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+1,61 @@ + + + + +## 线性表的基础概念和基本操作 + +> 强调线性表是一种逻辑结构,不是存储结构 + + +### 定义 + +线性表是具有相同数据类型的n(n≥0)个数据元素的有限序列。一般表示: + +L=(a1,a2,a3......an) 其中n可以理解为表长(线性表的长度),n=0时候,即表空 + + +- `表头元素`:线性表中唯一的“第一个”数据元素,例如a1 +- `表尾元素`:线性表中唯一的“最后一个”数据元素,例如an + + +重要逻辑特性: + +- 除表头元素外,线性表中每个元素有且仅有一个`直接前驱` +- 除表尾元素外,线性表中每个元素有且仅有一个`直接后继` + +基于此,这种**线性有序的逻辑结构**,使得线性表的特点如下: + +- 元素的**个数有限**(强调有限序列) +- 元素在逻辑上具有**顺序性**,在序列中每个元素都是都有先后次序的 +- 元素都数据元素,**每个元素都是单个元素** +- 元素的**数据类型都相同**(强调相同数据类型),每个数据元素占用相同大小的存储空间 +- 元素具有**抽象性**,仅仅讨论元素之间的逻辑关系,不需要去考虑元素究竟表示的什么内容 + + +> Tips: **线性表是一种逻辑结构**,表示元素之间一对一的相邻关系。**顺序表和链表则指的是存储结构** + + + +### 基本操作 + +- `InitList(&L)`: **初始化表**。构造空的线性表 +- `Length(L)`:**获取表的长度**。返回线性表L的长度,即表中的数据元素个数 +- `LocateElem(L,e)`:**按值查找操作**。在表L中国查找具有给定关键字的元素 +- `GetElem(L,i)`:**按位查找操作**。获取表中第i个位置的元素的值 +- `ListInsert(&L,i,e)`:**插入操作**。在表的第i个位置上插入指定元素e +- `ListDelete(&L,i,&e)`:**删除操作**。删除表中第i个位置的元素,并用e返回删除元素的值 +- `PrintList(L)`:**输出操作**。按照前后顺序(如:1、2....n)输出线性表的所有元素值 +- `Empty(L)`:**判空操作**。当表L为空,则返回true,否则返回false +- `DestoryList(&L)`:**销毁操作**。将线性表销毁,释放线性表L所占用的内存空间(类似:释放内存) + + +线性表是具有相同的数据类型的有限个数据元素组成的,**数据元素是由数据项组成的** + + diff --git a/数据结构/线性表/2.线性表的顺序表示.md b/数据结构/线性表/2.线性表的顺序表示.md new file mode 100644 index 0000000..0a3f570 --- /dev/null +++ b/数据结构/线性表/2.线性表的顺序表示.md @@ -0,0 +1,274 @@ + + + +## 线性表的顺序表示 + + +### 定义 + +`顺序表`:顺序存储的线性表,**是用一组地址连续的存储单元,依次存储线性表中的数据元素,使得在逻辑上相邻的两个元素在物理位置上也相邻。** + + + + + + +顺序表中的元素的逻辑顺序与实际的物理位置相同 + + + +注意: + +- 线性表中的元素的位序是从1开始的,例如1、2、3... +- 数组中的元素的下标是从0开始的,例如0、1、2... + +```c +# define MaxSize 20 // 定义常量MaxSize 用来声明顺序表的最大长度 + +// 线性表结构体定义【ElemType用来代指顺序表中元素的类型,例如高级语言中的int、string....】 +typedef struct{ + ElemType data[MaxSize]; // 顺序表的元素 + int length; // 顺序表的长度 +}SqList + +``` + +#### 存储分配 + +`静态分配`:数组的大小和空间都是实现确定好的,一旦存储空间占满就会产生溢出,直接导致程序崩溃。(有点内存不够,宕机重启的意思....) + +`动态分配`:存储数据的空间在程序执行过程中通过`动态存储分配语句`分配的,即便是数据空间占满,也可以另外开辟一块更大的空间,来替换原来的存储空间,满足扩充数据空间的目的。(有点动态规划的意思....)最重要的是:**不需要像静态分配那样,一次性地固定线性表的空间和大小** + + + +```c +#define InitSize 100 // 表长度初始化 + + +// 动态分配数组顺序表的结构体定义 +typedef struct{ + ElemType *data; // 动态分配数组的指针 + int MaxSize,length; // 数组的最大容量和当前元素个数 +}SqList; + +``` + +动态分配语句 + +```C +// C语言中 + +L.data=(ElemType*)malloc(sizeof(ElemType)*InitSize); + + +// C++ 中 + +L.data=new ElemType[InitSize]; + +``` + +`malloc()函数`: 指针型函数,返回的指针指向该分配域的开头的位置。