Problem:
Score : 300 points

Problem Statement

We have two desks: A and B. Desk A has a vertical stack of N books on it, and Desk B similarly has M books on it.

It takes us A_i minutes to read the i-th book from the top on Desk A (1 \leq i \leq N), and B_i minutes to read the i-th book from the top on Desk B (1 \leq i \leq M).

Consider the following action:

Choose a desk with a book remaining, read the topmost book on that desk, and remove it from the desk.

How many books can we read at most by repeating this action so that it takes us at most K minutes in total? We ignore the time it takes to do anything other than reading.

Constraints

1 \leq N, M \leq 200000

1 \leq K \leq 10^9

1 \leq A_i, B_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M K
A_1 A_2 \ldots A_N
B_1 B_2 \ldots B_M

Output

Print an integer representing the maximum number of books that can be read.

Sample Input 1

3 4 240
60 90 120
80 150 80 150

Sample Output 1

3

In this case, it takes us 60, 90, 120 minutes to read the 1-st, 2-nd, 3-rd books from the top on Desk A, and 80, 150, 80, 150 minutes to read the 1-st, 2-nd, 3-rd, 4-th books from the top on Desk B, respectively.

We can read three books in 230 minutes, as shown below, and this is the maximum number of books we can read within 240 minutes.

Read the topmost book on Desk A in 60 minutes, and remove that book from the desk.

Read the topmost book on Desk B in 80 minutes, and remove that book from the desk.

Read the topmost book on Desk A in 90 minutes, and remove that book from the desk.

Sample Input 2

3 4 730
60 90 120
80 150 80 150

Sample Output 2

7

Sample Input 3

5 4 1
1000000000 1000000000 1000000000 1000000000 1000000000
1000000000 1000000000 1000000000 1000000000

Sample Output 3

0

Watch out for integer overflows.

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

open Batteries
open Printf

let ($) g f x = g (f x)

let swap = Tuple2.swap

let s2i = int_of_string
let i2s = string_of_int
let s2f = float_of_string
let f2s = string_of_float
let c2i = int_of_char
let i2c = char_of_int
let f2i = int_of_float
let i2f = float_of_int 

let c2s = String.of_char
let s2cl = String.to_list
let cl2s = String.of_list

let puts s = print_endline s
let putyn b = puts (if b then "Yes" else "No")
let puti i = puts @@ i2s i

let rep lb bnd ub f = match bnd with
  | `To -> for i = lb to ub do f i done
  | `Til -> for i = lb to (ub - 1) do f i done

let rep2 lb1 bnd1 ub1 () lb2 bnd2 ub2 f = 
  match (bnd1, bnd2) with
  | (`To, `To) -> for i = lb1 to ub1 do for j = lb2 to ub2 do f i j done done
  | (`To, `Til) -> for i = lb1 to ub1 do for j = lb2 to (ub2 - 1) do f i j done done
  | (`Til, `To) -> for i = lb1 to (ub1 - 1) do for j = lb2 to ub2 do f i j done done
  | (`Til, `Til) -> for i = lb1 to (ub1 - 1) do for j = lb2 to (ub2 - 1) do f i j done done

module JunkList = struct
  include List

  let nmem x l = not @@ mem x l

  let len = length
  let size = length
  let nempty l = not @@ is_empty l

  let hdop l = if nempty l then Some (hd l) else None

  (* let taker = drop *)
  let takewh = take_while

  (* let dropr = take *)
  let dropwh = drop_while

  let split_by = span

  let chunk = ntake

  let foldl = fold_left
  let foldli = fold_lefti
  let foldr f z l = fold_right f l z
  let foldri f z l = fold_righti f l z

  let partition_to_map f l =
    foldr (fun x m -> Map.modify_opt (f x) (function
      | None -> Some [x] 
      | Some [] -> Some [x] 
      | Some vl -> Some (x::vl)) m) Map.empty l;;

  (* let scanl = scan_left *)
  (* let scanli = scan_lefti *)
  (* let scanr = scan_right *)
  (* let scanri = scan_righti *)
  
  let flatmap = concat_map

  let zipf = map2
  let zipfi = map2i

end

module L = JunkList

module JunkArray = struct
  include Array 

  let nmem x l = not @@ mem x l

  let len = length
  let size = length
  let is_empty r = len r = 0
  let nempty r = len r > 0

  let take n r = left r n
  let taker n r = right r n

  let takeb x r =
    let l = len r in
    let rec loop i =
      if i >= l then r
      else if r.(i) = x then take i r
      else loop (i + 1)
    in loop 0
     
