An O(nlogn) algorithm for the zoo-keeper's problem

The zoo-keeper's problem is to nd a shortest closed zoo-keeper's path starting at a point of the zoo and touching all cages. We present O(nlogn) algorithm for zoo-keeper's problem where n is the input size. The algorithm is based on a new data structure called the oodlight tree. We extend the approach to queries to nd a zoo-keeper's path with arbitrary nal point. We show that the zoo-keeper's queries can be answered in optimal time O(logn + K) time where K is the output size.