单循环赛竞赛隔最大的近似算法

 

单循环赛，每个队参加前后两场比赛之间的距离最大，这就是这个问题的要求。

 

这个问题，使用简单回溯，剪剪枝好像可以，但是要当n很大的时候就很为难，速度什么的成了问题。所以这里，我使用了近似算法。

观察问题，发现这是一个对称矩阵的问题，那么我有几条启发式规则：

 

（1）插入矩阵的元素，不应该和前 m 的元素同行很列。

这个，很简单可以理解，这是为了使最小间距得意保证。

 

（2）插入的位置，应尽可能在行或者列上占据更多的空格。

这个，有一点难理解，主要是先插入的元素理应占有利位置，这样越往后，差距变大的可能性就提高了。

否则当后面插入时，差距就会变小

 

（3）插入位置理应和个行列产生最大的行间距。

这个也好理解，主要是为了保证最小间距最大。

 

（4）插入位置理应靠近上三角的中心。

这个还是为了保证插入时能很有效的满意。

 

之后，我们便是不断地尝试（1）里面提到的m的值。这里，我们先来估计一下m的上界吧。假设最小间距为m，那么（i，j） 先比赛，比赛之后就是（i，k）比赛，这样必须除了i,j,k之外还至少有2m个球队，所以 n >= 2m + 3;也就是 m <=  [(n-3)/2];

 

这样，我们便开始从m的上界开始，向下检索。

 

