Efficient skyline query processing with user-specified conditional preference

Construction of classification tree

A classification tree segmentally categorizes tuples based on a sequence of conditional checks. Its fundamental concept involves partitioning the feature space into rectangles. Starting from the root node, the tuple is progressively separated through a series of decisions. When the division is complete, and the tuple cannot be further separated, it forms a leaf node.

Constructed upon this foundation, the classification tree exhibits the following characteristics:

Root node. The root node, $District$ is chosen due to its independence from other attributes, eliminating the need for a CPT.

Internal nodes. Each root node value corresponds to an internal node within a sub-tree, representing geographical branches. For attributes like $Price$ and $Score$, internal nodes are determined by the associated CPT, reflecting user preferences under specific conditions.

Leaf nodes. Leaf nodes represent unique attribute configurations and detailed conditional preferences, indicating that a candidate tuple aligns with the user’s overall preference profile.

Edges. Each directed edge from node $v_i$ to node $v_j$, denoted as $(v_i,v_j)\in E_1$, shows that the optimal choice for the next related attribute can be inferred from the CPT of node $v_j$. These edges demonstrate the dependency structure in the decision process.

Upon constructing the classification tree, each $t_i\in S$ is traversed starting from the root node. At each internal node, the attribute value of tuple is checked against the current criterion. If it meets the condition, traversal continues along the relevant branch; otherwise, “No Valid Category” is returned (represented by F in classification tree), or the process halts. Tuples that satisfy the user’s conditional preferences are classified ( $o_i\in \mathrm{\Omega }$), while those failing the CP-Nets criteria are pruned. This completes the first pruning step, resulting in $CS=\mathrm{\Omega }$.

Using the CP-Net from Example 1 and tuples from Table 1, a classification tree is constructed as shown in Fig. 2. Tuples $h_1,h_{10}$ are excluded due to their $Price$ being outside $[80,140]$, and $h_1,h_6,h_8$ are excluded because their Scores are not greater than 5. Consequently, each tuple is assigned to a classification that meets the user’s conditional preferences.

According to the user’s CP-Nets, the most important attribute is the administrative $District$, followed by $Price$, with $Score$ being the least important. The classification tree removes tuples not meeting the preferences (e.g., $t_5$ does not satisfy: if $100<Price<120$, then $Score$ $7$). Tuples satisfying the preferences are assigned to specific categories ( $o_i$), such as $t_2\in o_4$ and $t_3,t_4\in o_2$. At this stage, non-compliant tuples are removed from the candidate set, resulting in $CS=\mathrm{\Omega }=\{o_1,o_2,o_3,o_4,o_5,o_6\}$.

Construction of induced graph

Presently, through the utilization of a classification tree, we can filtered out tuple that does not align with user’s preferences. Our current focus lies in sorting $o_i$ based on varying degrees of user’s satisfaction. As a precursor to pruning, we introduce the concept of the induced graph, outlining the procedure for attaining optimal results using CP-Nets.

Based on the current preference-induced graph, we can rank the importance of different outcome $o_i\in \mathrm{\Omega }$ to the user. Selecting the outcomes with the highest preference level $o_i$ as CS allows us to perform the second step of pruning.

According to Definition 7, the induced graph is constructed as shown in Fig. 3. The node set, $\mathrm{\Omega }$, represents different outcomes. The edge set, derived from the user’s CP-Nets, includes directed edges $<o_i,o_j>$, where $o_i$ and $o_j$ differ by only one attribute value. For example, the preference for $o_1(Dis{t}_D\wedge P<100\wedge S>5)$ over $o_2(Dis{t}_C\wedge P<100\wedge S>5)$, indicated by the directed arc, reflects the semantics of $CPT(District)$.

Conducting a topological sort on the preference-induced graph can determine the ranking of outcomes: $o_1\succ o_2\succ o_3\succ o_4\succ o_5\succ o_6$. The top-level node ( $o_6$) represents the least preferred outcome, while the bottom-level node ( $o_1$) represents the most preferred. We select $o_1$, the outcome to the user is most interested in, as CS. If $o_1$ is empty, the next best $o_2$ is chosen as CS. After determining CS, we proceed with the CP-Skyline calculation.

