大きなセットのパワーセット

Question

私は10^5まで多くの要素を持つかもしれないセットのパワーセットを計算しなければなりません。私はalgoと以下のコードを試しましたが、失敗しました（私はpow(2, size)の大きな値を引き起こすと思います）。

void printPowerSet(int *set, int set_size) 
{ 
unsigned int pow_set_size = pow(2, set_size); 
int counter, j,sum=0; 
for(counter = 0; counter < pow_set_size; counter++) 
{ 
    for(j = 0; j < set_size; j++) 
    { 
     if(counter & (1<<j)) 
    std::cout<<set[i]<<" "; 
    } 
    std::cout<<sum; 
    sum=0; 
    printf("\n"); 
} 
} 

他のアルゴリズムがありますか、またはこれを修正する方法はありますか？
OR
Can you suggest me how to do it i.e. finding subset of large set.

私が X-Y problemでこだわっているようだ答えで指摘したように。基本的には、どの集合のすべての部分集合の合計が必要です。今あなたが私に問題を解決するための他のアプローチを提案できるかどうか。
ありがとうございます。

Answer 1

それはあなた自身が指摘したように、パワーセットは要素のpow(2, size)なしが含まれている、ことはできません。 を使用してセットを生成することによってのみ行うことができるセット全体を印刷する必要があります。

---

しかし、与えられたセットのすべてのサブセットの合計を見つけることは、より簡単な問題です。

要素の数がnの場合、配列内の各要素は、出力セット内で2^(n-1)回発生します。

擬似コード：

int sum = 0; 

for(int i = 0;i < n;i++){ 
    sum += (set[i] * (1 << (n-1))); //pow(2, n-1) == 1 << (n-1) 
} 

cout << sum << endl; 

あなたはnの値が大きいほどBig Integer Libraryが必要になります。

Answer 2

セット内の要素の数は10^5になるので、セットのサイズは最大で2^(10^5)になります。これは巨大です。印刷することも、保存することもできます。ファイルに保存します。

Answer 3

ここでは、コンピュータのメモリに収まる任意のセットの電源セットを表示するアルゴリズムを示します。

十分な時間が与えられると、長さ10^5のセットの電力セットが印刷されます。

しかし、「十分な時間」とは、数十億年の何十億年というものです。

C++ 14

#include <iostream> 
#include <vector> 
#include <algorithm> 

using namespace std; 

#include <iostream> 

void printPowerset (const vector<int>& original_set) 
{ 
    auto print_set = [&original_set](auto first, auto last) -> ostream& 
    { 
     cout << '('; 
     auto sep = ""; 
     for (; first != last ; ++first, sep = ",") 
     { 
      cout << sep << original_set[(*first) - 1]; 
     } 
     return cout << ')'; 
    }; 

    const int n = original_set.size(); 
    std::vector<int> index_stack(n + 1, 0); 
    int k = 0; 


    while(1){ 
     if (index_stack[k]<n){ 
      index_stack[k+1] = index_stack[k] + 1; 
      k++; 
     } 

     else{ 
      index_stack[k-1]++; 
      k--; 
     } 

     if (k==0) 
      break; 

     print_set(begin(index_stack) + 1, begin(index_stack) + 1 + k); 
    } 
    print_set(begin(index_stack), begin(index_stack)) << endl; 
} 

int main(){ 

    auto nums = vector<int> { 2, 4, 6, 8 }; 
    printPowerset(nums); 

    nums = vector<int> { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 }; 
    printPowerset(nums); 

    return 0; 
} 

予想される結果：

第1のパワーセット（4つの項目）：

(2)(2,4)(2,4,6)(2,4,6,8)(2,4,8)(2,6)(2,6,8)(2,8)(4)(4,6)(4,6,8)(4,8)(6)(6,8)(8)() 

