Codeforces 1083E The Fair Nut and Rectangles dp + 斜率优化

The Fair Nut and Rectangles

很裸的斜率优化呀。  

维护直线

#include<bits/stdc++.h>
#define LL long long
#define LD long double
#define ull unsigned long long
#define fi first
#define se second
#define mk make_pair
#define PLL pair<LL, LL>
#define PLI pair<LL, int>
#define PII pair<int, int>
#define SZ(x) ((int)x.size())
#define ALL(x) (x).begin(), (x).end()
#define fio ios::sync_with_stdio(false); cin.tie(0);
 
using namespace std;
 
const int N = 1e6 + 7;
const int inf = 0x3f3f3f3f;
const LL INF = 0x3f3f3f3f3f3f3f3f;
const int mod = 1e9 + 7;
const double eps = 1e-8;
const double PI = acos(-1);
 
template<class T, class S> inline void add(T &a, S b) {a += b; if(a >= mod) a -= mod;}
template<class T, class S> inline void sub(T &a, S b) {a -= b; if(a < 0) a += mod;}
template<class T, class S> inline bool chkmax(T &a, S b) {return a < b ? a = b, true : false;}
template<class T, class S> inline bool chkmin(T &a, S b) {return a > b ? a = b, true : false;}
 
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
 
namespace LC {
/**
 * Description: Container where you can add lines of the form kx+m, and query maximum values at points x.
 * Time: O(log N)
 */
 
struct Line {
    mutable LL k, m, p;
    bool operator < (const Line& o) const { return k < o.k; }
    bool operator < (LL x) const { return p < x; }
};
 
struct LineContainer : multiset<Line, less<>> {
    // (for doubles, use inf = 1/.0, div(a,b) = a/b)
    const LL inf = LLONG_MAX;
    LL div(LL a, LL b) { // floored division
        return a / b - ((a ^ b) < 0 && a % b);
    }
    bool isect(iterator x, iterator y) {
        if (y == end()) { x->p = inf; return false; }
        if (x->k == y->k) x->p = x->m > y->m ? inf : -inf;
        else x->p = div(y->m - x->m, x->k - y->k);
        return x->p >= y->p;
    }
    void add(LL k, LL m) {
        auto z = insert({k, m, 0}), y = z++, x = y;
        while (isect(y, z)) z = erase(z);
        if (x != begin() && isect(--x, y)) isect(x, y = erase(y));
        while ((y = x) != begin() && (--x)->p >= y->p)
            isect(x, erase(y));
    }
    LL query(LL x) {
        assert(!empty());
        auto l = *lower_bound(x);
        return l.k * x + l.m;
    }
};
}
 
int n;
LL dp[N];
 
struct Node {
    int x, y;
    LL v;
    bool operator < (const Node &rhs) const {
        return x < rhs.x;
    }
} a[N];
 
int main() {
 
    scanf("%d", &n);
    for(int i = 1; i <= n; i++) {
        scanf("%d%d%lld", &a[i].x, &a[i].y, &a[i].v);
    }
 
    sort(a + 1, a + 1 + n);
 
    LC::LineContainer cont;
    cont.add(0, 0);
 
    LL ans = 0;
 
    for(int i = 1; i <= n; i++) {
        int x = a[i].x, y = a[i].y;
        LL v = a[i].v;
        dp[i] = cont.query(y) + 1LL * x * y - v;
        cont.add(-x, dp[i]);
        chkmax(ans, dp[i]);
    }
 
    printf("%lld
", ans);
 
    return 0;
}
 
/*
*/

维护凸包

#include<bits/stdc++.h>
#define LL long long
#define LD long double
#define ull unsigned long long
#define fi first
#define se second
#define mk make_pair
#define PLL pair<LL, LL>
#define PLI pair<LL, int>
#define PII pair<int, int>
#define SZ(x) ((int)x.size())
#define ALL(x) (x).begin(), (x).end()
#define fio ios::sync_with_stdio(false); cin.tie(0);

using namespace std;

const int N = 1e6 + 7;
const int inf = 0x3f3f3f3f;
const LL INF = 0x3f3f3f3f3f3f3f3f;
const int mod = 1e9 + 7;
const double eps = 1e-8;
const double PI = acos(-1);

template<class T, class S> inline void add(T &a, S b) {a += b; if(a >= mod) a -= mod;}
template<class T, class S> inline void sub(T &a, S b) {a -= b; if(a < 0) a += mod;}
template<class T, class S> inline bool chkmax(T &a, S b) {return a < b ? a = b, true : false;}
template<class T, class S> inline bool chkmin(T &a, S b) {return a > b ? a = b, true : false;}

mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());

int n;
int que[N], be, ed;
LL dp[N];

struct Node {
    int x, y;
    LL v;
    bool operator < (const Node &rhs) const {
        return x < rhs.x;
    }
} a[N];

LD calc(int i, int j) {
    return ((LD)dp[j] - dp[i]) / (a[j].x - a[i].x);
}

int main() {

    scanf("%d", &n);
    for(int i = 1; i <= n; i++) {
        scanf("%d%d%lld", &a[i].x, &a[i].y, &a[i].v);
        a[i].v = 1LL * a[i].x * a[i].y - a[i].v;
    }

    sort(a + 1, a + 1 + n);

    be = 1; ed = 0;
    que[++ed] = 0;

    LL ans = 0;
    for(int i = 1; i <= n; i++) {
        int x = a[i].x, y = a[i].y;
        LL v = a[i].v;
        while(ed - be >= 1 && calc(que[be + 1], que[be]) >= y) be++;
        int id = que[be];
        dp[i] = dp[id] + - 1LL * y * a[id].x + v;
        while(ed - be >= 1 && calc(que[ed], que[ed - 1]) <= calc(i, que[ed])) ed--;
        que[++ed] = i;
        chkmax(ans, dp[i]);
    }

    printf("%lld
", ans);

    return 0;
}

/*
*/