let is_empty = function
    | Void -> true
    | Winner (_, _, _, _) -> false

  let cardinal' = function
    | Start -> 0
    | LLoser (s, _, _, _, _, _)
    | RLoser (s, _, _, _, _, _) -> s

  let cardinal = function
    | Void -> 0
    | Winner (_, _, t, _) -> cardinal' t

  let omega = 4

  let left = function
    | Start -> raise (Invalid_argument "left")
    | LLoser (_, _, _, l, _, _)
    | RLoser (_, _, _, l, _, _) -> l

  let right = function
    | Start -> raise (Invalid_argument "right")
    | LLoser (_, _, _, _, _, r)
    | RLoser (_, _, _, _, _, r) -> r

  let max_key = function
    | Void -> raise (Invalid_argument "max_key")
    | Winner (_, _, _, m) -> m

  let lloser k p l m r = LLoser (1 + cardinal' l + cardinal' r, k, p, l, m, r)
  let rloser k p l m r = RLoser (1 + cardinal' l + cardinal' r, k, p, l, m, r)

  let lsingle_left k1 p1 t1 m1 = function
    | Start -> raise (Invalid_argument "lsingle_left")
    | LLoser (_, k2, p2, t2, m2, t3) ->
        if 0 <= Priority.compare p2 p1
        then lloser k1 p1 (rloser k2 p2 t1 m1 t2) m2 t3
        else lloser k2 p2 (lloser k1 p1 t1 m1 t2) m2 t3
    | RLoser (_, k2, p2, t2, m2, t3) ->
        rloser k2 p2 (lloser k1 p1 t1 m1 t2) m2 t3

  let rsingle_left k1 p1 t1 m1 = function
    | Start -> raise (Invalid_argument "rsingle_left")
    | LLoser (_, k2, p2, t2, m2, t3) ->
        rloser k1 p1 (rloser k2 p2 t1 m1 t2) m2 t3
    | RLoser (_, k2, p2, t2, m2, t3) ->
        rloser k2 p2 (rloser k1 p1 t1 m1 t2) m2 t3

  let lsingle_right k1 p1 t1 m2 t3 =
    match t1 with
    | Start -> raise (Invalid_argument "lsingle_right")
    | LLoser (_, k2, p2, t1, m1, t2) ->
        lloser k2 p2 t1 m1 (lloser k1 p1 t2 m2 t3)
    | RLoser (_, k2, p2, t1, m1, t2) ->
        lloser k1 p1 t1 m1 (lloser k2 p2 t2 m2 t3)

  let rsingle_right k1 p1 t1 m2 t3 =
    match t1 with
    | Start -> raise (Invalid_argument "rsingle_right")
    | LLoser (_, k2, p2, t1, m1, t2) ->
        lloser k2 p2 t1 m1 (rloser k1 p1 t2 m2 t3)
    | RLoser (_, k2, p2, t1, m1, t2) ->
        if 0 <= Priority.compare p2 p1
        then rloser k1 p1 t1 m1 (lloser k2 p2 t2 m2 t3)
        else rloser k2 p2 t1 m1 (rloser k1 p1 t2 m2 t3)

  let ldouble_left k1 p1 t1 m1 = function
    | Start -> raise (Invalid_argument "ldouble_left")
    | LLoser (_, k2, p2, t2, m2, t3) ->
        lsingle_left k1 p1 t1 m1 (lsingle_right k2 p2 t2 m2 t3)
    | RLoser (_, k2, p2, t2, m2, t3) ->
        lsingle_left k1 p1 t1 m1 (rsingle_right k2 p2 t2 m2 t3)

  let ldouble_right k1 p1 t1 m2 t3 =
    match t1 with
    | Start -> raise (Invalid_argument "ldouble_right")
    | LLoser (_, k2, p2, t1, m1, t2) ->
        lsingle_right k1 p1 (lsingle_left k2 p2 t1 m1 t2) m2 t3
    | RLoser (_, k2, p2, t1, m1, t2) ->
        lsingle_right k1 p1 (rsingle_left k2 p2 t1 m1 t2) m2 t3

  let rdouble_left k1 p1 t1 m1 = function
    | Start -> raise (Invalid_argument "rdouble_left")
    | LLoser (_, k2, p2, t2, m2, t3) ->
        rsingle_left k1 p1 t1 m1 (lsingle_right k2 p2 t2 m2 t3)
    | RLoser (_, k2, p2, t2, m2, t3) ->
        rsingle_left k1 p1 t1 m1 (rsingle_right k2 p2 t2 m2 t3)