第2のパワーセット（10項目）

(2)(2,4)(2,4,6)(2,4,6,8)(2,4,6,8,10)(2,4,6,8,10,12)(2,4,6,8,10,12,14)(2,4,6,8,10,12,14,16)(2,4,6,8,10,12,14,16,18)(2,4,6,8,10,12,14,16,18,20)(2,4,6,8,10,12,14,16,20)(2,4,6,8,10,12,14,18)(2,4,6,8,10,12,14,18,20)(2,4,6,8,10,12,14,20)(2,4,6,8,10,12,16)(2,4,6,8,10,12,16,18)(2,4,6,8,10,12,16,18,20)(2,4,6,8,10,12,16,20)(2,4,6,8,10,12,18)(2,4,6,8,10,12,18,20)(2,4,6,8,10,12,20)(2,4,6,8,10,14)(2,4,6,8,10,14,16)(2,4,6,8,10,14,16,18)(2,4,6,8,10,14,16,18,20)(2,4,6,8,10,14,16,20)(2,4,6,8,10,14,18)(2,4,6,8,10,14,18,20)(2,4,6,8,10,14,20)(2,4,6,8,10,16)(2,4,6,8,10,16,18)(2,4,6,8,10,16,18,20)(2,4,6,8,10,16,20)(2,4,6,8,10,18)(2,4,6,8,10,18,20)(2,4,6,8,10,20)(2,4,6,8,12)(2,4,6,8,12,14)(2,4,6,8,12,14,16)(2,4,6,8,12,14,16,18)(2,4,6,8,12,14,16,18,20)(2,4,6,8,12,14,16,20)(2,4,6,8,12,14,18)(2,4,6,8,12,14,18,20)(2,4,6,8,12,14,20)(2,4,6,8,12,16)(2,4,6,8,12,16,18)(2,4,6,8,12,16,18,20)(2,4,6,8,12,16,20)(2,4,6,8,12,18)(2,4,6,8,12,18,20)(2,4,6,8,12,20)(2,4,6,8,14)(2,4,6,8,14,16)(2,4,6,8,14,16,18)(2,4,6,8,14,16,18,20)(2,4,6,8,14,16,20)(2,4,6,8,14,18)(2,4,6,8,14,18,20)(2,4,6,8,14,20)(2,4,6,8,16)(2,4,6,8,16,18)(2,4,6,8,16,18,20)(2,4,6,8,16,20)(2,4,6,8,18)(2,4,6,8,18,20)(2,4,6,8,20)(2,4,6,10)(2,4,6,10,12)(2,4,6,10,12,14)(2,4,6,10,12,14,16)(2,4,6,10,12,14,16,18)(2,4,6,10,12,14,16,18,20)(2,4,6,10,12,14,16,20)(2,4,6,10,12,14,18)(2,4,6,10,12,14,18,20)(2,4,6,10,12,14,20)(2,4,6,10,12,16)(2,4,6,10,12,16,18)(2,4,6,10,12,16,18,20)(2,4,6,10,12,16,20)(2,4,6,10,12,18)(2,4,6,10,12,18,20)(2,4,6,10,12,20)(2,4,6,10,14)(2,4,6,10,14,16)(2,4,6,10,14,16,18)(2,4,6,10,14,16,18,20)(2,4,6,10,14,16,20)(2,4,6,10,14,18)(2,4,6,10,14,18,20)(2,4,6,10,14,20)(2,4,6,10,16)(2,4,6,10,16,18)(2,4,6,10,16,18,20)(2,4,6,10,16,20)(2,4,6,10,18)(2,4,6,10,18,20)(2,4,6,10,20)(2,4,6,12)(2,4,6,12,14)(2,4,6,12,14,16)(2,4,6,12,14,16,18)(2,4,6,12,14,16,18,20)(2,4,6,12,14,16,20)(2,4,6,12,14,18)(2,4,6,12,14,18,20)(2,4,6,12,14,20)(2,4,6,12,16)(2,4,6,12,16,18)(2,4,6,12,16,18,20)(2,4,6,12,16,20)(2,4,6,12,18)(2,4,6,12,18,20)(2,4,6,12,20)(2,4,6,14)(2,4,6,14,16)(2,4,6,14,16,18)(2,4,6,14,16