TreeSet中有序操作的时间复杂度是多少？

Question

What is the time complexity of the following operations in java.util.TreeSet ?

• first()
• last()
• lower()
• higher()

I would assume that these are constant time but the API makes no guarantees.

Answer 1

Actually, I'd have thought that those operations are all going to be O(logN) for a general implementation.

• For first() and last() to be O(1) the TreeSet implementation would need to maintain a pointer to the leftmost and rightmost leaf nodes in the tree respectively. Maintaining these adds a constant cost to every insertion and at least a constant cost to every deletion. In reality, the implementation will probably find the left / rightmost nodes on the fly ... which is an O(logN) operation.

• The lower() and higher() methods have to do the same work as get and are therefore O(logN) .

Of course, you can check the source-code yourself to see what actually happens. (As other people have done: see below.)

Answer 2

Looks like both first() and last() will be O(log n) and not O(1)based on Implentation(sun jdk 1.6.0_23) of TreeMap which is used by TreeSet by default:

 /**
 * Returns the first Entry in the TreeMap (according to the TreeMap's
 * key-sort function).  Returns null if the TreeMap is empty.
 */
final Entry<K,V> getFirstEntry() {
    Entry<K,V> p = root;
    if (p != null)
        while (p.left != null)
            p = p.left;
    return p;
}

/**
 * Returns the last Entry in the TreeMap (according to the TreeMap's
 * key-sort function).  Returns null if the TreeMap is empty.
 */
final Entry<K,V> getLastEntry() {
    Entry<K,V> p = root;
    if (p != null)
        while (p.right != null)
            p = p.right;
    return p;
}

Answer 3

I actually looked up the source code, in http://developer.classpath.org/doc/java/util/TreeSet-source.html , first() calls maps.firstKey() then in http://developer.classpath.org/doc/java/util/TreeMap-source.html

393: public K firstKey()
394: (
395: if (root == nil)
396: throw new NoSuchElementException();
397: return firstNode().key;
398: )

and in firstNode(), it does the while loop to go all the way to the left

952: final Node<K, V> firstNode()
953: (
954: // Exploit fact that nil.left == nil.
955: Node node = root;
956: while (node.left != nil)
957: node = node.left;
958: return node;
959: )

Answer 4

The API makes no guarantees because these are based on the standard model of a trie. The best case is in O(1), with the average case being O(log n) and worst case O(n).

From the documentation:

This implementation provides guaranteed log(n) time cost for the basic operations (add, remove and contains).

These aren't the functions you asked for, but think about how Java will traverse the TreeSet.

Answer 5

It's going to depend on the implementation. I'm not incredibly familiar with JAVA, but it seems that all of those operations are traversal operations (get lowest element, get highest element, get next higher or next lower).

If the Tree is implemented as a Self-Balancing Binary Search Tree like an AVL Tree , or any other sort of a balanced-tree structure, you're going to be looking at Average-Case and Worst-Case O(log n) time for each of the operations, and a best case of O(1).