[Algorithm]排序

一.排序算法

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1.插入排序

1) 直接插入排序:(插入类)

 1 void InsertSort( ElemType R[], int n )
 2 {
 3     for ( int i = 2; i <= n; i++ )
 4     {
 5         if ( R[i].key < R[i - 1].key )
 6         {
 7             R[0] = R[i];
 8             for ( int j = i - 1; j > 0 && ( R[0].key < R[j].key ); j-- )
 9                 R[j + 1] = R[j];
10             R[j + 1] = R[0];
11         }
12     }
13 }

最好情况(顺序有序):

　　1)比较次数: $\sum_{i=2}^{n} 1=n-1$

　　2)移动次数: 0

最坏情况(逆序有序):

　　1)比较次数: $\sum_{i=2}^{n} i=\frac {(n+2)(n-1)}{2}$

　　2)移动次数: $\sum_{i=2}^{n} (i+1)=\frac {(n+4)(n-1)}{2}$

2)折半插入排序:(插入类)

 1 void BiInsertSort( ElemType R[], int n )
 2 {
 3     for ( int i = 2; i <= n; i++ )
 4     {
 5         R[0] = R[i];
 6         int low = 1, high = i - 1;
 7         while ( low <= high )
 8         {
 9             int mid = ( low + high ) / 2;
10             if ( R[0].key < R[m].key ) high = mid - 1;
11             else low = mid + 1;
12         }
13         for ( int j = i - 1; j > high; j-- )
14             R[j + 1] = R[j];
15         R[j + 1] = R[0];
16     }
17 }

3)希尔排序(又称缩小增量排序)(插入类)

 1 // 当dk=1时,即为直接插入排序
 2 void ShellSort( ElemType R[], int n )
 3 {
 4     for ( int dk = n / 2; dk >= 1; dk /= 2 )
 5     {
 6         for ( int i = dk + 1; i <= n; i++ )
 7         {
 8             if ( R[i].key < R[i - dk].key )
 9             {
10                 R[0] = R[i];
11                 for ( j = i - dk; j > 0 && ( R[0].key < R[j].key ); j -= dk )
12                     R[j + dk] = R[j];
13                 R[j + dk] = R[0];
14             }
15         }
16     }
17 }

2.交换排序

1)起泡排序(冒泡排序)(交换类)

 1 void BubbleSort( ElemType R[], int n )
 2 {
 3     for ( int i = 1; i <= n - 1; i++ )
 4     {
 5         bool flag = false;
 6         for ( int j = n; j > i; j-- )
 7         {
 8             if (R[j].key < R[j-1].key )
 9             {
10                 swap( R[j], R[j - 1] );
11                 flag = true;
12             }
13         }
14         if ( !flag ) return;
15     }
16 }

2)快速排序:(交换类)

 1 void Partition( ElemType R[], int low, int high );
 2
 3 // 快排
 4 void QuickSort( ElemType R[], int low, int high )
 5 {
 6     if ( low >= high ) return;
 7     int pivotpos = Partition( R, low, high );
 8     QuickSort( R, low, pivotpos - 1 );
 9     QuickSort( R, pivotpos + 1, high );
10 }
11
12 // 划分
13 void Partition( ElemType R[], int low, int high )
14 {
15     ElemType pivot = R[low];
16     while ( low < high )
17     {
18         while ( low < high && R[high].key >= pivot.key ) high--;
19         R[low] = R[high];
20         while ( low < high && R[low].key <= pivot.key ) low++;
21         R[high] = R[low];
22     }
23     R[low] = pivot;
24     return low;
25 }

3.选择排序

1)简单选择排序(选择类)

 1 void SelectSort( ElemType R[], int n )
 2 {
 3     for ( int i = 0; i < n - 1; i++ )
 4     {
 5         int min = i;
 6         for ( int j = i + 1; j < n; j++ )
 7         {
 8             if ( R[j].key < R[min].key ) min = j;
 9         }
10         if ( min != i ) swap( R[i], R[min] );
11     }
12 }

2)堆排序(选择类)

 1 void AdjustDown( ElemType R[], int s, int n );
 2
 3 void HeapSort( ElemType R[], int n )
 4 {
 5     for ( int i = n / 2; i > 0; i-- )
 6         void AdjustDown( R, i, n );
 7     for ( int i = n; i > 1; i-- )
 8     {
 9         swap( R[i], R[1] );
10         AdjustDown( R, 1, i - 1 );
11     }
12 }
13
14 // 向下调整
15 void AdjustDown( ElemType R[], int s, int n )
16 {
17     R[0] = R[s];
18     for ( int i = 2 * s; i <= n; i *= 2 )
19     {
20         if ( i < n&&R[i].key < R[i + 1].key ) i++;
21         if (R[0].key  >=R[i].key ) break;
22         else
23         {
24             R[s] = R[i]; s = i;
25         }
26     }
27     R[s] = R[0];
28 }
29
30 // 向上调整
31 void AdjustUp( ElemType R[], int s )
32 {
33     R[0] = R[s];
34     int p = s / 2;
35     while ( p > 0 && R[p].key < R[0].key )
36     {
37         R[s] = R[p];
38         s = p;
39         p /= 2;
40     }
41     R[s] = R[0];
42 }

4.归并排序(归并类)

