[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 }