线性表&l_线性表怎么表示

一. 线性表基础算法

1.线性表插入操作

 1 // 线性表插入操作(在第i(1≤i≤L.length+1)个位置上插入新元素elem)
 2 bool InsertSeq( SeqList& L, int i, ElemType elem )
 3 {
 4     if ( i < 1 || i>L.length + 1 || L.length >= MAXSIZE )
 5         return false;
 6     for ( j = L.length - 1; j >= i - 1; j-- )
 7         L.elem[j + 1] = L.elem[j];
 8     L.elem[j + 1] = elem;
 9     L.length++;
10     return true;
11 }

说明:

• 插入操作: 可选位置为1≤i≤L.length+1
• 最好情况: 表尾(i=n+1)插入, O(1)
• 最坏情况: 表头(i=1)插入, O(n)
• 平均情况: 设 $Pi=\frac {1}{(n+1)}$ 是在第i个位置插入一个结点的概率,则在长度为n的线性表中插入一个结点所需的移动结点的平均次数为$\frac{n}{2}$次,即O(n): $$\sum_{i=1}^{n+1} Pi\cdot{(n+1-i)}=\frac{1}{n+1}\cdot\sum_{i=1}^{n+1} {(n-i+1)}=\frac{1}{n+1}\cdot\frac{n(n+1)}{2}=\frac{n}{2}$$

2.线性表删除操作

1 bool DeleteSeq( SeqList& L, int i, ElemType& elem ) 2 { 3 for ( i<1 || i>L.length ) return false; 4 elem = L.elem[i - 1]; 5 for ( j = i; j < L.length; j++ ) 6 L.elem[j - 1] = L.elem[j]; 7 L.length--; 8 return true; 9 }

• 最好情况: 删除表位(i=n),O(1)
• 最坏情况: 删除表头(i=1),O(n)
• 平均情况: 设$Pi=\frac{1}{n}$是删除第i个位置上结点的概率,则在长度为n的线性表中删除一个结点所需移动结点的平均次数为$\frac{n-1}{2}$次,即O(n):$$\sum_{i=1}^{n}Pi\cdot{(n-i)}=\frac{1}{n}\sum_{i=1}^{n}{(n-i)}=\frac{1}{n}\cdot\frac{n(n-1)}{2}=\frac{n-1}{2}$$

3.线性表查找操作

1 int LocateSeq( SeqList& L, ElemType elem ) 2 { 3 for ( i = 0; i < L.length; i++ ) 4 if ( L.elem[i].key == elem.key ) 5 return i + 1; 6 return 0; 7 }

• 最好情况: 查找到表头,O(1)
• 最坏情况: 查找到表尾,O(n)
• 平均情况: 设$Pi=\frac{1}{n}$是查找元素在第i(1≤i≤L.length)个位置上的概率,则在长度为n的线性表中查找值为elem的元素所需比较的平均次数为$\frac{n+1}{2}$次,O(n):$$\sum_{i=1}^{n}Pi\cdot{i}=\frac{1}{n}\cdot\sum_{i=1}^{n}{i}=\frac{1}{n}\cdot\frac{n(n+1)}{2}=\frac{n+1}{2}$$

二.线性表综合应用

1.删除线性表中所有值为x的数据元素

1 bool DeleteX( SeqList& L, ElemType x ) 2 { 3 int k = 1; 4 for ( i = 1; i <= L.length; i++ ) 5 if ( L.elem[i].key != x.key ) 6 L.elem[k++] = L.elem[i]; 7 L.length = k; 8 return true; 9 }

2.从有序顺序表中删除值在[s,t]的所有元素

1 bool DeleteS2TOrderedSeq( SeqList& L, int s, int t ) 2 { 3 for ( i = 1; i <= L.length&&L.elem[i].key < s; i++ ); // 找≥s的第一个元素 4 for ( j = i; j <= L.length&&L.elem[j].key <= t; j++ ); // 找>t的第一个元素 5 while ( j <= L.length ) 6 L.elem[i++] = L.elem[j++]; 7 L.length = i; 8 return true; 9 }

