### Table 3: Iterator functions Notice that in this implementation, the system buffer is shared between at most two operators. Next functions never run concurrently; when join is executed at one operator, only hashing is performed at the upper one. Thus, given a memory buffer of M pages, the operator which is currently performing a join uses M-S pages and the upper operator, which performs hashing, uses S pages, where S is the number of slots/buckets. In this way, the utilization of the memory buffer is maximized. In the rest of the section we provide methods that estimate the selectivity of multiway spatial joins involving skewed inputs, describe an optimization method based on dynamic programming, and evaluate our proposal, by comparing the integration of ST and pairwise algorithms with each individual method.

"... In PAGE 27: ... COMBINING ST WITH PAIRWISE JOIN ALGORITHMS Since ST is essentially a generalization of RJ it can be easily integrated with other pairwise join algorithms to effectively process complex spatial queries. We have implemented all algorithms ( Table3 ) as iterator functions [Graefe 1993] in an execution engine running on a centralized, uni-processor environment that applies pipelining. ST (RJ for two inputs) just executes the join and passes the results to the upper operator.... ..."

### Table 3 Number of solutions as a function of n Similar behaviour is expected for other CSP algorithms in- cluding backtracking-based and hybrid algorithms. As a con- clusion, hierarchical systematic search significantly outper- forms flat search when the domains are large and the number of variables is small (as in most spatial database applica- tions). Some preliminary experiments indicate that H-FC also outperforms methods based on pairwise join algorithms (Mamoulis and Papadias 1999a) for finding all solutions of multi-way intersection joins involving dense queries and da- tasets. The performance gain is higher when only a small subset of the solutions is required. In the next section we ap- ply hierarchical constraint satisfaction with local search.

"... In PAGE 5: ...ariables increases. For n gt;25, FC outperforms H-FC. As the number of variables increases, the performance of FC does not deteriorate significantly because most inconsis- tent instantiations are detected during the early check- forwards. On the other hand, as shown in Table3 , the number of intermediate level solutions explodes with n. In general, the percentage of combinations that constitute solutions in- creases as we go up the levels of the trees because of the large node extents.... ..."

### Table 5: Output size estimation of pairwise spatial joins using grids.

"... In PAGE 30: ...ig. 19: Skew in dataset T1: (a) T1 dataset. (b) Number of rectangles per cell in a 50x50 grid. Table5 presents the actual (column 1) and estimated output size of several join pairs using various grids. The average accuracy improves with the size of the grid used.... ..."

### Table 2. Iterator functions

"... In PAGE 24: ... Combining ST with Pairwise Join Algorithms Since ST is essentially a generalization of RJ, it easily can be integrated with other pairwise join algorithms to effectively process complex spatial queries. Table2 shows how ST, SISJ and SHJ can be implemented as iterator functions (Graefe, 1993) in an execution engine running on a centralized, uni-processor environment that applies pipelining. ST (RJ for two inputs) executes the join and passes the results to the upper operator.... ..."

### Table 1: Classification of Various Spatial Join Algorithms

1996

"... In PAGE 6: ... Such systems can eas- ily use these index based join algorithms. This study, is not a meant to be a comprehensive performance study of all possi- ble spatial join algorithms (refer to Table1 for a classification of spatial join algorithms). However, the algorithms that we study are alternatives that can be used in a spatial database system that does not transform approximations of spatial ob- jects into another domain (e.... ..."

Cited by 159

### Table 1: Classification of Various Spatial Join Algorithms

1996

"... In PAGE 9: ... Such systems can easily use these index based join al- gorithms. This study, is not a meant to be a comprehensive performance study of all possible spatial join algo- rithms (refer to Table1 for a classification of spatial join algorithms). However, the algorithms that we study are alternatives that can be used in a spatial database system that does not transform approximations of spatial objects into another domain (e.... ..."

Cited by 159

### Table 1: Classification of Various Spatial Join Algorithms

"... In PAGE 6: ... Such systems can eas- ily use these index based join algorithms. This study, is not a meant to be a comprehensive performance study of all possi- ble spatial join algorithms (refer to Table1 for a classification of spatial join algorithms). However, the algorithms that we study are alternatives that can be used in a spatial database system that does not transform approximations of spatial ob- jects into another domain (e.... ..."

### Table 1: Classification of Various Spatial Join Algorithms

### Table 1. Classification of spatial join methods

"... In PAGE 5: ... Spatial hash-join (SHJ) (Lo amp; Ravishankar, 1996) avoids duplicate results by performing an irregular decomposition of space based on the data distribution of the build input. Table1 summarizes the existing algorithms of all three classes. In general, indexing facilitates efficiency in spatial join processing; an algorithm that uses existing indexes is expected to be more efficient than one that does not consider them.... ..."

### Table 2. The spatial joins of the experiments

"... In PAGE 3: ... The file is derived from the total set of street lines of California. In Table2 some of the spa- tial joins are described which we performed in our experiments. The selectivity refers to the number of results divided by the product of the number of MBRs of the input relations.... In PAGE 5: ... Temporary storing result tuples and duplicate removal in a separate phase is avoided at the additional cost of at most six comparisons (when a result is detected) which are required for computing the reference point and test- ing the point being in the region of the partition. In order to show the impact of the new method for duplicate removal we present the results of the exper- iments of spatial joins J1-J4 (see Table2 ). We assume here that the size of the available main memory is 2.... ..."