作用是在内存的动态存储区中分配一个长度为size的连续空间。[百度百科](https://baike.baidu.com/item/malloc%E5%87%BD%E6%95%B0/8582146?fr=aladdin) + +**动态分配不是链式存储,而是属于顺序存储结构**,动态分配的物理结构没有改变,依然是随机存取的方式。只是分配的空间大小可以在运行时决定; + + +#### 顺序表的特点 + + +- 随机访问【这是最主要的特点】,通过存储起始地址和元素序号O(1)时间内访问指定元素。 +- 存储密度高,没有结点只存储数据元素,不像索引存储那样,还需要索引表什么的.. +- 逻辑上相邻的元素物理上也相邻,插入和删除需要移动大量元素 + + + + + +### 基本操作 + + +#### 插入 + +在顺序表L的第i(1≤i≤L.length+1)个位置插入新的元素e + +- 第一步:如果i非法,则直接返回false,插入失败,结束插入过程 +- 第二步:i正常,将表的第i个元素以及后面的所有元素都像有移动一个位置,在腾出来的空位置插入元素e +- 第三步:顺序表插入成功,返回true + +注意:先判空和临界值,提高算法健壮性 + +```C++ + +/* + * @Description: 顺序表的插入操作 + * @Version: Beta1.0 + * @Author: 【B站&公众号】Rong姐姐好可爱 + * @Date: 2020-02-23 07:48:26 + * @LastEditors: 【B站&公众号】Rong姐姐好可爱 + * @LastEditTime: 2020-02-23 07:48:26 + */ +bool ListInsert(SqList &L, int i, ElemType e){ + + // i非法 i=1 表头 i=L.length+1 表尾巴 + if(i<1||i>L.length+1){ + return false; + } + + // 存储空间满,无法插入 + if(L.length >= MaxSize){ + return false; + } + + // 遍历,将位置元素往后移动,注意从后往前循环,避免值被覆盖 + for(int j=L.length; j>=i;j--){ + L.data[j]=L.data[j-1]; + } + + // 此时,表L中的第i个元素和第i+1元素素值一样,将新元素存入i位置即可 + + // 第i个元素,对应的位置角标为i-1 + L.data[i-1]=e; + + // 表长度加1 + L.length++; + + // 返回插入成功 + return true; +} + +``` + +注意:区别顺序表中的位序和角标; + +**时间复杂度** + +- 最好情况:在表尾插入,元素向后移动循环没有执行,时间复杂度O(1); +- 最坏情况:在表头插入,元素后移循环执行n次,时间复杂度为O(n); +- 平均情况:随机插入,平均次数为:n/2,对应的平均复杂度为O(n); + + +**线性表插入算法的平均时间复杂度为:O(n)** + +> Tips: 需要根据实现代码理解循环为什么是从后往前来实现元素后移,通过for循环可以很明显的看出表尾插入快,表头插入慢 + +#### 删除 + +删除顺序表L中第i(1≤i≤L.length+1)个位置的元素 + +- 成功,返回true,将被删除的元素用引用变量返回; +- 失败,返回false + +```C++ + +/* + * @Description: 顺序表的删除操作 + * @Version: Beta1.0 + * @Author: 【B站&公众号】Rong姐姐好可爱 + * @Date: 2020-02-23 07:48:26 + * @LastEditors: 【B站&公众号】Rong姐姐好可爱 + * @LastEditTime: 2020-02-23 07:48:26 + */ +bool ListDelete(SqList &L, int i, ElemType &e){ + + // i非法 i=1 表头 i=L.length+1 表尾巴 + if(i<1||i>L.length+1){ + return false; + } + + // 存储空间满,无法插入 + if(L.length >= MaxSize){ + return false; + } + + // 引用变量e赋值 + e=L.data[i-1] + + // 遍历,第i个元素后面的往前移动 + for(int j=i; j<=L.length;j++){ + // 从第i个元素开始,角标从i-1开始 + L.data[j-1]=L.data[j]; + } + + // 此时,表L中的表尾元素和倒数第二个元素值一样,将表的长度-1 + + // 表长度减1 + L.length--; + + // 返回删除成功 + return