#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
template<class T> bool ckmin(T& a, const T& b) { return b < a ? a = b, 1 : 0; }
template<class T> bool ckmax(T& a, const T& b) { return a < b ? a = b, 1 : 0; }
#define sz(x) (int)x.size()
/*
Date: 2022/08/08 11:13
Problem Link: link
Topic(s):
Time Spent:
Solution Notes:
There is a simply N^2 dp:
dp[i][j] = max reward in [0, i] after using j segments
dp[i][j] = ps[i] + max(-ps[i'] + dp[i'][j-1]), where ps[i] is the prefix sums of the first i elements OR dp[i-1][j]
Using Alien's trick to optimize
Consider the "value" of each segment, which is the max reward we can get if we place this segment optimally. As we add more segments, the answer increases,
but the answer increases by less as we have more segments. In other words, the value of a segment always decreases. We can think about the value of a segment
as a concave function. The kth segment will correpsond to some value x. Our goal is to find the sum of the values of segments before k (i.e. segments with value
greater or equal to x).
We can do this by binary searching on x, but how do we know if we have used more/less than k segments? First, think what does it mean for each segment to have value
at least x. That means the sum over all segments we takes >= k*x. Let's subtract our x every time we take a new segment, now we want to make sure we only
take segments that result in a positive value. Now, we can utilize our dp formulation. Instead of keeping track of how many segments we used as part of the state,
subtracting by x will automatically "deter" us from taking too many segments (if x is bigger, that means we will select less elements, and vice versa).
dp[i] = max reward in [0, i] (offseting by x everytime we take a new segment)
dp[i] = ps[i] + max(-ps[i'] + dp[i']) - x OR dp[i-1]
Track for each dp[i], how many x's have we subtracted by (how many segments used). In the end, we will adjust our binary search on x with it.
In the end, we will have to add back the k*x we offsetted by. There might be multiple segments all with value == x. However, we don't have to worry about this,
as we offset by x every time we take a segment (meaning in the dp, segments == x will contribute 0), and we ONLY add back k*x regardless of what lo (in the binary search) is.
*/
const int MAXN = 3e5+5, INF = 1e9;
int n, k;
ll arr[MAXN], ps[MAXN];
pair<ll, ll> check(ll x){
vector<pair<ll, ll>> dp(n+1);
pair<ll, ll> mx;
for (int i=1; i<=n; i++){
dp[i] = dp[i-1];
if (ps[i] + mx.first - x > dp[i].first){
dp[i] = {ps[i] + mx.first - x, mx.second + 1};
}
if (dp[i].first - ps[i] > mx.first){
mx = {dp[i].first - ps[i], dp[i].second};
}
}
return dp[n];
}
int main(){
cin.tie(0); ios_base::sync_with_stdio(0);
// freopen("file.in", "r", stdin);
// freopen("file.out", "w", stdout);
cin >> n >> k;
for (int i=0; i<n; i++){
cin >> arr[i];
}
for (int i=0; i<n; i++){
ps[i+1] = ps[i] + arr[i];
}
// binary search on x
ll low = 0, high = 1e15;
while (low < high){
ll mid = (low + high)/2;
int seg = check(mid).second;
if (seg > k){
low = mid+1;
}else{
high = mid;
}
}
pair<ll, ll> ans = check(high);
cout << ans.first + k*high << "\n";
}
/**
* Debugging checklist:
* - Reset everything after each TC
* - Integer overflow, index overflow
* - Special cases?
