Codeforces Round #343 (Div. 2) D. Babaei and Birthday Cake（dp、BIT）

题意:

给定N≤105个蛋糕,编号为1∼N,每个都有体积Vi

任意一个蛋糕都可以放在桌子上,对于蛋糕对(i,j),i可以放在j上,当且仅当i>j且Vi≥Vj

求能摆放的蛋糕的最大体积和

分析:

LIS模型的经典水题了

f[i]表示第i个蛋糕在最上面的最大体积和

f[i]=max{f[j]}+Vi,j<i且Vj<Vi

考虑用BIT优化一下这个转移,离散化一下,记录当前体积蛋糕在最上面的最大体积和,然后就可以O(logn)转移了

ans=max{f[i]},时间复杂度为O(nlogn)

代码:

//
//  Created by TaoSama on 2016-02-23
//  Copyright (c) 2016 TaoSama. All rights reserved.
//
#pragma comment(linker, "/STACK:1024000000,1024000000")
#include <algorithm>
#include <cctype>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <iomanip>
#include <iostream>
#include <map>
#include <queue>
#include <string>
#include <set>
#include <vector>

using namespace std;
#define pr(x) cout << #x << " = " << x << "  "
#define prln(x) cout << #x << " = " << x << endl
const int N = 1e5 + 10, INF = 0x3f3f3f3f, MOD = 1e9 + 7;

typedef long long LL;
int n;
LL v
;

struct BIT {
int n; LL b
;
void init(int _n) {
n = _n;
memset(b, 0, sizeof b);
}
void add(int i, LL v) {
for(; i <= n; i += i & - i) b[i] = max(b[i], v);
}
LL sum(int i) {
LL ret = 0;
for(; i; i -= i & -i) ret = max(ret, b[i]);
return ret;
}
} bit;

int main() {
#ifdef LOCAL
freopen("C:\\Users\\TaoSama\\Desktop\\in.txt", "r", stdin);
//  freopen("C:\\Users\\TaoSama\\Desktop\\out.txt","w",stdout);
#endif
ios_base::sync_with_stdio(0);

while(scanf("%d", &n) == 1) {
vector<LL> xs;
for(int i = 1; i <= n; ++i) {
int r, h; scanf("%d%d", &r, &h);
v[i] = 1LL * r * r * h;
xs.push_back(v[i]);
}
sort(xs.begin(), xs.end());
xs.resize(unique(xs.begin(), xs.end()) - xs.begin());

LL ans = 0;
bit.init(xs.size());
for(int i = 1; i <= n; ++i) {
int idx = lower_bound(xs.begin(), xs.end(), v[i]) - xs.begin() + 1;
LL cur = bit.sum(idx - 1) + v[i];
ans = max(ans, cur);
bit.add(idx, cur);
}
printf("%.12f\n", ans * acos(-1));
}
return 0;
}