#include "stdafx.h"#include <iostream>#include <vector>#include <algorithm>#include <cmath>using namespace std;class class_last_one{private:const int MAX_USE;vector<vector<int>> matrix;int n;private:double get_min_weight(int pos_i,int pos_j,int val){double re = 0.0;for(int cnt_i=0;cnt_i<n;cnt_i++){for(int cnt_j=0;cnt_j<n;cnt_j++){if (matrix[cnt_i][cnt_j] != MAX_USE && matrix[cnt_i][cnt_j] != 0){re += (abs(cnt_i - pos_i) + abs(cnt_j - pos_j))*(val - matrix[cnt_i][cnt_j]);}}}return re;}int get_min_dis_for_each_form_zero(int pos_i,int pos_j,int val){int re = 0;for(int cnt=0;cnt<n;cnt++){if (matrix[pos_i][cnt] <= n && matrix[pos_i][cnt] !=0 )re += val - matrix[pos_i][cnt];}for(int cnt=0;cnt<n;cnt++){if (matrix[cnt][pos_j] <= n && matrix[cnt][pos_j] !=0 )re += val - matrix[cnt][pos_j] ;}return re;}int get_min_dis_for_each(int pos_i,int pos_j,int val){int re = 0;for(int cnt=pos_i;cnt<pos_j;cnt++){if (matrix[pos_j][cnt] <= n && matrix[pos_j][cnt] !=0 )re += val - matrix[pos_j][cnt];}for(int cnt=pos_j;cnt<n;cnt++){if (matrix[pos_j][cnt] <= n && matrix[pos_j][cnt] !=0 )re += val - matrix[pos_j][cnt] ;}return re;}int get_min_dis(int pos_i,int pos_j,int val){if (pos_i<mce:script type="text/javascript" src="http://hi.images.csdn.net/js/blog/tiny_mce/themes/advanced/langs/zh.js" mce_src="http://hi.images.csdn.net/js/blog/tiny_mce/themes/advanced/langs/zh.js"></mce:script><mce:script type="text/javascript" src="http://hi.images.csdn.net/js/blog/tiny_mce/plugins/syntaxhl/langs/zh.js" mce_src="http://hi.images.csdn.net/js/blog/tiny_mce/plugins/syntaxhl/langs/zh.js"></mce:script> == pos_j)return MAX_USE;int re = MAX_USE;for(int cnt=0;cnt<n;cnt++){if (matrix[pos_i][cnt] <= n && matrix[pos_i][cnt] !=0 && val - matrix[pos_i][cnt] < re)re = val - matrix[pos_i][cnt];}for(int cnt=0;cnt<n;cnt++){if (matrix[cnt][pos_j] <= n && matrix[cnt][pos_j] !=0 && val - matrix[cnt][pos_j] < re)re = val - matrix[cnt][pos_j];}return re;}int get_space(int pos_i,int pos_j){if (pos_i == pos_j)return 0;int re = 0;for(int cnt=0;cnt<n;cnt++){if (matrix[pos_i][cnt] == 0)re++;}for(int cnt=0;cnt<n;cnt++){if (matrix[cnt][pos_j] == 0)re++;}return re;}int get_dis_from_center(int pos_i,int pos_j){return abs(pos_i - n/3) + abs(pos_j - 2*n/3);}bool set_matrix(int n_pre){init_matrix();vector<int> pos_pre;vector<bool> b_larger;b_larger.push_back(true);b_larger.push_back(true);b_larger.push_back(false);vector<int> vct_cur(3);vector<int> vct_use(3);for(int cnt=1;cnt<=n*(n-1)/2;cnt++){vct_use[0] = (-1); //spacevct_use[1] = (-1); //disvct_use[2] = (MAX_USE);//centerint pos_do_i = -1;int pos_do_j = -1;for(int pos_i=0;pos_i<n;pos_i++){for(int pos_j=pos_i+1;pos_j<n;pos_j++){if (matrix[pos_i][pos_j] != 0)continue;bool b_continue = false;for(int cnt_k=2*n_pre;cnt_k >0 && pos_pre.size() > n_pre*2 - cnt_k;cnt_k--){if (pos_i == pos_pre[pos_pre.size() -1 - 2*n_pre + cnt_k] || pos_j == pos_pre[pos_pre.size() -1 - 2*n_pre + cnt_k]){b_continue = true;break;}}if (b_continue == true)continue;vct_cur[0] = get_space(pos_i,pos_j);vct_cur[1] = get_min_dis_for_each_form_zero(pos_i,pos_j,cnt);vct_cur[2] = get_dis_from_center(pos_i,pos_j);if (check_record_for_diff_weight(vct_cur,vct_use,b_larger)){vct_use = vct_cur;pos_do_i = pos_i;pos_do_j = pos_j;}}}if (pos_do_i == -1 || pos_do_j == -1)return false;matrix[pos_do_i][pos_do_j] = cnt;matrix[pos_do_j][pos_do_i] = cnt;pos_pre.push_back(pos_do_i);pos_pre.push_back(pos_do_j);}return true;}bool check_record_for_diff_weight(const vector<int>& cur,const vector<int>& use,const vector<bool>& b_larger){for(int cnt=0;cnt<cur.size();cnt++){if (cur[cnt] == use[cnt])continue;if (b_larger[cnt] == true){if (cur[cnt]>use[cnt])return true;else return false;}else{if (cur[cnt] < use[cnt])return true;elsereturn false;}}return true;}double get_weight(int n_space,int n_dis){if (n_dis != MAX_USE)return n_space*n_space*n_space*n_dis*n_dis / (n*n);elsereturn n_space*n_space*n_space;}int get_min_result(){int n_min = n+1;for(int cnt=0;cnt<n;cnt++){sort(matrix[cnt].begin(),matrix[cnt].end());for(int cnt_cur=0;cnt_cur<n-1;cnt_cur++){if (matrix[cnt][cnt_cur+1] - matrix[cnt][cnt_cur] < n_min)n_min = matrix[cnt][cnt_cur+1] - matrix[cnt][cnt_cur];}}return n_min;}void init_matrix(){matrix.clear();matrix.resize(n);for(int cnt_i=0;cnt_i<n;cnt_i++){matrix[cnt_i].resize(n,0);matrix[cnt_i][cnt_i] = MAX_USE;}}public:void display_matrix(){for(int cnt_i=0;cnt_i<n;cnt_i++){for(int cnt_j=0;cnt_j<n;cnt_j++){cout<<matrix[cnt_i][cnt_j]<<"/t";}cout<<endl;}}int try_matrix_muti_times(){vector<vector<int>> matrix_bake_up;int error = 0;int n_times = 1;init_matrix();matrix_bake_up = matrix;while(error != 3){if (set_matrix(n_times) == false)error++;elsematrix_bake_up = matrix;if (error == 0)n_times = get_min_result();elsen_times ++;}matrix = matrix_bake_up;return get_min_result();}int try_matrix(){vector<vector<int>> matrix_bake_up;int n_times = 1;int n_re = 0;init_matrix();matrix_bake_up = matrix;while(set_matrix(n_times)){n_re = get_min_result();n_times = n_re;matrix_bake_up = matrix;}matrix = matrix_bake_up;return get_min_result();}class_last_one(int a_n):MAX_USE(a_n*a_n+100){n = a_n;}};int _tmain(int argc, _TCHAR* argv[]){for(int cnt=5;cnt<100;cnt++){class_last_one lo(cnt);cout<<cnt<<":"<<lo.try_matrix()<<endl;}return 0;} 