3.从顺序表中删除值在[s,t]的所有元素

1 bool DeleteS2TSeq( SeqList& L, int s, int t ) 2 { 3 int k = 1; 4 for ( i = 1; i <= L.length; i++ ) 5 if ( L.elem[i].key<s || L.elem[i].key>t ) 6 L.elem[k++] = L.elem[i]; 7 L.length = k; 8 return true; 9 }

4.从有序顺序表中删除所有值重复的元素

1 bool DeleteSameOrderedSeq( SeqList& L ) 2 { 3 int k = 1; 4 for ( i = 2; i <= L.length; i++ ) 5 if ( L.elem[i].key != L.elem[k].key ) 6 L.elem[++k] = L.elem[i]; 7 L.length = k; 8 return true; 9 }

5.将两个有序顺序表合并为一个新的有序顺序表

 1 bool Merge( SeqList A, SeqList B, SeqList& C ) 2 { 3 int i = 1, j = 1, k = 1; 4 while ( i<=A.length&&j<=B.length ) 5  { 6 if ( A.elem[i].key <= B.elem[j].key ) 7 C.elem[k++] = A.elem[i++]; 8 else 9 C.elem[k++] = B.elem[j++]; 10  } 11 while ( i <= A.length ) C.elem[k++] = A.elem[i++]; 12 while ( j <= B.length ) C.elem[k++] = B.elem[j++]; 13 C.length = k - 1; 14 return true; 15 }

6.原数组A[m+n]={a1,a2,…,am,b1,b2,…,bn},现要求转变为A[m+n]={b1,b2,…,bn,a1,a2,…,am}

 1 // 元素倒置
 2 void Reverse( ElemType A[], int s, int e ) 3 { 4 for ( i = s; i < ( s + e ) / 2; i++ ) 5 swap( A[i], A[s + e - i - 1] ); 6 } 7 8 void ExChange( ElemType A[], int m, int n ) 9 { 10 Reverse( A, 0, m ); 11 Reverse( A, m, m + n ); 12 Reverse( A, 0, m + n ); 13 } 

7.线性表(a1,a2,…,an)递增有序,设计算法花最少时间找到数值为x的元素:

1)找到,则与其后继元素位置互换

2)未找到,将其插入表中并使表中元素仍然递增有序

 1 // 使用折半查找的方法
 2 void SearchExchangeInsert( ElemType A[], int n, ElemType x ) 3 { 4 int low = 1, high = n; 5 while ( low <= high ) 6  { 7 mid = ( low + high ) / 2; 8 if ( x.key == A[mid].key ) 9  { 10 if ( mid != n ) 11 swap( A[mid], A[mid + 1] ); 12 return; 13  } 14 else if ( x.key < A[mid].key ) high = mid - 1; 15 else low = mid + 1; 16  } 17 for ( j = n; j >= high + 1; j-- ) 18 A[j + 1] = A[j]; 19 A[j + 1] = x; 20 }

8.设计算法将一维数组R中的序列循环左移p(0<p<n)个位置(算法思想和⑥相同)

 1 // 元素倒置
 2 void Reverse( ElemType A[], int s, int e ) 3 { 4 for ( i = s; i < ( s + e ) / 2; i++ ) 5 swap( A[i], A[s + e - i - 1] ); 6 } 7 8 void ShiftLeft( ElemType R[], int n, int p ) 9 { 10 Reverse( R, 0, p ); 11  Reverse( R, p, n ); 12 Reverse( R, 0, n ); 13 }

9.长度为L(L≥1)的升序序列S,处在$\lceil\frac{L}{2}\rceil$个位置的数成为S的中位数,设计一个在时空都尽量高效的算法找出两个等长序列A和B的中位数