  let rdouble_right k1 p1 t1 m2 t3 =
    match t1 with
    | Start -> raise (Invalid_argument "rdouble_right")
    | LLoser (_, k2, p2, t1, m1, t2) ->
        rsingle_right k1 p1 (lsingle_left k2 p2 t1 m1 t2) m2 t3
    | RLoser (_, k2, p2, t1, m1, t2) ->
        rsingle_right k1 p1 (rsingle_left k2 p2 t1 m1 t2) m2 t3

  let lbalance_left k p l m r =
    if cardinal' (left r) < cardinal' (right r)
    then lsingle_left k p l m r
    else ldouble_left k p l m r

  let lbalance_right k p l m r =
    if cardinal' (left l) > cardinal' (right l)
    then lsingle_right k p l m r
    else ldouble_right k p l m r

  let rbalance_left k p l m r =
    if cardinal' (left r) < cardinal' (right r)
    then rsingle_left k p l m r
    else rdouble_left k p l m r

  let rbalance_right k p l m r =
    if cardinal' (left l) > cardinal' (right l)
    then rsingle_right k p l m r
    else rdouble_right k p l m r

  let lbalance k p l m r =
    if cardinal' l + cardinal' r < 2
    then lloser k p l m r
    else if cardinal' r > omega * cardinal' l
    then lbalance_left k p l m r
    else if cardinal' l > omega * cardinal' r
    then lbalance_right k p l m r
    else lloser k p l m r

  let rbalance k p l m r =
    if cardinal' l + cardinal' r < 2
    then rloser k p l m r
    else if cardinal' r > omega * cardinal' l
    then rbalance_left k p l m r
    else if cardinal' l > omega * cardinal' r
    then rbalance_right k p l m r
    else rloser k p l m r

  let rec play t t' =
    match t, t' with
    | Void, _ -> t'
    | _, Void -> t
    | Winner (k, p, t, m), Winner (k', p', t', m') ->
        if 0 <= Priority.compare p' p
        then Winner (k, p, rbalance k' p' t m t', m')
        else Winner (k', p', lbalance k p t m t', m')

  let find_opt k =
    let rec find_opt_aux k' p' = function
      | Start ->
          if Key.compare k k' = 0
          then Some p'
          else None
      | RLoser (_, k'', p'', l, m, r) ->
          if 0 <= Key.compare m k
          then find_opt_aux k' p' l
          else find_opt_aux k'' p'' r
      | LLoser (_, k'', p'', l, m, r) ->
          if 0 <= Key.compare m k
          then find_opt_aux k'' p'' l
          else find_opt_aux k' p' r in
    function
      | Void -> None
      | Winner (k', p', t, _) -> find_opt_aux k' p' t

  let mem k t = Option.is_some @@ find_opt k t

  let min_binding_opt = function
    | Void -> None
    | Winner (k, p, _, _) -> Some (k, p)

  let empty = Void

  let singleton k p = Winner (k, p, Start, k)

  let add k p =
    let rec add_aux k' p' m' = function
      | Start ->
          let cmp = Key.compare k k' in
          if cmp < 0
          then play (singleton k p) (singleton k' p')
          else if cmp = 0
          then singleton k p
          else play (singleton k' p') (singleton k p)
      | RLoser (_, k'', p'', l, m, r) ->
          if 0 <= Key.compare m k
          then play (add_aux k' p' m l) (Winner (k'', p'', r, m'))
          else play (Winner (k', p', l, m)) (add_aux k'' p'' m' r)
      | LLoser (_, k'', p'', l, m, r) ->
          if 0 <= Key.compare m k
          then play (add_aux k'' p'' m l) (Winner (k', p', r, m'))
          else play (Winner (k'', p'', l, m)) (add_aux k' p' m' r) in
    function
      | Void -> singleton k p
      | Winner (k', p', t, m) -> add_aux k' p' m t

  let update k f =
    let exception Not_modified in
    let rec update_aux k' p' m' = function
      | Start ->
          let cmp = Key.compare k k' in
          if cmp < 0
          then
            begin match f None with
            | None -> raise Not_modified
            | Some p -> play (singleton k p) (singleton k' p')
            end
          else if cmp = 0
          then
            begin match f (Some p') with
            | None -> empty
            | Some p -> if p == p' then raise Not_modified else singleton k p
            end
          else
            begin match f None with
            | None -> raise Not_modified
            | Some p -> play (singleton k' p') (singleton k p)
            end
      | RLoser (_, k'', p'', l, m, r) ->
          if 0 <= Key.compare m k
          then play (update_aux k' p' m l) (Winner (k'', p'', r, m'))
          else play (Winner (k', p', l, m)) (update_aux k'' p'' m' r)
      | LLoser (_, k'', p'', l, m, r) ->
          if 0 <= Key.compare m k
          then play (update_aux k'' p'' m l) (Winner (k', p', r, m'))
          else play (Winner (k'', p'', l, m)) (update_aux k' p' m' r) in
    function
      | Void as t0 ->
          begin match f None with
          | None -> t0
          | Some p -> singleton k p
          end
      | (Winner (k', p', t, m')) as t0 ->
          try update_aux k' p' m' t with Not_modified -> t0