  (* let takewh = take_while *)

  let drop n r = tail r n
  let dropr n r = take ((len r) - n) r
  (* let dropwh = drop_while *)

  let index_of x r =
    let l = len r in
    let rec loop i =
      if i >= l then None
      else if r.(i) = x then Some i
      else loop (i + 1)
    in loop 0

  (* let split_by = span *)

  (* let chunk = ntake *)

  let foldl = fold_left
  let foldli = fold_lefti
  let foldr f z l = fold_right f l z
  let foldri f z l = fold_righti f l z

  (* let partition_to_map f l = *)
  (*   foldr (fun x m -> Map.modify_opt (f x) (function *)
  (*     | None -> Some [x]  *)
  (*     | Some [] -> Some [x]  *)
  (*     | Some vl -> Some (x::vl)) m) Map.empty l;; *)

  (* let scanl = scan_left *)
  (* let scanli = scan_lefti *)
  (* let scanr = scan_right *)
  (* let scanri = scan_righti *)
  
  let concat2 = append
  
  let flatmap (f : 'a -> 'b array) a =
    map f a |> enum |> L.of_enum |> concat

  let zipf = map2
  (* let zipfi = map2i *)

end

module A = JunkArray

module JunkString = struct
  include String

  let len = length

  let replace_chars = map
  let map = replace

  let maptol f s = L.map f (s2cl s)
  let maptoli f s = L.map f (s2cl s)

  let zipf f a b = L.zipf f (s2cl a) (s2cl b)
  let zipfi f a b = L.zipfi f (s2cl a) (s2cl b)

  let mkstr f sep ls =
    L.foldl (fun acc x -> acc ^ sep ^ (f x)) (f @@ L.hd ls) (List.tl ls)

end

module S = JunkString

let spsp s = S.split_on_char ' ' s


module JunkSortedSet = struct
  include Set

  module Make (Ord : Set.OrderedType) = struct
    include Set.Make(Ord)

    let nmem x st = not @@ mem x st

    let foldl f s z = fold (fun x a -> f a x) z s

    let size = cardinal
  end

  let nmem x st = not @@ mem x st

  let foldl f s z = fold (fun x a -> f a x) z s

  let size = cardinal
end

module Ss = JunkSortedSet

module JunkSortedMap  = struct
  include Map

  module Make (Ord : Map.OrderedType) = struct
    include Map.Make(Ord)

    let nmem x st = not @@ mem x st

    let get = find
    let get_or x d m = find_default d x m

    let size = cardinal
  end

  let nmem x st = not @@ mem x st

  let get = find
  let get_or x d m = find_default d x m

  let size = cardinal
end

module Ms = JunkSortedMap


let set_power ls =
  let rec loop ls pls =
    match ls with
    | [] -> pls
    | hd::tl -> loop tl (L.flatmap (fun p -> [p; hd::p]) pls)
  in loop ls [[]]


exception ArithmeticException of string

module JunkIntArithmetic = struct

  let rec power x e =
    if e = 0 then x
    else 
      let rec loop r x2 e' =
        if e' = 0 then r
        else if e' land 1 = 0 then loop r (x2 * x2) (e' lsr 1)
        else loop (r * x2) (x2 * x2) (e' lsr 1)
      in loop 1 x e

  let rec powers_le_1000 x =
    if x > 32 then [x]
    else
      L.init 9 (fun e -> power x (e + 1))
      |> L.filter (fun y -> y < 1000)

  let primes_le_1000 = [2;3;5;7;11;13;17;19;23;29;31;37;41;
    43;47;53;59;61;67;71;73;79;83;89;97;101;
    103;107;109;113;127;131;137;139;149;151;157;163;167;
    173;179;181;191;193;197;199;211;223;227;229;233;239;
    241;251;257;263;269;271;277;281;283;293;307;311;313;
    317;331;337;347;349;353;359;367;373;379;383;389;397;
    401;409;419;421;431;433;439;443;449;457;461;463;467;
    479;487;491;499;503;509;521;523;541;547;557;563;569;
    571;577;587;593;599;601;607;613;617;619;631;641;643;
    647;653;659;661;673;677;683;691;701;709;719;727;733;
    739;743;751;757;761;769;773;787;797;809;811;821;823;
    827;829;839;853;857;859;863;877;881;883;887;907;911;
    919;929;937;941;947;953;967;971;977;983;991;997]