 1 int FindMidFromABOrderedSeq( int A[], int B[], int n ) 2 { 3 int s1, s2, e1, e2, m1, m2; 4 s1 = s2 = 0; 5 e1 = e2 = n - 1; 6 while ( s1 != e1 || s2 != e2 ) 7  { 8 m1 = ( s1 + e1 ) / 2; 9 m2 = ( s2 + e2 ) / 2; 10 if ( A[m1] == B[m2] ) 11 return A[m1]; 12 else if ( A[m1] < B[m2] ) 13  { 14 if ( !( ( s1 + e1 ) % 2 ) ) 15 s1 = m1, e2 = m2; 16 else 17 s1 = m1 + 1, e2 = m2; 18  } 19 else 20  { 21 if ( !( ( s2 + e2 ) % 2 ) ) 22 s2 = m2, e1 = m1; 23 else 24 s2 = m2 + 1, e1 = m1; 25  } 26  } 27 return A[s1] < B[s2] ? A[s1] : B[s2]; 28 }

三.线性表的链式表示

1.采用头插法建立单链表

 1 LinkList CreateList( LinkList& L ) 2 { 3 L = ( LinkList ) malloc( sizeof( LNode ) ); 4 L->next = NULL; 5 scanf( "%d", &x ); 6 while ( x != 9999 ) 7  { 8 s = ( LNode* ) malloc( sizeof( LNode ) ); 9 s->data = x; 10 s->next = L->next; 11 L->next = s; 12 scanf( "%d", &x ); 13  } 14 return L; 15 }

2.采用尾插法建立单链表

 1 LinkList CreateList( LinkList& L ) 2 { 3 L = ( LinkList ) malloc( sizeof( LNode ) ); 4 L->next = NULL; 5 r = L; 6 scanf( "%d", &x ); 7 while ( x != 9999 ) 8  { 9 s = ( LNode* ) malloc( sizeof( LNode ) ); 10 s->data = x; 11 r->next=s; 12 r = s; 13 scanf( "%d", &x ); 14  } 15 r->next = NULL; 16 return L; 17 }

四.线性表相关综合算法

1.递归删除不带头结点的单列表L中所有值为x的结点

1 void DeleteX( LinkList& L, ElemType x ) 2 { 3 if ( !L ) return; 4 if ( L->data == x ) 5  { 6 q = L; 7 L = L->next; 8 free( q ); 9  DeleteX( L, x ); 10  } 11 else 12 DeleteX( L->next, x ); 13 } 

2.删除带头结点的单链表L中所有值为x的结点

 1 void DeleteX( LinkList& L, ElemType x ) 2 { 3 pre = L; 4 p = L->next; 5 while ( p ) 6  { 7 if ( p->data == x ) 8  { 9 q = p; 10 pre->next = p->next; 11 p = p->next; 12 free( q ); 13  } 14 else 15  { 16 pre = p; p = p->next; 17  } 18  } 19 }

3.反向输出带头结点的单链表L的每个结点的值

1 void PrintX( LinkList L ) 2 { 3 if ( !L )return; 4 PrintX( L->next ); 5  visit( L ); 6 }

4.删除带头结点单链表L中最小值结点

 1 LinkList DeleteMin( LinkList& L ) 2 { 3  LinkList p, s, pre, q; 4 p = s = L->next; 5 pre = q = L; 6 while ( p ) 7  { 8 if(p->data<s->data ) 9  { 10 s = p; q = pre; 11  } 12 pre = p; 13 p = p->next; 14  } 15 q->next = s->next; 16 free( s ); 17 return L; 18 }

5.将带头结点的单链表就地逆置,”就地”指辅助空间复杂度为O(1)

 1 LinkList Reverse( LinkList L ) 2 { 3  LinkList p, q; 4 p = L->next; 5 L->next = NULL; 6 while ( p ) 7  { 8 q = p->next; 9 p->next = L->next; 10 L->next = p; 11 p = q; 12  } 13 return L; 14 }

