数据结构实现之Splay伸展树

Splay Tree 是二叉查找树的一种，它与平衡二叉树、红黑树不同的是，Splay Tree从不强制地保持自身的平衡，每当查找到某个节点n的时候，在返回节点n的同时，Splay Tree会将节点n旋转到树根的位置，这样就使得SplayTree天生有着一种类似缓存的能力，因为每次被查找到的节点都会被搬到树根的位置，所以当80%的情况下我们需要查找的元素都是某个固定的节点，或者是一部分特定的节点时，那么在很多时候，查找的效率会是O(1)的效率！当然如果查找的节点是很均匀地分布在不同的地方时，Splay Tree的性能就会变得很差了，但Splay Tree的均摊时间复杂度还是O（logn）的。
摘自http://www.nocow.cn/index.php/Splay_Tree

PPT来自清华邓俊辉数据结构-BST-Splay(学堂在线有对应视频讲解)


先zig(g)再zig(p)的要比先zig(p)再zig(g)的局部性效果要好，前者会使p成为g的左子结点，从而不能很好发挥SplayTree利用局部性的目的。

Splay Tree 结点定义

//Splay结点定义    private class Node {        public Node left;        public Node right;        public Node parent;        public Key key;        public Value val;        public Node(Key k,Value v) {            key = k; val = v; parent = null; left = null; right = null;        }    }

splay

private Node splay(Node v) {        if(v == null) return null;        Node p,g;//v的父亲与祖父        //自下而上，反复对v做双层伸展        while((p=v.parent)!=null && (g = p.parent)!=null) {            Node gg = g.parent;//每轮之后v都以原曾祖父（great-grand parent）为父            if(isLChild(v)) {                if(isLChild(p)) {//zig-zig，zig（g）先g结点右旋，zig(p)再p右旋                    attachAsLChild(g, p.right);                    attachAsRChild(p,g);                    attachAsLChild(p, v.right);                    attachAsRChild(v,p);                }                else {//zig-zag,zig(p)先p右旋，再g左旋zag(g)                    attachAsLChild(p,v.right);                    attachAsRChild(v,p);                    attachAsRChild(g,v.left);                    attachAsLChild(v,g);                }            }            else {                if(isRChild(p)) {//zag-zag,zag(g)先g结点左旋，zip(p)再p左旋                    attachAsRChild(g,p.left);                    attachAsLChild(p,g);                    attachAsRChild(p,v.left);                    attachAsLChild(v,p);                }                else {//zag-zig,先p左旋,zag(p)，再g右旋zig(g)                    attachAsRChild(p,v.left);                    attachAsLChild(v,p);                    attachAsLChild(g,v.right);                    attachAsRChild(v,g);                }            }            //若v原先的曾祖父gg不存在，则v现在应为树根            if(gg==null)v.parent = null;            else {//否则，gg此后应该以v作为左或右孩子                if(gg.left == g) attachAsLChild(gg,v);                else attachAsRChild(gg,v);            }        }//双层伸展结束时，必有g == NULL，但p可能非空        //若p果真非空，则额外再做一次单旋        if(p!=null && p == v.parent) {            if(isLChild(v)) {//zig                attachAsLChild(p,v.right);                attachAsRChild(v,p);            }            else {//zag                attachAsRChild(p,v.left);                attachAsLChild(v,p);            }        }        v.parent = null;        return v;    }

查找

//查找    private Node find(Node p,Key key) {        if(p == null || key == null) return null;        Node prep = null;        while(p != null) {            prep = p;            int cmp = key.compareTo(p.key);            if(cmp<0) p = p.left;            else if(cmp>0) p = p.right;            else break;        }        root = splay(prep);        return root;    }

插入

//插入    public void insert(Key key,Value val) {        if(key == null) return;        if(val == null) delete(key);        if(root == null) {            root = new Node(key,val);            return;        }        Node t = find(root,key);        int cmp = key.compareTo(t.key);        if(cmp == 0) {//更新value            root.val = val; return;        }        root = new Node(key,val);        //先把t分为两个左右子树，再合并        if(cmp<0) {//插入新根，以t->lChild和t为左、右孩子            Node x = t.left;            root.left = x;            root.right = t;            t.parent = root;            if(x != null) {                x.parent = root;                t.left = null;            }        }        else if(cmp>0) {//插入新根，以t和t->rChild为左、右孩子            Node x = t.right;            root.left = t;            root.right = x;            t.parent = root;            if(x != null) {                x.parent = root;                t.right = null;            }        }    }

删除

//删除    public void delete(Key key) {        if(key == null) return;        Node t = find(root,key);        if(t == null || key.compareTo(t.key) != 0)            return;        if(t.right == null) {//右子树为空            root = root.left; root.parent = null; return;        }        //右子树不空        //1.删除t结点，得到子树tl,tr         Node tl = t.left,tr = t.right;        tr.parent = null;        //2.将右子树最小结点min伸展至右子树树根，min一定没有左子结点        Node min = min(tr);        //3.将tl作为左子树链接到min上        min.left = tl;        if(tl != null) tl.parent = min;        root = min;    }