*/
| # |
결과 |
실행 시간 |
메모리 |
Grader output |
| 1 |
Correct |
81 ms |
9540 KB |
Output is correct |
| 2 |
Correct |
73 ms |
12436 KB |
Output is correct |
| 3 |
Correct |
74 ms |
12528 KB |
Output is correct |
| 4 |
Correct |
71 ms |
12492 KB |
Output is correct |
| 5 |
Correct |
76 ms |
12436 KB |
Output is correct |
| 6 |
Correct |
74 ms |
12208 KB |
Output is correct |
| 7 |
Correct |
84 ms |
12176 KB |
Output is correct |
| 8 |
Correct |
73 ms |
12404 KB |
Output is correct |
| 9 |
Correct |
71 ms |
12288 KB |
Output is correct |
| 10 |
Correct |
78 ms |
12376 KB |
Output is correct |
| # |
결과 |
실행 시간 |
메모리 |
Grader output |
| 1 |
Correct |
68 ms |
9460 KB |
Output is correct |
| 2 |
Correct |
84 ms |
10796 KB |
Output is correct |
| 3 |
Correct |
68 ms |
10556 KB |
Output is correct |
| 4 |
Correct |
81 ms |
10756 KB |
Output is correct |
| 5 |
Correct |
67 ms |
12228 KB |
Output is correct |
| 6 |
Correct |
72 ms |
10572 KB |
Output is correct |
| 7 |
Correct |
67 ms |
10776 KB |
Output is correct |
| 8 |
Correct |
75 ms |
12516 KB |
Output is correct |
| 9 |
Correct |
73 ms |
12156 KB |
Output is correct |
| 10 |
Correct |
77 ms |
10780 KB |
Output is correct |
| # |
결과 |
실행 시간 |
메모리 |
Grader output |
| 1 |
Correct |
106 ms |
9592 KB |
Output is correct |
| 2 |
Correct |
99 ms |
12448 KB |
Output is correct |
| 3 |
Correct |
114 ms |
12508 KB |
Output is correct |
| 4 |
Correct |
106 ms |
12420 KB |
Output is correct |
| 5 |
Correct |
98 ms |
12580 KB |
Output is correct |
| 6 |
Correct |
102 ms |
12664 KB |
Output is correct |
| 7 |
Correct |
103 ms |
12708 KB |
Output is correct |
| 8 |
Correct |
98 ms |
12584 KB |
Output is correct |
| 9 |
Correct |
115 ms |
12748 KB |
Output is correct |
| 10 |
Correct |
101 ms |
12668 KB |
Output is correct |
| # |
결과 |
실행 시간 |
메모리 |
Grader output |
| 1 |
Correct |
0 ms |
212 KB |
Output is correct |
| 2 |
Correct |
1 ms |
340 KB |
Output is correct |
| 3 |
Correct |
0 ms |
340 KB |
Output is correct |
| 4 |
Correct |
1 ms |
340 KB |
Output is correct |
| 5 |
Correct |
0 ms |
212 KB |
Output is correct |
| 6 |
Correct |
1 ms |
340 KB |
Output is correct |
| 7 |
Correct |
1 ms |
340 KB |
Output is correct |
| 8 |
Correct |
1 ms |
212 KB |
Output is correct |
| 9 |
Correct |
1 ms |
340 KB |
Output is correct |
| 10 |
Correct |
1 ms |
212 KB |
Output is correct |
| # |
결과 |
실행 시간 |
메모리 |
Grader output |
| 1 |
Correct |
0 ms |
212 KB |
Output is correct |
| 2 |
Correct |
1 ms |
340 KB |
Output is correct |
| 3 |
Correct |
0 ms |
340 KB |
Output is correct |
| 4 |
Correct |
1 ms |
340 KB |
Output is correct |
| 5 |
Correct |
0 ms |
212 KB |
Output is correct |
| 6 |
Correct |
1 ms |
340 KB |
Output is correct |
| 7 |
Correct |
1 ms |
340 KB |
Output is correct |
| 8 |
Correct |
1 ms |
212 KB |
Output is correct |
| 9 |
Correct |
1 ms |
340 KB |
Output is correct |
| 10 |
Correct |
1 ms |
212 KB |
Output is correct |
| 11 |
Correct |
1 ms |
340 KB |
Output is correct |
| 12 |
Correct |
