伸展树也是一种平衡二叉搜索树，但它的平衡是在平均意义下的。伸展树中n次操作，每次的均摊复杂度是O(logn) 。而单独的一次操作，可能达到O(n)的复杂度。那么，与AVL相比，伸展树的优势又在哪里呢？
伸展树中，每次会把访问到的结点通过单旋和双旋移动到树的根部，这样下次在访问该结点时，就可以极大地节省时间。也就是说，伸展树可以很好地满足数据访问的局部性。假设在一段时间内，访问的元素有k个，k << n ，则伸展树会达到比AVL更好的性能。
此外，伸展树与AVL相比，对平衡的约束较小，因此更加容易实现。
查找和伸展
伸展树中，查找到相应元素后，通过旋转把该结点移动到根部，这个过程为splay(x) 。
这个过程也要用到AVL中的旋转，如果待上升结点为x，要看x的父节点和爷爷结点。具体有2种情况：
1. 
这种情况逐层旋转，x绕p旋转，然后绕g旋转。这和AVL的双旋一致。
2.
这种情况先绕G旋转，再绕P旋转。这种情况和AVL中不一样。
c++实现
https://github.com/yalewoo/cpp-data-structure
SplayBinTree.h
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#include "BST.h" template <typename T> class SplayBinTree : public BST<T> { protected: using BST<T>::updateHeightAbove; using BST<T>::_root; using BST<T>::_hot; BinNodePosi(T) splay(BinNodePosi(T) v); void rotateLL(BinNodePosi(T) p, BinNodePosi(T) q); void rotateRR(BinNodePosi(T) p, BinNodePosi(T) q); void connect34(BinNodePosi(T) t1, BinNodePosi(T) t2, BinNodePosi(T) t3, BinNodePosi(T) st1, BinNodePosi(T) st2, BinNodePosi(T) st3, BinNodePosi(T) st4); public: BinNodePosi(T) search(const T & e); BinNodePosi(T) insert(const T & e); bool remove(const T & e); }; template <typename T> BinNodePosi(T) SplayBinTree<T>::splay(BinNodePosi(T) v) { BinNodePosi(T) p; BinNodePosi(T) mid; BinNodePosi(T) q; BinNodePosi(T) t1; BinNodePosi(T) t2; BinNodePosi(T) t3; BinNodePosi(T) st1; BinNodePosi(T) st2; BinNodePosi(T) st3; BinNodePosi(T) st4; BinNodePosi(T) par; enum {ROOT, LEFT, RIGHT} ptoc; while ((q = v) && (mid = v->parent) && (p = v->parent->parent)) { par = p->parent; if (par) { if (p == par->lchild) ptoc = LEFT; else ptoc = RIGHT; } else ptoc = ROOT; if (p->lchild && q == p->lchild->lchild) { t1 = q; t2 = q->parent; t3 = p; st1 = q->lchild; st2 = q->rchild; st3 = mid->rchild; st4 = p->rchild; t1->rchild = t2; t2->parent = t1; t2->lchild = st2; if (st2) st2->parent = t2; t2->rchild = t3; t3->parent = t2; t3->lchild = st3; if (st3) st3->parent = t3; switch (ptoc) { case ROOT : _root = t1; break; case LEFT : par->lchild = t1; break; case RIGHT : par->rchild = t1; break; } t1->parent = par; updateHeightAbove(t3); } else if (p->rchild && q == p->rchild->rchild) { t1 = p; t2 = mid; t3 = q; st1 = p->lchild; st2 = mid->lchild; st3 = q->lchild; st4 = q->rchild; t1->rchild = st2; if (st2) st2->parent = t1; t2->lchild = t1; t1->parent = t2; t2->rchild = st3; if (st3) st3->parent = t2; t3->lchild = t2; t2->parent = t3; switch (ptoc) { case ROOT : _root = t3; break; case LEFT : par->lchild = t3; break; case RIGHT : par->rchild = t3; break; } t3->parent = par; updateHeightAbove(t1); } else if (p->lchild && q == p->lchild->rchild) { t1 = mid; t2 = q; t3 = p; st1 = mid->lchild; st2 = q->lchild; st3 = q->rchild; st4 = p->rchild; switch (ptoc) { case ROOT : _root = t2; break; case LEFT : par->lchild = t2; break; case RIGHT : par->rchild = t2; break; } t2->parent = par; this->connect34(t1, t2, t3, st1, st2, st3, st4); } else { t1 = p; t2 = q; t3 = mid; st1 = p->lchild; st2 = q->lchild; st3 = q->rchild; st4 = mid->rchild; switch (ptoc) { case ROOT : _root = t2; break; case LEFT : par->lchild = t2; break; case RIGHT : par->rchild = t2; break; } t2->parent = par; this->connect34(t1, t2, t3, st1, st2, st3, st4); } } if ((q = v) && (mid = v->parent) && !