Marrying Top-k with Skyline Queries: Operators with Relaxed Preference Input and Controllable Output Size

Properties of Candidate Records: Without loss of generality, assume that no two records coincide or score the same for the seed vector \({\boldsymbol{ w}}\). Unless a record belongs to the traditional k-skyband, it cannot belong to the \(\text{top-}k\) result for any preference vector [17]. Hence, for any \(\rho\), the \(\rho\)-skyband is a subset of the k-skyband. Therefore, the latter includes all the candidates we may need for \(\mathsf {ORD}\). Consider a record \({\boldsymbol{ r}}_i\) among them. The remaining candidates fall into three categories regarding their potential to \(\rho\)-dominate \({\boldsymbol{ r}}_i\):

Consider a record \({\boldsymbol{ r}}_j\) that falls in the third category. As such, it does not dominate \({\boldsymbol{ r}}_i\). Also, since \(U_{ {\boldsymbol{ w}}}( {\boldsymbol{ r}}_j) \gt U_{ {\boldsymbol{ w}}}( {\boldsymbol{ r}}_i)\), record \({\boldsymbol{ r}}_j\) is not dominated by \({\boldsymbol{ r}}_i\) either. Every pair of records that do not dominate each other define a hyper-plane with equation \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) = U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\) that cuts through the preference domain; i.e., it divides \(\Delta ^{d-1}\) into two non-empty parts. Since \({\boldsymbol{ r}}_j\) scores higher than \({\boldsymbol{ r}}_i\) for the seed \({\boldsymbol{ w}}\), it holds that \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) \lt U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\) in the entire part that includes \({\boldsymbol{ w}}\), while \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) \gt U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\) in the other part. Assuming \(d=3\), Figure 1(b) illustrates in gray a hyper-plane with equation \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) = U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\).

Let \({\boldsymbol{ s}}_{i,j}\) be the intersection of hyper-plane \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) = U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\) with the preference domain \(\Delta ^{d-1}\). Geometrically, \({\boldsymbol{ s}}_{i,j}\) is a \((d-2)\)-simplex, e.g., for \(d=3\) it is a line segment, shown in bold in Figure 1(b). Let also \(\rho _{i,j}\) be the minimum distance between \({\boldsymbol{ w}}\) and any \({\boldsymbol{ v}}\in {\boldsymbol{ s}}_{i,j}\). For any preference vector in \(\Delta ^{d-1}\) that is within distance \(\rho _{i,j}\) from \({\boldsymbol{ w}}\), record \({\boldsymbol{ r}}_j\) scores higher than \({\boldsymbol{ r}}_i\); i.e., \({\boldsymbol{ r}}_j\) \(\rho\)-dominates \({\boldsymbol{ r}}_i\) for every \(\rho \le \rho _{i,j}\). In contrast, it does not \(\rho\)-dominate \({\boldsymbol{ r}}_i\) for \(\rho \gt \rho _{i,j}\). In implementation terms, we can compute the mindist \(\rho _{i,j}\) using a quadratic programming solver [37, 62] with (squared) distance as the minimization objective, subject to the linear constraints that define \({\boldsymbol{ s}}_{i,j}\), i.e., \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) = U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\) and \(\sum _{i=1}^d w_i =1\).

Inflection Radius: By considering all records \({\boldsymbol{ r}}_j\) in the third category against \({\boldsymbol{ r}}_i\), they are each mapped into an interval of \(\rho\) values where they \(\rho\)-dominate it. Figure 2(a) offers an example. Assume that \(k=5\) and that the 5-skyband includes eight records that score higher than \({\boldsymbol{ r}}_i\) for \({\boldsymbol{ w}}\). Out of these, \({\boldsymbol{ r}}_i\) is dominated in the traditional sense by three (i.e., \({\boldsymbol{ r}}_3\), \({\boldsymbol{ r}}_4\), \({\boldsymbol{ r}}_6\)), thus their infinite intervals. The remaining five do not dominate \({\boldsymbol{ r}}_i\); hence, they fall in the third category; they each have a finite mindist \(\rho _{i,j}\) mapped to intervals as illustrated. By sweeping the intervals from left to right, we can easily identify the \(\rho\) value past which \({\boldsymbol{ r}}_i\) is dominated by fewer than k others; i.e., it becomes part of the \(\rho\)-skyband. We call that value the inflection radius of \({\boldsymbol{ r}}_i\) and denote it as \(\rho _i\). In our example, \(\rho _i = \rho _{i,7}\).

A Preliminary Approach: Based on the above, a first-cut \(\mathsf {ORD}\) solution is to compute the entire k-skyband, and for each record in it, to derive the inflection radius, and then, to output the m of them with the smallest inflection radii. To exemplify, assume that the 5-skyband includes 12 records; in Figure 2(b), we map each of them to an interval of \(\rho\) values where it belongs to the \(\rho\)-skyband, according to its inflection radius. These intervals have different meaning from Figure 2(a). In Figure 2(a), all intervals refer to \({\boldsymbol{ r}}_i\), helping to compute its own inflection radius \(\rho _i\). In contrast, Figure 2(b) is a global representation of the \(\rho\)-skyband for different \(\rho\) values. Specifically, if we sweep the chart with a vertical line, the intervals that intersect the line at any position indicate the \(\rho\)-skyband members for the \(\rho\) value that corresponds to that position. In our example, assuming that \(m=8\), the \(\mathsf {ORD}\) output is set \(\lbrace {\boldsymbol{ r}}_1, {\boldsymbol{ r}}_2, {\boldsymbol{ r}}_3, {\boldsymbol{ r}}_4, {\boldsymbol{ r}}_5, {\boldsymbol{ r}}_7, {\boldsymbol{ r}}_{10}, {\boldsymbol{ r}}_{12}\rbrace\), which corresponds to the \(\rho\)-skyband for \(\rho = \rho _{12}\).

An interesting insight is that the \(\mathsf {ORD}\) output may vary from standard ranking by utility (\(\text{top-}k\)) all the way to traditional dominance-based querying (k-skyband), depending on m. On the one hand, by definition, the \(\text{top-}k\) records are the only members of the \(\rho\)-skyband for \(\rho = 0\), which corresponds to the smallest possible m (i.e., \(m=k\)). On the other hand, every k-skyband member will appear in the \(\rho\)-skyband for a sufficiently large \(\rho\) or, equivalently, for a sufficiently large m. To visualize these extremes, at the leftmost position in Figure 2(b) the sweeping line intersects only the intervals of the \(\text{top-}k\) records (\({\boldsymbol{ r}}_1\) to \({\boldsymbol{ r}}_5\)), while at the rightmost² it intersects the entire k-skyband. That said, although this generality is welcome, the practical strength of \(\mathsf {ORD}\) is for m values in between these extremes.

The \(\mathsf {ORD}\) processing idea described so far is a foundation that offers an abstract-level understanding of our method. However, an efficient solution must address several performance issues. Primarily, we would want to avoid computing the entire k-skyband in the beginning of the process. Indeed, the k-skyband may include numerous records, many times more than the m required [15, 36]. Ideally, we want to limit the number of considered candidates to as tight a superset of the \(\mathsf {ORD}\) output as possible. The algorithm we present next serves that objective.

We first invoke a progressive k-skyband retrieval that fetches its members one by one and place them into a candidate set. Importantly, unlike standard k-skyband computation, we enforce that its members are fetched in decreasing score order for \({\boldsymbol{ w}}\) (we will explain how shortly). This retrieval order is essential, because when a new candidate \({\boldsymbol{ r}}_i\) is fetched, we can definitively compute its inflection radius \(\rho _i\) already, without having to derive the entire k-skyband. The rationale is that only k-skyband records with higher score may \(\rho\)-dominate \({\boldsymbol{ r}}_i\), and these are guaranteed to be fetched before it.

We keep fetching new k-skyband members in that fashion until the candidate set reaches size \((m+1)\). At that stage, we evict the candidate with the largest inflection radius. Also, an important algorithmic shift takes place. Let \(\bar{\rho }\) be the maximum inflection radius of the remaining m candidates. The \(\bar{\rho }\)-skyband is guaranteed to include at least the m existing candidates, and thus \(\bar{\rho }\) upper bounds the eventual stopping radius of the algorithm. Therefore, from this point onward, we switch to fetching \(\bar{\rho }\)-skyband members (instead of k-skyband members), still in decreasing order of score for \({\boldsymbol{ w}}\). The switch can be performed transparently, as we elaborate later.

To exemplify, consider Figure 2(b). Assume that \(k = 5\) and \(m = 11\) and that we have fetched the \(m+1 = 12\) depicted records, in the order indicated by their subscripts; i.e., \({\boldsymbol{ r}}_i\) was fetched i-th. To bring the candidates down to \(m=11\), we discard \({\boldsymbol{ r}}_8\), as it has the largest inflection radius. Practically, that sets \(\bar{\rho }\) to \(\rho _8\).

As new candidates are fetched and evictions are made to keep their total number to m, the current \(\bar{\rho }\) keeps shrinking. Meanwhile, as \(\bar{\rho }\) shrinks, records tend to \(\rho\)-dominate more others. This implies that the \(\bar{\rho }\)-skyband retrieval becomes increasingly more selective, thus filtering out more aggressively regular k-skyband members that cannot participate in the \(\mathsf {ORD}\) result. When the \(\bar{\rho }\)-skyband module cannot fetch any more records, the candidate set is finalized as the \(\mathsf {ORD}\) result. The latter corresponds to the \(\rho\)-skyband for \(\rho\) equal to the maximum inflection radius across its members.

