hdu 4701 Game 博弈论

思路：

▶ 设 win(i,x,y) 表示当前可以买的物品是 i，先手有 x 元，后 手有 y 元时，先手是否必胜

▶ win(i,x,y) ⇐⇒∃j((j > i)∧(x ≥ si−sj)∧¬win(j,y,x−si +sj))

▶ 其中 si = Ci + Ci+1 +···+ CN

▶ 注意到 x + y = A + B−s1 + si，即 win(i,x) := win(i,x,y)

▶ win(i,x) =⇒ win(i,x + 1)

▶ 设 m(i) = min{x : win(i,x)}，则 ¬win(i,x) ⇐⇒ x ≤ m(i)−1

▶ 令 D = A + B−s1 + si

▶           m(i) =min{x : ∃j((j > i)∧(x ≥ si −sj)∧¬win(j,D−x))}

          =min{x : ∃j((j > i)∧(x ≥ si −sj)∧D−x ≤ m(j)−1}

          =min{max{si −sj,D−m(j) + 1} : j > i}

          =min{max{si −sj,A + B−s1 + si −m(j) + +1} : j > i}

          =si + min{max{−sj,A + B−s1 −m(j) + 1} : j > i}

▶ 只要测试 A ≥ m(1)

代码如下：

#include<cstdio>
#include<cstring>
#include<algorithm>
#define ll __int64
#define M 1000002
#define inf 1e15
using namespace std;
int c[M];
ll s[M],m[M],cal,mi;
int main()
{
    int n,q,i,a,mm,b,j;
    while(scanf("%d%d%d",&n,&a,&b)!=EOF)
    {
        for(i=0;i<n;i++) scanf("%d",&c[i]);
        s[n]=0;
        for(i=n-1;i>=0;i--) s[i]=s[i+1]+c[i];
        m[n]=inf;mi=inf;
        cal=a+b-s[0]+1;
        for(i=n;i>=1;i--){
            mi=min(mi,max(-s[i],cal-m[i]));
            m[i-1]=s[i-1]+mi;
        }
        puts(a>=m[0]?"ALICE":"BOB");
    }
    return 0;
}