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Fig. 1. Example dataset and skyline. 1. INTRODUCTION The skyline operator is important for several applications involving multicriteria decision making. Given a set of objects p1 , p2 , . . , pN , the operator returns all objects pi such that pi is not dominated by another object p j . Using the common example in the literature, assume in Figure 1 that we have a set of hotels and for each hotel we store its distance from the beach (x axis) and its price ( y axis). The most interesting hotels are a, i, and k, for which there is no point that is better in both dimensions.
While the static structure of such documents can be described by some DTD or XML Schema, our extension of XML Schema with function types is a first step toward a more precise description of XML documents embedding computation. Further work in that direction is clearly needed to better understand this powerful paradigm. There are a number of other proposals for typing XML documents, for example, Makoto , Hosoya and Pierce , and Cluet et al. . We selected XML Schema (see footnote 10) for several reasons.
Note that the NN query in Px is empty because there is no other point whose x coordinate is below nx . On the other hand, the expected volume of P y (Pz ) is 1/2 (assuming unit axis length on all dimensions), because the nearest neighbor is decided solely on x coordinates, and hence n y (nz ) distributes uniformly in [0, 1]. Following the same reasoning, a NN in P y finds the second skyline point that introduces three new partitions such that one partition leads to an empty query, while the volumes of the other two are 1/4.