 1 void Merge( ElemType R[], int low, int mid, int high );
 2
 3 void MergeSort( ElemType R[], int low, int high )
 4 {
 5     if ( low >= high ) return;
 6     int mid = ( low + high ) / 2;
 7     MergeSort( R, low, mid );
 8     MergeSort( R, mid + 1, high );
 9     Merge( R, low, mid, high );
10 }
11
12 ElemType B[MAXSIZE];
13 void Merge( ElemType R[], int low, int mid, int high )
14 {
15     int i,j,k;
16     for ( i = low; i <= high; i++ )
17         B[i] = R[i];
18     i = k = low, j = mid + 1;
19     while ( i <= mid && j <= high )
20     {
21         if ( B[i].key <= B[j].key )
22             R[k++] = B[i++];
23         else
24             R[k++] = B[j++];
25     }
26     while ( i <= mid ) R[k++] = B[i++];
27     while ( j <= high ) R[k++] = B[j++];
28 }

二.综合题(算法)

1.设顺序表用数组R[]表示,表中存储在数组下标1~m+n的范围内,前m个元素递增有序,后n个元素递增有序,设计一个算法,使得整个顺序表有序

 1 void InsertSort( ElemType R[], int m, int n )
 2 {
 3     for ( int i = m + 1; i <= m + n; i++ )
 4     {
 5         if ( R[i].key < R[i - 1].key )
 6         {
 7             R[0] = R[i];
 8             for ( int j = i - 1; j > 0 && ( R[0].key < R[j].key ); j-- )
 9                 R[j + 1] = R[j];
10             R[j + 1] = R[0];
11         }
12     }
13 }

2.计数排序:对表进行排序并将结果放到另一个新的表中,要求表中所有关键码互不相同

 1 void CountSort( ElemType A[], ElemType B[], int n )
 2 {
 3     for ( int i = 0; i < n; i++ )
 4     {
 5         int cnt = 0;
 6         for ( int j = 0; j < n; j++ )
 7             if ( A[i].key > A[j].key )cnt++;
 8         B[cnt] = A[i];
 9     }
10 }

3.双向冒泡排序

 1 // 思想:第一趟通过交换把最大的放最后,第二趟通过交换把最小的放最前,反复进行
 2 void BubbleSort( ElemType A[], int n )
 3 {
 4     int low = 0, high = n - 1, i;
 5     bool flag = true;
 6     while ( low < high && flag )
 7     {
 8         flag = false;
 9         for (i = low; i < high; i++ )
10         {
11             if (A[i]>A[i+1] )
12             {
13                 swap( A[i], A[i + 1] ); flag = true;
14             }
15         }
16         high--;
17         for ( i = high; i > low; i-- )
18         {
19             if ( A[i] < A[i - 1] )
20             {
21                 swap( A[i], A[i - 1] ); flag = true;
22             }
23         }
24         low++;
25     }
26 }

4.单链表的简单选择排序(假设不带表头结点)

 1 void SelectSort( LinkList& L )
 2 {
 3     LinkList h, p, s, pre, r;
 4     h = L;
 5     while ( h )
 6     {
 7         p = s = h; pre = r = NULL;
 8         // 找最大结点s
 9         while ( p )
10         {
11             if (p->data>s->data )
12             {
13                 s = p; r = pre;
14             }
15             pre = p;
16             p = p->next;
17         }
18         // 脱链
19         if ( s == h ) h = h->next;
20         else r->next = s->next;
21         // 头插法
22         s->next = L; L = s;
23     }
24 }

5.顺序表中有n个不同整数(下标1~n),设计算法把所有奇数移动到偶数前面(时,空都最少)

 1 void Move( ElemType A[], int n )
 2 {
 3     int low = 1, high = n;
 4     while ( low < high )
 5     {
 6         while ( low < high&&A[low] % 2 ) low++;
 7         while ( low < high && A[high] % 2 == 0 ) high--;
 8         if ( low < high )
 9         {
10             swap( A[low], A[high] );
11             low++; high--;
12         }
13     }
14 }

6.在顺序表中找出第k小的元素(时空最少)

 1 // 思想:划分
 2 int Partition( ElemType R[], int low, int high )
 3 {
 4     int pivot = R[low];
 5     while ( low < high )
 6     {
 7         while ( low < high && R[high].key >= pivot.key ) high--;
 8         R[low] = R[high];
 9         while ( low < high&& R[low].key <= pivot.key ) low++;
10         R[high] = R[low];
11     }
12     R[low] = pivot;
13     return low;
14 }
15
16 ElemType Kth_elem( ElemType R[], int low, int high, int k )
17 {
18     int pivotpos = Partition( R, low, high );
19     if ( pivotpos == k ) return R[pivotpos];
20     else if ( pivotpos > k ) return Kth_elem( R, low, pivotpos - 1, k );
21     else return Kth_elem( R, pivotpos + 1, high, k );
22 }

7.n个正整数构成的集合A,将其划分为两个不相交的子集$A1,A2$,元素个数分别是n1和n2.A1和A2中元素之和分别为S1和S2.设计一个时空高效算法,使|n1-n2|最小且|s1-s1|最大.(下标从1开始)

 1 int Partition( ElemType R[], int low, int high )
 2 {
 3     int pivot = R[low];
 4     while ( low < high )
 5     {
 6         while ( low < high && R[high].key >= pivot.key ) high--;
 7         R[low] = R[high];
 8         while ( low < high&& R[low].key <= pivot.key ) low++;
 9         R[high] = R[low];
10     }
11     R[low] = pivot;
12     return low;
13 }
14
15 int SetPartition( ElemType R[], int n, int low, int high )
16 {
17     int k = n / 2, s1, s2, i;
18     int pivotpos = Partition( R, low, high );
19     if ( pivotpos == k )
20     {
21         s1 = s2 = 0;
22         for ( i = 1; i <= k; i++ ) s1 += R[i];
23         for ( j = k + 1; j <= n; j++ ) s2 += R[j];
24         return s2 - s1;
25     }
26     else if ( pivotpos > k )
27         return SetPartition( R, n, low, pivotpos - 1 );
28     else return SetPartition( R, n, pivotpos + 1, high );
29 }