The \(\mathsf {ORD}\) algorithm relies on a progressive k-skyband module, with the extra requirement to fetch records in decreasing score according to \({\boldsymbol{ w}}\). We use an adaptation of BBS [69] where we visit index nodes and records in decreasing (upper bound of) score for \({\boldsymbol{ w}}\), using a max-heap. Once the \((m+1)\)-th record is fetched, we shift to \(\bar{\rho }\)-skyband computation using the exact same heap as per normal, but replace the regular dominance tests of BBS with \(\bar{\rho }\)-dominance for the current \(\bar{\rho }\) value. The visiting order by score and the use of \(\bar{\rho }\)-dominance tests instead of regular dominance are permissible modifications to vanilla BBS, because its correctness is guaranteed as long as no record \({\boldsymbol{ r}}_j\) fetched after another \({\boldsymbol{ r}}_i\) may dominate \({\boldsymbol{ r}}_i\) [69]. That property is upheld by our visiting order (by score), both initially for regular dominance and after the shift to \(\bar{\rho }\)-dominance. Indeed, \(U_{ {\boldsymbol{ w}}}( {\boldsymbol{ r}}_i) \gt U_{ {\boldsymbol{ w}}}( {\boldsymbol{ r}}_j)\) ensures that \({\boldsymbol{ r}}_j\) cannot dominate or \({\rho }\)-dominate \({\boldsymbol{ r}}_i\) for any \(\rho\). An implementation note on the adapted BBS regards its \({\rho }\)-dominance building block. That block tests whether an already-fetched \(\bar{\rho }\)-skyband record \({\boldsymbol{ r}}_i\) \(\bar{\rho }\)-dominates a not-yet-fetched record \({\boldsymbol{ r}}_j\) (or an unvisited index node whose top corner is \({\boldsymbol{ r}}_j\)). The test is performed as explained in Section 4.1, by comparing the mindist \(\rho _{i,j}\) with \(\bar{\rho }\).

The abstractions and techniques used in \(\mathsf {BORU}\) have the notion of the convex hull at their core [16]. The convex hull of D is the smallest convex polytope that encloses all its records. It comprises facets, each defined by d extreme vertices (records) in general position. The outer polygon in Figure 3(a) is the convex hull of an example dataset. Facet \({\boldsymbol{ r}}_1 {\boldsymbol{ r}}_2\) is defined by extreme vertices \({\boldsymbol{ r}}_1\) and \({\boldsymbol{ r}}_2\), and so forth.

A vector is normal to a hyper-plane when its direction is perpendicular to the hyper-plane. The norm of a facet on the hull is the normal vector to that facet whose sum of coordinates is 1 and is directed toward the exterior of the hull. In our example, the norm of \({\boldsymbol{ r}}_1 {\boldsymbol{ r}}_2\) is vector \({\boldsymbol{ v}}_1\). Effectively, the norm of a facet corresponds to a point in the preference domain \(\Delta ^{d-1}\).

The top record for a preference vector \({\boldsymbol{ v}}\) is the one met first by a hyper-plane normal to \({\boldsymbol{ v}}\) that sweeps the data space from the top corner to the origin [28, 33]. Hence, the top record in D is guaranteed to lie on its convex hull [20, 59]. Since in our case the weights are non-negative, the top record is among the extreme vertices of facets with non-negative norms. We call upper hull the part that corresponds to these facets. In Figure 3(a), the upper hull is bold (and the rest of the convex hull is dashed). For the shown w, the top record is \({\boldsymbol{ r}}_3\), as it is met first by the sweeping line normal to w.

To explain the fundamentals of our methodology, assume that we have already computed the first k upper hull layers: the first layer, \(L_1\), includes the upper hull of D; the second, \(L_2\), includes the upper hull of \(D - L_1\); and, generally, layer \(L_i\) includes the upper hull of D after subtracting layers \(L_1\) to \(L_{i-1}\). In Figure 3(a), layer \(L_1\) includes records \({\boldsymbol{ r}}_1\) to \({\boldsymbol{ r}}_5\), and \(L_2\) records \({\boldsymbol{ r}}_6\) to \({\boldsymbol{ r}}_{10}\). Note that in reality the complete \(\mathsf {BORU}\) algorithm does not require such precomputation but instead builds on the fly (i.e., at query time) only parts of the necessary layers, thus applying to arbitrary k, avoiding precomputation costs (time and space), and extending transparently to dynamic datasets, i.e., to cases where record insertions/deletions may occur in D and invalidate precomputed information. Also, assume that we already know the necessary radius \(\rho\) for \(\mathsf {ORU}\) to output m records. Of course, this too is an assumption we will drop later (in Section 5.3). Adjacent Set \(\mathcal {A}( {\boldsymbol{ r}})\): Consider a record \({\boldsymbol{ r}}\) in layer \(L_i\). We denote by \(\mathcal {F}( {\boldsymbol{ r}})\) the set of \(L_i\) facets with \({\boldsymbol{ r}}\) as one of their extreme vertices, and by \(\mathcal {A}( {\boldsymbol{ r}})\) the records adjacent to \({\boldsymbol{ r}}\), i.e., the \(L_i\) records (other than \({\boldsymbol{ r}}\)) that define facets in \(\mathcal {F}( {\boldsymbol{ r}})\). In Figure 3(a), for example, \(\mathcal {F}( {\boldsymbol{ r}}_3) = \lbrace {\boldsymbol{ r}}_2 {\boldsymbol{ r}}_3, {\boldsymbol{ r}}_3 {\boldsymbol{ r}}_4\rbrace\) and \(\mathcal {A}( {\boldsymbol{ r}}_3) = \lbrace {\boldsymbol{ r}}_2, {\boldsymbol{ r}}_4\rbrace\). The following Lemmas 1, 2, and 3 are crucial. Note that they refer to records within the same layer \(L_i\).

By Lemma 1, if w shifts clockwise/anticlockwise in Figure 3(a), \({\boldsymbol{ r}}_4\) and \({\boldsymbol{ r}}_2\), respectively, will be the first \(L_1\) records to outscore \({\boldsymbol{ r}}_3\).

Top-region \(\mathcal {C}( {\boldsymbol{ r}})\): Building on Lemma 1, our next proposition reveals an important property within \(L_i\) and helps define the top-region of a record \({\boldsymbol{ r}}\in L_i\), i.e., the region \(\mathcal {C}( {\boldsymbol{ r}})\) in the preference domain where every vector has r as its top record in \(L_i\).

By Lemma 2, \(\mathcal {C}( {\boldsymbol{ r}})\) can be seen as a dual representation of \(\mathcal {F}( {\boldsymbol{ r}})\), where the former refers to the preference domain and the latter to the data space. Consider \({\boldsymbol{ r}}_3\) in Figure 3(a). Facet set \(\mathcal {F}( {\boldsymbol{ r}}_3) = \lbrace {\boldsymbol{ r}}_2 {\boldsymbol{ r}}_3, {\boldsymbol{ r}}_3 {\boldsymbol{ r}}_4\rbrace\) translates to the top-region defined by their norms \({\boldsymbol{ v}}_2\) and \({\boldsymbol{ v}}_3\); i.e., \(\mathcal {C}( {\boldsymbol{ r}}_3)\) is segment \({\boldsymbol{ v}}_2 {\boldsymbol{ v}}_3\) in the preference domain. Note that for \(d=2\), the preference domain \(\Delta ^{d-1}\) is a line segment.

Order Continuity: Lemmas 1 and 2, in tandem, suggest a continuity in the score order among \(L_i\) records for every v. Specifically, the different top-regions for any given layer L define a partitioning of the preference domain, with adjacent records \({\boldsymbol{ r}}_i, {\boldsymbol{ r}}_j\) in L having neighboring top-regions \(\mathcal {C}( {\boldsymbol{ r}}_i), \mathcal {C}( {\boldsymbol{ r}}_j)\) in \(\Delta ^{d-1}\). Considering layer \(L_1\) in our running example, Figure 3(b) demonstrates the partitioning of the preference domain. The top-region of \({\boldsymbol{ r}}_1\) is the segment from point \((0,1)\) to \({\boldsymbol{ v}}_1\); of \({\boldsymbol{ r}}_2\) from \({\boldsymbol{ v}}_1\) to \({\boldsymbol{ v}}_2\); of \({\boldsymbol{ r}}_3\) from \({\boldsymbol{ v}}_2\) to \({\boldsymbol{ v}}_3\); and so forth. Lemma 3 establishes a property for every vector in a top-region.

In our example, Lemma 3 implies that the top-2-nd record in layer \(L_1\) for any \({\boldsymbol{ v}}\in \mathcal {C}( {\boldsymbol{ r}}_3)\) is either \({\boldsymbol{ r}}_2\) or \({\boldsymbol{ r}}_4\). Note that all three lemmas consider a layer in isolation. For instance, although \({\boldsymbol{ w}}\in \mathcal {C}( {\boldsymbol{ r}}_3)\), its top-2-nd record in the entire D is none of \({\boldsymbol{ r}}_2\) or \({\boldsymbol{ r}}_4\), but \({\boldsymbol{ r}}_8\) from \(L_2\).