true; +} + +``` + +从这里来看,删除、插入元素都会涉及到大量的元素的移动(最好情况例外),总结而言: +- 元素从后往前移,循环从前往后遍历 +- 元素从前往后移,循环从后往前遍历 + + +**时间复杂度:** + + +- 最好情况:删除表尾元素,不需要移动任何元素,时间复杂度为O(1); +- 最坏情况:删除表头元素,需要移动除第一个元素外的所有元素,时间复杂度为O(n); +- 平均情况:随机删除,平均需要(n-1)/2,对应的时间复杂度为O(n); + + + +**线性表删除算法的平均时间复杂度为O(n);** + + +#### 按值查找(顺序查找) + +在顺序表L中查找第一个元素值等于e的元素,并返回位序 + + +```C++ +/* + * @Description: 顺序表的按值查找(顺序查找) + * @Version: Beta1.0 + * @Author: 【B站&公众号】Rong姐姐好可爱 + * @Date: 2020-02-23 07:48:26 + * @LastEditors: 【B站&公众号】Rong姐姐好可爱 + * @LastEditTime: 2020-02-23 07:48:26 + */ +int LocateElem(SqList L,ElemType e){ + int i; + // 循环判断 + for(i=0;i + + + + +## 线性表的链式表示 + +顺序表的插入、删除操作需要移动大量元素,影响了运行效率(虽然时间复杂度为O(1)的情况也存在)。 + + +**链式存储线性表时,不需要使用连续的存储单元,不要求逻辑上相邻的两个元素在物理位置上也相邻** + + +理解“链”的含义,链条--->捆绑、指向------>指针 + + +**链式存储线性表时,对线性表的插入、删除元素是不需要移动元素的,只是需要修改指针** + + +- 单链表 +- 双链表 +- 循环链表 +- 静态链表 + + + +### 单链表 + +线性表的链式存储称作`单链表`,通过**一组任意的存储单元**来存储线性表中的数据元素。 + + +每个链表结点(node)除了存放元素自身的信息外,还需要存放一个指向其后继结点的指针。目的是:**通过指针建立起链表元素之间的线性关系** + + +单链表中结点类型的描述: +```C++ + +// 单链表结点类型定义 +typeof struct LNode{ + ElemType data; // 数据域 + struct LNode *next; // 指针域 +}LNode , *LinkList; + +``` + + +单链表可以解决顺序表需要大量连续存储空间的缺点,但是单链表在数据域的基础上附加了指针域,存在浪费存储空间的缺点; + + +单链表的元素是**离散地分布**在存储空间中的,因此**单链表是非随机存取的存储结构**,不能直接找到表中特定的结点,需要从头开始遍历、一次查找; + + +通常,**头指针用来标识一个单链表**。头指针指向`NULL`时,标识单链表为空。 + + + +#### 头结点 + +为了操作上的方便,在单链表第一个结点之前附加一个结点,叫做**头结点**。 + +- 头结点的数据域可以不存任何信息、也可以记录表长等基础信息 +- 头结点的指针域指向线性表的第一个元素结点; + + + + +**不论单链表是否带头结点(可选),头指针始终指向链表的第一个结点。** + +头结点是带头结点的链表中的第一个结点【重要】 + +- 头结点的数据域可以不存任何信息、也可以记录表长等基础信息 +- 头结点的指针域指向线性表的第一个元素结点; + + +头结点的优点: + +- 因为开始结点的位置被存放在头结点的指针域中,所以在链表的第一个位置上的操作和在表的其他位置上的操作一致,不需要进行特殊处理; +- 无论链表是否为空,头指针始终是指向头结点的头结点的非空指针【空表中,往往就只有头结点,此时头结点的指针域为空,可以有效避免头指针空指针异常问题】-----> **头结点的引入,很好的统一了空表和非空表的操作;** + + + +#### 头插法 + +> 从空表开始,生成新的结点,将读取的数据存放在新结点的数据域中,将新结点插入到当前链表的表头【头结点之后】 + + +```C++ +/* + * @Description: 单链表头插法创建 + * @Version: Beta1.0 + * @Author: 【B站&公众号】Rong姐姐好可爱 + * @Date: 2020-03-04 23:38:04 + * @LastEditors: 【B站&公众号】Rong姐姐好可爱 + * @LastEditTime: 2020-03-04 23:39:16 + */ +LinkList CreateListWithStartNode(LinkList &L){ + + LNode *s; + int x; + L=(LinkList)malloc(sizeof(LNode)); // 创建头结点L + L->next=NULL; // 初始化空链表 + + // 控制台输入值 + scanf("%d",&x); + + // 输入9999 表示结束 + while(x!