Pruning

Pruning based on the above two process are presented in Algorithm 1 (Line 4). The algorithm aims to systematically derive a candidate set (CS) from an acyclic CP-Net G and a set of tuples T. The algorithm commences by initializing CS and Hierarchies as empty sets. Subsequently, it constructs a classification tree C and a preference induced graph I based on the characteristics of the given acyclic CP-Net (line 5). The algorithm then enters a loop that iterates through tuples $t_i$ in T which align with the branches of the classification tree C. For each tuples, it categorizes it into various outcomes ( $o_i\in \mathrm{\Omega }$). The identified $\mathrm{\Omega }$ that satisfy the user’s conditional preferences are subsequently appended to the Hierarchies set. The hierarchies are further sorted in accordance with a topological reverse sort of the preference-induced graph I, since the CP-Nets discussed are acyclic, their derivation is also acyclic and therefore does not form a cycle (Boutilier et al., 2004). For example, Fig. 3 shows the acyclic derived graph obtained through acyclic CP-Nets. The topological sorting result is $o_1\succ o_2\succ o_3\succ o_4\succ o_5\succ o_6$, that is, the hierarchies set $H=h_1\succ h_2\succ h_3\succ h_4\succ h_5\succ h_6$. According to definition 3, it can be seen that no cycle will be formed. Following this, the algorithm iterates through each hierarchy in $Hierarchies$, commencing with the highest hierarchy. If the hierarchy is not empty, it sets CS to the current hierarchy. Otherwise, it advances to the next hierarchy. Finally, the algorithm concludes by returning the resulting candidate set CS. This descriptive passage outlines the steps and purpose of the “Pruning” algorithm, emphasizing its operations on acyclic CP-Nets and tuples to generate a refined candidate set in accordance with user’s preferences.

Size of the skyline result sets

The size of the skyline result set is an important metric for evaluating query performance and efficiency according to the standards of scientific articles (Benouaret et al., 2021). A smaller result set typically indicates faster query processing, as the system returns only the most significant or valuable data set. By controlling the size of the skyline result set, it is possible to ensure that the returned result contains the most valuable and crucial information, thereby assisting users in making more accurate decisions and enhancing their overall experience.

We first compare the result sets of several skylines on the real data set QWS (Fig. 4A) and NBA players (Fig. 4B). Because the data set of the real data set is too small, we then analyze the changes of several skyline result sets with the size of the data set by synthesizing the data set (Fig. 5). Finally, we analyze the changes of various skyline result sets with the query attribute by changing the query attribute 2–5 on QWS (Fig. 6A) and NBA players (Fig. 6B).

Figures 4A and 4B depicts the size of result set of different skyline query methods on QWS and NBA players, where in this case, the entire dataset was used as the candidate set. Traditional skyline queries were conducted with a full-dimensional query space, while personalized skyline queries were performed with a five-dimensional query space. Obviously, the result set obtained by the traditional skyline query method has the largest number, while the remaining personalized skyline algorithms have a smaller result set, because they exclude tuples that does not meet the personalized requirements. In Fig. 5, synthetic data was utilized to examine the effect of candidate set size on the result set size with a five-dimensional query space. In all cases, the size of the result set will increase with the size of the dataset. However, the size of the result set of the personalized skyline algorithm is smaller than the size of the result set obtained by the traditional skyline query. As illustrated in the figure, CP-Skyline eliminates a lot of tuples that does not meet the user’s personalized requirements compared to the traditional skyline query method. Compared with the user-defined skyline, it also has a certain improvement. As the the decisive skyline defined as the subset of skylines that balances all attributes, its size is smaller than CP-Skyline.

Next, in Figs. 6A and 6B, the impact of the query dimension (attribute) involved in the skyline on the size of the result set is examined on QWS and NBA players. In all cases, the size of the result set will increase as the number of tuple’s attribute increases. Since the skyline is a set that is not dominated by other tuples on at least one dimension, increasing the query dimension means that tuples are more difficult to be dominated. Due to the large volume and dense distribution of data in the NBAPlayers dataset, there may be multiple instances of the same data in low-dimensional cases. The user-defined skyline exclude those instances that can dominate these tuples, resulting in a higher number of undominated tuples, Which leads to the the situation in Fig. 6B. Overall, the personalized skyline query method has a smaller change in the result set. For users, when the query dimension increases, The user’s effort to locate the most preferred tuple intensifies, and the more specific the user’s preference is, the more tuples that do not meet user preferences are removed through pruning, the less likely these tuples are to become part of the Skyline. As a result, the smaller the result set is, and the less effort is spent.