In this section, we prove an important theorem and provide an algorithmic basis to \(\mathsf {BORU}\). Recall that we assumed we already know the minimum radius \(\rho\) required to produce m records and that the first k upper hull layers are precomputed. We do not drop these assumptions yet. Here we focus on determining the \(\text{top-}k\) result for any possible preference vector within radius \(\rho\) from the seed w in order to form the \(\mathsf {BORU}\) output.

First, we find all records in layer \(L_1\) whose top-region has mindist to w no greater than \(\rho\). Let C be one of these regions. We already know the top record in it, say, r. Considering C in isolation, our next task is to determine the top-2-nd record anywhere in it (i.e., for any possible preference vector \({\boldsymbol{ v}}\in C\)) and to partition C accordingly. By Lemma 3, if we only considered \(L_1\), the top-2-nd record for any \({\boldsymbol{ v}}\in C\) would be among those adjacent to r. On the other hand, in the remaining dataset (i.e., if we ignored the records in \(L_1\)), the top-2-nd record would be in \(L_2\) and, more specifically, by Lemma 2, among the \(L_2\) records \({\boldsymbol{ r}}_i\) whose top-region \(\mathcal {C}( {\boldsymbol{ r}}_i)\) overlaps C. Theorem 1 generalizes this key observation.

Returning to our processing description for region C, the top record (the order-sensitive top-i result, in the general case) is already known and fixed anywhere in it, and thus we can readily determine Set (i). We can also extract from \(L_2\) (from \(L_{t+1}\), in the general case) the part of the upper hull that corresponds to records in Set (ii); let us denote that part as \(L_{prt}\). We update the upper hull \(L_{prt}\) to also cover Set (i) records and denote its updated version as \(L_{upd}\). Next, we apply Lemma 2 to \(L_{upd}\) to identify the top-2-nd records (the top-\((i+1)\)-th, in the general case) for any \({\boldsymbol{ v}}\in C\), and we partition C accordingly. We continue this process recursively in each produced partition until the full, order-sensitive \(\text{top-}k\) result is known anywhere in C. Repeating that process for all \(L_1\) top-regions with mindist up to \(\rho\), we derive all the required \(\text{top-}k\) results.

In our example, assume that \(\rho\) is as shown in Figure 3(b). The \(L_1\) top-regions with mindist from w up to \(\rho\) correspond to \({\boldsymbol{ r}}_2\), \({\boldsymbol{ r}}_3\), and \({\boldsymbol{ r}}_4\). Focusing on \(\mathcal {C}( {\boldsymbol{ r}}_3)\) (i.e., segment \({\boldsymbol{ v}}_2 {\boldsymbol{ v}}_3\)), we know already that the top record is \({\boldsymbol{ r}}_3\) and seek to find the top-2-nd. Set (i) includes \({\boldsymbol{ r}}_2\) and \({\boldsymbol{ r}}_4\). To determine Set (ii), we refer to \(L_2\). Figure 4(a) illustrates the \(L_2\) top-regions. Among them, those that overlap \(\mathcal {C}( {\boldsymbol{ r}}_3)\) are \(\mathcal {C}( {\boldsymbol{ r}}_7)\), \(\mathcal {C}( {\boldsymbol{ r}}_8)\), and \(\mathcal {C}( {\boldsymbol{ r}}_9)\). Thus, we form \(L_{prt}\) as the part of \(L_2\) that corresponds to \({\boldsymbol{ r}}_7, {\boldsymbol{ r}}_8, {\boldsymbol{ r}}_9\). Updating \(L_{prt}\) to also cover Set (i) (i.e., \({\boldsymbol{ r}}_2, {\boldsymbol{ r}}_4\)) results in the upper hull \(L_{upd}\) in Figure 4(b). \(L_{upd}\) suggests that the top-2-nd record is one of \({\boldsymbol{ r}}_2, {\boldsymbol{ r}}_8, {\boldsymbol{ r}}_4\). Furthermore, by Lemma 2, their \(L_{upd}\) top-regions (determined by facet norms \({\boldsymbol{ v}}_a\) and \({\boldsymbol{ v}}_b\)) help partition \(\mathcal {C}( {\boldsymbol{ r}}_3)\) according to which exactly among them is the top-2-nd. Figure 4(c) presents the induced partitioning. The top-2 result is \(\lbrace {\boldsymbol{ r}}_3, {\boldsymbol{ r}}_2\rbrace\) for preference vectors in \({\boldsymbol{ v}}_2 {\boldsymbol{ v}}_a\); \(\lbrace {\boldsymbol{ r}}_3, {\boldsymbol{ r}}_8\rbrace\) in \({\boldsymbol{ v}}_a {\boldsymbol{ v}}_b\); and \(\lbrace {\boldsymbol{ r}}_3, {\boldsymbol{ r}}_4\rbrace\) in \({\boldsymbol{ v}}_b {\boldsymbol{ v}}_3\). The process repeats recursively in order to determine the top-3-rd record in each of the three partitions, and so on.

Observe that we may not need to reach as deep as the k-th layer, since members of Set (i) could prevent those of Set (ii) from entering the result, thus giving more “width” to the \(\mathsf {BORU}\) search (in the data space) than “depth.” For instance, in Figure 4(b), it could be the case that none of the \(L_2\) records belongs to \(L_{upd}\), i.e., that the top-2-nd record comes from \(L_1\) (i.e., \({\boldsymbol{ r}}_2\) or \({\boldsymbol{ r}}_4\)) for any \({\boldsymbol{ v}}\in \mathcal {C}( {\boldsymbol{ r}}_3)\).

In this section, we describe our complete \(\mathsf {BORU}\) algorithm. So far, we have made two impractical assumptions, i.e., that we have already computed the first k upper hull layers and that we know in advance the necessary radius \(\rho\) to output m records. Here we drop these assumptions, the first so that our algorithm is precomputation-free, and the second because it defies our problem formulation.

Without any precomputed layers or known \(\rho\), our first step is to produce an overestimate of \(\rho\), denoted as \(\bar{\rho }\), which ensures an output size of at least m. That overestimate can be the radius required so that \(\mathsf {ORU}\)’s output for \(k=1\) includes m records. This radius, for any k, is guaranteed to produce at least as many records.

A straightforward approach to derive \(\bar{\rho }\) (based on \(k=1\)) would be to compute the upper hull of the entire dataset D, get the m top-regions with the smallest mindist to w, and use the largest mindist among them as \(\bar{\rho }\). That, however, may be too costly. Ideally, we would want to localize the upper hull computation to just the vicinity of w. To achieve this, we exploit the fact that the \(\rho\)-skyline is a superset of \(\mathsf {ORU}\)’s output for the same \(\rho\) and \(k=1\), as follows directly from the definition of \(\rho\)-dominance.

We use an incremental \(\rho\)-skyline algorithm, which supports “get next” calls to extend a \(\rho\)-skyline for the immediately larger \(\rho\) around w that admits exactly one new record to it. Details on that technique (for its general, \(\rho\)-skyband version) are presented in Section 5.3.2. We initialize that algorithm and prompt it until the \(\rho\)-skyline includes m records. Next, we compute their upper hull \(L_{tmp}\). In general, not all \(\rho\)-skyline records will make it to \(L_{tmp}\). If that is indeed the case, we keep prompting the \(\rho\)-skyline algorithm and updating the upper hull \(L_{tmp}\) to cover the additional records until \(L_{tmp}\) includes m extreme vertices (records). We use as \(\bar{\rho }\) the final radius reported by the \(\rho\)-skyline algorithm. On top of this, note that the final \(L_{tmp}\) is guaranteed to include all the parts of layer \(L_1\) that \(\mathsf {BORU}\) could possibly need for an output of size m. In other words, the final \(L_{tmp}\) can serve already as layer \(L_1\) in subsequent processing.

Using the obtained \(\bar{\rho }\), we compute the \(\rho\)-skyband (for the actual k specified in the input). That can be done with a standard k-skyband algorithm by simply replacing regular dominance tests with \(\bar{\rho }\)-dominance ones (described in the last paragraph of Section 4.2). The derived \(\bar{\rho }\)-skyband is a guaranteed superset of the \(\mathsf {ORU}\) output, and thus we place its members into a candidate set M.

Even with \(\bar{\rho }\) available, we have only an overestimate of the actual radius required to output m records. Thus, computing upper hull layers directly on M would be computationally wasteful. To circumvent this, we employ an adaptive technique that progressively outputs confirmed candidates (i.e., records guaranteed to be in the \(\mathsf {ORU}\) result) while the search is ongoing. Effectively, this enables the tightening of \(\bar{\rho }\) on the fly, and hence the shrinking of the \(\bar{\rho }\)-skyband and the elimination of candidates from M, so that the layer (i.e., upper hull) computations execute on increasingly fewer records. This improved implementation is presented next.

In the context of \(\mathsf {FORU}\), any reference to \(\text{top-}k\) result/set refers to its order-insensitive definition. Also, in case of utility ties in the top-k-th position, the \(\text{top-}k\) set is assumed to include all tied records (i.e., its cardinality may exceed k). A central concept in \(\mathsf {FORU}\) is the result region.

