Algorithms-3.2 Binary Search Trees 二叉查找树
2024-06-01 13:15:36
Binary Search Trees
- 1 BSTs 二叉查找树
- 2 ordered operations in BSTs 有序性相关方法
- 2.1 最大键和最小键
- 2.2 向上取整和向下取整
- 2.3 排名
- 2.3.1 size()
- 2.3.2 rank()
- 2.3.3 中序遍历
- 2.3.4 summary
- 3 deletion 删除操作
- 3.1 删除最小键
- 3.2 Hibbard deletion
- 3.3 summary
1 BSTs 二叉查找树
- no more tree (search miss)
package Chapter03;//基于二叉查找树的符号表
public class BST<Key extends Comparable<Key>, Value>{private Node root; //二叉查找树的根结点//内部类private class Node{private Key key;private Value val;private Node left, right; //指向子树的链接//构造器public Node(Key key, Value val){this.key = key;this.val = val;}}public void put(Key key, Value val){//查找key，找到则更新它的值，否则为它创造一个新的结点root = put(root, key, val);//root根结点}//cost: number of compare is equal to 1 + depth of nodeprivate Node put(Node x, Key key, Value val){//如果key存在于以x为根结点的子树中则更新它的值，否则将以key和val为键值对的新结点插入到该子树中if (x == null) return new Node(key,val);int cmp = key.compareTo(x.key);if (cmp < 0) x.left = put(x.left,key,val);//递归else if (cmp > 0) x.right = put(x.right,key,val);else x.val = val;return x;//最终回到一个node}//return value corresponding to given key, or null if no such key//cost: number of compare is equal to 1 + depth of nodepublic Value get(Key key){Node x = root;while (x != null){int cmp = key.compareTo(x.key); //key和key比较if (cmp < 0) x = x.left;else if (cmp > 0) x = x.right;else return x.val;}return null;}
}
- worst case和链表一样，按字母从小到大的顺序输入的
2 ordered operations in BSTs 有序性相关方法
2.1 最大键和最小键
- 从root开始一直往右，直到找到一个null的右链接，就找到了最大键；同理，一直向左，可找到最小键
public Key min(){return min(root).key;}private Node min(Node x){if (x.left == null) return x;return min(x.left);}
- max()将left和right调换
2.2 向上取整和向下取整
//寻找小于等于key的最大的Keypublic Key floor(Key key){Node x = floor(root, key);if (x == null) return null;return x.key;}private Node floor(Node x, Key key){if (x == null) return null;int cmp = key.compareTo(x.key);if (cmp == 0) return x;if (cmp < 0) return floor(x.left,key);//cmp > 0Node t = floor(x.right,key);if (t != null) return t;//在x.right子树中找到一个nodeelse return x;//如果是空，回到上一级Key，上一级Key即为最终结果，说明右边子树不符合小于等于key的条件}
- 把所有的左变右（同时将小于变成大于）就能得到ceiling()的算法
2.3 排名
2.3.1 size()
package Chapter03;//基于二叉查找树的符号表
public class BST<Key extends Comparable<Key>, Value>{private Node root; //二叉查找树的根结点//内部类private class Node{private Key key;private Value val;private Node left, right; //指向子树的链接private int count;//number of nodes in subtrees//构造器public Node(Key key, Value val, int count){this.key = key;this.val = val;this.count = count;}}public void put(Key key, Value val){//查找key，找到则更新它的值，否则为它创造一个新的结点root = put(root, key, val);//root根结点}//cost: number of compare is equal to 1 + depth of nodeprivate Node put(Node x, Key key, Value val){//如果key存在于以x为根结点的子树中则更新它的值，否则将以key和val为键值对的新结点插入到该子树中if (x == null) return new Node(key,val,1);int cmp = key.compareTo(x.key);if (cmp < 0) x.left = put(x.left,key,val);//递归else if (cmp > 0) x.right = put(x.right,key,val);else x.val = val;x.count = 1 + size(x.left) + size(x.right);//递归return x;//最终回到一个node}public int size(){return size(root);}private int size(Node x){if (x == null) return 0;return x.count;}
}
2.3.2 rank()
- how many keys < k ?
public int rank(Key key){return rank(key, root);}private int rank(Key key, Node x){//返回以x为根结点的子树中小于x.key的键的数量if (x == null) return 0;int cmp = key.compareTo(x.key);if (cmp < 0) return rank(key, x.left);//若key小于根结点，返回key在左子树的排名（递归）else if (cmp > 0) return 1 + size(x.left) + rank(key, x.right);else return size(x.left);//如果key和根结点的key相等，rank就是左边子树的结点总数}
2.3.3 中序遍历
遍历过程如下：
- 访问左子树，
- 访问根节点，
- 访问右子树。
- 队列(queue)是一种特殊的线性表，它只允许在表的前端进行删除操作，而在表的后端进行插入操作。
//中序遍历：迭代public Iterable<Key> keys(){Queue<Key> q = new Queue<Key>();inorder(root, q);return q;}private void inorder(Node x, Queue<Key> q){if (x == null) return;inorder(x.left, q);//递归q.enqueue(x.key);//把根结点加入队列inorder(x.right, q);}
2.3.4 summary
3 deletion 删除操作
- tombstone 墓碑
- 如果有大量的删除操作，内存过载
3.1 删除最小键
//删除最小键public void deleteMin(){root = delete(root);}private Node deleteMin(Node x){if (x.left == null) return x.right;x.left = deleteMin(x.left);//x.left被重新赋值，新的left是原来的left的rightx.count = size(x.left) + size(x.right) + 1;return x;}
3.2 Hibbard deletion
//删除键值对public void delete(Key key){root = delete(root, key);}private Node delete(Node x, Key key){if (x == null) return null;int cmp = key.compareTo(x.key);if (cmp < 0) x.left = delete(x.left, key);//递归else if (cmp > 0) x.right = delete(x.right, key);else{//已找到要删除的键if (x.right == null) return x.left;//no right childif (x.left == null) return x.right;//no left child//replace with successor，有两个子树的情况Node t = x;//x为即将被删除的结点x = min(t.right);//x更新为x的右子树的最小结点，作为新"根"结点x.right = deleteMin(t.right);//将x的右链接x.left = t.left;//把t的左子树安装到后继结点（新x）的左边}x.count = size(x.left) + size(x.right) + 1;//update subtree countsreturn x;}
- 删除会使得二叉树变得不对称
- after a sequence of insert and delete, 树的高度变为square root of N --> 根号N，比期待的logN大很多
3.3 summary
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