POJ 1759 Garland

题目链接：https://vjudge.net/problem/POJ-1759

题目大意

　　有一个长度为N 的数列 H，满足：

1. H[1] = A。
2. H[N] = B。
3. H[i] = (H[i - 1] + H[i + 1]) / 2 - 1，1 < i < N。
4. H[i] >= 0，1 <= i <= N。

　　现已知 A，问 B 最小是多少？

分析

　　先推一下公式，由题意可得：

$$
\begin{align*}
\sum_{k = i}^{j} H[k] = H[i] + H[j] - (j - i - 1) + \sum_{k = i + 1}^{j - 1} \frac{H[i - 1] + H[i + 1]}{2} \\
化简后即：H[i + 1] + H[j - 1] = H[i] + H[j] - 2 * (j - i - 1) \\
令 i = 1, j = N：H[2] + H[N - 1] = H[1] + H[N] - 2 * (N - 2) 
\end{align*} \tag{1}
$$

　　再利用数学归纳法可以得到下面的式子：

$$
\begin{align*}
H[i] = (i - 1) * H[2] - (i - 2) * H[1] + (i - 1) * (i - 2)
\end{align*} \tag{2}
$$

 　　联立(1), (2) 可得

代码如下

  1 #include <cmath>
  2 #include <ctime>
  3 #include <iostream>
  4 #include <string>
  5 #include <vector>
  6 #include <cstdio>
  7 #include <cstdlib>
  8 #include <cstring>
  9 #include <queue>
 10 #include <map>
 11 #include <set>
 12 #include <algorithm>
 13 #include <cctype>
 14 #include <stack>
 15 #include <deque>
 16 #include <list>
 17 #include <sstream>
 18 using namespace std;
 19  
 20 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
 21 #define Rep(i,n) for (int i = 0; i < (n); ++i)
 22 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
 23 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
 24 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
 25 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
 26 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
 27 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
 28  
 29 #define pr(x) cout << #x << " = " << x << "  "
 30 #define prln(x) cout << #x << " = " << x << endl
 31  
 32 #define LOWBIT(x) ((x)&(-x))
 33  
 34 #define ALL(x) x.begin(),x.end()
 35 #define INS(x) inserter(x,x.begin())
 36 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
 37 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // 删去 x 中所有 c 
 38 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
 39 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
 40  
 41 #define ms0(a) memset(a,0,sizeof(a))
 42 #define msI(a) memset(a,inf,sizeof(a))
 43 #define msM(a) memset(a,-1,sizeof(a))
 44 
 45 #define MP make_pair
 46 #define PB push_back
 47 #define ft first
 48 #define sd second
 49  
 50 template<typename T1, typename T2>
 51 istream &operator>>(istream &in, pair<T1, T2> &p) {
 52     in >> p.first >> p.second;
 53     return in;
 54 }
 55  
 56 template<typename T>
 57 istream &operator>>(istream &in, vector<T> &v) {
 58     for (auto &x: v)
 59         in >> x;
 60     return in;
 61 }
 62  
 63 template<typename T1, typename T2>
 64 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
 65     out << "[" << p.first << ", " << p.second << "]" << "\n";
 66     return out;
 67 }
 68 
 69 inline int gc(){
 70     static const int BUF = 1e7;
 71     static char buf[BUF], *bg = buf + BUF, *ed = bg;
 72     
 73     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
 74     return *bg++;
 75 } 
 76 
 77 inline int ri(){
 78     int x = 0, f = 1, c = gc();
 79     for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
 80     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
 81     return x*f;
 82 }
 83 
 84 template<class T>
 85 inline string toString(T x) {
 86     ostringstream sout;
 87     sout << x;
 88     return sout.str();
 89 }
 90 
 91 inline int toInt(string s) {
 92     int v;
 93     istringstream sin(s);
 94     sin >> v;
 95     return v;
 96 }
 97 
 98 //min <= aim <= max
 99 template<typename T>
100 inline bool BETWEEN(const T aim, const T min, const T max) {
101     return min <= aim && aim <= max;
102 }
103  
104 typedef long long LL;
105 typedef unsigned long long uLL;
106 typedef pair< double, double > PDD;
107 typedef pair< int, int > PII;
108 typedef pair< int, PII > PIPII;
109 typedef pair< string, int > PSI;
110 typedef pair< int, PSI > PIPSI;
111 typedef set< int > SI;
112 typedef set< PII > SPII;
113 typedef vector< int > VI;
114 typedef vector< double > VD;
115 typedef vector< VI > VVI;
116 typedef vector< SI > VSI;
117 typedef vector< PII > VPII;
118 typedef map< int, int > MII;
119 typedef map< int, string > MIS;
120 typedef map< int, PII > MIPII;
121 typedef map< PII, int > MPIII;
122 typedef map< string, int > MSI;
123 typedef map< string, string > MSS;
124 typedef map< PII, string > MPIIS;
125 typedef map< PII, PII > MPIIPII;
126 typedef multimap< int, int > MMII;
127 typedef multimap< string, int > MMSI;
128 //typedef unordered_map< int, int > uMII;
129 typedef pair< LL, LL > PLL;
130 typedef vector< LL > VL;
131 typedef vector< VL > VVL;
132 typedef priority_queue< int > PQIMax;
133 typedef priority_queue< int, VI, greater< int > > PQIMin;
134 const double EPS = 1e-8;
135 const LL inf = 0x3fffffff;
136 const LL infLL = 0x3fffffffffffffffLL;
137 const LL mod = 1e9 + 7;
138 const int maxN = 1e5 + 7;
139 const LL ONE = 1;
140 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
141 const LL oddBits = 0x5555555555555555;
142 
143 int N, r;
144 double A, B; 
145 
146 // 二分查找[2, N]中最后一个满足 (r - 2) * (r - 1) <= A 的 r 
147 void getR(int L, int R, int &r) {
148     if(L >= R) {
149         if((R - 2) * (R - 1) <= A) r = R;
150         return;
151     }
152     int mid = (L + R) >> 1;
153     
154     if((mid - 2) * (mid - 1) <= A) {
155         if(mid * (mid - 1) <= A) getR(mid + 1, R, r);
156         else {
157             r = mid;
158             return;
159         }
160     }
161     else getR(L, mid, r);
162 }
163 
164 int main(){
165     //freopen("MyOutput.txt","w",stdout);
166     //freopen("input.txt","r",stdin);
167     //INIT();
168     cin >> N >> A;
169     getR(2, N, r);
170     
171     B = (N - r) * (N - 1 - 1.0 * A / (r - 1));
172     printf("%.2f\n", B + EPS);
173     return 0;
174 }

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