Result Region \(\mathcal {R}(S)\): Given a \(\text{top-}k\) set S, the result region \(\mathcal {R}(S)\) is the preference region that includes those and only those vectors \({\boldsymbol{ v}}\) whose \(\text{top-}k\) set is S. Following directly from its definition, \(\mathcal {R}(S)\) is the intersection of half-spaces \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}^{\prime })\) for each pair of \({\boldsymbol{ r}}\in S\) and \({\boldsymbol{ r}}^{\prime } \in D-S\). Formally,

If \(\mathcal {R}(S) = \emptyset\), set S is not a possible \(\text{top-}k\) result and we call it infeasible. On the other hand, if \(\mathcal {R}(S) \ne \emptyset\), set S is a feasible \(\text{top-}k\) result and, geometrically, \(\mathcal {R}(S)\) (as the intersection of a finite number of half-spaces) is a convex polytope [16].

Let \(S_{ {\boldsymbol{ w}}}\) be the \(\text{top-}k\) set for the seed vector w and \(\mathcal {R}_{ {\boldsymbol{ w}}}\) be the respective result region. For now, assume that we derive \(\mathcal {R}_{ {\boldsymbol{ w}}}\) by Equation (1) using, however, the k-skyband instead of D, since it includes every record that could appear in any \(\text{top-}k\) result (and thus any record that may affect \(\mathcal {R}_{ {\boldsymbol{ w}}}\)). Still, the k-skyband may be large and, hence, the number of intersected half-spaces excessive, but this can be improved as we explain later, in Section 6.2. A high-level outline of \(\mathsf {FORU}\) is the following:

With \(\mathcal {R}_{ {\boldsymbol{ w}}}\) at hand, \(\mathsf {FORU}\) initially determines every possible \(\text{top-}k\) set when the preference vector moves infinitesimally outside \(\mathcal {R}_{ {\boldsymbol{ w}}}\) and derives accordingly the result regions for these \(\text{top-}k\) sets. After computing all the neighbor regions to \(\mathcal {R}_{ {\boldsymbol{ w}}}\), the process is repeated to determine their own neighbor regions, thus progressively tessellating the preference domain around w until the union of encountered \(\text{top-}k\) sets hits the desired output size m.

Neighbor Region/Set: Consider a non-empty result region \(\mathcal {R}(S)\). Formally, we define as neighbor region every non-empty result region \(\mathcal {R}(S^{\prime })\) that shares at least one extreme vertex with \(\mathcal {R}(S)\). Similarly, we refer to \(S^{\prime }\) as a neighbor set to S. Figure 6 illustrates the result region \(\mathcal {R}_{ {\boldsymbol{ w}}}\) that (includes the seed vector \({\boldsymbol{ w}}\) and) corresponds to \(S_{ {\boldsymbol{ w}}}\) and the result region \(\mathcal {R}(S)\) of another \(\text{top-}k\) set, S. As \(\mathcal {R}(S)\) has a common extreme vertex with \(\mathcal {R}_{ {\boldsymbol{ w}}}\) (e.g., \({\boldsymbol{ v}}_1\)), it is a neighbor region to \(\mathcal {R}_{ {\boldsymbol{ w}}}\), and set S is a neighbor set to \(S_{ {\boldsymbol{ w}}}\).

Referring to Figure 6, assume that \(k=3\), that \(S_{ {\boldsymbol{ w}}} = \lbrace {\boldsymbol{ r}}_1, {\boldsymbol{ r}}_2, {\boldsymbol{ r}}_3\rbrace\), and that the \(\text{top-}k\) set for \({\boldsymbol{ v}}_1\) is \(S_{ {\boldsymbol{ v}}_1} = \lbrace {\boldsymbol{ r}}_1, {\boldsymbol{ r}}_2, {\boldsymbol{ r}}_3, {\boldsymbol{ r}}_4, {\boldsymbol{ r}}_5 \rbrace\). According to Lemma 4, for any neighbor region that has \({\boldsymbol{ v}}_1\) as an extreme vertex (like \(\mathcal {R}(S)\) in the figure), the respective \(\text{top-}k\) set S is a subset of \(S_{ {\boldsymbol{ v}}_1}\). \(\mathsf {FORU}\) uses that fact to discover each and every neighbor region to \(\mathcal {R}_{ {\boldsymbol{ w}}}\) that shares \({\boldsymbol{ v}}_1\) as an extreme vertex; recall that, although \(S_{ {\boldsymbol{ w}}}\) (and thus \(\mathcal {R}_{ {\boldsymbol{ w}}}\) and its extreme vertices) is clear how to derive, the neighbor regions are unknown. To compute the neighbor regions (and thus expand the tessellation around \({\boldsymbol{ w}}\)) we exploit the fact that \(S \subset S_{ {\boldsymbol{ v}}_1}\) and consider every possible k-combination from \(S_{ {\boldsymbol{ v}}_1}\) as a candidate \(\text{top-}k\) set, e.g., \(S = \lbrace {\boldsymbol{ r}}_1, {\boldsymbol{ r}}_4, {\boldsymbol{ r}}_5 \rbrace\). For each of the candidate \(\text{top-}k\) sets, we apply Equation (1), using, however, \(S_{ {\boldsymbol{ v}}_1}\) in place of D; i.e., in our example where \(S = \lbrace {\boldsymbol{ r}}_1, {\boldsymbol{ r}}_4, {\boldsymbol{ r}}_5 \rbrace\), the only non-result records \({\boldsymbol{ r}}^{\prime }\) we need to consider in Equation (1) are \({\boldsymbol{ r}}_2\) and \({\boldsymbol{ r}}_3\). Those feasible among the candidate sets, i.e., those where \(\mathcal {R}(S) \ne \emptyset\), lead to the neighbor regions that share \({\boldsymbol{ v}}_1\) with \(\mathcal {R}_{ {\boldsymbol{ w}}}\), like the one illustrated in Figure 6.

An important point is that, as we explained in the beginning of the section, the \(\text{top-}k\) set for a neighbor region could include more than k records. Although we consider combinations of exactly k records from \(S_{ {\boldsymbol{ v}}_1}\), even if there are ties in the top-k-th position and some tying records are cut out from a k-combination, such records will not be missed by the overall process, because they are bound to appear in another feasible k-combination.

By repeating the above investigation for each extreme vertex of \(\mathcal {R}_{ {\boldsymbol{ w}}}\), we can derive all neighbor regions/sets to \(\mathcal {R}_{ {\boldsymbol{ w}}}\), completing one hop in the expansion. The tessellation could continue in a “breadth-first” manner, completing one hop after the other. This, however, is not the most efficient approach, since expansion is bounded by distance (i.e., the stopping radius \(\rho\)) instead of number of hops. For that reason, \(\mathsf {FORU}\) employs a min-heap Q that organizes encountered extreme vertices and result regions by minimum distance to \({\boldsymbol{ w}}\). This way, it incorporates their \(\text{top-}k\) sets in the overall output in the appropriate order and prioritizes the derivation of new result regions in increasing distance from \({\boldsymbol{ w}}\) to save computations. In the following section, we present the technical specifics of \(\mathsf {FORU}\) in the form of pseudo-code.

Algorithm 1 summarizes the \(\mathsf {FORU}\) algorithm. Line 1 performs a regular \(\text{top-}k\) computation using a standard technique, like BBR [79] (the same holds for Lines 9 and 34). Line 3 initializes \(\mathcal {G}\), an array of size m, which at the end of the process will hold the output of \(\mathsf {FORU}\). \(\mathcal {G}\) maintains records sorted in ascending order of a key (a distance to w, in particular), initialized to \(\infty\) for its unoccupied slots.

Line 4 initializes \(T_s\), whose purpose is to avoid examining the same k-combination of records multiple times. Specifically, its elements are \(\text{top-}k\) sets, starting with the seed’s \(\text{top-}k\) set (i.e., \(S_{ {\boldsymbol{ w}}}\)). Each k-combination encountered by \(\mathsf {FORU}\) (in Line 19) is book-kept as a separate element in \(T_s\). That is essential, since the same k-combination could occur while examining different extreme vertices whose \(\text{top-}k\) sets happen to overlap. Similarly, the role of \(T_v\) is to avoid considering anew extreme vertices that have already been accounted for (in Line 30) in the context of another neighbor result region.

As explained previously, the min-heap Q (initialized in Line 6) organizes the encountered extreme vertices and result regions. If a heap element corresponds to an extreme vertex \({\boldsymbol{ v}}\), the element includes the respective \(\text{top-}k\) set \(S_{ {\boldsymbol{ v}}}\), with key the distance between \({\boldsymbol{ v}}\) and \({\boldsymbol{ w}}\). If the heap element corresponds to a result region \(\mathcal {R}(S)\), it includes the respective k-combination S, with key the mindist between \(\mathcal {R}(S)\) and w. When we pop the heap (Lines 13 and 27), the element with the smallest key is removed from the heap, and its record set (\(S_{ {\boldsymbol{ v}}}\) or S) and key value (\(dist( {\boldsymbol{ v}}, {\boldsymbol{ w}})\) or \(mindist(\mathcal {R}(S), {\boldsymbol{ w}})\), respectively) become available.

