[Bzoj1010] [hnoi2008] toy movement gauge for toy packing _ slope Optimization

The solution to the first slope optimization question is still not very thorough.

I can recommend a blog here.

Http://blog.sina.com.cn/s/blog_5f5353cc0100jx41.html

Well written, but it may not be clear in some places.

Let me explain my understanding of slope optimization:

It is an optimization that finds a dynamic planning question, TLE, and then it happens to be in line with some crazy nature.

In essence, it is limited to create a property, so that the dynamic equation can satisfy certain evil monotonicity and thus be greatly optimized.

How can this monotonicity be achieved?

That is to say, let's get the original transfer status and make a conversion, for example, split the square. There are two types of transfer: J-> I and K-> I, we make it have a size relationship DP [J-> I] <= DP [k-> I], and finally converts it to a formula with J and K on the left, there is an I-only expression on the right side, so that two transfer directions can be judged at high speed.Which one is strong?.

In terms of ideology, WYF gave me a well-understood version as follows.

It is it! We need to maintain a Convex Shell. When I want to join the priority queue to judge the slope of J-> I and the slope of K-> I, then K is obviously going out, then Judge H-> I. Well, I stayed. That's it.

Well, let's talk about some specific things related to this question.

First, define f [I] = sum [I] + I. This sum [I] is the prefix and.

Then, when G (j, k)/S (j, k) <= I, j is better than K.

#define G(x,y) (dp[x]-dp[y]+(f[x]+c)*(f[x]+c)-(f[y]+c)*(f[y]+c))#define S(x,y) (2*(f[x]-f[y]))

Code:

#include <cstdio>#include <cstring>#include <iostream>#include <algorithm>#define N 55000#define G(x,y) (dp[x]-dp[y]+(f[x]+c)*(f[x]+c)-(f[y]+c)*(f[y]+c))#define S(x,y) (2*(f[x]-f[y]))using namespace std;long long c,sum[N],f[N],dp[N];int n,q[N];int main(){//freopen("test.in","r",stdin);scanf("%d%lld",&n,&c);c++;for(int i=1;i<=n;i++){scanf("%lld",&sum[i]);sum[i]+=sum[i-1];f[i]=sum[i]+i;}int head=1,tail=1;q[1]=0;for(int i=1;i<=n;i++){while(head<tail&&G(q[head+1],q[head])<=S(q[head+1],q[head])*f[i]) head++;dp[i]=dp[q[head]]+(f[i]-f[q[head]]-c)*(f[i]-f[q[head]]-c);while(head<tail&&G(i,q[tail])*S(q[tail],q[tail-1])<=G(q[tail],q[tail-1])*S(i,q[tail]))tail--;q[++tail]=i;}printf("%lld\n",dp[n]);return 0;}

[Bzoj1010] [hnoi2008] toy movement gauge for toy packing _ slope Optimization