Contest Results

Intuitively, Farmer John would always want to add milk to the bottles currently with the most milk. This is because

Similarly, Farmer Nhoj would intuitively want to remove milk from the highest bottles he can remove the milk from. This, by the same logic, decreases the amount of profit Farmer John can make by the highest amount. A proof sketch is included at the bottom of this analysis.

Thus, each day, it turns out that if we sort the array, the bottles indexed from

$N-A+1$ to

$N-B$ will gain one unit of milk. We will refer to this as the interval from now on.

Subtask 1:

It remains to efficiently simulate this process. When $N$ and $D$ are sufficiently small, as in this subtask, we can simply sort the array in each iteration and simulate Farmer John and Farmer Nhoj's moves.

Subtask 2:

To optimize this, let's consider the structure of the sorted array during each day.

We can observe that most of the time, the sorted order of the array does not change much. In fact, the only times the sorted order of the array changes is when the rightmost part of the interval ``catches up" with the element to the right of the interval.

To characterize this, let's call the group of elements on the right of our interval whose values are the same the $\textbf{border}$ , and we say that $k$ of them are activated if the intersection of the border and our interval as length $k$ .

Then, each day,

All values within our interval AND to the left of the border will increase by one.
The $k$ smallest elements within the border increase by $1$ .
The border will expand to include all elements to the left and right whose values are the same as the smallest element in the border. Specifically, if an element to the left (within the interval) is added, $k$ increases by $1$ .

To implement this, we just need to keep track of the left and right endpoints of the border, the smallest element of the border (we can call it $num$ ), and the number of elements in the border whose value is $num+1$ (we can call it $p$ ). We can implicitly store all of the values within our interval and to the left of the border as $m[i]+t$ , where $t$ is the time that has already elapsed. We can increase the $k$ smallest elements by simply updating our values for $num$ and $p$ , and we can update our border by simply checking to the left and right of our border.

The runtime becomes $O(N \log N+D)$ with sorting.

Larry's Code:

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
const int MAX_N = 2e5+5;
const ll MOD = 1e9+7;

ll n, d, a, b, l, r;
ll c[MAX_N];

int main(int argc, const char * argv[]){
    cin.tie(0)->sync_with_stdio(0);
    cin.exceptions(cin.failbit);
    cin >> n >> d >> a >> b;
    for(int i = 0; i < n; i++) cin >> c[i];
    sort(c, c+n);
    l = n-a, r = n-b-1;
    ll ql = r, qr = r;
    while(ql > 0 && c[ql-1] == c[r]) ql--;
    while(qr+1 < n && c[qr+1] == c[r]) qr++;
    ll p = 0, num = c[r];
    for(int k = 0; k < d; k++){
        ll x = (r-ql+1)+p;
        num += x/(qr-ql+1), p = x%(qr-ql+1);
        while(qr+1 < n && c[qr+1] == num) qr++;
        while(ql > l && c[ql-1]+k+1 == num) ql--;
    }
    for(int i = l; i < ql; i++) c[i] += d;
    for(int i = ql; i <= qr; i++) c[i] = qr-i < p ? num+1 : num;
    ll ans = 0;
    for(int i = 0; i < n; i++)
        ans = (ans+(c[i]%MOD)*(c[i]%MOD))%MOD;
    cout << ans << '\n';
    return 0;
}

Full Solution

Instead of iterating over each day individually, let's calculate the next time our border changes.

Our border can only ever expand, so we can simply calculate the time when the border will expand to the left, and when our border would expand to the right, and take the earlier one.

To do this, let's take another look at how the border's values grow. Clearly, the sum of the values within the border grows by $k$ each day; in fact, it is pretty easy to then derive that

$num = \lfloor \Sigma/\text{length(border)}\rfloor, p = \Sigma \mod{\text{length(border)}}$

Thus, the first time $num$ will exceed a certain theshold $M$ (the element to the right of the border) is

$\left\lceil \frac{(M-num)\cdot \text{length(border)}-p}{k}\right\rceil$

Similarly, we can do some algebra to obtain the minimum time for the border to expand on the left:

$\left\lfloor \frac{(num-M-t)\cdot \text{length(border)}+p}{\text{length(border)}-k} \right\rfloor$

Once we've calculated these times, we can update our values accordingly and expand the border.

