import java.util.*;
import java.util.stream.Stream;

// Answering LCA queries in O(log(n)) with O(n) preprocessing
public class Lca {

	int[] depth;
	int[] dfs_order;
	int cnt;
	int[] first;
	int[] minPos;
	int n;

	void dfs(List<Integer>[] tree, int u, int d) {
		depth[u] = d;
		dfs_order[cnt++] = u;
		for (int v : tree[u])
			if (depth[v] == -1) {
				dfs(tree, v, d + 1);
				dfs_order[cnt++] = u;
			}
	}

	void buildTree(int node, int left, int right) {
		if (left == right) {
			minPos[node] = dfs_order[left];
			return;
		}
		int mid = (left + right) >> 1;
		buildTree(2 * node + 1, left, mid);
		buildTree(2 * node + 2, mid + 1, right);
		minPos[node] = depth[minPos[2 * node + 1]] < depth[minPos[2 * node + 2]] ? minPos[2 * node + 1] : minPos[2 * node + 2];
	}

	public Lca(List<Integer>[] tree, int root) {
		int nodes = tree.length;
		depth = new int[nodes];
		Arrays.fill(depth, -1);

		n = 2 * nodes - 1;
		dfs_order = new int[n];
		cnt = 0;
		dfs(tree, root, 0);

		minPos = new int[4 * n];
		buildTree(0, 0, n - 1);

		first = new int[nodes];
		Arrays.fill(first, -1);
		for (int i = 0; i < dfs_order.length; i++)
			if (first[dfs_order[i]] == -1)
				first[dfs_order[i]] = i;
	}

	public int lca(int a, int b) {
		return minPos(Math.min(first[a], first[b]), Math.max(first[a], first[b]), 0, 0, n - 1);
	}

	int minPos(int a, int b, int node, int left, int right) {
		if (a == left && right == b)
			return minPos[node];
		int mid = (left + right) >> 1;
		if (a <= mid && b > mid) {
			int p1 = minPos(a, Math.min(b, mid), 2 * node + 1, left, mid);
			int p2 = minPos(Math.max(a, mid + 1), b, 2 * node + 2, mid + 1, right);
			return depth[p1] < depth[p2] ? p1 : p2;
		} else if (a <= mid) {
			return minPos(a, Math.min(b, mid), 2 * node + 1, left, mid);
		} else if (b > mid) {
			return minPos(Math.max(a, mid + 1), b, 2 * node + 2, mid + 1, right);
		} else {
			throw new RuntimeException();
		}
	}

	// Random test
	public static void main(String[] args) {
		Random rnd = new Random(1);
		for (int step = 0; step < 1000; step++) {
			int n = rnd.nextInt(50) + 1;
			List<Integer>[] tree = getRandomTree(n, rnd);
			int[] depth = new int[n];
			Arrays.fill(depth, -1);
			int root = rnd.nextInt(n);
			calcDepth(tree, depth, root, 0);
			Lca q = new Lca(tree, root);
			for (int i = 0; i < 1000; i++) {
				int a = rnd.nextInt(n);
				int b = rnd.nextInt(n);
				List<Integer> path = new ArrayList<>();
				getPathFromAtoB(tree, a, b, -1, path);
				int res1 = q.lca(a, b);
				int res2 = a;
				for (int u : path)
					if (depth[res2] > depth[u])
						res2 = u;
				if (res1 != res2)
					throw new RuntimeException();
			}
		}
		System.out.println("Test passed");
	}

	static List<Integer>[] getRandomTree(int n, Random rnd) {
		List<Integer>[] t = Stream.generate(ArrayList::new).limit(n).toArray(List[]::new);
		int[] p = new int[n];
		for (int i = 0, j; i < n; j = rnd.nextInt(i + 1), p[i] = p[j], p[j] = i, i++) ; // random permutation
		for (int i = 1; i < n; i++) {
			int parent = p[rnd.nextInt(i)];
			t[parent].add(p[i]);
			t[p[i]].add(parent);
		}
		return t;
	}

	static void calcDepth(List<Integer>[] tree, int[] depth, int u, int d) {
		depth[u] = d;
		for (int v : tree[u])
			if (depth[v] == -1)
				calcDepth(tree, depth, v, d + 1);
	}

	static boolean getPathFromAtoB(List<Integer>[] tree, int a, int b, int p, List<Integer> path) {
		path.add(a);
		if (a == b)
			return true;
		for (int u : tree[a])
			if (u != p && getPathFromAtoB(tree, u, b, a, path))
				return true;
		path.remove(path.size() - 1);
		return false;
	}
}