,18,20)(2,4,6,14,16,20)(2,4,6,14,18)(2,4,6,14,18,20)(2,4,6,14,20)(2,4,6,16)(2,4,6,16,18)(2,4,6,16,18,20)(2,4,6,16,20)(2,4,6,18)(2,4,6,18,20)(2,4,6,20)(2,4,8)(2,4,8,10)(2,4,8,10,12)(2,4,8,10,12,14)(2,4,8,10,12,14,16)(2,4,8,10,12,14,16,18)(2,4,8,10,12,14,16,18,20)(2,4,8,10,12,14,16,20)(2,4,8,10,12,14,18)(2,4,8,10,12,14,18,20)(2,4,8,10,12,14,20)(2,4,8,10,12,16)(2,4,8,10,12,16,18)(2,4,8,10,12,16,18,20)(2,4,8,10,12,16,20)(2,4,8,10,12,18)(2,4,8,10,12,18,20)(2,4,8,10,12,20)(2,4,8,10,14)(2,4,8,10,14,16)(2,4,8,10,14,16,18)(2,4,8,10,14,16,18,20)(2,4,8,10,14,16,20)(2,4,8,10,14,18)(2,4,8,10,14,18,20)(2,4,8,10,14,20)(2,4,8,10,16)(2,4,8,10,16,18)(2,4,8,10,16,18,20)(2,4,8,10,16,20)(2,4,8,10,18)(2,4,8,10,18,20)(2,4,8,10,20)(2,4,8,12)(2,4,8,12,14)(2,4,8,12,14,16)(2,4,8,12,14,16,18)(2,4,8,12,14,16,18,20)(2,4,8,12,14,16,20)(2,4,8,12,14,18)(2,4,8,12,14,18,20)(2,4,8,12,14,20)(2,4,8,12,16)(2,4,8,12,16,18)(2,4,8,12,16,18,20)(2,4,8,12,16,20)(2,4,8,12,18)(2,4,8,12,18,20)(2,4,8,12,20)(2,4,8,14)(2,4,8,14,16)(2,4,8,14,16,18)(2,4,8,14,16,18,20)(2,4,8,14,16,20)(2,4,8,14,18)(2,4,8,14,18,20)(2,4,8,14,20)(2,4,8,16)(2,4,8,16,18)(2,4,8,16,18,20)(2,4,8,16,20)(2,4,8,18)(2,4,8,18,20)(2,4,8,20)(2,4,10)(2,4,10,12)(2,4,10,12,14)(2,4,10,12,14,16)(2,4,10,12,14,16,18)(2,4,10,12,14,16,18,20)(2,4,10,12,14,16,20)(2,4,10,12,14,18)(2,4,10,12,14,18,20)(2,4,10,12,14,20)(2,4,10,12,16)(2,4,10,12,16,18)(2,4,10,12,16,18,20)(2,4,10,12,16,20)(2,4,10,12,18)(2,4,10,12,18,20)(2,4,10,12,20)(2,4,10,14)(2,4,10,14,16)(2,4,10,14,16,18)(2,4,10,14,16,18,20)(2,4,10,14,16,20)(2,4,10,14,18)(2,4,10,14,18,20)(2,4,10,14,20)(2,4,10,16)(2,4,10,16,18)(2,4,10,16,18,20)(2,4,10,16,20)(2,4,10,18)(2,4,10,18,20)(2,4,10,20)(2,4,12)(2,4,12,14)(2,4,12,14,16)(2,4,12,14,16,18)(2,4,12,14,16,18,20)(2,4,12,14,16,20)(2,4,12,14,18)(2,4,12,14,18,20)(2,4,12,14,20)(2,4,12,16)(2,4,12,16,18)(2,4,12,16,18,20)(2,4,12,16,20)(2,4,12,18)(2,4,12,18,20)(2,4,12,20)(2,4,14)(2,4,14,16)(2,4,14,16,18)(2,4,14,16,18,20)(2,4,14,16,20)(2,4,14,18)(2,4,14,18,20)(2,4,14,20)(2,4,16)(2,4,16,18)(2,4,16,18,20)(2,4,16,20)(2,4,18)(2,4