1. CSIS 7101: Spatial Data (Part 3) Distance Browsing in Spatial Database GÍSLI R. HJALTASON and HANAN SAMET
Rollo Chan
Chu Chung Man
Mak Wai Yip
Vivian Lee
Eric Lo
Sindy Shou
Hugh Wang

2. What is Distance Browsing?
A collection of spatial objects stored in an R-tree spatial data structure
Browsing through the database on the basis of distances from an arbitrary spatial query object
Ranking data objects in their order of distance from a given query object
E.g. Find the nearest person to me who is sleeping.
2 different techniques:
k-nearest neighbor algorithm (k-NN)
Incremental nearest neighbor algorithm (INN)

3. Before All of Them Requirement - Consistency Definition:
Let d be the combination of functions d0 and dn, and e  N denote
the fact that item e is contained in exactly set of nodes N. The function
d0 and dn are consistent iff for any query object q and any object or node
e in the hierarchical data structure there exists n in N, where e  N, such
that d(q, n)  d(q, e) q o
The circle around query object q depicts
search region after reporting o as next
nearest object.

4. Example R0 R1 R2 R3 R4 R5 R6
R0 (0) e f c i a b R5 R6 R3 R4 R1 R2 g h d
R0:
R1:
R2:
R3:
R4:
R5:
R6: f c g d h b a e i q
Find the THREE nearest neighbors to query point q in the R-tree given.
k-Nearest Neighbor Search
Incremental Nearest Neighbor Search

5. k-Nearest Neighbor Search
Applicable only when k is fixed in advance
Maintain a global list of candidate k nearest neighbors as traverse in depth-first manner
Only make local decisions
Next node to visit must be the child node
Make use of nearest list
Comparing with the max. value in the list

6. Pruning Strategies b o a q r
MINDIST (optimistic)
MINMAXDIST (pessimistic)
Strategy 1: prunes an entry whose bounding rectangle r1 is such that MINDIST(q, r1) > MINMAXDIST(q, r2), where r2 is some other bounding rectangle
Strategy 2: prunes an object o when DIST(q, o) > MINMAXDIST(q, r), where r is some bounding rectangle.

7. Pruning Strategies (con’t)
Strategies 1 & 2 are useful only when k=1
Strategy 3: prunes any node whose bounding rectangle r is such that MINDIST(q, r) > NearestList.MaxDist
Only MINDIST() is sufficient for pruning

8. Example – k-NN ∞ k = 3 q f c g d h b a e i R0 (0) R0: R1 R2 a b R4 R3
Nearest List
Max Dist. a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist. ∞
k = 3

9. Example – k-NN ∞ k = 3 q f c g d h b a e i R0: R0 (0) R1 R2 a b R4 R3
Nearest List
Max Dist. a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist. ∞
i(21)
b(48)
d(59) 59 21 48 81
a(17)
g(81)
h(17)
k = 3

10. Problems with k-NN Nodes/objects are not visited by order of distance.
May access non-optimal objects, and need to prune them.
Need to know k in advance, difficult to combine with other predicates.

11. Incremental Nearest Neighbor Search
Top-down manner tree traversal
Depth-first traversal
Breadth-first traversal

12. Incremental Nearest Neighbor Search
INN use Best-first traversal
Pick the node with least distance in the set of all nodes that have yet to be visited
Use a priority queue
Distance from the query object is the key
Makes global decisions (k-NN make local decisions)
Based on priority queue
Choose among the child nodes of all visited nodes

13. Example – INN q f c g d h b a e i R0: R0 (0) R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.

14. Example – INN a b h g e d q i f c R0 (0) R0: R1 R2 R1: R1 (0) R3 R4
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
R0 (0)

15. Example – INN a b h g e d q i f c R0 (0) R0: R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
R1 (0)
R2 (0)

16. Example – INN a b h g e d q i f c R0: R0 (0) R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
R2 (0)
R4 (11)
R3 (13)

17. Example – INN a b h g e d q i f c R0: R0 (0) R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
R5 (0)
R4 (11)
R3 (13)
R6 (44)

18. Example – INN a b h g e d q i f c R0: R0 (0) R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
[i](0)
R4 (11)
R3 (13)
R6 (44)
[c](53)
i (21)

19. Example – INN a b h g e d q i f c R0 (0) R0: R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
R4 (11)
R3 (13)
i (21)
R6 (44)
[c](53)

20. Example – INN a b h g e d q i f c R0: R0 (0) R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
R3 (13)
[h](17)
i (21)
R6 (44)
[c](53)
[d](30)
[g](74)

21. Example – INN a b h g e d q i f c R0 (0) R0: R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
[a](13)
[h](17)
i (21)
[b](27)
[d](30)
a (17)
R6 (44)
[c](53)
[g](74)

22. Example – INN a b h g e d q i f c R0 (0) R0: R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
[h](17)
i (21)
[b](27)
[d](30)
R6 (44)
a (17)
[c](53)
h (17)
[g](74)

23. Example – INN a b h g e d q i f c R0 (0) R0: R1 R2 R1: R3 R4 R2: R5 R6
Priority Queue a b c d e f g h i 17 48 57 59 86 81 21 13 27 53 30 45 74
Seg.
Dist.
BR Dist. R0 R1 R2 R3 R4 R5 R6 13 11 44 BR
Dist.
i (21)
[b](27)
[d](30)
R6 (44)
[c](53)
a (17)
[g](74)
h (17)
i (21)

24. Variants Find Farthest Object: Min and Max Distance:
Queue sorted in descending order of distance
Replace <= by >=
Min and Max Distance:
E.g. Find all Cities distanced from Hongkong for 100 Miles to 200 Miles
Prune unqualified nodes
Solve the Traditional k-NN Problem

25. Priority Queue Play a key role in performance In 2-dimension:
worst case unlikely to arise in practice
expected number of points in queue = O( )
usually fit in memory
In higher-dimension:
Higher dimension, larger queue size

26. Priority Queue (con’t)
Idea:
priority queue will be split into three-tiers
first tier in memory, 2nd and 3rd in a disk file
a set of ranges, first tier stores the nearest range, 3rd tier stores the farthest
when 1st tier exhausted, move elements from 2nd tier
when 2nd tier exhausted, scan elements and rebuild 1st and 2nd tier with new ranges

27. Comparison of k-NN and INN
Depth-first recursion
Make local decision
k is fixed
If used with k unknown,
Pick a fixed K’, do k-NN
If k gradually > K’, pick a m>=k and re-apply k-NN
Drawback: waste computational power if chosen m too large
INN
Priority queue
Make global decision
Number of neighbors not known in advanced

28. Experiment Dataset Hierarchical data structure: R*-tree
Real-world data: TIGER/Line File
Howard: 17,421 line segments
Water: 37,495 line segments
PG: 59,551 line segments
Roads: 200,482 line segments
Synthetic data
Hierarchical data structure: R*-tree
Utilizing buffered I/O
Three measures: execution time, R-tree node I/O,
object distance calculations

29. Cumulative Cost of Distance Browsing

30. Incremental Cost of Distance Browsing

31. k-Nearest Neighbor Queries

32.  Experimental Result INN outperforms k-NN in distance browsing
In k-NN queries, INN algorithm is better than k-NN algorithm
For large number of neighbor, priority queue for INN is smaller than the NearestList maintained by k-NN 
k-Nearest Neighbor Search
Incremental Nearest Neighbor Search

33. References
Gisli R. Hjaltason, Hanan Samet, “Distance Browsing in Spatial Databases”, ACM TODS, Volume 24, Number 1, pp , March 1999
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