  let remove k =
    let exception Not_modified in
    let rec remove_aux k' p' m' = function
      | Start ->
          if Key.compare k k' = 0
          then empty
          else raise Not_modified
      | RLoser (_, k'', p'', l, m, r) ->
          if 0 <= Key.compare m k
          then play (remove_aux k' p' m l) (Winner (k'', p'', r, m'))
          else play (Winner (k', p', l, m)) (remove_aux k'' p'' m' r)
      | LLoser (_, k'', p'', l, m, r) ->
          if 0 <= Key.compare m k
          then play (remove_aux k'' p'' m l) (Winner (k', p', r, m'))
          else play (Winner (k'', p'', l, m)) (remove_aux k' p' m' r) in
    function
      | Void as t0 -> t0
      | (Winner (k', p', t, m')) as t0 ->
          try remove_aux k' p' m' t with Not_modified -> t0

  let rec second_best m' = function
    | Start -> Void
    | LLoser (_, k, p, l, m, r) -> play (Winner (k, p, l, m)) (second_best m' r)
    | RLoser (_, k, p, l, m, r) -> play (second_best m l) (Winner (k, p, r, m'))

  let remove_min_binding = function
    | Void as t -> t
    | Winner (_, _, t, m) -> second_best m t

  let of_list = List.fold_left (fun t (k, p) -> add k p t) empty

  let fold f =
    let rec fold_aux k p acc = function
      | Start -> f k p acc
      | RLoser (_, k', p', l, _, r) ->
          fold_aux k p (fold_aux k' p' acc r) l
      | LLoser (_, k', p', l, _, r) ->
          fold_aux k' p' (fold_aux k p acc r) l in
    fun t acc ->
      match t with
      | Void -> acc
      | Winner (k, p, t, _) -> fold_aux k p acc t

  let bindings t = fold (fun k p -> List.cons (k, p)) t []
end

end

module Int = struct
  type t = int
  let zero = 0
  let compare = compare
end

module PSQ = PrioritySearchQueue.Make (struct
  type t = int array
  let compare = compare
end) (Int)

module G = WeightedDirectedGraph (Int)
  (struct
    type t = PSQ.t ref
    type elt = int array
    type key = int
    let take_min_bindings q =
      match PSQ.min_binding_opt !q with
      | None -> None
      | Some (v, w) -> q := PSQ.remove_min_binding !q; Some (w, [v])
    let add q w v = q := PSQ.add v w !q
  end)
  (struct
    type t = (int, Bigarray.int_elt, Bigarray.c_layout) Bigarray.Genarray.t
    type elt = int
    type key = int array
    let get = Bigarray.Genarray.get
    let set = Bigarray.Genarray.set
  end)

let () = Scanf.scanf "%d %d %d\n" @@ fun n m s ->
  let es = Array.make n [] in
  for i = 0 to m - 1 do
    Scanf.scanf "%d %d %d %d\n" @@ fun u v a b ->
      es.(u - 1) <- (v - 1, a, b) :: es.(u - 1);
      es.(v - 1) <- (u - 1, a, b) :: es.(v - 1);
  done;
  let vs = Array.init n @@ fun _ -> Scanf.scanf "%d %d\n" @@ fun c d -> (c, d) in
  let d =
    G.shortest_path (ref PSQ.empty)
      (let d = Bigarray.Genarray.create Bigarray.int Bigarray.c_layout [| n; 2501 |] in Bigarray.Genarray.fill d max_int; d)
      (fun [| v; w |] f ->
        List.iter (fun (u, a, b) ->
          if a <= w then
            f [| u; w - a |] (( + ) b)) es.(v);
        let (c, d) = vs.(v) in
        f [| v; min 2500 (w + c) |] (( + ) d))
      [| 0; min 2500 s |] in
  for v = 1 to n - 1 do
    Printf.printf "%d\n" @@
    Array.fold_left min max_int @@
    Array.init 2501 @@ fun w -> d [| v; w |]
  done