,18,20)(2,4,20)(2,6)(2,6,8)(2,6,8,10)(2,6,8,10,12)(2,6,8,10,12,14)(2,6,8,10,12,14,16)(2,6,8,10,12,14,16,18)(2,6,8,10,12,14,16,18,20)(2,6,8,10,12,14,16,20)(2,6,8,10,12,14,18)(2,6,8,10,12,14,18,20)(2,6,8,10,12,14,20)(2,6,8,10,12,16)(2,6,8,10,12,16,18)(2,6,8,10,12,16,18,20)(2,6,8,10,12,16,20)(2,6,8,10,12,18)(2,6,8,10,12,18,20)(2,6,8,10,12,20)(2,6,8,10,14)(2,6,8,10,14,16)(2,6,8,10,14,16,18)(2,6,8,10,14,16,18,20)(2,6,8,10,14,16,20)(2,6,8,10,14,18)(2,6,8,10,14,18,20)(2,6,8,10,14,20)(2,6,8,10,16)(2,6,8,10,16,18)(2,6,8,10,16,18,20)(2,6,8,10,16,20)(2,6,8,10,18)(2,6,8,10,18,20)(2,6,8,10,20)(2,6,8,12)(2,6,8,12,14)(2,6,8,12,14,16)(2,6,8,12,14,16,18)(2,6,8,12,14,16,18,20)(2,6,8,12,14,16,20)(2,6,8,12,14,18)(2,6,8,12,14,18,20)(2,6,8,12,14,20)(2,6,8,12,16)(2,6,8,12,16,18)(2,6,8,12,16,18,20)(2,6,8,12,16,20)(2,6,8,12,18)(2,6,8,12,18,20)(2,6,8,12,20)(2,6,8,14)(2,6,8,14,16)(2,6,8,14,16,18)(2,6,8,14,16,18,20)(2,6,8,14,16,20)(2,6,8,14,18)(2,6,8,14,18,20)(2,6,8,14,20)(2,6,8,16)(2,6,8,16,18)(2,6,8,16,18,20)(2,6,8,16,20)(2,6,8,18)(2,6,8,18,20)(2,6,8,20)(2,6,10)(2,6,10,12)(2,6,10,12,14)(2,6,10,12,14,16)(2,6,10,12,14,16,18)(2,6,10,12,14,16,18,20)(2,6,10,12,14,16,20)(2,6,10,12,14,18)(2,6,10,12,14,18,20)(2,6,10,12,14,20)(2,6,10,12,16)(2,6,10,12,16,18)(2,6,10,12,16,18,20)(2,6,10,12,16,20)(2,6,10,12,18)(2,6,10,12,18,20)(2,6,10,12,20)(2,6,10,14)(2,6,10,14,16)(2,6,10,14,16,18)(2,6,10,14,16,18,20)(2,6,10,14,16,20)(2,6,10,14,18)(2,6,10,14,18,20)(2,6,10,14,20)(2,6,10,16)(2,6,10,16,18)(2,6,10,16,18,20)(2,6,10,16,20)(2,6,10,18)(2,6,10,18,20)(2,6,10,20)(2,6,12)(2,6,12,14)(2,6,12,14,16)(2,6,12,14,16,18)(2,6,12,14,16,18,20)(2,6,12,14,16,20)(2,6,12,14,18)(2,6,12,14,18,20)(2,6,12,14,20)(2,6,12,16)(2,6,12,16,18)(2,6,12,16,18,20)(2,6,12,16,20)(2,6,12,18)(2,6,12,18,20)(2,6,12,20)(2,6,14)(2,6,14,16)(2,6,14,16,18)(2,6,14,16,18,20)(2,6,14,16,20)(2,6,14,18)(2,6,14,18,20)(2,6,14,20)(2,6,16)(2,6,16,18)(2,6,16,18,20)(2,6,16,20)(2,6,18)(2,6,18,20)(2,6,20)(2,8)(2,8,10)(2,8,10,12)(2,8,10,12,14)(2,8,10,12,14,16)(2,8,10,12,14,16,18)(2,8,10,12,14,16,18,20)(2,8,10,12,14,16,20