6.将带头结点的单链表L排序,使其递增有序

 1 void InsertSort( LinkList& L ) 2 { 3  LinkList p, pre, r; 4 p = L->next; r = p->next; 5 p->next = NULL; p = r; 6 while ( p ) 7  { 8 r = p->next; 9 pre = L; 10 while ( pre->next&&pre->next->data < p->data ) 11 pre = pre->next; 12 p->next = pre->next; 13 pre->next = p; 14 p = r; 15  } 16 }

7.在带头结点的单链表中,删除值介于(s,t)之间的元素

 1 void DeleteS2T( LinkList& L, int s, int t ) 2 { 3  LinkList pre, p; 4 pre = L; p = pre->next; 5 while ( p ) 6  { 7 if ( p->data > s && p->data < t ) 8  { 9 pre->next = p->next; 10 free( p ); 11 p = pre->next; 12  } 13 else 14  { 15 pre = p; 16 p = p->next; 17  } 18  } 19 }

8.找出两个单链表的公共结点

 1 LinkList SearchCommon( LinkList L1, LinkList L2 ) 2 { 3  LinkList pA, pB; 4 int lenA, lenB, dist; 5 pA = L1->next, pB = L2->next; 6 lenA = lenB = 0; 7 while ( pA ) { pA = pA->next; lenA++; } 8 while ( pB ) { pB = pB->next; lenB++; } 9 pA = L1->next, pB = L2->next; 10 if ( lenA > lenB ) 11  { 12 dist = lenA - lenB; 13 while ( dist-- ) pA = pA->next; 14  } 15 else 16  { 17 dist = lenB - lenA; 18 while ( dist-- ) pB = pB->next; 19  } 20 while ( pA ) 21  { 22 if ( pA == pB ) return pA; 23 pA = pA->next, pB = pB->next; 24  } 25 return NULL; 26 }

 9.带表头结点的单链表,按递增次序输出单链表中各结点的数据元素,并释放空间

 1 void AscDelete( LinkList& L ) 2 { 3  LinkList p, s, pre, r; 4 while ( L->next ) 5  { 6 s = p = L->next; r = pre = L; 7 while ( p ) 8  { 9 if ( p->data < s->data ) 10  { 11 s = p; r = pre; 12  } 13 pre = p; 14 p = p->next; 15  } 16 r->next = s->next; 17  visit( s ); 18 free( s ); 19  } 20 free( L ); 21 }

 10.将带头结点的单链表A分解成两个带头结点的单链表A和B,A中含有奇数序号元素,B中含有偶数序号元素且相对位置不变

 1 // 法一
 2 LinkList Split( LinkList& A ) 3 { 4  LinkList p, B, rA, rB; 5 int i = 0; 6 p = A->next; 7 B = ( LinkList ) malloc( sizeof( LNode ) ); 8 rA = A; A->next = NULL; 9 rB = B; B->next = NULL; 10 while ( p ) 11  { 12 i++; 13 if (i%2) 14  { 15 rA->next = p; rA = p; 16  } 17 else 18  { 19 rB->next = p; rB = p; 20  } 21 p = p->next; 22  } 23 rA->next = NULL; 24 rB->next = NULL; 25 return B; 26 } 27 28 // 法二 29 LinkList Split( LinkList& A ) 30 { 31  LinkList p, B, rB, pre; 32 int i = 0; 33 B = ( LinkList ) malloc( sizeof( LNode ) ); 34 rB = B; 35 pre = A; p = pre->next; 36 while ( p ) 37  { 38 i++; 39 if ( i % 2 == 0 ) 40  { 41 pre->next = p->next; 42 rB->next = p; 43 rB = p; 44 p = pre->next; 45  } 46 else 47  { 48 pre = p; 49 p = p->next; 50  } 51  } 52 return B; 53 }

