USACO Postal Vans 解题报告

这道题我是用偷懒的方法做的。在topcoder论坛上面看到大名鼎鼎的msg555也是用的这种方法，于是我就放心了。首先，用DFS跑n比较小的情况，可以跑到n=10，如果你足够耐心的话，也可以得到更多结果。于是就有了下面的结果，对n=1~11:int arr[] = {0, 2, 4, 12, 28, 74, 184, 472, 1192, 3034, 7692}; 在这之后就是找规律了。https://oeis.org/（我是给系统发邮件）和http://www.wolframalpha.com/网站都可以得到生成函数，后者还可以以极短的时间算出来后续的序列（很好奇是如何做到的，还是高精度运算）。由于数学功底已经不如之前了，不知道如何从生成函数导出递推公式。但知道有递推公式的存在，找到规律还是比较简单的。

我们有:a[n] = 2 * a[n - 1] + 2 * a[n - 2] - 2 * a[n - 3] + a[n - 4]

之后就是如何根据这个递推式导出结果，及超过int的范围怎么办（观察增长速度，突破32位int是很快的事情）。

我用的是最naive的方法，写了个1000位的加法和减法，然后把乘法当做多次加法（或减法）。于是就有了下面的解法：

USER: chen chen [thestor1]TASK: vansLANG: C++Compiling...Compile: OKExecuting...   Test 1: TEST OK [0.008 secs, 3504 KB]   Test 2: TEST OK [0.008 secs, 3504 KB]   Test 3: TEST OK [0.003 secs, 3504 KB]   Test 4: TEST OK [0.008 secs, 3504 KB]   Test 5: TEST OK [0.005 secs, 3504 KB]   Test 6: TEST OK [0.008 secs, 3504 KB]   Test 7: TEST OK [0.005 secs, 3504 KB]   Test 8: TEST OK [0.016 secs, 3504 KB]   Test 9: TEST OK [0.032 secs, 3504 KB]   Test 10: TEST OK [0.043 secs, 3504 KB]   Test 11: TEST OK [0.057 secs, 3504 KB]All tests OK.

/* ID: thestor1 LANG: C++ TASK: vans */#include <iostream>#include <fstream>#include <cmath>#include <cstdio>#include <cstring>#include <climits>#include <cassert>#include <string>#include <vector>#include <list>#include <set>#include <map>#include <queue>#include <stack>#include <algorithm>#include <cassert>using namespace std;const int MAXN = 1000;int N;int dx[] = {-1, 0, 1, 0}, dy[] = {0, 1, 0, -1};bool isIn(int r, int c){return 0 <= r && r < 4 && 0 <= c && c < N; }vector<pair<int, int> > neighbors(int r, int c){vector<pair<int, int> > nbs;int nr, nc;for (int d = 0; d < 4; ++d){nr = r + dx[d], nc = c + dy[d];if (isIn(nr, nc)){nbs.push_back(make_pair(nr, nc));}}return nbs;}void DFS(vector<vector<bool> > &visited, int r, int c, int cnt, int &total){if (visited[0][0]){if (cnt == 4 * N){total++;}return;}vector<pair<int, int> > nbs = neighbors(r, c);int nr, nc;for (int i = 0; i < nbs.size(); ++i){nr = nbs[i].first, nc = nbs[i].second;if (!visited[nr][nc]){visited[nr][nc] = true;DFS(visited, nr, nc, cnt + 1, total);visited[nr][nc] = false;}}}void zero(std::vector<int> &num){for (int i = 0; i < num.size(); ++i){num[i] = 0;}}void assign(std::vector<int> &lhs, std::vector<int> &rhs){assert (lhs.size() == rhs.size());for (int i = 0; i < lhs.size(); ++i){lhs[i] = rhs[i];}}void add(std::vector<int> &lhs, std::vector<int> &rhs){int carry = 0, result;for (int i = lhs.size() - 1; i >= 0; --i){result = lhs[i] + rhs[i] + carry;lhs[i] = result % 10;carry = result / 10;}assert(carry == 0);}void sub(std::vector<int> &lhs, std::vector<int> &rhs){int borrow = 0, result;for (int i = lhs.size() - 1; i >= 0; --i){result = lhs[i] - rhs[i] - borrow;if (result < 0){borrow = 1;lhs[i] = result + 10;}else{borrow = 0;lhs[i] = result;}}assert (borrow == 0);}void print(const std::vector<int> &lhs){int begin = 0;while (begin < MAXN && lhs[begin] == 0){begin++;}for (int i = begin; i < lhs.size(); ++i){cout << lhs[i];}cout << endl;}int main(){ifstream fin("vans.in");fin >> N;fin.close();if (N <= 4){int arr[] = {0, 2, 4, 12, 28, 74, 184, 472, 1192, 3034, 7692};ofstream fout("vans.out");fout << arr[N - 1] << endl;fout.close();return 0;}// vector<vector<bool> > visited(4, vector<bool>(N, false));// int total = 0;// DFS(visited, 0, 0, 0, total);// a[n] = 2 * a[n - 1] + 2 * a[n - 2] - 2 * a[n - 3] + a[n - 4]vector<vector<int> > a(5, vector<int>(MAXN, 0));// a[0]a[1][MAXN - 1] = 2;a[2][MAXN - 1] = 4;a[3][MAXN - 2] = 1;a[3][MAXN - 1] = 2;int a0 = 0, a1 = 1, a2 = 2, a3 = 3;for (int i = 4; i < N; ++i){a0 = i % 4, a1 = (a0 + 1) % 4, a2 = (a1 + 1) % 4, a3 = (a2 + 1) % 4;zero(a[4]);// cout << "[debug]0" << endl;// print(a[4]);add(a[4], a[a3]);// cout << "[debug]1" << endl;// print(a[4]);add(a[4], a[a3]);// cout << "[debug]2" << endl;// print(a[4]);add(a[4], a[a2]);// cout << "[debug]3" << endl;// print(a[4]);add(a[4], a[a2]);// cout << "[debug]4" << endl;// print(a[4]);sub(a[4], a[a1]);// cout << "[debug]5" << endl;// print(a[4]);sub(a[4], a[a1]);// cout << "[debug]6" << endl;// print(a[4]);add(a[4], a[a0]);// cout << "[debug]7" << endl;// print(a[4]);assign(a[a0], a[4]);}ofstream fout("vans.out");int begin = 0;while (begin < MAXN && a[4][begin] == 0){begin++;}for (int i = begin; i < a[4].size(); ++i){fout << a[4][i];}fout << endl;fout.close();return 0;  }