【HDOJ】4579 Random Walk

1. 题目描述
一个人沿着一条长度为n个链行走，给出了每秒钟由i到j的概率（$i,j \in [1,n]$）。求从1开始走到n个时间的期望。

2. 基本思路
显然是个DP。公式推导也相当容易。不妨设$dp[i], i \in [1,n]$表示由i到n的期望时间。
\begin{align}
    dp[i] &= \Sigma_{j=1}^{n} p(i, j) (dp[j] + 1),    &j<n\\
    dp[i] &= 0 &i=n
\end{align}
显然这是一个n元方程组，可以高斯消元解。但是因为n很大，因此不能直接套用高斯消元。但是通过观察系数矩阵可以发现规律。
以$n=7, m=2$为例，$\times$表示$A_{ij}$不为0。

发现每个行向量最多包含$2m+1$个非零向量，即$[i-m, i+m]$。因此，在高斯消元的过程中，实际每次需要减掉的系数最多也就$2m+1$个。
因为$m \in [1,5]$，可以直接模拟非零系数的消元。因为$i-m$有可能小于0，因此，将$[i-m,i+m]$映射到$[0,2m]$的区间内，$A_{ii}$恰好映射到$P_{im}$。
最后可以生成几个小规模的n，与高斯消元对拍一下。

3. 代码

 /* 4579 */
 #include <iostream>
 #include <sstream>
 #include <string>
 #include <map>
 #include <queue>
 #include <set>
 #include <stack>
 #include <vector>
 #include <deque>
 #include <algorithm>
 #include <cstdio>
 #include <cmath>
 #include <ctime>
 #include <cstring>
 #include <climits>
 #include <cctype>
 #include <cassert>
 #include <functional>
 #include <iterator>
 #include <iomanip>
 using namespace std;
 //#pragma comment(linker,"/STACK:102400000,1024000")
 
 #define sti                set<int>
 #define stpii            set<pair<int, int> >
 #define mpii            map<int,int>
 #define vi                vector<int>
 #define pii                pair<int,int>
 #define vpii            vector<pair<int,int> >
 #define rep(i, a, n)     for (int i=a;i<n;++i)
 #define per(i, a, n)     for (int i=n-1;i>=a;--i)
 #define clr                clear
 #define pb                 push_back
 #define mp                 make_pair
 #define fir                first
 #define sec                second
 #define all(x)             (x).begin(),(x).end()
 #define SZ(x)             ((int)(x).size())
 #define lson            l, mid, rt<<1
 #define rson            mid+1, r, rt<<1|1
 
 const double eps = 1e-;
 const int maxn = ;
 const int maxm = ;
 double g[maxn][maxm], p[maxn][maxm];
 double v[maxn], x[maxn];
 int C[maxn][];
 int n, m;
 
 void solve() {
     int i, j, k;
 
     memset(p, , sizeof(p));
     for (i=; i<n; ++i) {
         int tot = ;
         double tmp = 0.0;
         for (j=; j<=m; ++j)
             tot += C[i][j];
 
         for (j=; j<=m; ++j) {
             if (i-j >= ) {
                 p[i][m-j] = 0.3 * C[i][j] / tot;
                 tmp += p[i][m-j];
             }
             if (i+j <= n) {
                 p[i][m+j] = 0.7 * C[i][j] / tot;
                 tmp += p[i][m+j];
             }
         }
         p[i][m] = -tmp;
         v[i] = -;
     }
     p[n][m] = ;
     v[n] = ;
 
     memcpy(g[], p[], sizeof(p[]));
     for (i=,k=; i<=n; ++i,++k) {
         int l = max(k-m, );
         int r = min(k+m, n);
         for (j=i; j<=n&&j-k<=m; ++j) {
             if (fabs(p[k][m]) < eps)
                 continue;
             double t = p[j][k-j+m] / p[k][m];
             for (int kk=k+; kk<=n&&kk-k<=m; ++kk)
                 p[j][kk-j+m] -= t * p[k][kk-k+m];
             v[j] -= t * v[k];
         }
 
         l = max(i-m, );
         r = min(i+m, n);
         for (j=l; j<=r; ++j)
             g[i][j-i+m] = p[i][j-i+m];
     }
 
     x[n] = ;
     for (i=n-,k=n; i>; --i,--k){
         for (j=i; j>&&k-j<=m; --j)
             v[j] -= x[k] * g[j][k-j+m];
         x[i] = v[i] / g[i][m];
     }
 
     printf("%.2lf\n", x[]);
 }
 
 int main() {
     ios::sync_with_stdio(false);
     #ifndef ONLINE_JUDGE
         freopen("data.in", "r", stdin);
         freopen("data.out", "w", stdout);
     #endif
 
     while (scanf("%d%d",&n,&m)!=EOF && (n||m)) {
         rep(i, , n+)
             rep(j, , m+)
                 scanf("%d", &C[i][j]);
         solve();
     }
 
     #ifndef ONLINE_JUDGE
         printf("time = %d.\n", (int)clock());
     #endif
 
     return ;
 }

4. 数据生成器

 import sys
 import string
 from random import randint
 
 def GenData(fileName):
     with open(fileName, "w") as fout:
         t = 10
         for tt in xrange(t):
             n = randint(1, 200)
             m = randint(1, 5)
             fout.write("%d %d\n" % (n, m))
             L = [0] * m
             for i in xrange(n):
                 for j in xrange(m):
                     L[j] = randint(1, 9)
                 fout.write(" ".join(map(str, L)) + "\n")
         fout.write("0 0\n")
 
 def MovData(srcFileName, desFileName):
     with open(srcFileName, "r") as fin:
         lines = fin.readlines()
     with open(desFileName, "w") as fout:
         fout.write("".join(lines))
 
 def CompData():
     print "comp"
     srcFileName = "F:\Qt_prj\hdoj\data.out"
     desFileName = "F:\workspace\cpp_hdoj\data.out"
     srcLines = []
     desLines = []
     with open(srcFileName, "r") as fin:
         srcLines = fin.readlines()
     with open(desFileName, "r") as fin:
         desLines = fin.readlines()
     n = min(len(srcLines), len(desLines))-1
     for i in xrange(n):
         ans2 = int(desLines[i])
         ans1 = int(srcLines[i])
         if ans1 > ans2:
             print "%d: wrong" % i
 
 if __name__ == "__main__":
     srcFileName = "F:\Qt_prj\hdoj\data.in"
     desFileName = "F:\workspace\cpp_hdoj\data.in"
     GenData(srcFileName)
     MovData(srcFileName, desFileName)