In-Place Algorithms for Computing (Layers of) Maxima

Abstract

We describe space-efficient algorithms for solving problems related to finding maxima among points in two and three dimensions. Our algorithms run in optimal O(n log n) time and occupy only constant extra space in addition to the space needed for representing the input. Keywords In-place algorithms • Pareto-optimal points • Computational geometry 1 Introduction Space-efficient solutions for fundamental algorithmic problems such as merging, sorting, or partitioning have been studied over a long period of time; see [12, 13, 15, 18, 25, 28]. The advent of small-scale, handheld computing devices and an increasing interest in utilizing fast but limited-size memory, e.g., caches, recently led to a renaissance of space-efficient computing with a focus on processing geometric data. Brönnimann et al. [6] were the first to consider space-efficient geometric algorithms and showed how to optimally compute 2d-convex hulls using constant extra space.

Finding Pareto optimal groups

“…Since a dataset may exhibit a linear number of layers, this leads to an O(n 2 log n) worst-case running time. Existing work proposed space-efficient algorithms for computing all skyline layers simultaneously with O(n log n) time complexity for two dimensions [4] and O(n 2 ) for higher dimensions [23]. In this paper, we present an efficient O(n log n) algorithm for two dimensional space based on the ideas briefly mentioned in [4] (they did not provide any algorithm details as their focus is on designing in-place algorithms).…”

“…Existing work proposed space-efficient algorithms for computing all skyline layers simultaneously with O(n log n) time complexity for two dimensions [4] and O(n 2 ) for higher dimensions [23]. In this paper, we present an efficient O(n log n) algorithm for two dimensional space based on the ideas briefly mentioned in [4] (they did not provide any algorithm details as their focus is on designing in-place algorithms). In addition, since we only need the first k skyline layers for computing k-point G-Skyline groups, as we have shown in Theorem 1, we present more efficient output-sensitive algorithms with O(n + S k log k) time complexity for two-and O(nS k ) for higher-dimensional space after the points are sorted.…”

“…Example 8. We show a running example of Algorithm last candidate group of its subtree has |u3 ∪ u11 ∪ u6 ∪ u1|p = |u3 ∪ u11 ∪ u1|p = 4, so it is a G-Skyline (4) group. Based on the Subset Pruning strategy, the algorithm is terminated because there are no more candidate groups that need to be checked.…”

Flexible and Efficient Resolution of Skyline Query Size Constraints

“…Inplace algorithms for computing layers of maxima in [4] are focused on 2D data sets only. This paper's setting poses no in-place requirement, and our algorithms are not limited to two dimensions.…”

Management of historical information and preference in massive datasets

“…This is because, one often faces the situation of having to strike a balance between a pair of naturally contradicting factors (e.g., price vs quality, space vs query time). Finally, the algorithms are based on a novel restricted dynamization of layers of minima [16]. This is of independent interest in case only the first k layers of minima are of interest.…”