![]() ![]() Data is partitioned into several clusters and each partition has a reference point. I-Distance ( Yu et al., 2001) is an efficient method for k-NN search in a high-dimensional space. The iMinMax( θ) ( Ooi et al., 2000) method which maps points from high-dimensional to single-dimensional space has been shown in experiments to outperform the Pyramid scheme for range queries. The Pyramid tree ( Berchtold et al., 1998) is not affected by the curse of dimensionality, a term coined by Richard Bellman best known for dynamic programming ( Bellman, 2003). X-trees ( Berchtold et al., 1996 ) combine linear and hierarchical structures using supernodes to avoid overlaps to attain improved performance for NN search in higher dimensions. ![]() SR-trees outperform both R ⁎-trees and SS-trees. SS-trees ( White and Jain, 1996) use hyperspheres to partition the space, while SR-trees ( Katayama and Satoh, 1997) use the intersection of hyperspheres and hyperrectangles to represent regions. R +-trees ( Sellis et al., 1987) and R ⁎-trees ( Beckmann et al., 1990) are improved version of R-trees. R-trees were designed to handle spatial data on secondary storage for CAD applications ( Guttman, 1984). k-d-b trees combine k-d trees and one-dimensional B-trees ( Ramakrishnan and Gehrke, 2002) to build a disk based multidimensional index ( Robinson, 1981). Quad trees ( Finkel and Bentley, 1974) and k-d trees ( Bentley, 1975) are two main memory resident indexing structures. Points are best represented by PAMs, since with 2-d and 3-d SAMs the space required to represent points is doubled and tripled, since corner coordinates are the same. SAMs represent objects in 2-d with Minimum Bounding Rectangles - MBRs, i.e., the coordinates of left lower corner and right upper corner ( Faloutsos, 1996). PAMs index high dimensional data points representing feature vectors, which are used for similarity search via nearest neighbor queries. We are concerned with Point Access Methods - PAMs rather than Spatial Access Methods - SAMs, which in two or three dimensions are of interest in Computer Aided Design - CAD. ![]() The number of pages accessed from disk and CPU processing time are the major performance metrics, but the major cost difference between sequential and random disk accesses should be noted. High dimensional indices pose challenging problems in storing them on disks and DRAM for efficient processing various queries, e.g., Nearest Neighbor - NN queries used in similarity search applied to feature vectors ( Faloutsos, 1996), ( Gaede and Günther, 1998), ( Castelli, 2002) in ( Castelli and Bergman, 2002), ( Yu, 2002), ( Samet, 2006). G N′ is a ( n − 1)-dimensional pancake graph embedding many-to-one dilation 3 into G N′ (hypothesis of induction).Īlexander Thomasian, in Storage Systems, 2022 8.15.1 Survey of multi-dimensional indices In other words, we use only four super nodes among the k projections or super nodes. The Embed-node( A) algorithm, as shown in Table 6.1, embeds all nodes A = 00 Prefa N − 3 a N − 2 a N − 1 ( A Є V) into the first super node or into the projection G N′, all nodes A = 10 Prefa N − 3 a N − 2 a N − 1 into G N′, all node A = 01 Prefa N − 3 a N − 2 a N − 1 into G N′, and all nodes A = 11 Prefa N − 3 a N − 2 a N − 1 into G N′. Let k = 2 m + 1, where m ∈ ℕ, and TQ n be obtained from two copies of 0 TQ 2 m′ and 1 TQ 2 m′ and suppose that for N = 2 m we have 0 TQ N′ and 1 TQ N′, that is to say, 00 TQ N − 1′, 01 TQ N − 1′ and 10 TQ N − 1′, 11 TQ N − 1.′ However, the dilation of embedding into G k − 1′ is 3 (hypothesis of induction). That is to say, they are embedded into G k − 1′. All nodes A Є V, that is, A = 1 prefa k − 3 a k − 2 a k − 1 or A = Pref 2 a k − 3 a k − 2 a k − 1 are embedded into the second super node or into the projection G k′, as shown in Table 6.5. All nodes A Є V, such that, A = 0 Prefa n − 3 a n − 2 a n − 1 = Pref 1 a k − 3 a k − 2 a k − 1, are embedded by Embed_node( A) algorithm, as shown in Table 6.4, into the first super node or into the projection G k′. TQ n′ = ( V, U1) is constructed by two copies of TQ n − 1′: one copy is prefixed by 0(0 TQ k − 1′) and the second one is prefixed by 1(1 TQ k − 1′). Let us now prove that it is true for k = n. Suppose that for k ≤ n − 1, TQ k − 1′ embedding many-to-one dilation 3 into Q k − 1′ is true. P roof.– We prove lemma 6.1 by induction.įor n = 3, Table 6.1 presents all paths between the embedded nodes of TQ 3 into G 3 with dilation 3. The n-dimensional twisted hypercube TQ n′ = ( V, U1) has many-to-one dilation 3 embedding into G n′ = ( P n′, E n′) for any n > 3. Zerarka, in Building Wireless Sensor Networks, 2017 6.4.1 Lemma 6.1 ![]()
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