)(2,8,10,12,14,18)(2,8,10,12,14,18,20)(2,8,10,12,14,20)(2,8,10,12,16)(2,8,10,12,16,18)(2,8,10,12,16,18,20)(2,8,10,12,16,20)(2,8,10,12,18)(2,8,10,12,18,20)(2,8,10,12,20)(2,8,10,14)(2,8,10,14,16)(2,8,10,14,16,18)(2,8,10,14,16,18,20)(2,8,10,14,16,20)(2,8,10,14,18)(2,8,10,14,18,20)(2,8,10,14,20)(2,8,10,16)(2,8,10,16,18)(2,8,10,16,18,20)(2,8,10,16,20)(2,8,10,18)(2,8,10,18,20)(2,8,10,20)(2,8,12)(2,8,12,14)(2,8,12,14,16)(2,8,12,14,16,18)(2,8,12,14,16,18,20)(2,8,12,14,16,20)(2,8,12,14,18)(2,8,12,14,18,20)(2,8,12,14,20)(2,8,12,16)(2,8,12,16,18)(2,8,12,16,18,20)(2,8,12,16,20)(2,8,12,18)(2,8,12,18,20)(2,8,12,20)(2,8,14)(2,8,14,16)(2,8,14,16,18)(2,8,14,16,18,20)(2,8,14,16,20)(2,8,14,18)(2,8,14,18,20)(2,8,14,20)(2,8,16)(2,8,16,18)(2,8,16,18,20)(2,8,16,20)(2,8,18)(2,8,18,20)(2,8,20)(2,10)(2,10,12)(2,10,12,14)(2,10,12,14,16)(2,10,12,14,16,18)(2,10,12,14,16,18,20)(2,10,12,14,16,20)(2,10,12,14,18)(2,10,12,14,18,20)(2,10,12,14,20)(2,10,12,16)(2,10,12,16,18)(2,10,12,16,18,20)(2,10,12,16,20)(2,10,12,18)(2,10,12,18,20)(2,10,12,20)(2,10,14)(2,10,14,16)(2,10,14,16,18)(2,10,14,16,18,20)(2,10,14,16,20)(2,10,14,18)(2,10,14,18,20)(2,10,14,20)(2,10,16)(2,10,16,18)(2,10,16,18,20)(2,10,16,20)(2,10,18)(2,10,18,20)(2,10,20)(2,12)(2,12,14)(2,12,14,16)(2,12,14,16,18)(2,12,14,16,18,20)(2,12,14,16,20)(2,12,14,18)(2,12,14,18,20)(2,12,14,20)(2,12,16)(2,12,16,18)(2,12,16,18,20)(2,12,16,20)(2,12,18)(2,12,18,20)(2,12,20)(2,14)(2,14,16)(2,14,16,18)(2,14,16,18,20)(2,14,16,20)(2,14,18)(2,14,18,20)(2,14,20)(2,16)(2,16,18)(2,16,18,20)(2,16,20)(2,18)(2,18,20)(2,20)(4)(4,6)(4,6,8)(4,6,8,10)(4,6,8,10,12)(4,6,8,10,12,14)(4,6,8,10,12,14,16)(4,6,8,10,12,14,16,18)(4,6,8,10,12,14,16,18,20)(4,6,8,10,12,14,16,20)(4,6,8,10,12,14,18)(4,6,8,10,12,14,18,20)(4,6,8,10,12,14,20)(4,6,8,10,12,16)(4,6,8,10,12,16,18)(4,6,8,10,12,16,18,20)(4,6,8,10,12,16,20)(4,6,8,10,12,18)(4,6,8,10,12,18,20)(4,6,8,10,12,20)(4,6,8,10,14)(4,6,8,10,14,16)(4,6,8,10,14,16,18)(4,6,8,10,14,16,18,20)(4,6,8,10,14,16,20)(4,6,8,10,14,18)(4,6,8,10,14,18,20)(4,6,8,10,14,20)(4,6,8,10,16)(4,6,8,10,16,18)(4,6,8,