BZOJ1911: [Apio2010]特别行动队(dp 斜率优化)

题意

题目链接

sol

裸的斜率优化，注意推导过程中的符号问题。

#include<bits/stdc++.h> 
#define pair pair<int, int>
#define mp(x, y) make_pair(x, y)
#define fi first
#define se second
#define int long long 
#define ll long long 
#define fin(x) {freopen(#x".in","r",stdin);}
#define fout(x) {freopen(#x".out","w",stdout);}
using namespace std;
const int maxn = 1e6 + 10, mod = 1e9 + 7, inf = 6e18 + 10;
const double eps = 1e-9;
template <typename y> inline void chmin(y &a, y b){a = (a < b ? a : b);}
template <typename y> inline void chmax(y &a, y b){a = (a > b ? a : b);}
template <typename y> inline void debug(y a){cout << a << '\n';}
template <typename y> inline ll sqr(y x){return 1ll * x * x;}
int add(int x, int y) {if(x + y < 0) return x + y + mod; return x + y >= mod ? x + y - mod : x + y;}
void add2(int &x, int y) {if(x + y < 0) x = x + y + mod; else x = (x + y >= mod ? x + y - mod : x + y);}
int mul(int x, int y) {return 1ll * x * y % mod;}
inline int read() {
    char c = getchar(); int x = 0, f = 1;
    while(c < '0' || c > '9') {if(c == '-') f = -1; c = getchar();}
    while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar();
    return x * f;
}
int n, s[maxn], a, b, c, f[maxn], q[maxn];
int f(int x) {
    return a * sqr(x) + b * x + c;
}
double y(int x) {
    return a * sqr(s[x]) - b * s[x] + f[x];
}
double x(int x) {
    return s[x];
}
double slope(int a, int b) {
//  debug((y(b) - y(a)) / (x(b) - x(a)));
    return double(y(b) - y(a)) / (x(b) - x(a));
}
signed main() {
    n = read(); a = read(); b = read(); c = read();
    for(int i = 1; i <= n; i++) s[i] = s[i - 1] + read();
    q[1] = 0;
    for(int i = 1, h = 1, t = 1; i <= n; i++) {
        while(h < t && slope(q[h], q[h + 1]) > 2 * a * s[i]) h++;
        f[i] = f[q[h]] + f(s[i] - s[q[h]]);
        //printf("%d\n", q[h]);
        while(h < t && slope(q[t - 1], q[t]) < slope(q[t], i)) t--;
        q[++t] = i;
    }
    cout << f[n];
    return 0;
}
/*
7
-1 160 -2000
14 82 61 85 41 10 34
*/