完整代码

public class SplayTree <Key extends Comparable<Key>,Value>{    private Node root;    //Splay结点定义    private class Node {        public Node left;        public Node right;        public Node parent;        public Key key;        public Value val;        public Node(Key k,Value v) {            key = k; val = v; parent = null; left = null; right = null;        }    }    public SplayTree() {        root = null;    }    public Value find(Key key) {        root = find(root,key);        if(root == null || root.key.compareTo(key) != 0) return null;        return root.val;    }    //查找    private Node find(Node p,Key key) {        if(p == null || key == null) return null;        Node prep = null;        while(p != null) {            prep = p;            int cmp = key.compareTo(p.key);            if(cmp<0) p = p.left;            else if(cmp>0) p = p.right;            else break;        }        root = splay(prep);        return root;    }    //插入    public void insert(Key key,Value val) {        if(key == null) return;        if(val == null) delete(key);        if(root == null) {            root = new Node(key,val);            return;        }        Node t = find(root,key);        int cmp = key.compareTo(t.key);        if(cmp == 0) {//更新value            root.val = val; return;        }        root = new Node(key,val);        //先把t分为两个左右子树，再合并        if(cmp<0) {//插入新根，以t->lChild和t为左、右孩子            Node x = t.left;            root.left = x;            root.right = t;            t.parent = root;            if(x != null) {                x.parent = root;                t.left = null;            }        }        else if(cmp>0) {//插入新根，以t和t->rChild为左、右孩子            Node x = t.right;            root.left = t;            root.right = x;            t.parent = root;            if(x != null) {                x.parent = root;                t.right = null;            }        }    }    //删除    public void delete(Key key) {        if(key == null) return;        Node t = find(root,key);        if(t == null || key.compareTo(t.key) != 0)            return;        if(t.right == null) {//右子树为空            root = root.left; root.parent = null; return;        }        //右子树不空        //1.删除t结点，得到子树tl,tr         Node tl = t.left,tr = t.right;        tr.parent = null;        //2.将右子树最小结点min伸展至右子树树根，min一定没有左子结点        Node min = min(tr);        //3.将tl作为左子树链接到min上        min.left = tl;        if(tl != null) tl.parent = min;        root = min;    }    /***************************************************************************        *  Splay tree helper functions.     ***************************************************************************/    private Node splay(Node v) {        if(v == null) return null;        Node p,g;//v的父亲与祖父        //自下而上，反复对v做双层伸展        while((p=v.parent)!=null && (g = p.parent)!=null) {            Node gg = g.parent;//每轮之后v都以原曾祖父（great-grand parent）为父            if(isLChild(v)) {                if(isLChild(p)) {//zig-zig，zig（g）先g结点右旋，zig(p)再p右旋                    attachAsLChild(g, p.right);                    attachAsRChild(p,g);                    attachAsLChild(p, v.right);                    attachAsRChild(v,p);                }                else {//zig-zag,zig(p)先p右旋，再g左旋zag(g)                    attachAsLChild(p,v.right);                    attachAsRChild(v,p);                    attachAsRChild(g,v.left);                    attachAsLChild(v,g);                }            }            else {                if(isRChild(p)) {//zag-zag,zag(g)先g结点左旋，zip(p)再p左旋                    attachAsRChild(g,p.left);                    attachAsLChild(p,g);                    attachAsRChild(p,v.left);                    attachAsLChild(v,p);                }                else {//zag-zig,先p左旋,zag(p)，再g右旋zig(g)                    attachAsRChild(p,v.left);                    attachAsLChild(v,p);                    attachAsLChild(g,v.right);                    attachAsRChild(v,g);                }            }            //若v原先的曾祖父gg不存在，则v现在应为树根            if(gg==null)v.parent = null;            else {//否则，gg此后应该以v作为左或右孩子                if(gg.left == g) attachAsLChild(gg,v);                else attachAsRChild(gg,v);            }        }//双层伸展结束时，必有g == NULL，但p可能非空        //若p果真非空，则额外再做一次单旋        if(p!=null && p == v.parent) {            if(isLChild(v)) {//zig                attachAsLChild(p,v.right);                attachAsRChild(v,p);            }            else {//zag                attachAsRChild(p,v.left);                attachAsLChild(v,p);            }        }        v.parent = null;        return v;    }    //判断结点n是否为父节点的左孩子    private boolean isLChild(Node n) {        Node p = n.parent;        return p.left == n;    }    //判断结点n是否为父节点的右孩子    private boolean isRChild(Node n) {        Node p = n.parent;        return p.right == n;    }    //在节点p与rc（可能为空）之间建立父（左）子关系    private void attachAsLChild(Node parent,Node lchild) {        parent.left = lchild;         if(lchild!=null)             lchild.parent = parent;    }    //在节点p与rc（可能为空）之间建立父（右）子关系    private void attachAsRChild(Node parent,Node rchild) {        parent.right = rchild;        if(rchild!=null)            rchild.parent = parent;    }    /***************************************************************************     *  Utility functions.     ***************************************************************************/    public Value min() {        if(root == null) return null;        root = min(root);        return root.val;    }    private Node min(Node x) {        if(x == null) return null;        while(x.left!=null)            x = x.left;        x = splay(x);        return x;    }    public Value max() {        if(root == null) return null;        root = max(root);        return root.val;    }    private Node max(Node x) {        if(x == null) return null;        while(x.right!=null)            x = x.right;        x = splay(x);        return x;    }    //中序遍历    public void preorder() {        System.out.println("preorder: ");        preorder(root);        System.out.println();    }    private void preorder(Node p) {        if(p == null) return;        System.out.print(p.key+" ");        preorder(p.left);        preorder(p.right);    }    //中序遍历    public void inorder() {        System.out.println("inorder: ");        inorder(root);        System.out.println();    }    private void inorder(Node p) {        if(p == null) return;        inorder(p.left);        System.out.print(p.key+" ");        inorder(p.right);    }}