Skyline query response time

In the skyline query algorithm, because it deals with large-scale multi-dimensional data sets, the running time of the algorithm will affect the efficiency of the entire query process. In practical applications, the shorter running time means that the algorithm can respond to user’s query requests faster and improve the responsiveness and user’s experience of the system. And the size of the data set may continue to expand, so the running time of the algorithm will be affected by the size of the data. If the running time of the algorithm increases significantly with the increase of data size, the scalability of the algorithm may be affected and cannot meet the needs of future data size growth.

We first compare the query time of several skylines on the real data set QWS (Fig. 7A) and NBA players (Fig. 7B). Then we analyze the changes of query time of several with the size of the data set by synthesizing the data set (Fig. 8). We analyze the changes of query time of various skyline with the query dimension by changing the query dimension 2–5 on QWS (Fig. 9A) and NBA players (Fig. 9B). Finally, we analyze the changes of several query times with the size of the data set in the synthetic anti-correlation data set(Fig. 10).

Figures 7A and 7B shows the query time of different types of skyline query methods on QWS and NBA players, where in this case, the entire dataset was used as the candidate set. Traditional skyline queries were conducted with a full-dimensional query space, while personalized skyline queries were performed with a five-dimensional query space. The traditional skyline query does not have any preference, so it needs to calculate the dominance relationship of all candidate data, so the query time is much higher than other personalized skyline query methods. Next, since the query time of subspace skyline and decisive skyline is much longer than that of CP-Skyline and user-defined skyline, Fig. 8A only describes the query time of decisive skyline and subspace skyline with a five-dimensional query space, Fig. 8B describes the query time of CP-Skyline and user-defined skyline with a five-dimensional query space. As the dataset size changes, we can see that CP-Skyline not only provides more detailed user’s preferences than user-defined, but also spends less query time.

Figures 9A and 9B show the impact of the query dimension (attribute) involved in the skyline on the query time is examined on QWS and NBA players; it can be found that the query time of subspace skyline and decisive skyline is not much different, and the query time increases with the increase of query dimension. The reason is that the decisive skyline is based on the result of subspace skyline. The user-defined skyline and CP-Skyline reduce the query time with the increase of the dimension. The reason for this result is that with the increase of the query dimension, when the candidate data is pruned by the user’s preference, a large number of data that the user is not interested in can be eliminated, which reduces the time spent on the skyline calculation, because the time complexity of the pruning is $O(n\cdot log(n))$, and the time complexity of the skyline query is $O(n^2)$. Due to the small amount of QWS data, less time is spent in the Skyline calculation phase at low latitudes, while the user-defined skyline and CP-Skyline require additional processing of user preferences, and then pruning candidate data based on user preferences. Pruning takes a relatively large proportion of time, and as the dimension increases, the time spent on pruning gradually decreases, and the advantages of using preference for pruning are gradually revealed, so there is a situation as shown in Fig. 9A.

Finally, we compared the query times of several methods as the data increases on the synthetic anti-correlated data set. Given that the query times for subspace skyline and decisive skyline are significantly longer than those for CP-Skyline and user-defined skyline, Fig. 10A focuses on the query times for decisive skyline and subspace skyline within a 5-dimensional query space. Meanwhile, Fig. 10B illustrates the query times for CP-Skyline and user-defined skyline within the same 5-dimensional query space. Similarly, because the advantage of CP-Skyline lies in its excellent pruning ability, CP-Skyline still has better performance. In general, a substantial portion of the computational time in skyline queries is allocated to the dominance calculation of the skyline. In personalized skyline queries, after incorporating user preferences, we can prune many tuples that do not meet the user’s preferences. These tuples do not need to be included in the dominance calculation, thus reducing query time. Therefore, the more specific the user’s preferences, the more tuples are removed during the pruning stage, and the less time it takes to obtain the Skyline set.

Precision

Precision refers to the proportion of the number of documents that match the required information to the total number of documents retrieved. The precision indicates how many data points in the skyline results returned by the query do meet the query conditions. High precision means that the returned results are credible and can be safely used for decision-making and analysis. High precision query results help users make more accurate decisions. Especially in important decision-making scenarios, achieving high precision in personalized skyline queries is critical for providing tailored and effective solutions. Since the all personalized skyline meets the user’s preferences, the precision of the personalized skyline is 1. The difference is that CP-Skyline provides more refined preferences.