10,16,18,20)(4,6,8,10,16,20)(4,6,8,10,18)(4,6,8,10,18,20)(4,6,8,10,20)(4,6,8,12)(4,6,8,12,14)(4,6,8,12,14,16)(4,6,8,12,14,16,18)(4,6,8,12,14,16,18,20)(4,6,8,12,14,16,20)(4,6,8,12,14,18)(4,6,8,12,14,18,20)(4,6,8,12,14,20)(4,6,8,12,16)(4,6,8,12,16,18)(4,6,8,12,16,18,20)(4,6,8,12,16,20)(4,6,8,12,18)(4,6,8,12,18,20)(4,6,8,12,20)(4,6,8,14)(4,6,8,14,16)(4,6,8,14,16,18)(4,6,8,14,16,18,20)(4,6,8,14,16,20)(4,6,8,14,18)(4,6,8,14,18,20)(4,6,8,14,20)(4,6,8,16)(4,6,8,16,18)(4,6,8,16,18,20)(4,6,8,16,20)(4,6,8,18)(4,6,8,18,20)(4,6,8,20)(4,6,10)(4,6,10,12)(4,6,10,12,14)(4,6,10,12,14,16)(4,6,10,12,14,16,18)(4,6,10,12,14,16,18,20)(4,6,10,12,14,16,20)(4,6,10,12,14,18)(4,6,10,12,14,18,20)(4,6,10,12,14,20)(4,6,10,12,16)(4,6,10,12,16,18)(4,6,10,12,16,18,20)(4,6,10,12,16,20)(4,6,10,12,18)(4,6,10,12,18,20)(4,6,10,12,20)(4,6,10,14)(4,6,10,14,16)(4,6,10,14,16,18)(4,6,10,14,16,18,20)(4,6,10,14,16,20)(4,6,10,14,18)(4,6,10,14,18,20)(4,6,10,14,20)(4,6,10,16)(4,6,10,16,18)(4,6,10,16,18,20)(4,6,10,16,20)(4,6,10,18)(4,6,10,18,20)(4,6,10,20)(4,6,12)(4,6,12,14)(4,6,12,14,16)(4,6,12,14,16,18)(4,6,12,14,16,18,20)(4,6,12,14,16,20)(4,6,12,14,18)(4,6,12,14,18,20)(4,6,12,14,20)(4,6,12,16)(4,6,12,16,18)(4,6,12,16,18,20)(4,6,12,16,20)(4,6,12,18)(4,6,12,18,20)(4,6,12,20)(4,6,14)(4,6,14,16)(4,6,14,16,18)(4,6,14,16,18,20)(4,6,14,16,20)(4,6,14,18)(4,6,14,18,20)(4,6,14,20)(4,6,16)(4,6,16,18)(4,6,16,18,20)(4,6,16,20)(4,6,18)(4,6,18,20)(4,6,20)(4,8)(4,8,10)(4,8,10,12)(4,8,10,12,14)(4,8,10,12,14,16)(4,8,10,12,14,16,18)(4,8,10,12,14,16,18,20)(4,8,10,12,14,16,20)(4,8,10,12,14,18)(4,8,10,12,14,18,20)(4,8,10,12,14,20)(4,8,10,12,16)(4,8,10,12,16,18)(4,8,10,12,16,18,20)(4,8,10,12,16,20)(4,8,10,12,18)(4,8,10,12,18,20)(4,8,10,12,20)(4,8,10,14)(4,8,10,14,16)(4,8,10,14,16,18)(4,8,10,14,16,18,20)(4,8,10,14,16,20)(4,8,10,14,18)(4,8,10,14,18,20)(4,8,10,14,20)(4,8,10,16)(4,8,10,16,18)(4,8,10,16,18,20)(4,8,10,16,20)(4,8,10,18)(4,8,10,18,20)(4,8,10,20)(4,8,12)(4,8,12,14)(4,8,12,14,16)(4,8,12,14,16,18)(4,8,12,14,16,18,20)(4,8,12,14,16,20)(4,8,12,14,18)(4,8,12,14,18,20)(4,8,12,14,20)