P1404 平均数（二分/斜率优化）
先说说二分的思路：
对数列中每个数字都减去当前二分的答案，然后求出前缀和，如果前缀和在某个位置加上前M个的最小值大于0，那么就可以更新答案
事实上，减去了当前二分的答案之后，就相当于在与这一段区间都为二分答案的序列互相比较了，剩下只需要维护前M个最小值即可
注意输出答案不要四舍五入，并且要乘以1000，那么就给所有数字乘上10000，最后除以10即可
代码：
#include <bits/stdc++.h> #define int long long #define sc(a) scanf("%lld",&a) #define scc(a,b) scanf("%lld %lld",&a,&b) #define sccc(a,b,c) scanf("%lld %lld %lld",&a,&b,&c) #define scs(a) scanf("%s",a) #define schar(a) scanf("%c",&a) #define pr(a) printf("%lld",a) #define fo(i,a,b) for(int i=a;i<b;++i) #define re(i,a,b) for(int i=a;i<=b;++i) #define rfo(i,a,b) for(int i=a;i>b;--i) #define rre(i,a,b) for(int i=a;i>=b;--i) #define prn() printf("\n") #define prs() printf(" ") #define mkp make_pair #define pii pair<int,int> #define pub(a) push_back(a) #define pob() pop_back() #define puf(a) push_front(a) #define pof() pop_front() #define fst first #define snd second #define frt front() #define bak back() #define mem0(a) memset(a,0,sizeof(a)) #define memmx(a) memset(a,0x3f3f,sizeof(a)) #define memmn(a) memset(a,-0x3f3f,sizeof(a)) #define debug #define db double #define yyes cout<<"YES"<<endl; #define nno cout<<"NO"<<endl; #define all(i,a) for(auto i=a.begin();i!=a.end();++i) using namespace std; typedef vector<int> vei; typedef vector<pii> vep; typedef map<int,int> mpii; typedef map<char,int> mpci; typedef map<string,int> mpsi; typedef deque<int> deqi; typedef deque<char> deqc; typedef priority_queue<int> mxpq; typedef priority_queue<int,vector<int>,greater<int> > mnpq; typedef priority_queue<pii> mxpqii; typedef priority_queue<pii,vector<pii>,greater<pii> > mnpqii; const int maxn=500005; const int inf=0x3f3f3f3f3f3f3f3f; const int MOD=100000007; const db eps=1e-10; const db pi=3.1415926535; int qpow(int a,int b){int tmp=a%MOD,ans=1;while(b){if(b&1){ans*=tmp,ans%=MOD;}tmp*=tmp,tmp%=MOD,b>>=1;}return ans;} int lowbit(int x){return x&-x;} int max(int a,int b){return a>b?a:b;} int min(int a,int b){return a<b?a:b;} int mmax(int a,int b,int c){return max(a,max(b,c));} int mmin(int a,int b,int c){return min(a,min(b,c));} void mod(int &a){a+=MOD;a%=MOD;} bool chk(int now); int half(int l,int r){while(l<=r){int m=(l+r)/2;if(chk(m))l=m+1;else r=m-1;}return l;} int ll(int p){return p<<1;} int rr(int p){return p<<1|1;} int mm(int l,int r){return (l+r)/2;} int lg(int x){if(x==0) return 1;return (int)log2(x)+1;} bool smleql(db a,db b){if(a<b||fabs(a-b)<=eps)return true;return false;} bool bigeql(db a,db b){if(a>b||fabs(a-b)<=eps)return true;return false;} bool eql(db a,db b){if(fabs(a-b)<eps) return 1;return 0;} db len(db a,db b,db c,db d){return sqrt((a-c)*(a-c)+(b-d)*(b-d));} bool isp(int x){if(x==1)return false;if(x==2)return true;for(int i=2;i*i<=x;++i)if(x%i==0)return false;return true;} inline int read(){ char ch=getchar();int s=0,w=1; while(ch<48||ch>57){if(ch=='-')w=-1;ch=getchar();} while(ch>=48&&ch<=57){s=(s<<1)+(s<<3)+ch-48;ch=getchar();} return s*w; } inline void write(int x){ if(x<0)putchar('-'),x=-x; if(x>9)write(x/10); putchar(x%10+48); } int gcd(int a, int b){ if(a==0) return b; if(b==0) return a; if(!(a&1)&&!(b&1)) return gcd(a>>1,b>>1)<<1; else if(!(b&1)) return gcd(a,b>>1); else if(!(a&1)) return gcd(a>>1,b); else return gcd(abs(a-b),min(a,b)); } int lcm(int x,int y){return x*y/gcd(x,y);} int n,m,a[maxn],s[maxn]; bool chk(int now){ int mn=inf; re(i,1,n){ s[i]=s[i-1]+a[i]-now; if(i>=m){ mn=min(mn,s[i-m]); if(s[i]>mn) return 1; } } return 0; } signed main(){ ios_base::sync_with_stdio(0); cin.tie(0),cout.tie(0); cin>>n>>m; re(i,1,n) cin>>a[i],a[i]*=10000; cout<<half(1,1e9)/10; return 0; }
事实上本题还可以利用斜率，考虑到平均数可以写成：(s[j]-s[i-1])/(j-(i-1))的形式，其中s是前缀和