(p = v->parent->parent)) { if (q == mid->lchild) this->rotateLL(q, mid); else this->rotateRR(q, mid); updateHeightAbove(mid); } return v; } template <typename T> BinNodePosi(T) SplayBinTree<T>::search(const T & e) { BinNodePosi(T) x = searchIn(_root, e, _hot); if (!x) splay(_hot); else splay(x); return x; } template <typename T> BinNodePosi(T) SplayBinTree<T>::insert(const T & e) { BinNodePosi(T) x = search(e); if (x) return x; BinNodePosi(T) v = new BinNode<T>(e, NULL); BinNodePosi(T) p = _root; _root = v; if (p && p->data < e) { p->parent = v; v->lchild = p; v->rchild = p->rchild; if (p->rchild) p->rchild->parent = v; p->rchild = 0; } else if (p) { p->parent = v; v->rchild = p; v->lchild = p->lchild; if (p->lchild) p->lchild->parent = v; p->lchild = 0; } updateHeightAbove(p); updateHeightAbove(v); return v; } template <typename T> bool SplayBinTree<T>::remove(const T & e) { BinNodePosi(T) v = search(e); if (!v) return false; BinNodePosi(T) p = v->succ(); BinNodePosi(T) oldleft = v->lchild; if (v->lchild) v->lchild->parent = 0; if (v->rchild) v->rchild->parent = 0; delete v; if (!p) { _root = oldleft; return true; } splay(p); _root = p; p->parent = 0; p->lchild = oldleft; if (p->lchild) p->lchild->parent = p; return true; } template <typename T> void SplayBinTree<T>::rotateLL(BinNodePosi(T) p, BinNodePosi(T) q) { q->lchild = p->rchild; if (p->rchild) p->rchild->parent = q; p->rchild = q; if (q->parent == NULL) { p->parent = NULL; this->_root = p; q->parent = p; return; } p->parent = q->parent; if (q->parent->lchild == q) q->parent->lchild = p; else q->parent->rchild = p; q->parent = p; } template <typename T> void SplayBinTree<T>::rotateRR(BinNodePosi(T) p, BinNodePosi(T) q) { q->rchild = p->lchild; if (p->lchild) p->lchild->parent = q; p->lchild = q; if (q->parent == NULL) { this->_root = p; p->parent = NULL; q->parent = p; return; } p->parent = q->parent; if (q->parent->lchild == q) q->parent->lchild = p; else q->parent->rchild = p; q->parent = p; } template <typename T> void SplayBinTree<T>::connect34(BinNodePosi(T) t1, BinNodePosi(T) t2, BinNodePosi(T) t3, BinNodePosi(T) st1, BinNodePosi(T) st2, BinNodePosi(T) st3, BinNodePosi(T) st4) { t1->lchild = st1; if (st1) st1->parent = t1; t1->rchild = st2; if (st2) st2->parent = t1; t3->lchild = st3; if (st3) st3->parent = t3; t3->rchild = st4; if (st4) st4->parent = t3; t2->lchild = t1; t1->parent = t2; t2->rchild = t3; t3->parent = t2; updateHeightAbove(t1); updateHeightAbove(t3); 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SplayBinTree.cpp
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#include <iostream> #include <cstdlib> #include "SplayBinTree.h" using namespace std; int main() { SplayBinTree<int> b; for (int i = 32; i >=1; --i) { b.insert(i); } // b.remove(32); // b.remove(4); char c; for (int i = 32; i >=1; --i) { b.display(); cout << "Press Enter to Search " << i; while ((c = getchar()) != '\n') ; b.search(i); system("cls"); } b.search(1); b.display(); return 0; } |
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