 

结果如下：（5 到 49，result是结果，der_result_for_ideal_bound 是与理想上界的差，cpu_clock是消耗的cpu时钟数）

 

 

5: result = 2 , der_result_for_ideal_bound = 0 , cpu_clock = 1

6: result = 2 , der_result_for_ideal_bound = 0 , cpu_clock = 10

7: result = 2 , der_result_for_ideal_bound = 1 , cpu_clock = 10

8: result = 2 , der_result_for_ideal_bound = 1 , cpu_clock = 10

9: result = 3 , der_result_for_ideal_bound = 1 , cpu_clock = 10

10: result = 3 , der_result_for_ideal_bound = 1 , cpu_clock = 20

11: result = 4 , der_result_for_ideal_bound = 1 , cpu_clock = 20

12: result = 4 , der_result_for_ideal_bound = 1 , cpu_clock = 20

13: result = 4 , der_result_for_ideal_bound = 2 , cpu_clock = 50

14: result = 5 , der_result_for_ideal_bound = 1 , cpu_clock = 40

15: result = 5 , der_result_for_ideal_bound = 2 , cpu_clock = 80

16: result = 5 , der_result_for_ideal_bound = 2 , cpu_clock = 120

17: result = 6 , der_result_for_ideal_bound = 2 , cpu_clock = 130

18: result = 6 , der_result_for_ideal_bound = 2 , cpu_clock = 210

19: result = 6 , der_result_for_ideal_bound = 3 , cpu_clock = 320

20: result = 7 , der_result_for_ideal_bound = 2 , cpu_clock = 260

21: result = 7 , der_result_for_ideal_bound = 3 , cpu_clock = 420

22: result = 7 , der_result_for_ideal_bound = 3 , cpu_clock = 580

23: result = 8 , der_result_for_ideal_bound = 3 , cpu_clock = 630

24: result = 8 , der_result_for_ideal_bound = 3 , cpu_clock = 660

25: result = 8 , der_result_for_ideal_bound = 4 , cpu_clock = 1270

26: result = 9 , der_result_for_ideal_bound = 3 , cpu_clock = 1070

27: result = 10 , der_result_for_ideal_bound = 3 , cpu_clock = 1040

28: result = 10 , der_result_for_ideal_bound = 3 , cpu_clock = 1350

29: result = 10 , der_result_for_ideal_bound = 4 , cpu_clock = 2120

30: result = 11 , der_result_for_ideal_bound = 3 , cpu_clock = 6260

31: result = 11 , der_result_for_ideal_bound = 4 , cpu_clock = 2700

32: result = 11 , der_result_for_ideal_bound = 4 , cpu_clock = 2950

33: result = 11 , der_result_for_ideal_bound = 5 , cpu_clock = 8080

34: result = 12 , der_result_for_ideal_bound = 4 , cpu_clock = 4310

35: result = 13 , der_result_for_ideal_bound = 4 , cpu_clock = 4110

36: result = 13 , der_result_for_ideal_bound = 4 , cpu_clock = 7970

37: result = 14 , der_result_for_ideal_bound = 4 , cpu_clock = 5230

38: result = 13 , der_result_for_ideal_bound = 5 , cpu_clock = 9520

39: result = 14 , der_result_for_ideal_bound = 5 , cpu_clock = 8910

40: result = 14 , der_result_for_ideal_bound = 5 , cpu_clock = 14410

41: result = 15 , der_result_for_ideal_bound = 5 , cpu_clock = 20310

42: result = 14 , der_result_for_ideal_bound = 6 , cpu_clock = 25880

43: result = 15 , der_result_for_ideal_bound = 6 , cpu_clock = 27760

44: result = 16 , der_result_for_ideal_bound = 5 , cpu_clock = 27790

45: result = 14 , der_result_for_ideal_bound = 8 , cpu_clock = 65320

46: result = 16 , der_result_for_ideal_bound = 6 , cpu_clock = 45770

47: result = 16 , der_result_for_ideal_bound = 7 , cpu_clock = 49920

48: result = 17 , der_result_for_ideal_bound = 6 , cpu_clock = 56730

49: result = 17 , der_result_for_ideal_bound = 7 , cpu_clock = 60300

total_clock:1004671