==9999){ + // 开辟新结点存储空间 + s=(LNode*)malloc(sizeof(LNode)); + // 结点数据域赋值 + s->data=x; + // 修改指针,新结点插入表中【注意:L->next为头结点的指针域】 + s->next=L->next; + L->next=s; + scanf("%d",&x); + } + + // 返回单链表 + return L; +} + + +``` + +特点: + +- 读入数据的顺序与生成的链表中的元素顺序是相反的【结合队列先进先出思考】 +- 每个结点插入的时间复杂度为O(1),单链表长度为n时,头插法的时间复杂度为O(n)【结合算法中的while循环,可以很明确看出时间复杂度】 + + +#### 尾插法 + +头插法建立的单链表,链表中结点的次序和输入数据的顺序不一致【相反】,尾插法则很好的避免了这个问题; + +>新结点插入到当前链表的表尾上,必须增加一个尾指针r,始终指向当前链表的尾结点; + + +```C++ + +/* + * @Description: 单链表尾插法创建 + * @Version: Beta1.0 + * @Author: 【B站&公众号】Rong姐姐好可爱 + * @Date: 2020-03-04 23:38:04 + * @LastEditors: 【B站&公众号】Rong姐姐好可爱 + * @LastEditTime: 2020-03-04 23:39:16 + */ +LinkList CreateListWithEndNode(LinkList &L){ + + + int x; // 输入结点值 + L=(LinkList)malloc(sizeof(LNode)); + LNode *s; // 新结点s + LNode *r=L; // r为尾指针 + + // 控制台输入值 + scanf("%d",&x); + + while(x!==9999){ + // 开辟新结点存储空间 + s=(LNode *)malloc(sizeof(LNode)); + + // 新结点s的数据域赋值为x + s->data=x; + // 单链表L的尾指针指向新的结点s + r->next=s; + + // 指针r指向新的表尾结点 + r=s; + + scanf("%d",&x); + } + + // 表尾指针置空【重要】 + r->next=NULL; + + // 返回单链表 + return L; + +} + +``` + +特点: + +- 读入数据的顺序与生成的链表中的元素顺序完全一致 +- 每个结点插入的时间复杂度为O(1),单链表长度为n时,尾巴插法的时间复杂度为O(n)【结合算法中的while循环,可以很明确看出时间复杂度】 +- 相比头插法附设了一个指向表尾结点的指针,但时间复杂度与头插法相同 + + + +#### 按序号查找 + +> 在单链表中从第一个结点出发,顺指针next域逐个往下搜索、遍历,直到找出第i个结点为止,否则返回最后一个结点指针域NULL + + +```C++ + +/* + * @Description: 单链表按序号查找 + * @Version: Beta1.0 + * @Author: 【B站&公众号】Rong姐姐好可爱 + * @Date: 2020-03-04 23:38:04 + * @LastEditors: 【B站&公众号】Rong姐姐好可爱 + * @LastEditTime: 2020-03-04 23:39:16 + */ +LNode *GetElem(LinkList L,int i){ + int j=1; // 查询计数,初始为1 + LNode *p=L->next; // 单链表头结点指针赋值给指针p + + + // 第0个元素,则指向头结点,返回头结点 + if(i==0){ + // 头结点包含数据域和指针域 + return L; + } + + // 不等于0,却小于1,则i为负数无效,直接返回NULL,查询结果空; + if(i<1){ + return NULL; + } + + // p存在且计数没有走到初始i的位置 + while(p&&jnext; + + // 计数标记+1 + j++; + } + + // 注意: 当p不存在时, 跳出循环,p=NULL; 当p存在但是j大于等于i,跳出循环,返回查找的结果,返回p + // 从跳出循环上来分析,p要么存在即:找到的结点元素,要么为空即NULL + + // 跳出循环,返回第i个结点的指针 + return p; + +} +``` + +需要遍历(扫描)单链表,时间复杂度为:O(n) + + + +#### 按值查找 + +> 从单链表的第一个结点开始,从前往后依次比较表中个结点数据域的值,等于给定值e,则返回该结点的指针;若整个单链表【遍历完】中没有数据域值为e的结点,则返回NULL; + + +```C++ + +/* + * @Description: 单链表按值查找 + * @Version: Beta1.0 + * @Author: 【B站&公众号】Rong姐姐好可爱 + * @Date: 2020-03-04 23:38:04 + * @LastEditors: 【B站&公众号】Rong姐姐好可爱 + * @LastEditTime: 2020-03-04 23:39:16 + */ +LNode *LocateElem(LinkList L,ElemType e){ + + // 指针【哨兵】 + LNode *p=L->next; + // 从第1个结点开始查找数据域(data)为e的结点 + while(p!