(4,8,12,16)(4,8,12,16,18)(4,8,12,16,18,20)(4,8,12,16,20)(4,8,12,18)(4,8,12,18,20)(4,8,12,20)(4,8,14)(4,8,14,16)(4,8,14,16,18)(4,8,14,16,18,20)(4,8,14,16,20)(4,8,14,18)(4,8,14,18,20)(4,8,14,20)(4,8,16)(4,8,16,18)(4,8,16,18,20)(4,8,16,20)(4,8,18)(4,8,18,20)(4,8,20)(4,10)(4,10,12)(4,10,12,14)(4,10,12,14,16)(4,10,12,14,16,18)(4,10,12,14,16,18,20)(4,10,12,14,16,20)(4,10,12,14,18)(4,10,12,14,18,20)(4,10,12,14,20)(4,10,12,16)(4,10,12,16,18)(4,10,12,16,18,20)(4,10,12,16,20)(4,10,12,18)(4,10,12,18,20)(4,10,12,20)(4,10,14)(4,10,14,16)(4,10,14,16,18)(4,10,14,16,18,20)(4,10,14,16,20)(4,10,14,18)(4,10,14,18,20)(4,10,14,20)(4,10,16)(4,10,16,18)(4,10,16,18,20)(4,10,16,20)(4,10,18)(4,10,18,20)(4,10,20)(4,12)(4,12,14)(4,12,14,16)(4,12,14,16,18)(4,12,14,16,18,20)(4,12,14,16,20)(4,12,14,18)(4,12,14,18,20)(4,12,14,20)(4,12,16)(4,12,16,18)(4,12,16,18,20)(4,12,16,20)(4,12,18)(4,12,18,20)(4,12,20)(4,14)(4,14,16)(4,14,16,18)(4,14,16,18,20)(4,14,16,20)(4,14,18)(4,14,18,20)(4,14,20)(4,16)(4,16,18)(4,16,18,20)(4,16,20)(4,18)(4,18,20)(4,20)(6)(6,8)(6,8,10)(6,8,10,12)(6,8,10,12,14)(6,8,10,12,14,16)(6,8,10,12,14,16,18)(6,8,10,12,14,16,18,20)(6,8,10,12,14,16,20)(6,8,10,12,14,18)(6,8,10,12,14,18,20)(6,8,10,12,14,20)(6,8,10,12,16)(6,8,10,12,16,18)(6,8,10,12,16,18,20)(6,8,10,12,16,20)(6,8,10,12,18)(6,8,10,12,18,20)(6,8,10,12,20)(6,8,10,14)(6,8,10,14,16)(6,8,10,14,16,18)(6,8,10,14,16,18,20)(6,8,10,14,16,20)(6,8,10,14,18)(6,8,10,14,18,20)(6,8,10,14,20)(6,8,10,16)(6,8,10,16,18)(6,8,10,16,18,20)(6,8,10,16,20)(6,8,10,18)(6,8,10,18,20)(6,8,10,20)(6,8,12)(6,8,12,14)(6,8,12,14,16)(6,8,12,14,16,18)(6,8,12,14,16,18,20)(6,8,12,14,16,20)(6,8,12,14,18)(6,8,12,14,18,20)(6,8,12,14,20)(6,8,12,16)(6,8,12,16,18)(6,8,12,16,18,20)(6,8,12,16,20)(6,8,12,18)(6,8,12,18,20)(6,8,12,20)(6,8,14)(6,8,14,16)(6,8,14,16,18)(6,8,14,16,18,20)(6,8,14,16,20)(6,8,14,18)(6,8,14,18,20)(6,8,14,20)(6,8,16)(6,8,16,18)(6,8,16,18,20)(6,8,16,20)(6,8,18)(6,8,18,20)(6,8,20)(6,10)(6,10,12)(6,10,12,14)(6,10,12,14,16)(6,10,12,14,16,18)(6,10,12,14,16,18,20)(6,10,12,14,16,20)(6,10,12,14,18)(6,10,12,