Efficient Randomized Incremental Algorithm For The Closest Pair Problem Using Leafary Trees

1. O(n) expected time randomized algorithms are presented in [4, 5, 7]. They use hashing and assumes the floor function as a unit cost operation. 2. An O(n log log n) time deterministic algorithm is presented in [3] which uses hashing and assumes floor function as a unit cost operation. 3. In [1] an O(n) expected time randomized algorithm using the real-RAM model is presented. This algorithm assumes that the input coordinates are in a range [0..U 1] for some constant U. In a restricted real-RAM model where LOG and EXP are not allowed they have shown that the running time of the algorithm is O(n + log log U) if the input coordinates are integers and O(n + log log(6,na~/6rnin)) if the input coordinates are reals, where ~rnax is the largest and 6,nin is the shortest distance between the points. They further assume that all the numbers manipulated by these algorithms contain O(log n + log U) bits.