Since the border can only expand at most $O(N)$ times, this process can only occur at most $O(N)$ times. So, with sorting, the overall runtime is $O(N \log N)$ .

Larry's Code:

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
#define ff first
#define ss second
const int MAX_N = 2e5+5;
const ll MOD = 1e9+7;
const ll INF = 1e18;

ll n, d, a, b, l, r;
ll c[MAX_N];

ll ceil(ll a, ll b){
    return (a+b-1)/b;
}

int main(int argc, const char * argv[]){
    cin.tie(0)->sync_with_stdio(0);
    cin.exceptions(cin.failbit);
    cin >> n >> d >> a >> b;
    for(int i = 0; i < n; i++) cin >> c[i];
    sort(c, c+n);
    l = n-a, r = n-b-1;
    ll ql = r, qr = r;
    while(ql > l && c[ql-1] == c[r]) ql--;
    while(qr+1 < n && c[qr+1] == c[r]) qr++;
    ll k = 0, p = 0, num = c[r];
    while(true){
        ll numx = qr+1 < n ? ceil((c[qr+1]-num)*(qr-ql+1)-p, r-ql+1) : INF;
        ll numy = ql > l && r != qr ? ((qr-ql+1)*(num-(c[ql-1]+k))+p)/(qr-r) : INF;
        if(k+min(numx, numy) > d) break;
        ll x = min(numx, numy)*(r-ql+1)+p;
        num += x/(qr-ql+1), p = x%(qr-ql+1), k += min(numx, numy);
        while(qr+1 < n && c[qr+1] == num) qr++;
        while(ql > l && c[ql-1]+k == num) ql--;
    }
    ll x = ((d-k)*(r-ql+1)+p)/(qr-ql+1), y = (d-k)*(r-ql+1)+p-x*(qr-ql+1);
    num += ((d-k)*(r-ql+1)+p)/(qr-ql+1), p = y, k = d;
    for(int i = l; i < ql; i++) c[i] += d;
    for(int i = ql; i <= qr; i++) c[i] = qr-i < p ? num+1 : num;
    ll ans = 0;
    for(int i = 0; i < n; i++)
        ans = (ans+(c[i]%MOD)*(c[i]%MOD))%MOD;
    cout << ans << '\n';
    return 0;
}

Alternatively, you can model the values within the border as fractions. At any time, the values within the interval and not in the border increase at a steady rate of $1$ unit of milk per day, and the values within the border increase at a steady rate of $k/\text{length(border)}$ per day.

Storing each of these as fractions, we can greatly simplify our transitions.

Eric's Code:

import sys
from fractions import Fraction
import math
 
input = sys.stdin.readline
 
N, D = map(int, input().split())
A, B = map(int, input().split())
M = list(sorted(map(int, input().split())))
 
constant = M[:N-A]
V = M[N-A:]
 
C = A-B
 
assert C > 0
 
border_value = V[C-1]
border_min = C-1
border_max = C-1
 
ct = 0
 
def expand_border():
    global border_min, border_max
    while border_min > 0 and V[border_min-1] + ct == border_value:
        border_min -= 1
    while border_max < A-1 and V[border_max+1] == border_value:
        border_max += 1
 
while ct < D:
    expand_border()
    delta_to_min = Fraction(1000000000, 1)
    delta_to_max = Fraction(1000000000, 1)
    num_border = border_max - border_min + 1
    alloc_to_border = C - border_min
    border_rate = Fraction(alloc_to_border, num_border)
    if border_min > 0:
        val = V[border_min-1] + ct
        if val == border_value:
            delta_to_min = Fraction(0, 1)
        elif border_rate < 1:
            delta_to_min = Fraction((border_value - val), 1 - border_rate)
    if border_max < A-1:
        val = V[border_max+1]
        if val == border_value:
            delta_to_max = Fraction(0, 1)
        elif border_rate > 0:
            delta_to_max = Fraction((val - border_value), border_rate)
    delta = min(delta_to_min, delta_to_max)
    if ct + delta >= D:
        delta = D - ct
    border_value += border_rate * delta
    ct += delta
 