Let \(S_{ {\boldsymbol{ v}}}\) be the \(\text{top-}k\) set and \(dist( {\boldsymbol{ v}}, {\boldsymbol{ w}})\) be the key of the popped heap entry. For every record r in \(S_{ {\boldsymbol{ v}}}\), we update \(\mathcal {G}\) (in Lines 15-18). Specifically, if r is not already in \(\mathcal {G}\) and \(dist( {\boldsymbol{ v}}, {\boldsymbol{ w}})\) is smaller than the m-th key in \(\mathcal {G}\), we insert r into \(\mathcal {G}\) with key \(dist( {\boldsymbol{ v}}, {\boldsymbol{ w}})\) (evicting the previous m-th record from \(\mathcal {G}\) to keep its size to m). If r is already in \(\mathcal {G}\) with a key greater than \(dist( {\boldsymbol{ v}}, {\boldsymbol{ w}})\), we update its key to \(dist( {\boldsymbol{ v}}, {\boldsymbol{ w}})\). The latter situation may occur if r was previously encountered in the \(\text{top-}k\) set of another popped heap entry. As an example, consider Figure 6, where \(\mathcal {R}\) is pushed into the heap only after vertex \({\boldsymbol{ v}}_1\) has been popped, despite the fact that \(\mathcal {R}\) is actually nearer to \({\boldsymbol{ w}}\). If a record \({\boldsymbol{ r}}\) in \(S_{ {\boldsymbol{ v}}_1}\) had been inserted into \(\mathcal {G}\), and the same record is now encountered in the \(\text{top-}k\) set of \(\mathcal {R}\), the key of \({\boldsymbol{ r}}\) in \(\mathcal {G}\) will be updated to \(mindist(\mathcal {R}, {\boldsymbol{ w}})\). The process to update \(\mathcal {G}\) when the popped heap entry corresponds to a result region is similar (Line 29).

\(\mathsf {FORU}\) terminates when the m-th key in \(\mathcal {G}\) is no greater than the key at the top of the heap, at which point \(\mathcal {G}\) is finalized and output. The radius \(\rho\) in Definition 2 corresponds to the m-th key in the finalized \(\mathcal {G}\). Another termination scenario is when the heap becomes empty, which means that the entire preference domain has been explored. The latter case may occur when the possible \(\text{top-}k\) records throughout the preference domain are fewer than m.

A low-level optimization regards Lines 23 and 24. To save computations, we may first check the feasibility of S via linear programming [16] before spending the (typically higher) cost to derive the exact geometry of \(\mathcal {R}(S)\) via half-space intersection. In Line 25, we compute \(mindist(\mathcal {R}(S), {\boldsymbol{ w}})\) using quadratic programming [37, 62].

As a note on Line 23, recall that \(\mathcal {R}(S)\) is defined by Equation (1), using \(S_{ {\boldsymbol{ v}}}\) in place of D for the half-space intersection. \(S_{ {\boldsymbol{ v}}}\), being a \(\text{top-}k\) set, is expected to be small, i.e., to include just slightly over k records. Unlike Line 23, however, the derivation of \(\mathcal {R}_{ {\boldsymbol{ w}}}\) in Line 2 is more complex. Specifically, \(\mathcal {R}_{ {\boldsymbol{ w}}}\) can still be computed by Equation (1), but the non-result records we need to consider are all the k-skyband records (i.e., by using the k-skyband in place of D). Surely, using the k-skyband instead of the entire D in Equation (1) is a relief. However, the k-skyband may still include numerous records, implying a large number of half-spaces to intersect. We may reduce the number of half-spaces based on the following two observations.

First, on closer inspection, even the entire k-skyband is not necessary (when replacing D in Equation (1)). In particular, let \(S_{NR}\) be the skyline of set \(D-S_{ {\boldsymbol{ w}}}\), i.e., the skyline of the dataset after we exclude the \(\text{top-}k\) records for w. The records in \(S_{NR}\) are the only possible candidates to enter the \(\text{top-}k\) result when the preference vector moves infinitesimally outside \(\mathcal {R}_{ {\boldsymbol{ w}}}\). To see this, from the definition of the skyline [17], it follows that for every non-result record \({\boldsymbol{ r}}_j \notin S_{NR}\), there is a record \({\boldsymbol{ r}}_i \in S_{NR}\) that dominates it. For any preference vector \({\boldsymbol{ v}}\), it holds that \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\). Therefore, referring to Equation (1), the condition that a result record \({\boldsymbol{ r}}\in S_{ {\boldsymbol{ w}}}\) stays ahead of \({\boldsymbol{ r}}_i\) utility-wise (i.e., the condition \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i)\)) is stricter, and hence subsumes the condition that \({\boldsymbol{ r}}\) stays ahead of \({\boldsymbol{ r}}_j\). Formally, the half-space \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\) is not a bounding half-space of \(\mathcal {R}_{ {\boldsymbol{ w}}}\) and can thus be ignored in Equation (1).

Second, consider a result record \({\boldsymbol{ r}}_i \in S_{ {\boldsymbol{ w}}}\) that dominates another result record \({\boldsymbol{ r}}_j \in S_{ {\boldsymbol{ w}}}\). For any preference vector \({\boldsymbol{ v}}\) it holds that \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j)\). Therefore, the condition that \({\boldsymbol{ r}}_j\) stays ahead of a non-result record \({\boldsymbol{ r}}^{\prime }\) utility-wise (i.e., the condition \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_j) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}^{\prime })\)) is stricter, and thus subsumes the condition that \({\boldsymbol{ r}}_i\) stays ahead of \({\boldsymbol{ r}}^{\prime }\). In other words, the half-space \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}^{\prime })\) is not a bounding half-space of \(\mathcal {R}_{ {\boldsymbol{ w}}}\) and can be ignored in Equation (1). Letting \(S_R\) be the set of result records \({\boldsymbol{ r}}_j \in S_{ {\boldsymbol{ w}}}\) that do not dominate any other result record \({\boldsymbol{ r}}_i \in S_{ {\boldsymbol{ w}}}\), and using set \(S_{NR}\) from the previous observation, \(\mathcal {R}_{ {\boldsymbol{ w}}}\) is the intersection \(\bigcap _{ {\boldsymbol{ r}}\in S_R, {\boldsymbol{ r}}^{\prime } \in S_{NR}}\lbrace {\boldsymbol{ v}}\in \Delta ^{d-1}| U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}^{\prime })\rbrace\).

As explained in the beginning of Section 6, a key difference between \(\mathsf {BORU}\) and \(\mathsf {FORU}\) is that the latter centers its execution around order-insensitive \(\text{top-}k\) results. Although order sensitivity is not required by Definition 2, if it is desired, \(\mathsf {FORU}\) can still be used.

Specifically, consider the definition of a result region \(\mathcal {R}(S)\) and Equation (1). The order-sensitive result region for a specific (ordered) \(\text{top-}k\) result is the intersection of \(\mathcal {R}(S)\) from Equation (1) with a half-space \(U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_{i}) \ge U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_{i+1})\) for each pair of consecutive records \({\boldsymbol{ r}}_{i}, {\boldsymbol{ r}}_{i+1}\) in the (ordered) \(\text{top-}k\) result. Effectively, (1) the half-spaces involved in Equation (1) ensure that the result records stay ahead of the non-result ones, and (2) the newly introduced half-spaces (henceforth referred to as order-preserving half-spaces) ensure that the record order within the \(\text{top-}k\) result stays the same. It follows that the order-sensitive result region for a \(\text{top-}k\) result is contained in its order-insensitive counterpart.

Now, recall that \(\mathsf {FORU}\) offers a tessellation of the preference domain around w into (order-insensitive) result regions \(\mathcal {R}(S)\), and for each of them it reports the respective \(\text{top-}k\) set S. Given such a region \(\mathcal {R}(S)\), we may partition it into order-sensitive regions by considering every possible ranking within S and intersecting \(\mathcal {R}(S)\) with the corresponding order-preserving half-spaces. If the intersection is empty, it means that the considered ordering is infeasible; otherwise, it is feasible and we know its (order-sensitive) result region. With \(\mathsf {FORU}\)’s stopping radius \(\rho\) at hand, we may simply report the (order-sensitive) result regions that are within distance \(\rho\) from w, together with their respective (ordered) \(\text{top-}k\) sets.

In this section, we perform an analytical comparison between \(\mathsf {BORU}\) and \(\mathsf {FORU}\) to get an indication of the relative performance we expect in practice.

The complexity of \(\mathsf {BORU}\) is dominated by the geometric computations spent on the candidate set M, and specifically in the computation of the preference regions C across the nodes of the implicit tree (elaborated in Section 5.3.1 and exemplified in Figure 5). The candidate set M is a subset of the k-skyband, whose expected cardinality is \(\frac{k\ln ^{d-1} |D|}{d!}\) [36]. Hence, \(|M|\) is in \(O(\frac{k\ln ^{d-1} |D|}{d!})\). Deriving the region (convex polytope) C in a node \(N_j\) of the implicit tree is equivalent to intersecting the preference regions that correspond to all the nodes along the path from \(N_j\) to the root of the implicit tree and therefore (1) the time spent in region computations is dominated by those for the leaves of the implicit tree and (2) the time spent for each of these regions is upper bounded by the time to intersect \(|M|\) half-spaces, that is, \(O(|M|^{\lfloor {d/2} \rfloor })\) time [11]. Now, the number of leaves in the implicit tree is in the order of possible order-sensitive \(\text{top-}k\) results within radius \(\rho\), which in turn is upper bounded by the permutations selecting k records from the m in the \(\mathsf {ORU}\) output, i.e., \(\frac{m!}{(m-k)!}\). Overall, the time complexity of \(\mathsf {BORU}\) is \(O(\frac{m!}{(m-k)!}|M|^{{\lfloor {d/2} \rfloor }})\). The latter complexity also applies to the space requirements of \(\mathsf {BORU}\), since the geometric representation of each node’s preference region takes \(O(|M|^{{\lfloor {d/2} \rfloor }})\) space [16]. Note that the derived time/space complexities are greater than (and thus subsume) the computation time/space required for the k upper hull layers, i.e., \(O(k|M|^{{\lfloor {d/2} \rfloor }})\) [21].

