Lowest Common Ancestor implementations - what's the difference?

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);
    }
}

Solution

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)