, Volume 16, Issue 1, pp 5-28

Algorithms and analyses for maximal vector computation

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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 $\mathcal{O}(KN)$ 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 $\mathcal{O}(KN)$ average-case. 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 $\mathcal{O}(KN)$ .

Part of this work was conducted at William & Mary where Ryan Shipley was a student and Parke Godfrey was on faculty while on leave of absence from York.