=NULL&&p->data!=e){ + // 无法匹配,指针后移 + p=p->next; + } + + // 注意:p为NULL的时候,说明单链表已经遍历的尾结点了,跳出循环,没有找到目标结点; + + // 查找到第1个匹配的结点,跳出循环,返回结点指针 + return p; + // +} + +``` +链表遍历无法匹配,会返回NULL,因为在尾结点无法匹配的时候,直接返回尾结点指针域 + +需要遍历(扫描)单链表,时间复杂度为:O(n) + + +#### 结点插入 + +> 单链表中,将值为x的新结点插入到单链表的第i个位置上 + +- 第一步: 检查插入位置的合法性; +- 第二步: 找到待插入位置的前驱结点,即第(i-1)个结点; +- 第三部: 在前驱结点后插入新结点; + +```C++ + // 循环遍历,时间复杂度O(n) + p=GetElem(L,i-1); + + // 移动指针,时间复杂度O(1) + s->next=p->next; + p->next=s; +``` + +结合上面的代码可以看出,将元素x插入到单链表L的第i个元素上,必须先找到单链表L的i个结点的前驱结点,即(i-1)的位置,需要采用`GetElem()`函数,按照序号查找; + +如果返回的前驱结点不为空,则说明插入的位置i合法,否则位置非法,插入失败; + +找到前驱结点p后,最重要的是移动指针,将新的结点s的指针域指向结点p的指针域,也就是s的指针域指向元素p的后继结点,第i个结点元素 + +原来的(i-1)位置上的元素,也就是前驱结点p的指针域则必须指向新的结点元素; + +**上面的过程不能更换,避免后继指针不存在的问题** + + +最后的最后,一定要注意将s的数据域赋值x + + +插入结点的时间复杂度集中在查找第(i-1)个元素,时间复杂度为O(n);如果在给定结点的后面插入新结点,只需要执行`p->next=s`操作,时间复杂度为O(1) + + + +##### 前插操作 + +> 在某结点的前面插入一个新的结点 + +**对结点的前插操作都可以转化为后插操作,前提:需要从单链表的头结点开始顺序查找到其前驱结点;时间复杂度为O(n)。** + +##### 后插操作 + +> 在某结点的后面插入一个新的结点,单链表插入算法中,通常采用后插操作的 + + +```C++ + +// 结点s插入到结点p的前面,修改指针域,顺序不能改变 +s->next=p->next; +p->next=s; + + +// 经典的借助变量,进行值交换 +temp=p->data; +p->data=s->data; +s->data=temp; + +``` + +上述借助临时变量`temp`来将结点s和结点p的数据域进行交换,需要开辟O(1)的空间复杂度,但是时间复杂度却从O(n)改变为O(1),典型的空间换时间策略 + + +#### 删除结点 + +> 将单链表L的第i个结点元素删除; + + +- 第一步: 先检查删除位置的合法性; +- 第二步: 查找表中的第(i-1)个结点,即被删结点的前驱结点; +- 第三步: 移动指针,删除结点元素; + + +```C++ + +// 获取删除位置结点元素的前驱结点 +p=GetElem(L,i-1); + +// 删除位置结点元素指针 +q=p->next; + +// 修改指针,将删除位置结点元素前驱结点的指针域指向其后继结点 +p->next=q->next; + +// 释放结点元素的内存控件 +free(q) + +``` + +和插入算法一样,时间都消耗在查询前驱结点上,时间复杂度为:O(n) + + + + +> 删除单链表L中给点结点元素*p,通常是按值查找获取到p结点的前驱元素,再执行删除操作,这样很明显会导致时间复杂度为:O(n),主要都消耗在`按值查找`上 + +这里可以利用p结点的后继结点将p结点删除 + +- 第一步:申请结点q,使其只想p结点的后继结点; +- 第二步:将p结点的数据域值换成其后继结点的数据域;【注意,交换没什么意义,最终p的后继结点会删除、释放】 +- 第三步:p的指针域指向q的指针域,q结点从链中“断开” +- 第四步:释放q的内存空间 + + +```C++ + // 存放p的后继结点指针 + q=p->next; + + // 结点p的后继结点元素赋值给结点p,避免后继结点的数据域丢失 + p->data=p->next->data; + p->next=q->next; + + // 此时q指向更换数据域后的p,即原来p的后继结点 + free(q) + +``` + +相比按值查找前驱结点来删除给定的结点p,利用后继结点来删除的时间复杂度更小,为:O(1) + + + +#### 计算表长 + +> 计算单链表中数据结点(不含头结点)的个数 + + +算法思路:从第一个结点开始顺序依次访问表中的每一个结点,为此需要设置一个`计数器变量`,每访问一个结点,计算器加1,直到访问到空结点为止。 + +算法时间复杂度:O(n) + + +**单链表的长度是不包括头结点的,不带头结点和带头结点的单链表在求表长操作上会略有不同。** + +不带头结点的单链表,当表为空时候,需要单独处理; + +```C++ +// 不带头结点的单链表L为空,判定条件是L=NULL。 +if(L===NULL){ +// 链表为空,表长为0 + return 0; +} + +// 带头结点的单链表L为空,判空条件:L->next=NULL; + +if(L->next===NULL){ + // 链表为空,不包含头结点,表长为0 + return 0; +} + +``` \ No newline at end of file diff --git a/数据结构/线性表/4.线性表的链式表示【双链表】.md b/数据结构/线性表/4.线性表的链式表示【双链表】.md new file mode 100644 index 0000000..d56e0f9 --- /dev/null +++ b/数据结构/线性表/4.线性表的链式表示【双链表】.md @@ -0,0 +1,92 @@ + + + + + +## 双链表 + + +从单链表的结构上来看 +- 访问特定结点的前驱结点需要遍历整个单链表,移动指针,时间复杂度为:O(n) +- 访问特定结点的后继结点只需要移动一次指针,时间复杂度为:O(1) + + +双链表的引入,很好的解决单链表访问前驱结点时间消耗大的问题。 + + +双链表结点由三部分组成: + +- `数据域` 存放数据信息 +- `prior指针域` 指向结点的前驱结点 +- `next指针域` 指向结点的后继结点 + + +```C++ + +// 双链表结点类型 +typedef struct DNode{ + ElemType data; // 结点的数据域 + struct DNode *prior; // 结点的前驱指针 + struct DNode *next; // 结点的后继指针 +}DNode, *DlinkList; + +``` + + +### 基本特点 + +- 双链表仅仅在单链表的结点中增加了一个指向结点前驱的`prior`指针; +- `按值查找`、`按序号查找`在单链表和双链表上的操作是相同的。 +- 和单链表不同,`插入`、`删除`操作除了修改`next`指针域,双链表还需要修改`prior`指针域,确保不断`链`,时间复杂度都为:O(1) + + + + +### 插入结点 + +> 在双链表中p所指的结点之后插入结点s + + +```C++ + +// 第一步 +s->next=p->next; + +// 第二步 +p->next->prior=s; + +// 第三步 +s->prior=p; + +// 第四步 +p->next=s + +``` + +**第一步和第二步必须再第四步之前**,整体时间复杂度为:O(1) + +### 删除结点 + +> 删除双链表中结点p的后继结点q + +```C++ + +// 第一步 +p->next=q->next; + +// 第二步 +q->next->prior=p; + +// 第三步 +free(q); + +``` + +**第一步和第二步顺序可换**,整体时间复杂度为:O(1) \ No newline at end of file diff --git a/数据结构/线性表/5.线性表的链式表示【循环链表】.md b/数据结构/线性表/5.线性表的链式表示【循环链表】.md new file mode 100644 index 0000000..8de1fa7 --- /dev/null +++ b/数据结构/线性表/5.线性表的链式表示【循环链表】.md @@ -0,0 +1,76 @@ + + + + + +## 循环链表 + +- 循环单链表 +- 循环双链表 + + +### 循环单链表 + +`循环单链表`是在单链表的基础上,将最后一个结点(尾结点)的指针由`NULL`改为指向`头结点`,形成`环`。【单链表----->循环单链表】 + + +```C++ +// 双链表结点类型 +typedef struct DNode{ + ElemType data; // 结点的数据域 + struct DNode *prior; // 结点的前驱指针 + struct DNode *next; // 结点的后继指针 +}DNode, *DlinkList; + +``` + +#### 判空条件 + +**不是判断头结点的指针是否为空,而是需要判断是否等于头指针**,表为空时,头结点的next指针域其实是指向自己; + + + +#### 特点 + +- 在循环单链表中,尾结点*p的next指针域指向链表L(即:头结点),形成了`闭环`,不存在指针域为`NULL`的结点。 +- **由于循环单链表是个`环`,在任何位置上的插入、删除操作都是等价的,不需要去判断是否是表尾**。