    let primes_and_powers_le_1000 =
      [2;3;4;5;7;8;9;11;13;16;17;19;23;25;27;29;31;32;37;41;
      43;47;49;53;59;61;64;67;71;73;79;81;83;89;97;
      101;103;107;109;113;121;125;127;128;131;137;139;149;151;
      157;163;167;169;173;179;181;191;193;197;199;211;223;227;
      229;233;239;241;243;251;256;257;263;269;271;277;281;283;
      289;293;307;311;313;317;331;337;343;347;349;353;359;361;
      367;373;379;383;389;397;401;409;419;421;431;433;439;443;
      449;457;461;463;467;479;487;491;499;503;509;512;521;523;
      529;541;547;557;563;569;571;577;587;593;599;601;607;613;
      617;619;625;631;641;643;647;653;659;661;673;677;683;691;
      701;709;719;727;729;733;739;743;751;757;761;769;773;787;
      797;809;811;821;823;827;829;839;841;853;857;859;863;877;
      881;883;887;907;911;919;929;937;941;947;953;961;967;971;
      977;983;991;997]

  let factorize_with ps n =
    let rec loop acc ps e n' =
      if n' <= 1 then ([], n') 
      else
        match ps with
        | [] -> (acc, n')
        | p::t ->
          let r = n' mod p in
          if r = 0 then (
            if n' = p then ((p, e + 1) :: acc, 1)
            else loop acc ps (e + 1) (n' / p)
          ) else
            loop (if e = 0 then acc else (p, e) :: acc) t 0 n'
    in
    let (fs, r) = loop [] ps 0 n in
    (L.rev fs, r)

  let factorize_le_million n =
    factorize_with primes_le_1000 n

  let rec to_bits_le n =
    if n = 0 then []
    else (n mod 2) :: to_bits_le (n lsr 1)

  let rec gcd x y =
    match x mod y with
    | 0 -> y
    | z -> gcd y z

  let ext_euclid a b =
    let rec next (r0, s0, t0) (r1, s1, t1) =
      match r1 with
      | 0 -> (r0, s0, t0)
      | r2 ->
        let q = r0 / r1 in
        next (r1, s1, t1) (r0 - q * r1, s0 - q * s1, t0 - q * t1)
    in
    if a < b then
      let (g, y, x) = next (b, 1, 0) (a, 0, 1) in (g, x, y)
    else
      next (a, 1, 0) (b, 0, 1)

  let modinv x m =
    let (g, y, z) = ext_euclid x m in
    match g with
    | 1 -> y mod m
    | _ -> raise @@ ArithmeticException "Not relatively prime"
      

end

module N = JunkIntArithmetic


module JunkMatrix = struct

  type 'a t = { w : int; h : int; e : 'a array array }

  let make w h e0 =
    { w = w; h = h; e = A.make_matrix w h e0 }

  let from_columns w h e0 elems =
    let matr = make w h e0 in
    elems |> L.iteri (fun j col ->
      col |> L.iteri (fun i x -> matr.e.(i).(j) <- x));
    matr

  let blit (mats, rs, cs, w, h) (matd, rd, cd) =
    rep2 0 `Til w () 0 `Til h (fun i j ->
        matd.e.(rd + i).(cd + j) <- mats.e.(rs + i).(cs + j))

  let expand (w1, h1) (w2, h2) e0 mat =
    let matr = make w2 h2 e0 in
    blit (mat, 0, 0, w1, h1) (matr, 0, 0);
    matr

end

module M = JunkMatrix

module type FieldOps = sig
  type elt
  val zero : elt
  val one : elt
  val neg : elt -> elt
  val rcp : elt -> elt
  val add : elt -> elt -> elt
  val sub : elt -> elt -> elt
  val mul : elt -> elt -> elt
  val div : elt -> elt -> elt
end

module JunkLinearAlgebra (Fop : FieldOps) = struct

  type t = Fop.elt M.t

  let zero w h = M.make w h Fop.zero

  let prod (mat1 : t) (mat2 : t) =
    let matr = zero mat1.w mat2.h in
    rep2 0 `Til mat1.w () 0 `Til mat2.h (fun i j ->
        matr.e.(i).(j) <-
          mat1.e.(i) |> A.foldli (fun
            acc k x -> Fop.add acc (Fop.mul x mat2.e.(k).(i)))
            Fop.zero);
    matr

