2016 Multi-University Training Contest 1 H.Shell Necklace

Posted 2020-08-04 Stupidboy is here！

Shell Necklace

Time Limit: 16000/8000 MS (Java/Others)    Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 534    Accepted Submission(s): 227

Problem Description

Perhaps the sea‘s definition of a shell is the pearl. However, in my view, a shell necklace with n beautiful shells contains the most sincere feeling for my best lover Arrietty, but even that is not enough.

Suppose the shell necklace is a sequence of shells (not a chain end to end). Considering i continuous shells in the shell necklace, I know that there exist different schemes to decorate the i shells together with one declaration of love.

I want to decorate all the shells with some declarations of love and decorate each shell just one time. As a problem, I want to know the total number of schemes.

 

 

Input

There are multiple test cases(no more than 20 cases and no more than 1 in extreme case), ended by 0.

For each test cases, the first line contains an integer n, meaning the number of shells in this shell necklace, where 1≤n≤105. Following line is a sequence with nnon-negative integer a1,a2,…,an, and ai≤107 meaning the number of schemes to decorate i continuous shells together with a declaration of love.

 

 

Output

For each test case, print one line containing the total number of schemes module 313(Three hundred and thirteen implies the march 13th, a special and purposeful day).

 

 

Sample Input

3 1 3 7 4 2 2 2 2 0

 

 

Sample Output

14 54

Hint

For the first test case in Sample Input, the Figure 1 provides all schemes about it. The total number of schemes is 1 + 3 + 3 + 7 = 14.

 

 

Author

HIT

 

题意：
    一串由n颗珍珠组成的项链，连续的i个珍珠有ci种染色方式。
    问n颗珍珠有多少种染色方式？

题解：
    显然可知dp方程
    f[i] = sigma(f[i - j] * a[j])
    由于n很大，可以考虑分治fft解决。


    说说这个分治fft：
    如果我们需要计算出l～r的f的值。
    假设我们已经计算出（l，mid）的f的值
    观察方程
    f[i] = sigma(f[j] * a[i - j]) 1 <= j < i
        = sigma(f[j] * a[i - j]) + sigma(f[k] * a[i - k])   1<= j <= mid, mid < k < i
    所以可以分开计算f[l~mid]对f[mid+1~r]的影响。
    具体过程可以看代码。
    使用时记得调整一下下标。

　　

  1 class Complex {
  2     public :
  3         double real, image;
  4 
  5         Complex(double real = 0., double image = 0.):real(real), image(image) {}
  6         Complex(const Complex &t):real(t.real), image(t.image) {}
  7 
  8         Complex operator +(const Complex &t) const {
  9             return Complex(real + t.real, image + t.image);
 10         }
 11 
 12         Complex operator -(const Complex &t) const {
 13             return Complex(real - t.real, image - t.image);
 14         }
 15 
 16         Complex operator *(const Complex &t) const {
 17             return Complex(real * t.real - image * t.image, 
 18                         real * t.image + t.real * image);
 19         }
 20 };
 21 
 22 const int N = 300010, MOD = 313;
 23 const double PI = acos(-1.);
 24 
 25 class FFT {
 26     /**
 27      * 1. Need define PI
 28      * 2. Need define class Complex
 29      * 3. tmp is need for fft, so define a N suffice it
 30      * 4. dig[30] -> (1 << 30) must bigger than N
 31      * */
 32     private :
 33         static Complex tmp[N];
 34         static int revNum[N], dig[30];
 35 
 36     public :
 37         static void init(int n) {
 38             int len = 0;
 39             for(int t = n - 1; t; t >>= 1) ++len;
 40             for(int i = 0; i < n; i++) {
 41                 revNum[i] = 0;
 42                 for(int j = 0; j < len; j++) dig[j] = 0;
 43                 for(int idx = 0, t = i; t; t >>= 1) dig[idx++] = t & 1;
 44                 for(int j = 0; j < len; j++)
 45                     revNum[i] = (revNum[i] << 1) | dig[j];
 46             }
 47         }
 48 
 49         static int rev(int x) {
 50             return revNum[x];
 51         }
 52 
 53         static void fft(Complex a[], int n, int flag) {
 54             /**
 55              * flag = 1 -> DFT
 56              * flag = -1 -> IDFT
 57              * */
 58             for(int i = 0; i < n; ++i) tmp[i] = a[rev(i)];
 59             for(int i = 0; i < n; ++i) a[i] = tmp[i];
 60             for(int i = 2; i <= n; i <<= 1) {
 61                 Complex wn(cos(2 * PI / i), flag * sin(2 * PI / i));
 62                for(int k = 0, half = i / 2; k < n; k += i) {
 63                    Complex w(1., 0.);
 64                    for(int j = k; j < k + half; ++j) {
 65                        Complex x = a[j], y = w * a[j + half];
 66                        a[j] = x + y, a[j + half] = x - y;
 67                        w = w * wn;
 68                    }
 69                }
 70             }
 71             if(flag == -1) {
 72                 for(int i = 0; i < n; ++i) a[i].real /= n;
 73             }
 74         }
 75 
 76         static void dft(Complex a[], int n) {
 77             fft(a, n, 1);
 78         }
 79 
 80         static void idft(Complex a[], int n) {
 81             fft(a, n, -1);
 82         }
 83 };
 84 Complex FFT::tmp[N];
 85 int FFT::revNum[N], FFT::dig[30];
 86 
 87 int n, arr[N];
 88 int f[N];
 89 Complex A[N], B[N];
 90 
 91 inline void divideAndConquer(int lef, int rig) {
 92     if(lef >= rig) return;
 93     int mid = (lef + rig) >> 1;
 94     divideAndConquer(lef, mid);
 95 
 96     int m;
 97     for(m = 1; m < (rig - lef + 1) * 2; m <<= 1);
 98     for(int i = 0; i < m; ++i) A[i] = B[i] = Complex();
 99     for(int i = lef; i <= mid; ++i) A[i - lef] = Complex(f[i]);
100     int len = min(n, m - 1);
101     for(int i = 0; i < len; ++i) B[i + 1] = Complex(arr[i]);
102     FFT::init(m);
103     FFT::dft(A, m), FFT::dft(B, m);
104     for(int i = 0; i < m; ++i) A[i] = A[i] * B[i];
105     FFT::idft(A, m);
106 
107     for(int i = mid + 1; i <= rig; ++i)
108         f[i] = (f[i] + ((ll)(A[i - lef].real + 0.5))) % MOD;
109 
110     divideAndConquer(mid + 1, rig);
111 }
112 
113 inline void solve() {
114     for(int i = 0; i < n; ++i) arr[i] %= MOD;
115     for(int i = 0; i <= n; ++i) f[i] = 0;
116     f[0] = 1;
117     divideAndConquer(0, n);
118 
119     printf("%d\\n", f[n]);
120 }
121 
122 int main() {
123     while(scanf("%d", &n) == 1 && n) {
124         for(int i = 0; i < n; ++i) scanf("%d", &arr[i]);
125         solve();
126     }
127     return 0;
128 }

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