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1 Top-k Spatial Joins Po-Sungalowblow@hotmail.com

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2 Survey What ’ s top-k spatial joins

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3 Map overlays incur high execution cost Retrieve the k objects The processing of such this is expensive A B

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4 Top-k Spatial Joins Apply a conventional spatial join algorithm on the two data sets A and B Count the number of output pairs in which each object participates Return the k objects with the maximum intersection counts

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5 Top-1 join A1 A2 B1 B2 B3 a1 b1 b10 b5 {A1, 3, [B1, B2, B3]} {a1, 3, [b1, b5, b10]}

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6 Definition 1 E is an intermediate entry of R a C the node capacity e.level the level of the node that contains e Upper bound for the number of objects in the subtree of e

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7 Definition 2 If e is leaf entry of R a the number of objects of R b that intersect If e is intermediate entry upper bound of the actual count of any object in e

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8 Example A1.IL = [B1, B2, B5] A2.IL = [B5] B1.IL = [A1] B5.IL = [A1, A2] A1 A2 B1 B2 B5

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9 Example (cont.) Heap H E : e is the entry (of R a or R b ) list is e.IL

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10 Example (cont.) a1.IL= [b1, b5, b10] a1.key=3 A2.IL= [b5] a2.key=1 A1 A2 B1 B2 B3 b1 b10 b5 a2 a1

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11 Pseudocode TS (Rtree R a, Rtree R b, int k) Join RTa and RTb to get intersecting pair (e a,e b ) For each entry e that appears in a pair build e.IL, compute e.count and insert to a heap H (sorted by e.count) While number of reported objects < k e = de-heap(H) If e is a leaf entry // actual object –Report ( ) Else // e is an intermediate entry pointing to node n For each Join n and n i // n i is pointed by e i For each intersecting entry pair(e’, e’ i ) // Add e’ i to r’.IL Compute e’.count If e’.count > pruning condition //ie...count of the k-th best object found so far Insert to H If e’ is a leaf entry //object Update pruning condition return

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12 Algorithm Visiting order Pruning condition count

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13 Multiple Expansions Method (ME)

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14 Two binary search trees

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15 Full join VS. Semi join

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16 Comparison environment 1. MCB x LA returns 16,477,244 intersection pairs 2. SKEW x LA returns 19,657,973 intersection pairs

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17 Node accesses versus k (full join).

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18 CPU time versus k (full join).

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19 Total cost versus k (full join, 10 percent cache).

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20 Node accesses versus k (semijoin).

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21 CPU time versus k (semijoin).

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22 Total cost versus k (semijoin, 10 percent cache).

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23 Conclusions Bottom-k queries Top-k distance (semi) join Top-k nearest neighbor (semi) joins Computing the NN (in A) of all objects of B Sorting the resulting pairs (o b, o a ) where in the NN of with respect to o a Reporting the top-k objects of A

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