1 ms |
212 KB |
Output is correct |
| 13 |
Correct |
1 ms |
340 KB |
Output is correct |
| 14 |
Correct |
0 ms |
340 KB |
Output is correct |
| 15 |
Correct |
1 ms |
340 KB |
Output is correct |
| 16 |
Correct |
1 ms |
332 KB |
Output is correct |
| 17 |
Correct |
1 ms |
340 KB |
Output is correct |
| 18 |
Correct |
1 ms |
332 KB |
Output is correct |
| 19 |
Correct |
1 ms |
340 KB |
Output is correct |
| 20 |
Correct |
1 ms |
340 KB |
Output is correct |
| # |
결과 |
실행 시간 |
메모리 |
Grader output |
| 1 |
Correct |
0 ms |
212 KB |
Output is correct |
| 2 |
Correct |
1 ms |
340 KB |
Output is correct |
| 3 |
Correct |
0 ms |
340 KB |
Output is correct |
| 4 |
Correct |
1 ms |
340 KB |
Output is correct |
| 5 |
Correct |
0 ms |
212 KB |
Output is correct |
| 6 |
Correct |
1 ms |
340 KB |
Output is correct |
| 7 |
Correct |
1 ms |
340 KB |
Output is correct |
| 8 |
Correct |
1 ms |
212 KB |
Output is correct |
| 9 |
Correct |
1 ms |
340 KB |
Output is correct |
| 10 |
Correct |
1 ms |
212 KB |
Output is correct |
| 11 |
Correct |
1 ms |
340 KB |
Output is correct |
| 12 |
Correct |
1 ms |
212 KB |
Output is correct |
| 13 |
Correct |
1 ms |
340 KB |
Output is correct |
| 14 |
Correct |
0 ms |
340 KB |
Output is correct |
| 15 |
Correct |
1 ms |
340 KB |
Output is correct |
| 16 |
Correct |
1 ms |
332 KB |
Output is correct |
| 17 |
Correct |
1 ms |
340 KB |
Output is correct |
| 18 |
Correct |
1 ms |
332 KB |
Output is correct |
| 19 |
Correct |
1 ms |
340 KB |
Output is correct |
| 20 |
Correct |
1 ms |
340 KB |
Output is correct |
| 21 |
Correct |
1 ms |
340 KB |
Output is correct |
| 22 |
Correct |
2 ms |
340 KB |
Output is correct |
| 23 |
Correct |
1 ms |
340 KB |
Output is correct |
| 24 |
Correct |
1 ms |
340 KB |
Output is correct |
| 25 |
Correct |
1 ms |
344 KB |
Output is correct |
| 26 |
Correct |
1 ms |
340 KB |
Output is correct |
| 27 |
Correct |
2 ms |
368 KB |
Output is correct |
| 28 |
Correct |
1 ms |
340 KB |
Output is correct |
| 29 |
Correct |
1 ms |
340 KB |
Output is correct |
| 30 |
Correct |
1 ms |
340 KB |
Output is correct |
| # |
결과 |
실행 시간 |
메모리 |
Grader output |
| 1 |
Correct |
81 ms |
9540 KB |
Output is correct |
| 2 |
Correct |
73 ms |
12436 KB |
Output is correct |
| 3 |
Correct |
74 ms |
12528 KB |
Output is correct |
| 4 |
Correct |
71 ms |
12492 KB |
Output is correct |
| 5 |
Correct |
76 ms |
12436 KB |
Output is correct |
| 6 |
Correct |
74 ms |
12208 KB |
Output is correct |
| 7 |
Correct |
84 ms |
12176 KB |
Output is correct |
| 8 |
Correct |
73 ms |
12404 KB |
Output is correct |
| 9 |
Correct |
71 ms |
12288 KB |
Output is correct |
| 10 |
Correct |
78 ms |
12376 KB |
Output is correct |
| 11 |
Correct |
68 ms |
9460 KB |
Output is correct |
| 12 |
Correct |
84 ms |
10796 KB |
Output is correct |
| 13 |
Correct |
68 ms |
10556 KB |
Output is correct |
| 14 |
Correct |
81 ms |
10756 KB |
Output is correct |
| 15 |
Correct |
67 ms |
12228 KB |
Output is correct |
| 16 |
Correct |
72 ms |
