一 原题
Alice likes snow a lot! Unfortunately, this year's winter is already over, and she can't expect to have any more of it. Bob has thus bought her a gift — a large snow maker. He plans to make some amount of snow every day. On day i he will make a pile of snow of volume Vi and put it in her garden.
Each day, every pile will shrink a little due to melting. More precisely, when the temperature on a given day is Ti, each pile will reduce its volume by Ti. If this would reduce the volume of a pile to or below zero, it disappears forever. All snow piles are independent of each other.
Note that the pile made on day i already loses part of its volume on the same day. In an extreme case, this may mean that there are no piles left at the end of a particular day.
You are given the initial pile sizes and the temperature on each day. Determine the total volume of snow melted on each day.
The first line contains a single integer N (1 ≤ N ≤ 105) — the number of days.
The second line contains N integers V1, V2, ..., VN (0 ≤ Vi ≤ 109), where Vi is the initial size of a snow pile made on the day i.
The third line contains N integers T1, T2, ..., TN (0 ≤ Ti ≤ 109), where Ti is the temperature on the day i.
Output a single line with N integers, where the i-th integer represents the total volume of snow melted on day i.
3 10 10 5 5 7 2
5 12 4
5 30 25 20 15 10 9 10 12 4 13
9 20 35 11 25
In the first sample, Bob first makes a snow pile of volume 10, which melts to the size of 5 on the same day. On the second day, he makes another pile of size 10. Since it is a bit warmer than the day before, the first pile disappears completely while the second pile shrinks to 3. At the end of the second day, he has only a single pile of size 3. On the third day he makes a smaller pile than usual, but as the temperature dropped too, both piles survive till the end of the day.
二 分析
题意:每天给你体积为V(i)的一堆雪,每天所有的雪堆会融化T(i)(体积不会为负)。求1-N天内每天融化的雪的体积。1=<N<=1e5,0<=V(i), T(i) <= 1e9
思路:暴力的做法是O(n^2)的。一个更好的做法是:维护T的前缀和Pre,Pre(i)=T(1)+T(2)+...+T(i-1)。对于第i天新增的雪堆,不妨假设它第一天就存在,那么它的初始体积为V'(i)=V(i)+Pre(i-1),在第k天(k>=i)如果V'(i)<=Pre(k),那么这堆雪融化完,在第k天它融化的体积为V'(i)-Pre(k-1),否则就是T(k)。用一个multiset维护目前所有的V',总复杂度O(n*lgn)。
三 代码
#include <cstdio> #include <set> #define LL long long using std::multiset; const int maxn = 1e5 + 10; int n, v[maxn], t[maxn]; LL ans = 0, pre[maxn]; multiset<LL> s; int main() { scanf("%d", &n); for (int i = 1; i <= n; i++) scanf("%d", &v[i]); for (int i = 1; i <= n; i++) scanf("%d", &t[i]); for (int i = 1; i <= n; i++) { ans = 0; s.insert(v[i] + pre[i - 1]); pre[i] = pre[i - 1] + t[i]; while (!s.empty() && *s.begin() <= pre[i]) { ans += *s.begin() - pre[i - 1]; s.erase(s.begin()); } ans += s.size() * t[i]; printf("%lld ", ans); } return 0; }