Returning to the \(\mathsf {FORU}\) algorithm detailed in Section 6.2, its complexity too is dominated by the computational geometric primitives it calls on, and in particular by the computation of the different result regions \(\mathcal {R}(S)\) it needs to consider. Due to the best-first fashion of the tessellation/expansion of \(\mathsf {FORU}\), the total number of records it encounters is in \(O(m)\). This implies that \(\mathsf {FORU}\) examines \(O(\binom{m}{k}) = O(\frac{m!}{k!(m-k)!})\) k-combinations of records. For the feasible among them (i.e., those where the result region \(\mathcal {R}(S) \ne \emptyset\)), it needs to compute the exact geometry of \(\mathcal {R}(S)\) as the intersection of \(O(m)\) half-spaces, requiring \(O(m^{{\lfloor {d/2} \rfloor }})\) time [11]. Therefore, \(\mathsf {FORU}\)’s total time complexity is \(O(\frac{m!}{k!(m-k)!} m^{{\lfloor {d/2} \rfloor }})\). The same complexity applies for its space requirements, since they are dominated by the size of its heap; at any point the heap contains a maximum of \(O(\frac{m!}{k!(m-k)!})\) result regions, whose geometric representation requires \(O(m^{{\lfloor {d/2} \rfloor }})\) space each [16].

In summary, the time/space complexity of \(\mathsf {BORU}\) is \(O(\frac{m!}{(m-k)!}|M|^{{\lfloor {d/2} \rfloor }})\) vis-à-vis \(O(\frac{m!}{k!(m-k)!} m^{{\lfloor {d/2} \rfloor }})\) for \(\mathsf {FORU}\). These complexities involve two factors that reflect the core differences between the two algorithms. First, the permutation (for \(\mathsf {BORU}\)) instead of the combination (for \(\mathsf {FORU}\)) is aligned with \(\mathsf {BORU}\)’s order sensitivity versus \(\mathsf {FORU}\)’s order insensitivity in the \(\text{top-}k\) sets they consider/report. Second, factor \(O(|M|)\) (for \(\mathsf {BORU}\)) versus \(O(m)\) (for \(\mathsf {FORU}\)) represents \(\mathsf {BORU}\)’s exploration of a larger part of the preference domain due to its starting by an overestimate of \(\rho\), as opposed to \(\mathsf {FORU}\)’s starting from w and its tight incremental expansion until the desired output size is reached.

At the core of \(\mathsf {AORU}\) lies the drill operation. A drill is a regular \(\text{top-}k\) query for a preference vector, called the drill location. The general idea is to perform repetitive drills around w, with drill locations gradually moving farther from it, until the union of their \(\text{top-}k\) results includes at least m distinct records. The drill locations are determined by parameter \(\delta\).

In particular, we utilize an (implicit) partitioning of the preference domain according to d-dimensional hyper-cubes. The side-length of the hyper-cubes is \(\frac{\delta }{d}\), so that the main diagonal of each hyper-cube has an \(L^1\)-norm length of \(\delta\). The partitioning is centered around w. Note that since the preference domain is a \((d-1)\)-simplex and the hyper-cubes are d-dimensional, we only consider hyper-cubes that overlap with the preference domain. In the 2-dimensional example of Figure 7, the relevant hyper-cubes (squares, in \(d=2\)) are drawn with a solid boundary. Their side-length is \(\frac{\delta }{d} = \frac{\delta }{2}\), except for the border ones that may be smaller (since the partitioning is centered around w).

\(\mathsf {AORU}\) performs the first drill at w itself. Subsequently, it considers the hyper-cubes one by one, in increasing minimum distance to w. For each considered hyper-cube in the aforementioned order, it performs a drill at the hyper-cube’s centroid. The records retrieved by the repetitive drills are appended to the \(\mathsf {AORU}\) output (avoiding duplicates) until its size hits m and the algorithm terminates. In the event that the last drill introduces to the output more new records than necessary, we only include the higher-ranking among them to bring the output size to exactly m.

An implementation remark is that the partitioning of the preference domain into hyper-cubes is implicit, meaning that not all hyper-cubes need be exhaustively produced. Instead, since the partitioning is centered around w, we first produce the hyper-cubes that have one corner at w and whose open representation intersects the preference domain (i.e., hyper-cubes \(\mathcal {Q}_1\) and \(\mathcal {Q}_2\) in Figure 7), continue with their own neighboring hyper-cubes (\(\mathcal {Q}_3\) and \(\mathcal {Q}_4\) in our example), and so on incrementally, as needed by the \(\mathsf {AORU}\) process until its output is complete. Note that the incremental generation of hyper-cubes around w hints directly to their order of consideration (i.e., drilling) by \(\mathsf {AORU}\).

Another technical detail is that for \(d\gt 2\) the centroids of the hyper-cubes lie higher than the preference domain, i.e., have coordinates where \(\sum _{i=1}^d w_i \gt 1\). To demonstrate, a simple example for \(d=3\) is that the centroid of the unit cube is vector \((0.5, 0.5, 0.5)\), which lies above the respective preference domain, i.e., above \(\Delta ^{2} = \lbrace {\boldsymbol{ v}}\in {\rm I\!R}_+^3 | \sum _{i=1}^3 w_i =1\rbrace\). That fact, however, does not necessitate any normalization to the centroids prior to the drills; i.e., it does not affect the design of \(\mathsf {AORU}\) (nor the validity of its accuracy analysis that follows in Section 7.2), because the magnitude of the centroids does not affect the retrieved \(\text{top-}k\) results, as we established in the beginning of Section 3 for the preference vectors in general.

A high-level remark about \(\mathsf {AORU}\) regards the twofold purpose it serves in our study. First, as originally (and primarily) motivated, it complements our algorithmic suite with a faster alternative to exact \(\mathsf {ORU}\) algorithms. Second, the minimalism of its design enriches the performance evaluation of the exact \(\mathsf {ORU}\) algorithms, giving a better sense of how efficient they are compared to a very intuitive (yet inexact) \(\mathsf {ORU}\) processing approach. In particular, \(\mathsf {AORU}\) relies on repetitive \(\text{top-}k\) retrievals (drills) using the established BBR algorithm, which has trivial computational requirements [79]. Regarding the number of drills, if we assume that each drill is equally likely to contribute a new record to the output, \(\mathsf {AORU}\) requires \(O(m)\) calls to BBR.

In relation to the exact \(\mathsf {ORU}\) algorithms, aside from performance, we must mention that \(\mathsf {AORU}\) too executes an expansion around w, but instead of considering every possible preference vector along the way, it only accounts for some of them (namely, the drill locations). This implies that the extent of \(\mathsf {AORU}\)’s expansion is naturally greater. Specifically, its stopping radius corresponds to the distance between w and the farthest drill location, and that distance can be no smaller than the stopping radius of the exact algorithms. A corollary of this fact is that, up to the radius of the exact \(\mathsf {ORU}\) solution (i.e., up to radius \(\rho\) in Definition 2), all the drills, by definition, introduce actual \(\mathsf {ORU}\) results in \(\mathsf {AORU}\)’s output (but this may not hold for farther drill locations).

In this section, we analyze \(\mathsf {AORU}\)’s accuracy. To facilitate presentation, note that just like the magnitude of a preference vector, the scaling of dataset D by a constant factor does not affect the ordering of D by utility. In particular, letting \(\alpha\) be the maximum value across all attributes of any record in D, the \(\mathsf {ORU}\) result on D (for a given \({\boldsymbol{ w}}\) and m) is identical to the \(\mathsf {ORU}\) result if we scale (i.e., multiply) every record of D by constant factor \(\frac{1}{\alpha }\). Without loss of generality, we assume that \(\alpha = 1\).

Let \(\rho\) be the stopping radius of the exact \(\mathsf {ORU}\) operator and \({\boldsymbol{ v}}\) be the preference vector of a user where \(| {\boldsymbol{ v}}- {\boldsymbol{ w}}| \lt \rho\). By Definition 2, \(\mathsf {ORU}\)’s output includes the user’s \(\text{top-}k\) records in D, i.e., those with the maximum utility. We will center our analysis on the utility loss incurred in the \(\text{top-}k\) result of \({\boldsymbol{ v}}\) when instead of the exact \(\mathsf {ORU}\) output, the user is presented with \(\mathsf {AORU}\)’s output.