不难发现这就是两个坐标点的二维坐标，意味着我们现在要求出j-i+1>=m的最大斜率
考虑到这题不存在斜率无穷大的情况，实际处理起来还要方便许多
注意到两个步骤：删除上凸线、维护最大斜率，分别对应了出队尾和出队首
显然发现相对于XY段，YZ一定不能起到更新答案的作用，那么就弹出YZ段。
此外每次还要不断弹出队首，直到找到最大斜率点为止。
代码：
#include <bits/stdc++.h> #define int long long #define sc(a) scanf("%lld",&a) #define scc(a,b) scanf("%lld %lld",&a,&b) #define sccc(a,b,c) scanf("%lld %lld %lld",&a,&b,&c) #define scs(a) scanf("%s",a) #define schar(a) scanf("%c",&a) #define pr(a) printf("%lld",a) #define fo(i,a,b) for(int i=a;i<b;++i) #define re(i,a,b) for(int i=a;i<=b;++i) #define rfo(i,a,b) for(int i=a;i>b;--i) #define rre(i,a,b) for(int i=a;i>=b;--i) #define prn() printf("\n") #define prs() printf(" ") #define mkp make_pair #define pii pair<int,int> #define pub(a) push_back(a) #define pob() pop_back() #define puf(a) push_front(a) #define pof() pop_front() #define fst first #define snd second #define frt front() #define bak back() #define mem0(a) memset(a,0,sizeof(a)) #define memmx(a) memset(a,0x3f3f,sizeof(a)) #define memmn(a) memset(a,-0x3f3f,sizeof(a)) #define debug #define db double #define yyes cout<<"YES"<<endl; #define nno cout<<"NO"<<endl; #define all(i,a) for(auto i=a.begin();i!=a.end();++i) using namespace std; typedef vector<int> vei; typedef vector<pii> vep; typedef map<int,int> mpii; typedef map<char,int> mpci; typedef map<string,int> mpsi; typedef deque<int> deqi; typedef deque<char> deqc; typedef priority_queue<int> mxpq; typedef priority_queue<int,vector<int>,greater<int> > mnpq; typedef priority_queue<pii> mxpqii; typedef priority_queue<pii,vector<pii>,greater<pii> > mnpqii; const int maxn=500005; const int inf=0x3f3f3f3f3f3f3f3f; const int MOD=100000007; const db eps=1e-10; const db pi=3.1415926535; int qpow(int a,int b){int tmp=a%MOD,ans=1;while(b){if(b&1){ans*=tmp,ans%=MOD;}tmp*=tmp,tmp%=MOD,b>>=1;}return ans;} int lowbit(int x){return x&-x;} int max(int a,int b){return a>b?a:b;} int min(int a,int b){return a<b?a:b;} int mmax(int a,int b,int c){return max(a,max(b,c));} int mmin(int a,int b,int c){return min(a,min(b,c));} void mod(int &a){a+=MOD;a%=MOD;} bool chk(int now){} int half(int l,int r){while(l<=r){int m=(l+r)/2;if(chk(m))r=m-1;else l=m+1;}return l;} int ll(int p){return p<<1;} int rr(int p){return p<<1|1;} int mm(int l,int r){return (l+r)/2;} int lg(int x){if(x==0) return 1;return (int)log2(x)+1;} bool smleql(db a,db b){if(a<b||fabs(a-b)<=eps)return true;return false;} bool bigeql(db a,db b){if(a>b||fabs(a-b)<=eps)return true;return false;} bool eql(db a,db b){if(fabs(a-b)<eps) return 1;return 0;} db len(db a,db b,db c,db d){return sqrt((a-c)*(a-c)+(b-d)*(b-d));} bool isp(int x){if(x==1)return false;if(x==2)return true;for(int i=2;i*i<=x;++i)if(x%i==0)return false;return true;} inline int read(){ char ch=getchar();int s=0,w=1; while(ch<48||ch>57){if(ch=='-')w=-1;ch=getchar();} while(ch>=48&&ch<=57){s=(s<<1)+(s<<3)+ch-48;ch=getchar();} return s*w; } inline void write(int x){ if(x<0)putchar('-'),x=-x; if(x>9)write(x/10); putchar(x%10+48); } int gcd(int a, int b){ if(a==0) return b; if(b==0) return a; if(!(a&1)&&!(b&1)) return gcd(a>>1,b>>1)<<1; else if(!(b&1)) return gcd(a,b>>1); else if(!(a&1)) return gcd(a>>1,b); else return gcd(abs(a-b),min(a,b)); } int lcm(int x,int y){return x*y/gcd(x,y);} int n,m,a,s[maxn]; deque<int> q; db k(int x,int y){ return (s[y]-s[x]+0.0)/(y-x); } signed main(){ ios_base::sync_with_stdio(0); cin.tie(0),cout.tie(0); cin>>n>>m; re(i,1,n){ cin>>a; s[i]=s[i-1]+a; } db ans=-inf; re(i,m,n){ #define l q.size() while(l>=2&&k(i-m,q[l-1])<k(i-m,q[l-2])) q.pob(); q.pub(i-m); while(l>=2&&k(i,q[0])<k(i,q[1])) q.pof(); ans=max(ans,k(i,q.frt)); #undef l } cout<<(int)floor(ans*1000); return 0; }