11.C={a1,b1,a2,b2,…,an,bn}为线性表,带有头结点,设计一个就地算法将其拆分为两个线性表,使A={a1,a2,…,an},B={bn,…,b2,b1}

 1 LinkList Split( LinkList& A ) 2 { 3  LinkList B, pre, p; 4 int i = 0; 5 B = ( LinkList ) malloc( sizeof( LNode ) ); 6 pre = A; p = pre->next; 7 while ( p ) 8  { 9 i++; 10 if ( i % 2 == 0 ) 11  { 12 pre->next = p->next; 13 p->next = B->next; 14 B->next = p; 15 p = pre->next; 16  } 17 else 18  { 19 pre = p; 20 p = p->next; 21  } 22  } 23 return B; 24 }

12.在递增有序的带头结点的单链表中,数值相同的只保留一个,使表中不再有重复的元素

 1 void DeleteSame( LinkList& L ) 2 { 3  LinkList p, q; 4 p = L->next; 5 while ( p ) 6  { 7 q = p->next; 8 if ( q&&q->data == p->data ) 9  { 10 p->next = q->next; 11 free( q ); 12  } 13 else 14 p = p->next; 15  } 16 }

13.将两个按元素值递增的单链表合并为一个按元素值递减的单链表

 1 void MergeList( LinkList& LA, LinkList& LB ) 2 { 3  LinkList pA, pB, q; 4 pA = LA->next; pB = LB->next; 5 LA->next = NULL; 6 while ( pA&&pB ) 7  { 8 if ( pA->data <= pB->data ) 9  { 10 q = pA->next; 11 pA->next = LA->next; 12 LA->next = pA; 13 pA = q; 14  } 15 else 16  { 17 q = pB->next; 18 pB->next = LA->next; 19 LA->next = pB; 20 pB = q; 21  } 22  } 23 if ( pA ) 24 pB = pA; 25 while(pB ) 26  { 27 q = pB->next; 28 pB->next = LA->next; 29 LA->next = pB; 30 pB = q; 31  } 32 free( LB ); 33 }

14.A,B为两个元素递增有序的单链表(带头结点),设计算法从A,B中公共元素产生单链表C,要求

 1 void MergeList( LinkList& LA, LinkList& LB ) 2 { 3  LinkList pA, pB, q; 4 pA = LA->next; pB = LB->next; 5 LA->next = NULL; 6 while ( pA&&pB ) 7  { 8 if ( pA->data <= pB->data ) 9  { 10 q = pA->next; 11 pA->next = LA->next; 12 LA->next = pA; 13 pA = q; 14  } 15 else 16  { 17 q = pB->next; 18 pB->next = LA->next; 19 LA->next = pB; 20 pB = q; 21  } 22  } 23 if ( pA ) 24 pB = pA; 25 while ( pB ) 26  { 27 q = pB->next; 28 pB->next = LA->next; 29 LA->next = pB; 30 pB = q; 31  } 32 free( LB ); 33 }

15.求两个元素递增排列的链表(带头结点)A和B的交集并存放于A链表中,并释放其他结点

 1 void Intersect( LinkList& LA, LinkList& LB ) 2 { 3  LinkList pA, pB, r, q; 4 pA = LA->next; pB = LB->next; 5 r = LA; LA->next = NULL; 6 while ( pA&&pB ) 7  { 8 if ( pA->data == pB->data ) 9  { 10 r->next = pA; 11 r = pA; 12 pA = pA->next; 13 q = pB; 14 pB = pB->next; 15 free( q ); 16  } 17 else if ( pA->data < pB->data ) 18  { 19 q = pA; 20 pA = pA->next; 21 free( q ); 22  } 23 else 24  { 25 q = pB; 26 pB = pB->next; 27 free( q ); 28  } 29  } 30 r->next = NULL; 31 while ( pA ) 32  { 33 q = pA; 34 pA = pA->next; 35 free( q ); 36  } 37 while ( pB ) 38  { 39 q = pB; 40 pB = pB->next; 41 free( q ); 42  } 43 free( LB ); 44 }