14,18,20)(6,10,12,14,20)(6,10,12,16)(6,10,12,16,18)(6,10,12,16,18,20)(6,10,12,16,20)(6,10,12,18)(6,10,12,18,20)(6,10,12,20)(6,10,14)(6,10,14,16)(6,10,14,16,18)(6,10,14,16,18,20)(6,10,14,16,20)(6,10,14,18)(6,10,14,18,20)(6,10,14,20)(6,10,16)(6,10,16,18)(6,10,16,18,20)(6,10,16,20)(6,10,18)(6,10,18,20)(6,10,20)(6,12)(6,12,14)(6,12,14,16)(6,12,14,16,18)(6,12,14,16,18,20)(6,12,14,16,20)(6,12,14,18)(6,12,14,18,20)(6,12,14,20)(6,12,16)(6,12,16,18)(6,12,16,18,20)(6,12,16,20)(6,12,18)(6,12,18,20)(6,12,20)(6,14)(6,14,16)(6,14,16,18)(6,14,16,18,20)(6,14,16,20)(6,14,18)(6,14,18,20)(6,14,20)(6,16)(6,16,18)(6,16,18,20)(6,16,20)(6,18)(6,18,20)(6,20)(8)(8,10)(8,10,12)(8,10,12,14)(8,10,12,14,16)(8,10,12,14,16,18)(8,10,12,14,16,18,20)(8,10,12,14,16,20)(8,10,12,14,18)(8,10,12,14,18,20)(8,10,12,14,20)(8,10,12,16)(8,10,12,16,18)(8,10,12,16,18,20)(8,10,12,16,20)(8,10,12,18)(8,10,12,18,20)(8,10,12,20)(8,10,14)(8,10,14,16)(8,10,14,16,18)(8,10,14,16,18,20)(8,10,14,16,20)(8,10,14,18)(8,10,14,18,20)(8,10,14,20)(8,10,16)(8,10,16,18)(8,10,16,18,20)(8,10,16,20)(8,10,18)(8,10,18,20)(8,10,20)(8,12)(8,12,14)(8,12,14,16)(8,12,14,16,18)(8,12,14,16,18,20)(8,12,14,16,20)(8,12,14,18)(8,12,14,18,20)(8,12,14,20)(8,12,16)(8,12,16,18)(8,12,16,18,20)(8,12,16,20)(8,12,18)(8,12,18,20)(8,12,20)(8,14)(8,14,16)(8,14,16,18)(8,14,16,18,20)(8,14,16,20)(8,14,18)(8,14,18,20)(8,14,20)(8,16)(8,16,18)(8,16,18,20)(8,16,20)(8,18)(8,18,20)(8,20)(10)(10,12)(10,12,14)(10,12,14,16)(10,12,14,16,18)(10,12,14,16,18,20)(10,12,14,16,20)(10,12,14,18)(10,12,14,18,20)(10,12,14,20)(10,12,16)(10,12,16,18)(10,12,16,18,20)(10,12,16,20)(10,12,18)(10,12,18,20)(10,12,20)(10,14)(10,14,16)(10,14,16,18)(10,14,16,18,20)(10,14,16,20)(10,14,18)(10,14,18,20)(10,14,20)(10,16)(10,16,18)(10,16,18,20)(10,16,20)(10,18)(10,18,20)(10,20)(12)(12,14)(12,14,16)(12,14,16,18)(12,14,16,18,20)(12,14,16,20)(12,14,18)(12,14,18,20)(12,14,20)(12,16)(12,16,18)(12,16,18,20)(12,16,20)(12,18)(12,18,20)(12,20)(14)(14,16)(14,16,18)(14,16,18,20)(14,16,20)(14,18)(14,18,20)(14,20)(16)(16,18)(16,18,20)(16,20)(18)(18,20)(20)()