当其中的结点的next指针指向自己,也就能判断表为空 +- 单链表只能从头结点(表头结点)开始往后顺序遍历整个表,循环单链表可以从表中任意位置开始遍历整个链表,结点是等价的; +- **循环单链表可以抽象为时钟,形成的`环`是有顺序的;** +- 频繁的`表头`和`表尾`操作,可以对循环单链表设置`尾指针`,而不设置`头指针`,明确尾指针r后,头指针即为:`r->next` ,减少头指针到尾指针间的遍历,时间复杂度:O(n)---->O(1) + + + + +### 循环双链表 + +`循环双链表`是在双链表的基础上,将`尾结点`的`next`指针指向`头结点`,将`头结点`的`prior`指针指向`尾结点`。【双链表----->循环双链表】 + + +```C++ +// 双链表结点类型 +typedef struct DNode{ + ElemType data; // 结点的数据域 + struct DNode *prior; // 结点的前驱指针 + struct DNode *next; // 结点的后继指针 +}DNode, *DlinkList; + +``` + +#### 判空条件 + +循环双链表为空时,头结点*p的prior指针和next指针都指向L,即同时满足: + +- p->next=L +- p->prior=L + + +#### 基本特点 + +- 从双向链表中的任意一个结点开始,都可以很方便地访问它的`前驱结点`和`后继结点`。 \ No newline at end of file diff --git a/数据结构/线性表/6.线性表的链式表示【静态链表】.md b/数据结构/线性表/6.线性表的链式表示【静态链表】.md new file mode 100644 index 0000000..0ef36bf --- /dev/null +++ b/数据结构/线性表/6.线性表的链式表示【静态链表】.md @@ -0,0 +1,44 @@ + + + + + +## 静态链表 + +> 借助数组来描述线性表的链式存储结构,结点元素同样存在数据域`data`和指针域`next` + + +注意: 和普通的链表的指针域不同的是,静态链表的指针是结点元素的相对地址(数组下标),也称为`游标`,建议结合高级语言中数组的概念来理解; + + +**与顺序表一样,虽然静态链表属于链表,但是存储时需要预先分配一块连续的内存空间** + + +静态链表结构类型: + + +```C++ + +// 定义静态链表的最大长度 +# define MaxSize 50 + +typedef struct{ + ElemType data; // 存储数据元素,数据域 + int next; // 下个元素的相对地址,数组下标 +}SLinkList[MaxSize]; + +``` + +很显然,静态链表是通过`数组游标`来访问下一个结点元素,可以和`指针域`的相关概念结合理解; + + +- 静态链表以`next=-1`作为结束的标志【尾结点】 +- 和动态链表相同,**插入、删除操作不需要移动元素,只需要修改指针**; +- 总体来说,静态链表没有单链表使用方便,需要将整个链表存储在一块连续的内存空间中,内部的存储可以分散,通过指针构成`链`的关系 \ No newline at end of file diff --git a/数据结构/线性表/7.顺序表和链表的比较.md b/数据结构/线性表/7.顺序表和链表的比较.md new file mode 100644 index 0000000..67e5f9d --- /dev/null +++ b/数据结构/线性表/7.顺序表和链表的比较.md @@ -0,0 +1,72 @@ + + + + +## 顺序表和链表的比较 + + +### 存取方式 + +- 顺序表支持顺序存取和随机存取; +- 链表只能从表头顺序存取元素,支持顺序存取; + + +### 逻辑结构与物理结构 + + +- 顺序存储时,逻辑上相邻的元素,对应的物理存储位置也相邻【一定性】。 +- 链式存储时,逻辑上相邻的元素,对应的物理存储位置不一定相邻【可以相邻,也可以不相邻】。 +- 链式存储的逻辑关系通过指针链接表示; + + + +### 时间复杂度 + +#### 按值查找 + +- 顺序表无序的情况下,顺序表和链表的时间复杂度均为O(n) +- 顺序表有序的情况下,顺序表的时间复杂度为O(log2n),链表的时间复杂度为O(n); + + +**注意:O(log2n) < O(n)** + +#### 按序号查找 + +- 顺序表支持随机访问,时间复杂度为O(1); +- 顺序表不支持随机访问,时间复杂度为O(n); + +#### 插入、删除 + +- 顺序表平均需要移动半个表长的元素; +- 链表只需要修改相应结点的指针域,不需要移动元素; +- 链表结点除了数据域,还有指针域,在存储空间上比顺序存储需要更大的存储空间,付出更大的存储代价,存储密度不够大 + +### 空间分配 + +#### 顺序存储 + + +##### 静态分配 + +- 需要预先分配足够大的存储空间; +- 空间装满后不能扩充,存储新元素将出现`内存溢出`; +- 存储空间过大,顺序表后部闲置空间过多,造成`内部碎片` +- 存储空间过小,会造成`内存溢出` + +##### 动态分配 + +- 动态分配能够扩充存储空间,但需要移动大量元素,操作效率降低 +- 内存中没有更大块的连续存储空间,将会导致空间分配失败; + + +#### 链式存储 + +- 链式存储的结点空间只在需要的时候申请分配 +- 只要内存由空间就可以分配,操作灵活、高效 \ No newline at end of file diff --git a/数据结构/线性表/8.