16.判断单链表序列B是否是A的连续子序列(不带头结点)

 1 bool IsSubsequence( LinkList A, LinkList B ) 2 { 3  LinkList pA, pB, h; 4 pA = A; pB = B; 5 h = pA; 6 while ( pA&&pB ) 7  { 8 if ( pA->data == pB->data ) 9  { 10 pA = pA->next; 11 pB = pB->next; 12  } 13 else 14  { 15 h = h->next; 16 pA = h; 17 pB = B; 18  } 19  } 20 if ( pB ) return false; 21 return true; 22 }

17.判断带头结点的循环双链表是否对称

 1 bool IsSymmetry( DLinkList L ) 2 { 3  DLinkList p, q; 4 p = L->next; q = L->prior; 5 while ( p != q && q->next != p ) 6  { 7 if ( p->data != q->data ) 8 return false; 9 p = p->next; 10 q = q->next; 11  } 12 return true; 13 }

18.将循环单链表h2链接到h1之后

 1 LinkList Link( LinkList& h1, LinkList& h2 ) 2 { 3  LinkList p; 4 p = h1; 5 while ( p->next != h1 )p = p->next; 6 p->next = h2; 7 p = h2; 8 while ( p->next != h2 )p = p->next; 9 p->next = h1; 10 return h1; 11 }

19.带头结点的循环链表,按递增次序输出循环链表中各结点的数据元素,并释放空间

 1 void AscDelete( LinkList& L ) 2 { 3  LinkList p, s, r, pre; 4 while ( L->next != L ) 5  { 6 s = p = L->next; r = pre = L; 7 while ( p != L ) 8  { 9 if ( p->data < s->data ) 10  { 11 s = p; r = pre; 12  } 13 pre = p; 14 p = p->next; 15  } 16  visit( s ); 17 r->next = s->next; 18 free( s ); 19  } 20 free( L ); 21 }

20.查找单链表(带头结点)中倒数第k个位置的结点,成功:则输出并返回true,否则只返回false

 1 bool SearchBackwardK( LinkList L, int k ) 2 { 3  LinkList p, q; 4 int count; 5 p = q = L->next; 6 count = 0; 7 while (p) 8  { 9 if ( count < k ) count++; 10 else q = q->next; 11 p = p->next; 12  } 13 if ( count < k ) return false; 14  visit( q ); 15 return true; 16 }

21.链表中data绝对值相等的点,只保留第一次出现的结点($\vert{data}\vert\le{n}$)

 1 void DeleteSameAbs( LinkList L, int n ) 2 { 3  LinkList pre, p; 4 int *B, pos; 5 B = ( int * ) malloc( sizeof( int )*( n + 1 ) ); 6 for ( int i = 0; i < n + 1; i++ ) 7 B[i] = 0; 8 pre = L; p = L->next; 9 while ( p ) 10  { 11 pos = p->data > 0 ? p->data : -p->data; 12 if ( B[pos] == 0) 13  { 14 B[pos] = 1; pre = p; p = p->next; 15  } 16 else 17  { 18 pre->next = p->next; free( p ); p = pre->next; 19  } 20  } 21 free( B ); 22 }

22.带头结点的循环双链表递增排序

 1 void AscSort( DLinkList L ) 2 { 3  DLinkList p, q, r; 4 if ( !L ) return; 5 p = L->next; q = p->next; r = q->next; 6 while ( q!=L ) 7  { 8 while ( p != L && p->data > q->data ) 9 p = p->prior; 10 // 脱链结点p 11 q->prior->next = r; 12 r->prior = q->prior; 13 // 插入节点p 14 q->next = p->next; 15 q->prior = p; 16 p->next->prior = q; 17 p->next = q; 18 // 归位(相对位置) 19 q = r; 20 p = q->prior; 21 r = r->next; 22  } 23 }

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