## 4 Experimental Evaluation
### Join Query Performance and Storage Amount  
In our experiments, we compared our method with other popular existing methods. Our objective is to show that we can achieve excellent join query performance with a small amount of storage. In the experiments, our hybrid approach, referred to as HYBRID, was compared with QUAD [9], DARQ [14], and MIDAS-RDF (referred to as MIDAS) [18]. Figure 8: Two input sets of the nested-loop join algorithm. QUAD, DARQ, and MIDAS are existing local, live exploration, and index approaches, respectively. This evaluation is conducted on a server system with a 3.6-GHz Intel i7 CPU and 32 GB memory. All programs were written in the Java language on a server running Windows 10. In this work, we performed two sets of experiments; first, we measured the join query performance, and then we measured the storage amount. We used the Lehigh University Benchmark (LUBM) dataset [31] to obtain realistic results of the performance of large-scale RDF storage structures. The LUBM dataset is the most widely used benchmark dataset in the Semantic Web community. It contains 230,061 triples, 38,334 subjects, 17 predicates, and 29,635 objects. The size of the entire dataset is 36.7 MB. The original version of the LUBM queries has not been updated since 2009. With an increase in the query complexity, the older version cannot support all join query types. To address this issue, we modify the original version to support all join query types. We first design 15 queries from Q1 to Q15. Then, we enumerate all join query types and compare their join query performance. The SPARQL join query types are star (Q1, Q3, Q4, Q5, Q11, Q13), chain (Q6, Q10), directed cycle (Q2), cycle (Q15), tree (Q9), and complex (Q7, Q8, Q12) types. Figure 9 presents the three-dimensional points in the 3D R*-tree with respect to the LUBM dataset. All data are visualized through the interface provided by ELKI (Environment for Developing KDD-Applications Supported by Index Structures). This figure shows that all points are converted into the three-dimensional space. In the LOD cloud, the volume of data is very large, with some application areas containing a billion or more points. Such datasets usually far surpass the capabilities of a standard computing system, particularly if real-time interactions are desired. This may result in significant demand for memory and disk storage. Thus, the factors primarily influencing the system performance are the storage space and query performance. To evaluate the join query performance of the HYBRID method, we performed 15 SPARQL join queries that expressed the above join query types. Figure 10 illustrates the join query performance for various join query types. Experimental results show that the query performance of HYBRID is always superior to that of other methods. DARQ and MIDAS perform considerably worse than QUAD and HYBRID because these methods produce considerably larger intermediate results and require more time for I/O and CPU. QUAD shows slightly worse performance than HYBRID because QUAD needs to load six B+-trees to support all access query patterns and decompress intermediate results in a short time. However, HYBRID considers only a 3D R*-tree with \(k\)-d trees without storing all possible access combinations. It further reduces the intermediate results using the two-step query processing techniques described in Section 3.4. Figure 9: Three-dimensional points in the 3D R*-tree. The second set of experiments involves the storage amount. In Figure 11, the storage amounts are presented for HYBRID, QUAD, DARQ, and MIDAS. In terms of storage space, HYBRID is superior to QUAD for all datasets because QUAD maintains six indexes for all access patterns on the RDF triples with the context. DARQ requires the least amount of storage space because DARQ queries are processed at the SPARQL endpoints provided by remote publishers; hence, in DARQ, data need not be stored in a local server. HYBRID and MIDAS require almost the same amount of storage space.