Since (as elaborated at the end of Section 7.1) \(\mathsf {AORU}\)’s search extends at least up to radius \(\rho\) from \({\boldsymbol{ w}}\), it considers (i.e., drills) the hyper-cube \(\mathcal {Q}\) that includes vector \({\boldsymbol{ v}}\). Let c be the centroid of \(\mathcal {Q}\) and sets \(S_{ {\boldsymbol{ c}}}\) and \(S_{ {\boldsymbol{ v}}}\) be the \(\text{top-}k\) results for c and v, respectively (where possible ties in the top-k-th position are resolved arbitrarily). If a record \({\boldsymbol{ r}}^{out}\) belongs to \(S_{ {\boldsymbol{ v}}}\) but not in the \(\mathsf {AORU}\) output, this means that \({\boldsymbol{ r}}^{out} \notin S_{ {\boldsymbol{ c}}}\) and, consequently, that there must be a record \({\boldsymbol{ r}}^{in} \in S_{ {\boldsymbol{ c}}}\) that does not belong to \(S_{ {\boldsymbol{ v}}}\). The following theorem bounds (by \(\delta\)) the utility loss experienced by the user (i.e., preference vector \({\boldsymbol{ v}}\)) if \({\boldsymbol{ r}}^{in}\) takes the place of \({\boldsymbol{ r}}^{out}\) in the \(\text{top-}k\) result.

The \(\mathsf {ORU}\) operator (Definition 2) is centered on the relevant vectors’ entire \(\text{top-}k\) sets as opposed to individual records therein. Hence, a meaningful use of Theorem 2 is to bound the total (i.e., summed) utility loss experienced by v across its entire \(\text{top-}k\) set. Corollary 1 bounds that loss.

We note that several common definitions of utility for ranked lists (e.g., discounted cumulative gain and its variants [47, 86]) associate a decreasing, non-negative weight \(\lambda _i\) per rank position and report the weighted sum of the records’ utilities along the list as its overall utility. That is, if the (ordered) \(\text{top-}k\) result for vector v is \(\lbrace {\boldsymbol{ r}}_1, {\boldsymbol{ r}}_2, \ldots , {\boldsymbol{ r}}_k \rbrace\), the result’s overall utility is defined as \(\sum _{i=1}^k \lambda _i \cdot U_{ {\boldsymbol{ v}}}( {\boldsymbol{ r}}_i)\). For any aggregate measure of that type, following the same lines as Corollary 1, Theorem 2 guarantees that \(\mathsf {AORU}\) incurs an overall utility loss for v that cannot exceed \(\sum _{i=1}^k \lambda _i \delta\).

First, we perform a case study to visualize how the results of \(\mathsf {ORD}\) and \(\mathsf {ORU}\) differ from (1) an OSS skyline and (2) a top-m query for the same vector w. Focused on the results of our operators (rather than processing efficiency), note that \(\mathsf {ORD}\) and \(\mathsf {ORU}\) in this case study refer to the problem outputs in Definitions 1 and 2, respectively, and are orthogonal to the choice of processing algorithm (e.g., \(\mathsf {BORU}\) or \(\mathsf {FORU}\)). From the OSS skyline family, we use as representative the most cited full-dimensionality formulation [57], which reports the m skyline members that dominate the most non-skyline records. We use the NBA 2018–2019 season statistics for the total of its 708 players on Assists and Rebounds (in Figure 8(a)) and Points and Rebounds (in Figure 8(b)), normalized in the \([0,1]\) range. We set \(k=2\) and \(m=6\) and use \((0.49,0.51)\) and \((0.43, 0.57)\) as the seed w, respectively. The results of the methods are illustrated as differently oriented/colored triangles.

A first observation is that our operators report distinct results from past approaches (and from each other). For example, in Figure 8(a) only \(\mathsf {ORU}\) reports Trae Young (rising stars challenge player), while half of its output records are not in the top-m result, and one-third of them are not in the OSS skyline. The comparison of our operators with top-m reveals an even more interesting fact. In Figure 8(a), top-m misses Andre Drummond (the rebound leader in the 2018–2019 season), whom it would report if we only slightly revised w from \((0.49,0.51)\) to \((0.48,0.52)\). Similarly, in Figure 8(b), top-m misses James Harden (the season’s scoring leader), whom it would report if we revised w from \((0.43, 0.57)\) to \((0.44, 0.56)\). The inclusion of these players in the result of both of our operators (in the respective Figure 8(a) and 8(b) settings) confirms that they successfully employ some “width” in their search by reporting records that are particularly strong for alternative, very similar preferences to the seed w. Investigations in our full-scale experimental settings, using the Jaccard coefficient as the similarity measure, demonstrate that (1) the OSS skyline is more dissimilar to our operators than top-m, as it is not guided by w, and (2) top-m becomes increasingly more dissimilar to \(\mathsf {ORD}\)/\(\mathsf {ORU}\) as m grows. We omit the charts for brevity, but as an indication, for IND data and the default parameters in Table 2, the Jaccard similarity of the OSS skyline to \(\mathsf {ORD}\) is 0.25, and to \(\mathsf {ORU}\) it is 0.24. In the same setting, the Jaccard similarity of top-m to \(\mathsf {ORD}\) is 0.44, and to \(\mathsf {ORU}\) it is 0.32.

Next, we use real TripAdvisor data and reviews (TA) [6]. TA includes ratings for 1,850 hotels on \(d = 7\) aspects (value, room, location, etc.), forming dataset D. It also includes actual user reviews for these hotels, each comprising a comment and an overall score. The reviews offer a practical example of how preference vectors could be extracted for real users. Specifically, we employ [84], an established preference mining method that estimates a user’s weight vector based on his or her reviews. Applying [84], we get vectors for 137,563 users. The dataset and vectors from TA are used in the next experiment on fixed-region methods [24, 64], but at the same time they offer an end-to-end application example for our techniques.

The fixed-region methods require a preference polytope R to be specified as part of their input. We demonstrate that it is not feasible to estimate the size of R required to produce, even approximately, m records. Worse yet, even the same polytope R, when positioned at different parts of the preference domain, can produce vastly different output sizes. Specifically, we use \(k=5\) and vary m from 10 to 20 (since the dataset in TA is highly correlated and its 5-skyband includes only 61 hotels). We first execute \(\mathsf {ORU}\) for 50 randomly selected TA users (preference vectors) and record the average stopping radius, denoted as \(\rho ^*\). We then produce a hyper-cube R with volume equal to the hyper-sphere with radius \(\rho ^*\). Next, we count the number of distinct records (TA hotels) output by the fixed-region \(\text{top-}k\) operator in [64] when R is positioned around each of the 50 user vectors (we use [64] since [24] works only for \(k=1\)). Figure 9(a) presents in box-plot the observed variation in output size. To confirm, in Figure 9(b), we repeat this process for our full-scale setting, using random preference vectors, IND data, the default parameters, and standard range for m from Table 2. The box-plots indicate that even with knowledge of \(\rho ^*\), fixed-region techniques can produce dramatically different output sizes, which may hugely under- or over-shoot the target m. In contrast, our operators relieve the user/application from the need to meddle with the preference domain’s complex dynamics and abide strictly by the requested output size m. Note that the counterpart of this experiment, using \(\mathsf {ORD}\) to compute \(\rho ^*\) and producing the fixed-region R-skyband, demonstrates an even greater variability than Figure 9 for both TA and IND (charts omitted for brevity). For example, for target \(m=50\) in IND data, the fixed-region R-skyband outputs from 12 all the way to 269 records.

In this section, we evaluate the performance of our \(\mathsf {ORD}\) algorithm. For comparison, we adapted a fixed-region R-skyband technique (\(\mathsf {RSB}\)), as sketched in Section 2. In the adaptation, the initial hyper-cube R is sized so that its volume ratio to the preference domain equals the ratio of m to the expected k-skyband cardinality (i.e., \(\frac{k\ln ^{d-1} |D|}{d!}\), according to [36]). Based on the size of the R-skyband computed for the initial R, its volume is re-estimated proportionally to the desired m. The trials (R-skyband computation and R re-estimation) are repeated until the output is within 10% from m. For the implementation of \(\mathsf {RSB}\), we use the index-based R-skyband module of [64], as it is considerably faster than the no-index approach of [24]. We also include baseline ORD-BSL that computes the entire k-skyband according to Section 4.1, without the enhancements in Section 4.2.

In Figure 10, we present (in logarithmic scale) the running time for all competitors versus each problem parameter, using IND data. \(\mathsf {ORD}\) is 2 to 4 orders of magnitude faster than \(\mathsf {RSB}\), indicating the impracticality of the fixed-region approach to mimic \(\mathsf {ORD}\)’s output, despite the ample slack given. The main reason is the numerous trials required for \(\mathsf {RSB}\) to “converge” to an acceptable deviation from m. The runner-up is ORD-BSL, which is 1 to 2 orders of magnitude slower than the fully enhanced \(\mathsf {ORD}\). Moreover, the results also demonstrate \(\mathsf {ORD}\)’s ability to scale. Its running time grows almost linearly to k and m, and sub-linearly to \(|D|\). It increases more sharply with d, due to the growing complexity of its geometric building blocks. Still, even for \(d=7\) \(\mathsf {ORD}\) takes only 3.2 seconds.

Having established the superiority of our \(\mathsf {ORD}\) algorithm, in Figure 11(a) we plot its running time for different synthetic distributions, varying m. Processing in ANTI is the slowest. The reason is that dominance (and, by deduction, \(\rho\)-dominance too) is less common among ANTI records, and thus many candidates need to be considered before \(\mathsf {ORD}\) can terminate. COR exhibits the inverse effect.