10572 KB |
Output is correct |
| 17 |
Correct |
67 ms |
10776 KB |
Output is correct |
| 18 |
Correct |
75 ms |
12516 KB |
Output is correct |
| 19 |
Correct |
73 ms |
12156 KB |
Output is correct |
| 20 |
Correct |
77 ms |
10780 KB |
Output is correct |
| 21 |
Correct |
106 ms |
9592 KB |
Output is correct |
| 22 |
Correct |
99 ms |
12448 KB |
Output is correct |
| 23 |
Correct |
114 ms |
12508 KB |
Output is correct |
| 24 |
Correct |
106 ms |
12420 KB |
Output is correct |
| 25 |
Correct |
98 ms |
12580 KB |
Output is correct |
| 26 |
Correct |
102 ms |
12664 KB |
Output is correct |
| 27 |
Correct |
103 ms |
12708 KB |
Output is correct |
| 28 |
Correct |
98 ms |
12584 KB |
Output is correct |
| 29 |
Correct |
115 ms |
12748 KB |
Output is correct |
| 30 |
Correct |
101 ms |
12668 KB |
Output is correct |
| 31 |
Correct |
0 ms |
212 KB |
Output is correct |
| 32 |
Correct |
1 ms |
340 KB |
Output is correct |
| 33 |
Correct |
0 ms |
340 KB |
Output is correct |
| 34 |
Correct |
1 ms |
340 KB |
Output is correct |
| 35 |
Correct |
0 ms |
212 KB |
Output is correct |
| 36 |
Correct |
1 ms |
340 KB |
Output is correct |
| 37 |
Correct |
1 ms |
340 KB |
Output is correct |
| 38 |
Correct |
1 ms |
212 KB |
Output is correct |
| 39 |
Correct |
1 ms |
340 KB |
Output is correct |
| 40 |
Correct |
1 ms |
212 KB |
Output is correct |
| 41 |
Correct |
1 ms |
340 KB |
Output is correct |
| 42 |
Correct |
1 ms |
212 KB |
Output is correct |
| 43 |
Correct |
1 ms |
340 KB |
Output is correct |
| 44 |
Correct |
0 ms |
340 KB |
Output is correct |
| 45 |
Correct |
1 ms |
340 KB |
Output is correct |
| 46 |
Correct |
1 ms |
332 KB |
Output is correct |
| 47 |
Correct |
1 ms |
340 KB |
Output is correct |
| 48 |
Correct |
1 ms |
332 KB |
Output is correct |
| 49 |
Correct |
1 ms |
340 KB |
Output is correct |
| 50 |
Correct |
1 ms |
340 KB |
Output is correct |
| 51 |
Correct |
1 ms |
340 KB |
Output is correct |
| 52 |
Correct |
2 ms |
340 KB |
Output is correct |
| 53 |
Correct |
1 ms |
340 KB |
Output is correct |
| 54 |
Correct |
1 ms |
340 KB |
Output is correct |
| 55 |
Correct |
1 ms |
344 KB |
Output is correct |
| 56 |
Correct |
1 ms |
340 KB |
Output is correct |
| 57 |
Correct |
2 ms |
368 KB |
Output is correct |
| 58 |
Correct |
1 ms |
340 KB |
Output is correct |
| 59 |
Correct |
1 ms |
340 KB |
Output is correct |
| 60 |
Correct |
1 ms |
340 KB |
Output is correct |
| 61 |
Correct |
135 ms |
12448 KB |
Output is correct |
| 62 |
Correct |
135 ms |
12760 KB |
Output is correct |
| 63 |
Correct |
123 ms |
12416 KB |
Output is correct |
| 64 |
Correct |
134 ms |
12660 KB |
Output is correct |
| 65 |
Correct |
138 ms |
12652 KB |
Output is correct |
| 66 |
Correct |
145 ms |
12620 KB |
Output is correct |
| 67 |
Correct |
136 ms |
12492 KB |
Output is correct |
| 68 |
Correct |
118 ms |
12676 KB |
Output is correct |
| 69 |
Correct |
121 ms |
12436 KB |
Output is correct |
| 70 |
Correct |
118 ms |
12368 KB |
Output is correct |
