A Note on the Expected Time for Finding Maxima by List Algorithms

Abstract.

Maxima in R ^d are found incrementally by maintaining a linked list and comparing new elements against the linked list. If the elements are independent and uniformly distributed in the unit square [0,1] ^d , then, regardless of how the list is manipulated by an adversary, the expected time is O(n log ^d-2 n) . This should be contrasted with the fact that the expected number of maxima grows as log ^d-1 n , so no adversary can force an expected complexity of n log ^d-1 n . Note that the expected complexity is O(n) for d=2 . Conversely, there are list-manipulating adversaries for which the given bound is attained. However, if we naively add maxima to the list without changing the order, then the expected number of element comparisons is n +o(n) for any $d \ge 2$ . In the paper we also derive new tail bounds and moment inequalities for the number of maxima.