斜率优化
套路:把转移方程推到只有关\(i,j\),与\(i,j\)的乘积时(由\(j\)转移到\(i\)),把只与\(i\)有关的量放到一起作截距\(B\),只与\(j\)相关的量作已知点的纵坐标\(Y\),\(i,j\)的乘积中常量与\(i\)相关量的乘积作斜率\(K\),\(j\)相关量作已知点的横坐标\(X\),再进行线性规划
1.P2365任务安排\((O(n^2)\)可过题)
一批任务的准备时间会影响从这批任务开始的所有任务的结束时间
[BZOJ2726][SDOI2012]任务安排(斜率优化+\(cdq\)分治)cdq分治我们不熟
变形一下
令$$X_j=s_j$$ $$Y_j=f_j-as_j^2+bs_j$$
\(\because a < 0\) ,\(\therefore K_i < 0\), 且\(K\)递增,要使\(B\)最大,维护上凸包
double eps=1e-7;
int dcmp(double x){if(fabs(x) <= eps) return 0; return x > 0 ? 1 : -1;}
double Y(int x){return f[x]+a*s[x]*s[x]-b*s[x];}
double X(int x){return s[x];}
double slope(int i, int j){
double x1=X(i), y1=Y(i), x2=X(j), y2=Y(j);
return (y1-y2)/(x1-x2);
}
int q[Maxn], l, r;
void solve(){
n=read(); a=read(), b=read(), c=read(); for(int i=1; i <= n; i++) s[i]=s[i-1]+read();
l=r=1, q[1]=0;
for(int i=1; i <= n; i++){
while(l < r && dcmp(slope(q[l], q[l+1]) - 1.0*a*2*s[i]) > 0) l++;
f[i]=f[q[l]]+a*(s[i]-s[q[l]])*(s[i]-s[q[l]])+b*(s[i]-s[q[l]])+c;
while(l < r && dcmp(slope(q[r-1], q[r])-slope(q[r], i) <= 0)) r--;
q[++r]=i;
}
printf("%.0lf", f[n]);
}
3.[HNOI2008]玩具装箱TOY
令\(S_i=\sum_{j=1}^iC_j\),\(P_i=S_i+i\)
则$$f_i=min_{0\le j<i}{f_j+(P_i-P_j-1-L)^2}$$
令$$X_j=P_j$$ $$Y_j=f_j+P_j^2$$
\(\because P_i > 0,\therefore K_i\)递增,要使\(B_i\)最小,维护下凸包
#define a(i) (sum[i]+i)
#define b(i) (a(i)+L+1)
#define Y(i) (dp[i]+b(i)*b(i))
#define X(i) b(i)
#define slope(i, j) ((Y(i)-Y(j))/(X(i)-X(j)))
int q[N], head, tail;
void solve(){
n=read(), L=read(); head=tail=1;
for(int i=1; i <= n; i++){
sum[i]=sum[i-1]+read();
while(head < tail && slope(q[head], q[head+1]) < 2*a(i)) head++;
dp[i]=dp[q[head]]+(a(i)-b(q[head]))*(a(i)-b(q[head]));
while(head < tail && slope(i, q[tail-1]) < slope(q[tail-1], q[tail])) tail--;
q[++tail]=i;
}
printf("%lld\n", dp[n]);
}
4.[SDOI2016]征途
令\(S_i\)为每天的路程长度,\(sum\)为总路程,先推\(vm^2\)
所以令\(f_{(i, j)}\)为前\(i\)段路分为\(j\)天走的最小\(\sum_{i=1}^jS_i\)
不管\(j\),
令$$X_k=Sum_k$$ $$Y_k=f_k+Sum_k^2$$
令\(B_i\)最小,\(\because K_i\)递增,\(\therefore\)维护下凸包
int q[Maxn], l, r;
ll *f=f1, *g=g1;
double X(int i) {return S[i];}
double Y(int i) {return g[i]+S[i]*S[i];}
double slope(int i, int j){
double x1=X(i), y1=Y(i), x2=X(j), y2=Y(j);
return (y1-y2)/(x1-x2);
}
void solve(){