存储结构的选取.md b/数据结构/线性表/8.存储结构的选取.md new file mode 100644 index 0000000..c2b89c5 --- /dev/null +++ b/数据结构/线性表/8.存储结构的选取.md @@ -0,0 +1,45 @@ + + + + + +## 存储结构的选取 + + + +### 基于存储的考虑 + +- 对线性表的长度和存储规模难以估计时,不宜采用顺序表存储 +- 链表不用事先估计存储规模,但存储密度较低 +- 链式存储结构的存储密度小于1,不要求连续的存储空间 + + + +### 基于运算的考虑 + +- 顺序表支持随机存取,按序号查找顺序表的时间复杂度为O(1); +- 链表不支持随机存取,按序号查找链表的时间复杂度为O(n); +- 顺序表的插入、删除操作,平均需要移动表中一半的元素,当表的数据量较大时,这种情况需要重点考虑的。 +- 链表的插入、删除操作,也是需要找插入位置(前驱结点、后继结点),主要的操作还是比较操作,相对较好; + + + + +### 基于环境的考虑 + +- 顺序表容易实现,任何高级语言中都有数组类型; +- 链表操作是基于指针的,指针移动,相对复杂; + + + +综上比较 + +- 通常比较稳定的线性表选择顺序存储; +- 频繁进行插入、删除操作的线性表,应该选择链式存储,动态性较强 \ No newline at end of file diff --git a/数据结构/线性表/9.零碎知识补充.md b/数据结构/线性表/9.零碎知识补充.md new file mode 100644 index 0000000..0b6d07e --- /dev/null +++ b/数据结构/线性表/9.零碎知识补充.md @@ -0,0 +1,33 @@ + + + + +## 零碎知识补充 + +- 无论是链表的插入还是删除操作,必须保证不断链【重要】 +- 顺序存储结构可以随机存取也能顺序存取,链式结构只能进行顺序存取 +- 顺序存储方式同样适合图和树的存储,例如:满二叉树的顺序存储 +- 队列需要在表头删除元素,在表尾插入元素【先进先出】,采用带尾指针的循环单链表比较方便 +- 数组排序最少时间复杂度为O(nlog2n)【重要】 +- 静态链表中的指针称为`游标`,指示下一个元素在数组中的`下标` +- 静态链表用数组表示,需要预先分配较大的连续空间,同时具有一般链表的特点(插入、删除元素不需要移动元素) + + +### 单链表设置头结点 + + +目的 + +> 主要是方便运算。 + +好处 + +- 有头结点后,插入、删除数据元素的算法统一起来了,不需要判断是否在第一个元素之前插入或者删除第一个元素了。 +- 不论链表是否为空,头指针是指向头结点的`非空指针`,链表的头指针不变,因此空链表和非空链表的处理也就统一起来了。 diff --git a/数据结构/线性表/线性表.png b/数据结构/线性表/线性表.png new file mode 100644 index 0000000000000000000000000000000000000000..f6fdcdfbc6c3f06b59e37dfce5c25b6541b790f6 GIT binary patch literal 2234729 zcmeFZby(DE7dC3ZMn(yh5d|5rkPuM>lB=}{q|DAasx;M~+b3mz7jK za^wW|$dR*iC;kBca&OhT;>Z!EBlji6)Etlt!_O<)HwcX@w3M&KpS*lSqa_#E&Bevl z{0@cwY!>Z*nM>jl%_Zk2ODf!wm#%P(M#h*>PcN4X>FwxL&);aj%^zYY#!Nv@rHkX zR0}Q$Ie>s5v8}eTc)c=P3 zm+Sa{tNd5N=6`|ZS9|DxvHVxj`hS7t|6hUSb&;r=VG4tgeQOAVkXd60gM6R)qtz2k zVb&zwDfbObyVHsaO}jP9{QTY&+*0@a4T_2*1^M3iXoURbA2_RKkk&6CApi4c**nMB zjTvJFm|oP(l7s7=K7Bg!rf%7na!TVXzky_|Y5-|KT=Q<6{BwPIyt(moeMUvVdi$vy z^0K+a5f2)cFky*5< zfJqu_Jz@Acx~oUk&*L|5#k5Q~=7kub_z4`tvPpW_Smf}YL^vtRMI%dldYr@S_t=ua 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