Lowest Common Ancestor implementations - what'

I've been reading about the Lowest Common Ancestor algorithm on top coder and I can't understand why the RMQ algorithm is involved - the solution listed there is insanely complicated and has the following properties:

O(sqrt(n)) time complexity for searches, O(n) precalculation time complexity
O(n) space complexity for storing parents of each node
O(n) space complexity again, for storing precalculations of each node

My solution: given 2 integer values, find the nodes through a simple preorder traversal. Take one of the nodes and go up the tree and store the path into a Set. Take the other node and go up the tree and check each node as I go up: if the node is in the Set, stop and return the LCA. Full implementation.

O(n) time complexity for finding each of the 2 nodes, given the values (because it's a regular tree, not a BST -
O(log n) space complexity for storing the path into the Set
O(log n) time complexity for going up the tree with the second node

So given these two choices, is the algorithm on Top Coder better and if yes, why? That's what I can't understand. I thought O(log n) is better than O(sqrt(n)).

public class LCA {

    private class Node {

        int data;
        Node[] children = new Node[0];
        Node parent;

        public Node() {
        }

        public Node(int v) {
            data = v;
        }

        @Override
        public boolean equals(Object other) {
            if (this.data == ((Node) other).data) {
                return true;
            }
            return false;
        }
    }
    private Node root;

    public LCA() {
        root = new Node(3);

        root.children = new Node[4];
        root.children[0] = new Node(15);
        root.children[0].parent = root;
        root.children[1] = new Node(40);
        root.children[1].parent = root;
        root.children[2] = new Node(100);
        root.children[2].parent = root;
        root.children[3] = new Node(10);
        root.children[3].parent = root;

        root.children[0].children = new Node[3];
        root.children[0].children[0] = new Node(22);
        root.children[0].children[0].parent = root.children[0];
        root.children[0].children[1] = new Node(11);
        root.children[0].children[1].parent = root.children[0];
        root.children[0].children[2] = new Node(99);
        root.children[0].children[2].parent = root.children[0];

        root.children[2].children = new Node[2];
        root.children[2].children[0] = new Node(120);
        root.children[2].children[0].parent = root.children[2];
        root.children[2].children[1] = new Node(33);
        root.children[2].children[1].parent = root.children[2];

        root.children[3].children = new Node[4];
        root.children[3].children[0] = new Node(51);
        root.children[3].children[0].parent = root.children[3];
        root.children[3].children[1] = new Node(52);
        root.children[3].children[1].parent = root.children[3];
        root.children[3].children[2] = new Node(53);
        root.children[3].children[2].parent = root.children[3];
        root.children[3].children[3] = new Node(54);
        root.children[3].children[3].parent = root.children[3];

        root.children[3].children[0].children = new Node[2];
        root.children[3].children[0].children[0] = new Node(25);
        root.children[3].children[0].children[0].parent = root.children[3].children[0];
        root.children[3].children[0].children[1] = new Node(26);
        root.children[3].children[0].children[1].parent = root.children[3].children[0];

        root.children[3].children[3].children = new Node[1];
        root.children[3].children[3].children[0] = new Node(27);
        root.children[3].children[3].children[0].parent = root.children[3].children[3];
    }

    private Node findNode(Node root, int value) {
        if (root == null) {
            return null;
        }
        if (root.data == value) {
            return root;
        }
        for (int i = 0; i < root.children.length; i++) {
            Node found = findNode(root.children[i], value);
            if (found != null) {
                return found;
            }
        }
        return null;
    }

    public void LCA(int node1, int node2) {
        Node n1 = findNode(root, node1);
        Node n2 = findNode(root, node2);
        Set<Node> ancestors = new HashSet<Node>();
        while (n1 != null) {
            ancestors.add(n1);
            n1 = n1.parent;
        }
        while (n2 != null) {
            if (ancestors.contains(n2)) {
                System.out.println("Found common ancestor between " + node1 + " and " + node2 + ": node " + n2.data);
                return;
            }
            n2 = n2.parent;
        }
    }

    public static void main(String[] args) {
        LCA tree = new LCA();
        tree.LCA(33, 27);
    }
}

回答1:

The LCA algorithm works for any tree (not necessarily binary and not necessarily balanced). Your "simple algorithm" analysis breaks down since tracing a path to a root node is actually O(N) time and space instead of O(log N)

回答2:

Just want to point out that the problem is about the rooted tree and not binary search tree. So, in you algorithm the

O(n) time complexity for finding each of the 2 nodes, given the values O(n) space complexity for storing the path into the Set O(sqrt(n)) time complexity for going up the tree with the second node and searching in the first n-stored elements.

Checking of each node as we go up from the second node with take O(n), so for n nodes it will take O(sqrt(n)).

回答3:

The Harel and Tarjan LCA algorithm (reference in the link you gave) uses a pre-calculation with O(n) complexity, after which a lookup is O(1) (not O(sqrt(n) as you claim).