Fast Computation of Output-Sensitive Maxima in a Word RAM

Abstract

In this paper, we study the problem of computing the maxima of a set of n points in three dimensions with integer coordinates and show that in a word RAM, the maxima can be found in O n log log n/h n deterministic time in which h is the output size. For h = n 1−α this is O(n log(1/α)). This improves the previous O(n log log h) time algorithm and can be considered surprising since it gives a linear time algorithm when α > 0 is a constant, which is faster than the current best deterministic and randomized integer sorting algorithms. We observe that improving this running time is most likely difficult since it requires breaking a number of important barriers, even if randomization is allowed.Additionally, we show that the same deterministic running time could be achieved for performing n point location queries in an arrangement of size h. Finally, our maxima result can be extended to higher dimensions by paying a log n/h n factor penalty per dimension. This has further interesting consequences for example it preserves the linear running time when h ≤ n 1−α , for a constant α > 0, and thus it shows that for a variety of input distributions the maxima can be computed in linear expected time without knowing the distribution.

Skyline Computation with Noisy Comparisons

“…For any constant d, [20] show that skylines can be computed in O(nlog d−2 k). In the RAM model, the fastest algorithms we are aware of run in O(nlog d−3 n) expected time [10], and O(nloglog n/k n(log n/k n) d−3 deterministic time [2]. When d ∈ {2, 3}, the problem even admits "instance-optimal" algorithms [4].…”

Skyline Queries with Noisy Comparisons

“…Skylines can be computed faster in the RAM model [10,2] or when the dimension is very large [47]. In particular, the skyline can be computed in O(n log d−3 n) expected time [10] on RAM.…”

Output-sensitive Evaluation of Prioritized Skyline Queries

“…Kirkpatrick and Seidel [25] were the first to propose an output-sensitive procedure. More recently, [32] developed a divide-and-conquer algorithm that is efficient in external memory, while several results have been obtained using the word-RAM model [18,10,1]. Following [25] we denote by C d (v, n) the worst-case complexity of skyline queries, and by F d (b, w) the worst-case complexity of screening, assuming b = |B| and w = |W |.…”