Again varying m, in Figure 11(b), we use real data. NBA is smaller than HOTEL, but it has larger dimensionality, which explains why \(\mathsf {ORD}\) is slower on NBA in most cases. At the same time, NBA is significantly more correlated than HOTEL and HOUSE, which is why the running time increases less sharply with m compared to the other two datasets.

Turning to the \(\mathsf {ORU}\) problem (Definition 2), we first consider exact processing. Our suite includes two exact algorithms, i.e., \(\mathsf {BORU}\) (in Section 5) and \(\mathsf {FORU}\) (in Section 6). For comparison, we use an adaptation of a fixed-region \(\text{top-}k\) algorithm from [64], termed \(\mathsf {JAA}\), to simulate our problem’s output. Specifically, we employ a similar R estimation and trial approach as for \(\mathsf {RSB}\), allowing a deviation of up to 10% from the desired output size m. We also include \(\mathsf {ORU}\)-\(\mathsf {BSL}\), a baseline that utilizes the initial overestimate \(\bar{\rho }\) in Section 5.3, computes upper hull layers on the entire candidate set M, partitions the \(L_1\) top-regions by Theorem 1, and reports the m-sized union of the \(\text{top-}k\) records for the closest produced regions to w.

In Figure 12, we plot (in logarithmic scale) the running time of all competitors versus each problem parameter, using IND data. \(\mathsf {ORU}\)-\(\mathsf {BSL}\), although identical to \(\mathsf {BORU}\) for \(k=1\), generally performs very poorly, failing to terminate within reasonable time in most large settings. This demonstrates the vital role of the progressive tightening of the radius overestimate and the respective elimination of candidate records, presented in Section 5.3.1. Indeed, \(\mathsf {BORU}\) is 2 to 4 orders of magnitude faster than \(\mathsf {ORU}\)-\(\mathsf {BSL}\). It is also 12 to 134 times faster than \(\mathsf {JAA}\), confirming the general unsuitability of fixed-region approaches to our problems.

Having demonstrated the deficiencies of alternatives compared to \(\mathsf {BORU}\), we may now focus on the further gains achieved by \(\mathsf {FORU}\). Recall that \(\mathsf {FORU}\) is one of the two (and by far the most important) technical extensions in this journal article. In Figure 12(a) (varying k) \(\mathsf {FORU}\) is 6 to 56 times faster than \(\mathsf {BORU}\), with the margin generally growing with k. The main reason behind this drastic improvement lies in the fundamentals of \(\mathsf {FORU}\) expansion. Recall that \(\mathsf {BORU}\) starts with an overestimate of \(\rho\), which is conservatively set so that \(\mathsf {ORU}\)’s output for \(k=1\) includes m records. Since this radius determines the search in deeper upper hull layers, being conservative, it leads to the consideration of a larger part of the upper hulls than necessary. As k grows, more upper hull layers become relevant to the problem, which means that the original overestimate (for \(k=1\)) is even further off from the actual stopping radius; i.e., effectively the overestimate becomes looser. On the other hand, \(\mathsf {FORU}\) does not suffer from this issue, because it starts from w and tessellates the preference domain around it in an incremental, only-if-needed fashion.

In Figure 12(b) (varying m) \(\mathsf {FORU}\) is 1 order of magnitude faster than \(\mathsf {BORU}\), and the margin grows with m. The major factor at play for this behavior is again the overestimated starting radius in \(\mathsf {BORU}\). As m grows, the overestimate necessarily becomes larger, since more top-1-st records (m, to be exact) need be confirmed by its \(\mathsf {IRD}\) module (in Section 5.3.2). This means that it also becomes looser, leading to more unnecessary computations in the deeper upper hull layers. \(\mathsf {FORU}\), on the other hand, relies on no overestimate and avoids that problem.

In Figure 12(c) (varying \(|D|\)) \(\mathsf {FORU}\) is 3 to 89 times faster than \(\mathsf {BORU}\). Moreover, just as with k and m, \(\mathsf {FORU}\)’s gains over \(\mathsf {BORU}\) grow with \(|D|\) too, demonstrating that \(\mathsf {FORU}\) offers a general improved scalability to larger problem instances. The reason the gap gets wider with \(|D|\) lies in the way these algorithms fetch candidate records. \(\mathsf {BORU}\) fetches candidates using (partial) upper hull computations, whereas \(\mathsf {FORU}\) uses plain \(\text{top-}k\) search. Given that the former operation is significantly more complex, the advantage offered to \(\mathsf {FORU}\) is amplified as the total number of records increases. Comparison with \(\mathsf {BORU}\) aside, one of the boldest findings here is that \(\mathsf {FORU}\) delivers sub-second running time even for the largest dataset of 25.6M records!

In Figure 12(d) (varying d) \(\mathsf {FORU}\) is faster than \(\mathsf {BORU}\) in all dimensionalities, being 3 orders of magnitude faster for \(d=2\), with the gap progressively narrowing for higher d. The reason for this trend lies in the computational geometric primitives the two algorithms rely on, whose cost increases exponentially with d [16]. As d grows, these primitives consume a larger fraction of the total running time, until they completely dominate it, rendering largely inconsequential the individual optimizations and algorithmic distinctions of the two algorithms. The nullification of algorithmic ingenuity for large d befits the curse of dimensionality in multi-objective querying and its loss of meaning, discussed at the end of Section 3.

The next set of experiments (Figure 13) serves two purposes. The first is to conclude the comparison of the exact approaches by considering other datasets. Specifically, in Figures 13(a) through 13(e), we omit the problematic \(\mathsf {JAA}\) and \(\mathsf {ORU}\)-\(\mathsf {BSL}\) and focus on \(\mathsf {FORU}\) and \(\mathsf {BORU}\) (let us ignore the line of \(\mathsf {AORU}\) for now). We present their running time on the remaining synthetic datasets (i.e., COR and ANTI) and on the real datasets, as we vary m according to Table 2. An exception to this setting is NBA; its strong correlation leads to only 31 distinct \(\text{top-}k\) records (for the default \(k=5\)) throughout the entire preference domain, which are insufficient for the larger m values. To make the experiment meaningful, for NBA in particular, we use \(k=15\) and a different range of m values (i.e., 30, 35, 40, 45, 50). The big picture from the five datasets is similar to Figure 12(b) for IND data, with \(\mathsf {FORU}\) being several times (and up to 3 orders of magnitude) faster than \(\mathsf {BORU}\). The only exception is the \(m=10\) setting for ANTI data. That is due to \(m =10\) being very small (just twice of k) and the fact that in anti-correlated data just a minuscule expansion suffices to output that many records. In turn, this implies that one-off initialization costs dominate the total running time, suppressing the effects of \(\mathsf {FORU}\)’s more efficient expansion process.

The second purpose of Figure 13 is to shed light on approximate processing. In particular, we included \(\mathsf {AORU}\) in the first five charts (running time) and present its respective accuracy in a combined diagram (Figure 13(f)). To ensure homogeneity across the datasets, we set the \(\delta\) parameter of \(\mathsf {AORU}\) such that the side-length of the hyper-cubes is consistent (i.e., 0.035 in our experiments). That is, for a d-dimensional dataset, \(\delta\) is set to \(0.035\cdot d\) (recall that the synthetic datasets have the default \(d=4\), while the real ones have their own native dimensionality, as described in the beginning of the experimental section). To measure \(\mathsf {AORU}\)’s accuracy, we present its F1 score (with respect to the exact result) in the combined Figure 13(f) that covers all datasets and settings tested in this experiment. The F1 score is the harmonic mean of precision and recall (i.e., \(2\frac{precision \cdot recall}{precision + recall}\)), which balances them effectively [82].

\(\mathsf {AORU}\) is faster than the exact methods in all settings and offers a decent approximation accuracy. However, we observe a deviation from this norm for ANTI, where the running time is particularly short but the accuracy suffers. The reason for this behavior lies in the common hyper-cube side-length we employed across datasets, which is too large for ANTI but enables interesting insights. In particular, in anti-correlated data, there is a high density of different \(\text{top-}k\) results around w. Using the same hyper-cube side-length as COR, for instance, leads to earlier termination, because the drills fetch vastly different \(\text{top-}k\) results, and thus fewer drills are required to fill \(\mathsf {AORU}\)’s output with distinct records. On the other hand, the drill locations make large leaps in the preference domain, failing to account for many alternative \(\text{top-}k\) results in between, which has a negative effect on accuracy. Another observation is that the accuracy drops with m. A larger m means a wider expansion, i.e., that \(\mathsf {AORU}\) needs to drill farther from w. The farther we drill from w, the more likely it is to introduce non-result records to \(\mathsf {AORU}\)’s output.

In Figure 14, we investigate the effect of the hyper-cube side-length (effectively, of parameter \(\delta\)) on the running time and accuracy of \(\mathsf {AORU}\). We use the default IND dataset and consider side-lengths 0.01, 0.02, 0.03, and 0.04, as indicated by the suffix of the \(\mathsf {AORU}\) labels. In Figure 14(a), we also include the running times of \(\mathsf {BORU}\) and \(\mathsf {FORU}\) for reference. As expected, the side-length directly determines the performance of \(\mathsf {AORU}\), because it controls the density (and thus the number) of drills performed. On the other hand, it also determines accuracy; the interesting finding here is that even for the larger side-lengths tested, the accuracy remains reasonable, which suggests that the running time drops much more aggressively than the accuracy.