CareerCup

BFS in largest first order, and then reverse the result:

public static String bottomsUp(Node root){
		StringBuilder sb = new StringBuilder();
		Queue<Node> q = new LinkedList<>();
		q.add(root);
		while(!q.isEmpty()) {
			Node n = q.poll();
			sb.append(n.value);
			if (n.right != null) {
				q.add(n.right);
			}
			if (n.left != null) {
				q.add(n.left);
			}
		}		
		return sb.reverse().toString();
	}

So, if I read it correctly we are checking whether the cycle passes through the start index.
In which case this code should work:

static boolean cyclesThroughStart(int start, int... arr) {
		int slowIdx = start;
		int fastIdx = start;
		do {
			slowIdx = f(arr,slowIdx);
			fastIdx = f(arr,f(arr,fastIdx));
			if (slowIdx == start || fastIdx == start) {
				return true;
			}
		} while(slowIdx != fastIdx);
		return false;
	}
	
	static int f(int[] arr,int idx) {
		idx += arr[idx];
		return Math.abs(idx % arr.length);
	}