skip-list（跳表）原理及C++代码实现

跳表是一个很有意思的数据结构，它实现简单，但是性能又可以和平衡二叉搜索树差不多。

据MIT公开课上教授的讲解，它的想法和纽约地铁有异曲同工之妙，简而言之就是不断地增加“快线”，从而降低时间复杂度。

当“快线”的数量为lgn时，我们就得到了现在的快表——一个类似于平衡二叉搜索树的数据结构。

网上没有比较标准的实现方案，CLRS中也没有给出伪代码。（可能是因为他们觉得太容易实现了...）所以我给出是一个完全由我自己根据快表性质写的一个版本，如果大家有更好的实现版本欢迎和我分享。

增加层数的方案用的是抛硬币，即根据随机数来确定是否增加“快线”，这也是MIT公开课上给出的一个比较简单的实现方法。

代码如下：（仅供参考）

 class Skiplist {
 private :
     struct Node {
         int key;
         Node * prev;
         Node * next;
         Node * down;
         Node * top;
         Node() : key(), prev(nullptr), next(nullptr), down(nullptr), top(nullptr) {}
     };
 private :
     Node * head;
     int level;
     int size;
 private :
     void bindNewNode(Node * x, Node * p);
     void delNode(Node * x);
     Node * searchNode(int key);
 public :
     Skiplist() : head(new Node), level(),size()
         {head->key = INT_MIN; srand(static_cast<int>(time()));}
     ~Skiplist() {delete head;}
     void insert(int key);
     void remove(int key);
     bool search(int key) {return (searchNode(key) != nullptr);}
     void showSkiplist();
     int getLevel() {return level;}
     int getSize() {return size;}
 };

 void Skiplist::bindNewNode(Node * x, Node * p) {
     if (!x->next) {
         x->next = p;
         p->prev = x;
     }
     else {
         p->next = x->next;
         x->next->prev = p;
         p->prev = x;
         x->next = p;
     }
 }

 void Skiplist::insert(int key) {
     Node * p = new Node;
     p->key = key;

     Node * x = head;
     while () { //find the prev node of p, which represents the right insert place
         if (x->key <= key) {
             if (x->next)
                 x = x->next;
             else if (x->down)
                 x = x->down;
             else break;
         }
         else if (x->prev->down)
             x = x->prev->down;
         else {
             x = x->prev;
             break;
         }
     }
     bindNewNode(x, p);
     while (rand() % ) {  //throw the coin, then judge whether it needs to be higher according to the results
         Node * highp = new Node;
         highp->key = key;
         while (!x->top && x->prev)
             x = x->prev;
         if (x->top) {
             x = x->top;
             bindNewNode(x, highp);
             highp->down = p;
             p->top = highp;
         }
         else { //already the top, add a sentry
             Node * top = new Node;
             x = top;
             top->key = INT_MIN;
             top->down = head;
             head->top = top;
             head = top;
             bindNewNode(top, highp);
             highp->down = p;
             p->top = highp;
             ++level;
         }
         p = highp;
     }
     ++size;
 }

 void Skiplist::delNode(Node * x) {
     if (!x->next) { //node x is the last one
         if (x->prev == head && head->down) { //x is not at the bottom and x is the last one of this level
             head = head->down;
             head->top = nullptr;
             delete x->prev;
             --level;
         }
         else
             x->prev->next = nullptr;
     }
     else {
         x->prev->next = x->next;
         x->next->prev = x->prev;
     }
     delete x;
 }

 void Skiplist::remove(int key) {
     Node * x = searchNode(key);
     if (x) {
         while (x->down) {
             Node * y = x->down;
             delNode(x);
             x = y;
         }
         delNode(x);
         --size;
     }
 }

 Skiplist::Node * Skiplist::searchNode(int key) {
     Node * x = head;
     while () { //find the prev node of p, which represents the right insert place
         if (x->key == key)
             return x;
         else if (x->key < key) {
             if (x->next)
                 x = x->next;
             else if (x->down)
                 x = x->down;
             else
                 return nullptr;
         }
         else if (x->prev->down)
             x = x->prev->down;
         else {
             return nullptr;
         }
     }
 }

 void Skiplist::showSkiplist() {
     Node * x = head, * y = head;
     while (y) {
         x = y;
         while (x) {
             if (x->prev != nullptr)
                 cout << x->key << ' ';
             x = x->next;
         }
         cout << endl;
         y = y->down;
     }
 }