问题引出: 编号为1-N的N个小球，装入编号为1-N的N个盒子， 要求每个盒子装一个小球，并且盒子和小球的编号不相同。 问有几种排法？ 假设N个小球有D[N]种排法， 易得: D[1]=0, D[2]=1, D[3]=2; 容易推导关系式： D[n] = (n-1) * { D[n-1] + D[n-2] } ,其中n>=3; 代码展示:
方法1:公式递推法
def getN(N): assert type(N) == type(1) , "输入数据类型错误!!!" assert N > 0 , "输入数据取值范围错误!!!" if N == 1: return 0 if N == 2: return 1 if N > 2: return (N-1)*(getN(N-1)+getN(N-2)) for i in range(1,20): print("D{0} = {1}".format(i,getN(i))) 控制台下输出:
Windows PowerShell 版权所有 (C) Microsoft Corporation。保留所有权利。 尝试新的跨平台 PowerShell https://aka.ms/pscore6 加载个人及系统配置文件用了 851 毫秒。 (base) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> conda activate pytorch_1.7.1_cu102 (pytorch_1.7.1_cu102) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> & 'D:\Anaconda3\envs\pytorch_1.7.1_cu102\python.exe' 'c:\Users\chenxuqi\.vscode\extensions\ms-python.python-2021.6.\pythonFiles\lib\python\debugpy\launcher' '51893' '--' 'c:\Users\chenxuqi\Desktop\News4cxq\实验\test.py' D1 = 0 D2 = 1 D3 = 2 D4 = 9 D5 = 44 D6 = 265 D7 = 1854 D8 = 14833 D9 = D10 = D11 = D12 = D13 = D14 = D15 = 4 D16 = 45 D17 = 9664 D18 = 33953 D19 = (pytorch_1.7.1_cu102) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> 方法2:深度优先搜索实现暴力枚举:
# 深度优先搜索实现错位重排问题 count = 0 def getN(N): assert type(N) == type(1) , "输入数据类型错误!!!" assert N > 0 , "输入数据取值范围错误!!!" global count count = 0 count_shuffle_num([None],list(range(1,N+1)),N) def count_shuffle_num(prior,resNums,chosNums): global count if chosNums < 1: return elif chosNums == 1: prior = prior + resNums if len(prior)-1 != prior[-1]: count += 1 # print(prior) return else: for i in resNums: if i == len(prior): continue else: resNums_ = resNums[:] resNums_.remove(i) count_shuffle_num(prior+[i],resNums_,chosNums-1) if __name__ == '__main__': for i in range(1,12): getN(i) print("D{0} = {1}".format(i,count)) 控制台下输出:
Windows PowerShell 版权所有 (C) Microsoft Corporation。保留所有权利。 尝试新的跨平台 PowerShell https://aka.ms/pscore6 加载个人及系统配置文件用了 957 毫秒。 (base) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> conda activate pytorch_1.7.1_cu102 (pytorch_1.7.1_cu102) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> & 'D:\Anaconda3\envs\pytorch_1.7.1_cu102\python.exe' 'c:\Users\chenxuqi\.vscode\extensions\ms-python.python-2021.6.\pythonFiles\lib\python\debugpy\launcher' '52379' '--' 'c:\Users\chenxuqi\Desktop\News4cxq\实验\test.py' D1 = 0 D2 = 1 D3 = 2 D4 = 9 D5 = 44 D6 = 265 D7 = 1854 D8 = 14833 D9 = D10 = D11 = (pytorch_1.7.1_cu102) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> 打印各种组合(按照序列递增顺序枚举):
