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2D Nearest Neighbor Search
Preprocess S = {(x1, y1), . . . , (xn, yn)}. Query: q = (x, y) −
Simple and Fast Nearest Neighbor Search Marcel Birn, Manuel - - PowerPoint PPT Presentation
Simple and Fast Nearest Neighbor Search Marcel Birn, Manuel - - PowerPoint PPT Presentation
Birn et al.: Nearest Neighbors 1 Simple and Fast Nearest Neighbor Search Marcel Birn, Manuel Holtgrewe, Peter Sanders , Johannes Singler Birn et al.: Nearest Neighbors 2 2D Nearest Neighbor Search Preprocess S = { ( x 1 , y 1 ) , . . . , ( x
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Application Example
Geographic Information Systems, e.g., Route Planning
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A Greedy Algorithm
Preprocessing: Delaunay Triangulation (not Yao graph etc. !) Query: Start at any point in S. Follow any edge towards q. Iterate until no improvement possible. still linear time. . .
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Multiple Levels
Start with a sample of points from S. Closest point in sample → starting point in next level.
How many levels? Complicated data structure?
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n Levels?
Randomly number points in S. Function nlevelNN(q)
u:= s1
for i := 1 to n do
u:= greedyNN(u, q, DT(s1..k))
return u Sounds like quadratic space and at least linear time???
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Full Delaunay Hierarchy (FDH) G:= (1..n,
n
- i=1
Ek)
where Gk = (1..k, Ek):= Delaunay triangulation of points 1..k.
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G has a linear expected number of edges
e.g., [Guibas et al. 92].
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FDH-Based n Level Search!
Procedure nn(q, {s1, . . . , sn})
u:= s1
repeat foreach neighbor v of u in G with u > v in increasing order do invariant u is nearest neighbor of q wrt 1..v − 1 if ||q − sv||2 < ||q − su||2 then
u:= v
break for loop until no improvement found return u
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Analysis
Theorem: Expected number of nodes visited ≤ ln n + 1 Proof Idea: si is visited
⇔si is closest to q wrt 1..i.
This event has probability 1/i.
harmonic sum claim.
i
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Bad Inputs
# of inspected edges might be Ω (n)
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Implementation
Store coordinates redundantly in edge array.
Scan logarithmic number of pieces. Never worse than greedy scanning algorithm.
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4(6,0) 4(6,0) 7(8,0) 6(5,0) 8(3,0) 6(5,0) 3(2,0) 8(3,0) 2(7,0) 5(1,0)
x: #:
2 3 4 1 q=(2.8, 0)
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Other Approaches
Delaunay Hierarchy kd−Tree Voronoi Diagram
- ptimal
+point location in practice implementation? [CGAL, Mount et al.] currently preferred [Devillers92] relatively complicated
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Instances
[Devillers02]
5% 95% in square
- n circle
mixed
- n parabola
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Experiments: Square
0.1 1 10 128 512 211 213 215 217 219 221 223 time per query [µs] number of points (n) CGAL Delaunay hierarchy FDH exact ANN kd-tree CGAL kd-tree FDH array, inexact
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Experiments: Circle (parabola similar)
0.1 1 10 100 1000 104 128 512 211 213 215 217 219 221 223 time per query [µs] number of points (n) ANN kd-tree CGAL Delaunay hierarchy CGAL kd-tree FDH, exact FDH, inexact
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Experiments: Mixed
0.1 1 10 100 1000 128 512 211 213 215 217 219 221 223 time per query [µs] number of points (n) ANN kd-tree CGAL Delaunay hierarchy CGAL kd-tree FDH, exact FDH, inexact
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More Related Work Several n-level geom. data structures on more complex objects
(triangles. . . ) and for point location. [BoissonatTeillaud86], [Guibas et al. 92],. . .
Contraction Hierarchies – route planning with n levels
[Geisberger et al. 2008]
n-level Graph Partitioning [with Vitaly Osipov]
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Conclusions
Full Delaunay Hierarchies
(mostly) fast simple elegant
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Open Problems Comparison with worst case efficient algorithms Make worst case efficient,
e.g., replace high degree nodes? move high degree nodes in hierarchy?