n=read(), m=read(); for(int i=1; i <= n; i++) S[i]=S[i-1]+read(), f[i]=S[i]*S[i];
for(int i=1; i < m; i++){
l=r=1; q[1]=i; swap(f, g);
for(int j=i+1; j <= n; j++){
while(l < r && slope(q[l], q[l+1]) <= 2.0*S[j]) l++;
f[j]=g[q[l]]+(S[j]-S[q[l]])*(S[j]-S[q[l]]);
while(l < r && slope(q[r], q[r-1]) >= slope(q[r], j)) r--;
q[++r]=j;
}
}
printf("%lld", f[n]*m-S[n]*S[n]);
}
5.[CF311B]Cats Transport
令 $$d_i=\sum_{j=1}^iD_j, A_i=T_i-d_i$$
则\(A_i\)为接到猫\(i\)的最早出发时刻,若出发时刻为\(t\),则猫等待的时间为\(t-A_i\),把猫按\(A\)从小到大排序,令\(S_i=\sum_{j=1}^iA_i\)
,则有
每个\(feeder\)接到的为连续一段的猫最优
证明略(懒)
不管\(j\)
令$$X_k=k$$ $$Y_k=f'_k+S_k$$
要令\(B\)最小,\(\because K\)递增,\(\therefore\)维护下凸包
ll *f=f1, *g=g1;
int q[Maxn], l, r;
double X(int i){return i;}
double Y(int i){return g[i]+S[i];}
double slope(int i, int j){
double x1=X(i), x2=X(j), y1=Y(i), y2=Y(j);
return (y2-y1)/(x2-x1);
}
void solve(){
n=read(), m=read(), p=read();
for(int i=2; i <= n; i++) D[i]=D[i-1]+read();
for(int i=1, x, t; i <= m; i++) x=read(), t=read(), A[i]=t-D[x];
sort(A+1, A+1+m); for(int i=1; i <= m; i++) S[i]=S[i-1]+A[i];
for(int i=1; i <= m; i++) f[i]=A[i]*i-S[i];
for(int i=1; i < p; i++){
q[1]=0; l=r=1; swap(f, g);
for(int j=1; j <= m; j++){
while(l < r && slope(q[l], q[l+1]) <= A[j]) l++;
f[j]=g[q[l]]+(j-q[l])*A[j]+S[q[l]]-S[j];
while(l < r && slope(q[r], q[r-1]) >= slope(q[r], j)) r--;
q[++r]=j;
}
}
printf("%lld", f[m]);
}
6.[POJ3709]K-Anonymous Sequence
- 题意:求通过减小序列中元素的方式把给定序列\(a[]\)变为\(K-Anonymous\)序列的最小代价
- 定义:
1. \(K-Anonymous\):对于\(\forall s \in a[]\),至少存在其它\(k-1\)个元素与它相等
2. 代价:把元素\(s\)变为\(s'\)的代价为\(s-s'\)
令$$X_j=a_{j+1}$$ $$Y_j=f_j-S_j+ja_{j+1}$$
要使\(B\)最小,\(\because K\)递增,\(\therefore\)维护下凸包(我不知道为什么用\(double\)求\(slope\)会被卡)
ll Y(int i){return f[i]-S[i]+a[i]*i;}
int q[Maxn], l, r;
void solve(){
int T=read();
while(T--){
n=read(), k=read(); for(int i=0; i < n; i++) a[i]=read();
for(int i=1; i <= n; i++) S[i]=S[i-1]+a[i-1];
l=r=1, q[1]=0;
for(int i=1; i <= n; i++){
while(l < r && (a[q[l+1]]-a[q[l]])*i >= Y(q[l+1])-Y(q[l])) l++;
f[i]=f[q[l]]+S[i]-S[q[l]]+a[q[l]]*(q[l]-i);
if(i-k+1 >= k) {
while(l < r && (a[q[r-1]]-a[q[r]])*(Y(q[r])-Y(i-k+1)) <= (a[q[r]]-a[i-k+1])*(Y(q[r-1])-Y(q[r]))) r--;
q[++r]=i-k+1;
}
}
printf("%lld\n", f[n]);
}
}