# 深度优先搜索实现错位重排问题 count = 0 def getN(N): assert type(N) == type(1) , "输入数据类型错误!!!" assert N > 0 , "输入数据取值范围错误!!!" global count count = 0 count_shuffle_num([None],list(range(1,N+1)),N) def count_shuffle_num(prior,resNums,chosNums): global count if chosNums < 1: return elif chosNums == 1: prior = prior + resNums if len(prior)-1 != prior[-1]: count += 1 # print(prior) print(' ',prior[1:],'\tcount =',count) return else: for i in resNums: if i == len(prior): continue else: resNums_ = resNums[:] resNums_.remove(i) count_shuffle_num(prior+[i],resNums_,chosNums-1) if __name__ == '__main__': a = 5 print('位置序号标记->',list(range(1,a+1)),'<-位置序号标记') getN(a) print('综上所述:\tD{0} = {1}'.format(a,count)) 控制台下结果输出:
Windows PowerShell 版权所有 (C) Microsoft Corporation。保留所有权利。 尝试新的跨平台 PowerShell https://aka.ms/pscore6 加载个人及系统配置文件用了 971 毫秒。 (base) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> conda activate pytorch_1.7.1_cu102 (pytorch_1.7.1_cu102) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> & 'D:\Anaconda3\envs\pytorch_1.7.1_cu102\python.exe' 'c:\Users\chenxuqi\.vscode\extensions\ms-python.python-2021.6.\pythonFiles\lib\python\debugpy\launcher' '52632' '--' 'c:\Users\chenxuqi\Desktop\News4cxq\实验\test.py' 位置序号标记-> [1, 2, 3, 4, 5] <-位置序号标记 [2, 1, 4, 5, 3] count = 1 [2, 1, 5, 3, 4] count = 2 [2, 3, 1, 5, 4] count = 3 [2, 3, 4, 5, 1] count = 4 [2, 3, 5, 1, 4] count = 5 [2, 4, 1, 5, 3] count = 6 [2, 4, 5, 1, 3] count = 7 [2, 4, 5, 3, 1] count = 8 [2, 5, 1, 3, 4] count = 9 [2, 5, 4, 1, 3] count = 10 [2, 5, 4, 3, 1] count = 11 [3, 1, 2, 5, 4] count = 12 [3, 1, 4, 5, 2] count = 13 [3, 1, 5, 2, 4] count = 14 [3, 4, 1, 5, 2] count = 15 [3, 4, 2, 5, 1] count = 16 [3, 4, 5, 1, 2] count = 17 [3, 4, 5, 2, 1] count = 18 [3, 5, 1, 2, 4] count = 19 [3, 5, 2, 1, 4] count = 20 [3, 5, 4, 1, 2] count = 21 [3, 5, 4, 2, 1] count = 22 [4, 1, 2, 5, 3] count = 23 [4, 1, 5, 2, 3] count = 24 [4, 1, 5, 3, 2] count = 25 [4, 3, 1, 5, 2] count = 26 [4, 3, 2, 5, 1] count = 27 [4, 3, 5, 1, 2] count = 28 [4, 3, 5, 2, 1] count = 29 [4, 5, 1, 2, 3] count = 30 [4, 5, 1, 3, 2] count = 31 [4, 5, 2, 1, 3] count = 32 [4, 5, 2, 3, 1] count = 33 [5, 1, 2, 3, 4] count = 34 [5, 1, 4, 2, 3] count = 35 [5, 1, 4, 3, 2] count = 36 [5, 3, 1, 2, 4] count = 37 [5, 3, 2, 1, 4] count = 38 [5, 3, 4, 1, 2] count = 39 [5, 3, 4, 2, 1] count = 40 [5, 4, 1, 2, 3] count = 41 [5, 4, 1, 3, 2] count = 42 [5, 4, 2, 1, 3] count = 43 [5, 4, 2, 3, 1] count = 44 综上所述: D5 = 44 (pytorch_1.7.1_cu102) PS C:\Users\chenxuqi\Desktop\News4cxq\实验> 版权声明：本文内容由互联网用户自发贡献，该文观点仅代表作者本人。本站仅提供信息存储空间服务，不拥有所有权，不承担相关法律责任。如发现本站有涉嫌侵权/违法违规的内容， 请联系我们举报，一经查实，本站将立刻删除。
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