Before revision:  Initially, we preprocess the input orthogonal polygon with q vertices in O(q \lg{q}) time to construct data structures of size O(q\frac{\lg{q}}{\lg\lg{q}})}  .
After revision:  Initially, we preprocess the input orthogonal polygon with q vertices in O(q \lg{q}) time to construct data structures of size O(q\frac{\lg{q}}{\lg\lg{q}})} updates in a few structures maintained by our algorithm .