链接
题目传送门:http://www.lydsy.com/JudgeOnline/problem.php?id=4540
数据生成器:http://paste.ubuntu.com/23631812/
神犇题解-线段树:http://blog.csdn.net/qq_34637390/article/details/51313126
神犇题解-莫队:https://blog.sengxian.com/solutions/bzoj-4540
解题报告
1. 莫队做法
我们只需要考虑如何O(1)进行转移。
因为是转移,所以一端是固定的,不妨设固定的是左端。
现在从左向右考虑右端点,不难发现:最小值不增(如下图所示)
通过观察我们还可以发现:除了第④段,其他几段的长度都是固定的
于是我们先预处理一下相关信息,对于第④段这种进行特殊处理。
因为第④段这货是l~r中的最小值,所以我们搞一个RMQ就可以特殊处理啦!
这样一来,转移就可以做到O(n),于是就可以上O(n^1.5)的莫队辣!
2. 线段树做法
因为此题和BZOJ_4262 一毛一样,所以就不单独写了。
这里有一份很好的题解,强烈推荐:http://blog.csdn.net/qq_34637390/article/details/51313126
Code (莫队version)
#include<bits/stdc++.h> #define LL long long using namespace std; const int N = 100000+9; const int INF = 2e9; struct Query{ int l,r,blk,num; inline bool operator < (const Query &B) const { return blk < B.blk || (blk == B.blk && r < B.r); } }q[N]; int n,m,sqr,top,val[N],stk[N],pos[N],lft[N],rit[N]; LL vout[N],sum_right[N],sum_left[N]; inline int read(){ char c=getchar(); int ret=0,f=1; while (c<'0'||c>'9') {if(c=='-')f=-1;c=getchar();} while (c<='9'&&c>='0') {ret=ret*10+c-'0';c=getchar();} return ret*f; } namespace Sparse_Table{ #define ST Sparse_Table int sur[N*3][20],arr[N*3][20],POW[20],log[N]; inline void init() { POW[0] = 1; for (int i=1;i<=17;i++) POW[i] = POW[i-1] << 1; for (int i=1,w=1;i<=n;w=1,++i) while (w * 2 <= i) w <<= 1, log[i]++; for (int i=1;i<=n;i++) { arr[i][0] = val[i]; sur[i][0] = i; } for (int j=1,cur=1;j<=17;j++,cur<<=1) { for (int i=1;i<=n;i++) { if (arr[i][j-1] < arr[i+cur][j-1]) { arr[i][j] = arr[i][j-1]; sur[i][j] = sur[i][j-1]; } else { arr[i][j] = arr[i+cur][j-1]; sur[i][j] = sur[i+cur][j-1]; } } } } inline int query(int l, int r) { int w = log[r-l+1]; if (arr[l][w] < arr[r-POW[w]+1][w]) return sur[l][w]; else return sur[r-POW[w]+1][w]; } }; inline LL move_right(int l, int r) { int p = ST::query(l, r); return sum_left[r] - sum_left[p] + (LL)val[p] * (p - l + 1); } inline LL move_left(int l, int r) { int p = ST::query(l, r); return sum_right[l] - sum_right[p] + (LL)val[p] * (r - p + 1); } inline void prework() { stk[top = 0] = -INF; for (int i=1;i<=n;i++) { while (stk[top] > val[i]) top--; lft[i] = pos[top]; sum_left[i] = sum_left[pos[top]] + (LL)(i-pos[top]) * val[i]; stk[++top] = val[i]; pos[top] = i; } pos[top = 0] = n+1; for (int i=n;i;i--) { while (stk[top] >= val[i]) top--; rit[i] = pos[top]; sum_right[i] = sum_right[pos[top]] + (LL)(pos[top]-i) * val[i]; stk[++top] = val[i]; pos[top] = i; } ST::init(); } int main(){ n = read(); m = read(); sqr = sqrt(n); for (int i=1;i<=n;i++) val[i] = read(); prework(); for (int i=1;i<=m;i++) { q[i].num = i; q[i].l = read(); q[i].r = read(); q[i].blk = q[i].l / sqr; } sort(q+1, q+1+m); LL cur = val[1]; for (int k=1,l=1,r=1;k<=m;k++) { while (r < q[k].r) cur += move_right(l,r+1), r++; while (r > q[k].r) cur -= move_right(l,r), r--; while (l < q[k].l) cur -= move_left(l,r), l++; while (l > q[k].l) cur += move_left(l-1,r), l--; vout[q[k].num] = cur; } for (int i=1;i<=m;i++) printf("%lld\n",vout[i]); return 0; }
ps:这一份代码并不能处理值为负数的情况,至于为什么:我也普吉岛 QwQ
—————————— UPD 2017.2.2 ——————————
这题今天重新想的时候,想到了一个$O(nlog^2n)$的算法
虽然时间复杂度稍高,但思维复杂度和编程复杂度都相对较低
设$l_i$为第i个数离左边第一个比他小的数的距离,$r_i$同理
于是询问整个序列的话,答案就是 $\sum\limits_{i = 1}^n {{a_i} \times {l_i} \times {r_i}} $
然后仍然考虑离线,将询问排序依次做。
对于每一个数,其贡献有三种情况:
1. $l_i$,$r_i$都没有被询问限制
2. 有一个被限制
3. 有两个被限制
于是我们就可以搞一个树套树套树,把这三种情况分别计算即可
复杂度:$O(nlog^3n)$
但我们考虑离线之后,右端点逐渐右移,于是我们搞一个队列什么的维护一下
然后就搞一个树套树维护左端点就好,复杂度$O(nlog^2n)$
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