  exception NotZero of int
  let is_zero (mat : t) =
    try (
      rep 0 `Til mat.w (fun i ->
        if mat.e.(i) |> A.for_all (fun x -> x <> Fop.zero)
          then raise (NotZero 1)
      );
      true
    ) with _ -> false

  let col_addmul c1 (c2, a) (mat : t) =
    rep 0 `Til mat.h (fun i ->
      mat.e.(i).(c1) <- Fop.add mat.e.(i).(c1) (Fop.mul a mat.e.(i).(c2)))

  let col_inner_prod c1 c2 (mat : t) =
    let sum = ref Fop.zero in
    rep 0 `Til mat.h (fun i ->
      sum := Fop.add !sum @@ Fop.mul mat.e.(i).(c1) mat.e.(i).(c2));
    !sum

  (* let orth_extend k0 k1 mat = *)
  (*   let matx = extend (k0, h) ((k0 + k1), h) 0 mat in *)
  (*   rep k0 `Til (k0 + k1) (fun t -> *)
  (*     rep 0 `Til t (fun s ->  *)
  (*       let ip = col_inner_prod s t h matx in *)
  (*       col_addmul t (s, ip) h matx)); *)
  (*   submat (k0, 0) (k0 + k1 - 1, h - 1) matx *)

end

module type OrderedData = sig
  type t
  val compare : t -> t -> int
  val default : t
end

module JunkGraph = struct

  type 'a v_t = int * 'a
  type 'a t = { vs : 'a v_t array; nb : int Ss.t array }

  let empty nv vd =
    let vs0 = Array.make nv (0, vd) in
    for i = 0 to (nv - 1) do vs0.(i) <- (i, vd) done;
    {
      vs = vs0;
      nb = A.make nv Ss.empty
    }

  let add_edge (e : int * int) vd g =
    let (v1, v2) = e in
    g.vs.(v1 - 1) <- (v1, vd);
    g.vs.(v2 - 1) <- (v2, vd);
    g.nb.(v1 - 1) <- g.nb.(v1 - 1) |> Ss.add v2;
    g

  let add_uedge e vd g =
    add_edge e vd (add_edge (swap e) vd g)

  let from_edges nv vd es =
    es |> L.foldl (fun g e -> add_edge e vd g) (empty nv vd)

  let from_uedges nv vd es =
    es |> L.foldl (fun g e -> add_uedge e vd g) (empty nv vd)


end

module G = JunkGraph

module JunkGraphAlgorithms (Vdt : OrderedData) = struct
  
  type t = Vdt.t G.t

  module V = struct
    type t = int * Vdt.t
    let compare (vi1, vd1) (vi2, vd2) =
      let c1 = Vdt.compare vd1 vd2 in
      if c1 = 0 then compare vi1 vi2
      else c1

    let default i = (i, Vdt.default)
    let index (v : t) = fst v
  end

  module Vset = Ss.Make(V)

  let print_vertices label vs =
    printf "%s: " label;
    vs |> Vset.iter (fun (vi, _) -> printf "%d " vi);
    puts ""

  let nbiter (upd : V.t -> V.t -> t -> t * bool)
    (ui : int) (fr : Vset.t) (cls : int Ss.t) (g : t) =
    let nb = g.nb.(ui - 1) in
    let result = nb |> Ss.foldl (fun (ga, nxfr, nxcls) vi ->
    (* puts "nxcls: "; *)
      (* Ss.iter (fun x -> printf "%d " x) nxcls; puts ""; *)
      if ui = vi || Ss.mem vi nxcls then
        (ga, nxfr, nxcls)
      else
        let ue = g.vs.(ui - 1) and ve = g.vs.(vi - 1) in
        let (gu, close) = upd ue ve ga in
        (gu,
          nxfr |> Vset.add ve,
          if close then nxcls |> Ss.add vi else nxcls)
    ) (g, fr, cls)
    in (* puts "";  *)result

  let traverse (upd : V.t -> V.t -> t -> t * bool) 
    (fr0 : Vset.t) (cls0 : int Ss.t) (g : t) =

    let rec progress fr cls g' =
      let (ve, fr') = Vset.pop_min fr in
      let (vi, _) = ve in
      let (gu, nxfr, nxcls) = nbiter upd vi fr' cls g' in
      if Vset.is_empty nxfr then (gu, nxcls)
      else progress nxfr nxcls gu
    in
    progress fr0 cls0 g

