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

// Answering LCA queries in O(1) with O(n) preprocessing
public class LcaSchieberVishkin {
	int[] parent;
	int[] preOrder;
	int[] I;
	int[] head;
	int[] A;
	int time;

	void dfs1(List<Integer>[] tree, int u, int p) {
		parent[u] = p;
		I[u] = preOrder[u] = time++;
		for (int v : tree[u]) {
			if (v == p) continue;
			dfs1(tree, v, u);
			if (Integer.lowestOneBit(I[v]) > Integer.lowestOneBit(I[u])) {
				I[u] = I[v];
			}
		}
		head[I[u]] = u;
	}

	void dfs2(List<Integer>[] tree, int u, int p, int up) {
		A[u] = up | Integer.lowestOneBit(I[u]);
		for (int v : tree[u]) {
			if (v == p) continue;
			dfs2(tree, v, u, A[u]);
		}
	}

	public LcaSchieberVishkin(List<Integer>[] tree, int root) {
		int n = tree.length;
		preOrder = new int[n];
		I = new int[n];
		head = new int[n];
		A = new int[n];
		parent = new int[n];

		dfs1(tree, root, -1);
		dfs2(tree, root, -1, 0);
	}

	int enterIntoStrip(int x, int hz) {
		if (Integer.lowestOneBit(I[x]) == hz)
			return x;
		int hw = Integer.highestOneBit(A[x] & (hz - 1));
		return parent[head[I[x] & (~hw + 1) | hw]];
	}

	public int lca(int x, int y) {
		int hb = I[x] == I[y] ? Integer.lowestOneBit(I[x]) : Integer.highestOneBit(I[x] ^ I[y]);
		int hz = Integer.lowestOneBit(A[x] & A[y] & (~hb + 1));
		int ex = enterIntoStrip(x, hz);
		int ey = enterIntoStrip(y, hz);
		return preOrder[ex] < preOrder[ey] ? ex : ey;
	}

	// 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);
			LcaSchieberVishkin q = new LcaSchieberVishkin(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;
	}
}