Algorithms and analyses for maximal vector computation

Abstract

The maximal vector problem is to identify the maximals over a collection of vectors. This arises in many contexts and, as such, has been well studied. The problem recently gained renewed attention with skyline queries for relational databases and with work to develop skyline algorithms that are external and relationally well behaved.While many algorithms have been proposed, how they perform has been unclear. We study the performance of, and design choices behind, these algorithms. We prove runtime bounds based on the number of vectors n and the dimensionality k. Early algorithms based on divide-and-conquer established seemingly good average and worst-case asymptotic runtimes. In fact, the problem can be solved in O(n) average-case (holding k as fixed). We prove, however, that the performance is quite bad with respect to k. We demonstrate that the more recent skyline algorithms are better behaved, and can also achieve O(kn) averagecase. While k matters for these, in practice, its effect vanishes in the asymptotic. We introduce a new external algorithm, LESS, that is more efficient and better behaved. We evaluate LESS's effectiveness and improvement over the field, and prove that its average-case running time is O(kn).

Highly Scalable Multiprocessing Algorithms for Preference-Based Database Retrieval

“…In fact the basic operation of collecting maxima during a single scan of the input data can be found at the core of several state-of-the-art skyline algorithms [5,8,9,10,11] and is illustrated in Figure 2. The -block‖ in its name refers to the fact that it linearly scans over the data set and continuously maintains a block (or window) of data elements containing the maximal elements with respect to the data read so far.…”

“…Furthermore, a multitude of optimization techniques is applicable to BNL algorithms like e.g. dynamic sorting or indexing [5].…”

“…Otherwise, is split into two sets 1 and 2 and the algorithm applies itself recursively on both of them. The two results 1 and 2 are subsequently cleaned from local maxima by comparing each element of 1 to each element of 2 and removing all dominated elements Although the algorithm has excellent theoretical properties [3,17], there is no efficient implementation of this recursive process [5]. The main problem preventing an efficient implementation seems to be that the algorithm either requires massive disk IO, or needs to keep a large amount of intermediate results in main memory.…”

Efficient skyline evaluation over partially ordered domains

“…Sorting guarantees that an object cannot be dominated by objects that follow in the order. Optimized versions of SFS (i.e., LESS and SaLSa) were later proposed in [14] and [1]. Finally, the object-based space partitioning (OPS) skyline algorithm [31] operates in a similar fashion, but organizes the skyline found so far in a left-child/right-sibling tree, which accelerates the checking of whether the currently read point is already dominated by the found skyline.…”

“…To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Articles from this volume were presented at The 36th International Conference on Very Large Data Bases, September [13][14][15][16][17]2010, Singapore. values in a two-integer domain; then, a dominance check translates to a set of comparisons in the transformed space.…”