  let frontier (upd : V.t -> V.t -> t -> t * bool)
    (fr0 : Vset.t) (cls0 : int Ss.t) (g : t) =

    let rec progress fr cls g' =
      let (gu, nxfr, nxcls) =
        fr |> Vset.foldl (fun (ga, nxfra, nxclsa) ve ->
          let (vi, _) = ve in
          nbiter upd vi nxfra nxclsa ga
        ) (g', Vset.empty, cls)
      in
      if Vset.is_empty nxfr then (gu, nxcls)
      else progress nxfr nxcls gu
    in
    progress fr0 cls0 g

  let bfs (upd : V.t -> V.t -> t -> t) =
    frontier (fun fr cls g' -> (upd fr cls g', true))
  
end


let rds () = read_line ()

let rdi () = s2i @@ rds ()

let rdhs () = spsp @@ rds ()
let rdhi () = rdhs () |> L.map s2i

let rdhf () = rdhs () |> L.map s2f

let rdf2 () = match rdhf () with
| a::b::_ -> (a, b)
| _ -> (0., 0.)

let rdf3 () = match rdhf () with
| a::b::c::_ -> (a, b, c)
| _ -> (0., 0., 0.)

let rdf4 () = match rdhf () with
| a::b::c::d::_ -> (a, b, c, d)
| _ -> (0., 0., 0., 0.)

let rds2 () = match rdhs () with
  | a::b::_ -> (a, b)
  | _ -> ("", "")

let rdi2 () = match rdhi () with
  | a::b::_ -> (a, b)
  | _ -> (0, 0)

let rdi3 () = match rdhi () with
| a::b::c::_ -> (a, b, c)
| _ -> (0, 0, 0)

let rdi4 () = match rdhi () with
| a::b::c::d::_ -> (a, b, c, d)
| _ -> (0, 0, 0, 0)

let rdi5 () = match rdhi () with
| a::b::c::d::e::_ -> (a, b, c, d, e)
| _ -> (0, 0, 0, 0, 0)

let rdv n rf =
  let rec loop n =
    if n <= 0 then []
    else (
      let s = rf () in
      s::(loop (n - 1))
    )
  in loop n

let rdvi n = rdv n rdi
let rdvi2 n = rdv n rdi2
let rdvi3 n = rdv n rdi3
let rdvi4 n = rdv n rdi4


module Galg = JunkGraphAlgorithms (
  struct
    type t = int
    let compare = compare
    let default = 0
  end)

let print_list aa = 
  printf "[ "; aa |> L.iter (fun a -> printf "%d " a); puts "]"

let print_array aa = 
  printf "[ "; aa |> A.iter (fun a -> printf "%d " a); puts "]"

let rec bsearch_max_int f lb ub =
  if lb >= ub then (if f lb then lb else (lb - 1))
  else (
    let m = (lb + ub) / 2 in
    if f m then bsearch_max_int f (m + 1) ub
    else bsearch_max_int f lb m
  )

let partial_sum n ls =
  let rec loop s k = function
    | [] -> s
    | h::t ->
      if k >= n then s
      else loop (s + h) (k + 1) t
  in
  loop 0 0 ls

let scanl f z ls =
  let rec loop acl acc = function
    | [] -> L.rev acl
    | h::t ->
      let r = f acc h in
      loop (r::acl) r t
  in loop [] z ls

let _ =
  let (n, m, k) = rdi3 () in
  let aas = rdhi () in
  let bas = rdhi () in
  let aas_sums = A.of_list @@ 0::(scanl (+) 0 aas) in
  let bas_sums = A.of_list @@ 0::(scanl (+) 0 bas) in
  (* print_array aas_sums; *)
  (* print_array bas_sums; *)

  let can_read nbk =
    let rec test na =
      let nb = nbk - na in
      (* printf "%d %d\n" na nb; *)
      if na < 0 || na > n || nb < 0 || nb > m then false 
      else (
        (* print_list abk; print_list bbk; puts "----"; *)
        if aas_sums.(na) + bas_sums.(nb) <= k then true
        else test (na + 1)
      )
    in

    (* puti nbk; *)
    test (max 0 (nbk - m)) 
  in
  
  puti @@ bsearch_max_int can_read 0 (n + m)