assert ct == D
 
ans = 0
for m in constant:
    ans += m * m
for m in V[:border_min]:
    ans += (m+ct) * (m+ct)
for m in V[border_max+1:]:
    ans += m * m
 
num_border = border_max - border_min + 1
floor_value = math.floor(border_value)
num_ceil = 0
while floor_value < border_value:
    floor_value += Fraction(1, num_border)
    num_ceil += 1
 
assert floor_value == border_value
 
ans += pow(math.floor(border_value), 2) * (num_border - num_ceil)
ans += pow(math.ceil(border_value), 2) * num_ceil
 
print(ans % (10 ** 9 + 7))

Note: The complexity of this solution is difficult to analyze, since the size of the denominator may contribute to the time complexity. However, it is possible to show that the denominator is at most $N\cdot D$ , so the solution comfortably runs in time.

Proof Sketch

Model the problem as a two-player game. A game state consists of the array of milk values. A move consists of either Farmer John or Farmer Nhoj performing their action.

For two game states $x$ and $y$ , $x \succ y$ means $x$ majorizes $y$ , or that for all $k$ , the sum of the $k$ largest elements of $x$ is $\ge$ the $k$ largest elements of $y$ . Additionally, the sums of $x$ and $y$ must be equal. Equivalently, $x$ majorizes $y$ iff $x$ can be transformed into $y$ by repeatedly performing one of the following operations on $x$ :

Permute $x$ arbitrarily.
Move one unit of milk from a bottle with more milk to a bottle with less milk.

Our goal is to show for all $m$ that if $x \succ y$ , then $\text{final_profit}_m(x) \ge \text{final_profit}_m(y)$ . We proceed by induction on $m$ , the number of moves remaining ( $0\le m\le 2D$ ).

Base case: For $m=0$ , if $x \succ y$ , then by Karamata's inequality, $\text{final_profit}_0(x)=\sum x_i^2 \ge \sum y_i^2=\text{final_profit}_0(y)$ .

Inductive step: Assume that the inductive hypothesis holds if $m-1$ moves remain, for $1\le m\le 2D$ . We would like to show that the hypothesis holds for $m$ moves remaining.

Suppose it is Farmer John to move, and that the state before the move was $x$ . Let $\text{move_largest}(x)$ represent the state of $x$ after adding 1 to the $A$ largest bottles and $\text{move}(x)$ represent any possible state of $x$ after FJ moves. Clearly $\text{move_largest}(x) \succ \text{move}(x)$ , and for any state $y$ such that $x \succ y$ , $\text{move_largest}(x) \succ \text{move_largest}(y)$ . So
$\text{final_profit}_m(x)=\text{final_profit}_{m-1}(\text{move_largest}(x))\ge \text{final_profit}_{m-1}(\text{move_largest}(y))=\text{final_profit}_m(y),$
implying that the inductive hypothesis holds for $m$ .
On the other hand, suppose it is Farmer Nhoj to move. Similarly, let $\text{move_largest}$ represent subtracting 1 from the $B$ largest bottles and $\text{move}$ represent any possible move. It may be shown that $\text{move_largest}(x) \prec \text{move}(x)$ . Also, we claim that for any state $y$ such that $x\prec y$ , $\text{move_largest}(x)\prec \text{move_largest}(y)$ (left as an exercise to the reader). This claim is harder to show than in the case where it is Farmer John to move since the sorted order of the bottles may change after Farmer Nhoj moves. However, keep in mind that we only need to show that it holds when $x$ and $y$ differ by only a single operation of type 2, after which it naturally extends to the case where $x$ and $y$ differ by multiple operations of type 2.
Assuming the above,
$\text{final_profit}_m(x)=\text{final_profit}_{m-1}(\text{move_largest}(x))\le \text{final_profit}_{m-1}(\text{move_largest}(y))=\text{final_profit}_m(y),$
implying that the inductive hypothesis holds for $m$ .
(Note that it is always possible for Farmer Nhoj to make $\text{move_largest}$ , since $A > B$ .)