1. Outlier Respecting Points Approximation
Danny Z. Chen and Haitao Wang
Computer Science and Engineering
University of Notre Dame
Indiana, USA

2. The motivation Propose a new problem model for dealing with outliers
an optimal solution
data without outliers
data
many optimal solutions
outliers
a particular optimal solution
that respects outliers

3. Points approximation Input: A point set P in 2-D
Output: An approximation function f
Approximation error: Vertical distance
e(P,f)=max{error of each point}
error

4. The problem (min-#) Given: An allowed error ε≥ 0
Goal: an approximation function f of minimized size, such that e(p,f) ≤ε

5. Problem variations
Step function (SF)
error

6. Problem variations (cont.)
Piecewise-linear function (PF)
error

7. Problem variations (cont.)
Weighted versions for both SF and PF
Every point has a weight ui
Error of each point: ui×vertical distance
WSF and WPF
there is a weight ui
vertical distance

8. Problem variations (cont.)
Outlier versions:
Allow a given number g of outliers
e(P,f)=max{error of every non-outlier point}
error
an outlier

9. Problem variations (cont.)
Outlier versions of SF, WSF
OSF and OWSF
error
outlier

10. Problem variations (cont.)
Outlier versions of PF, WPF
OPF and OWPF
error
outlier

11. Outlier-respecting versions (new)
Allow a given number g of outliers
e(P,f)=max{errors of non-outlier points}
Outlier error: e’(P,f)=max{errors of outliers}
Goal:
minimize the size |f|, such that e(P,f)≤ε
among all optimal solutions for the minimized |f|, find the solution with minimum outlier error e’(P,f)

12. An example outlier an SF case: ɛ=1.5, g=1 (one outlier) y (4,5) ORSF
outlier error: 3.5
(3,2)
1.5
(2,1)
0.5
OSF
outlier error: 4.5 x
(1,0)

13. Outlier-respecting versions (cont.)
Step function: SF -> OSF -> ORSF
Weighted: WSF -> OWSF -> ORWSF
Piecewise linear function: PF->OPF->ORPF
Weighted: WPF -> OWPF -> ORWPF

14. Previous results problem versions results OSF
O(ng2) [Fournier and Vigneron,08’]
OWSF
O(ng2) [Chen and Wang 09’]
OPF
O(ng4log2n) [Chen and Wang 09’]
OWPF

15. Our new results on the outlier-respecting versions
problem versions
results
OSF
O(ng2)
ORSF
O(ng3logn)
OWSF
ORWSF
O(ng3lognlogg)
OPF
O(ng4log2n)
OWPF
ORPF
O(n2+δg δlogn)
ORWPF
O(glogn)
O(glogn)
O(n)

16. Our algorithms
A dynamic programming algorithmic scheme for all problems (ORSF, ORWSF, ORPF, ORWPF)
Different computational components for each specific problem

17. The computational components
The computational components of our algorithmic scheme
Compute w(i,j,q) for each query on (i,j,q)
1≤i ≤j≤n, for point index, 0 ≤q≤g, for the number of outliers
w(i,j,q): the outlier error of using one segment to approximate {pi,pi+1,…,pj} with q outliers
outliers
wijq pi pj

18. Computing w(i,j,q) for ORSF
Observation: Only need to consider the highest q points and the lowest q points
outlier
wijq pi pj
the case q=1

19. Computing w(i,j,q) for ORSF
Observation: Only need to consider the highest q points and the lowest q points
2 outliers
wijq pi pj
the case q=2

20. Computing w(i,j,q) for ORSF
Find the highest and lowest q points
Use a q-range-minima data structure
O(q) time, with O(nlognlogq) time preprocessing (Chen and Wang 09’)
Compute wijq: O(q) time

21. Computing w(i,j,q) for ORWSF
Observation: a point pt can be approximated within error ɛ by a segment if and only if the y-coordinate of the segment is in the interval [yt-ɛ/ut,yt+ɛ/ut]
yt-ɛ/ut is the lower end of the interval
yt+ɛ/ut is the upper end of the interval pi pj

22. Computing w(i,j,q) for ORWSF (cont.)
Observation: only need to consider the points with q highest lower ends and q lowest upper ends
q=3
outliers pi pj

23. Computing w(i,j,q) for ORWSF (cont.)
A difficulty: determine the optimal segment in the strip to minimize the outlier error
Model it as finding the lowest point in the common intersection of a set of upper half-planes
A naïve approach takes O(q2) time
O(qlogq) time: model it as updating the upper envelope for an offline sequence half-plane insertions and deletions
common intersection
lowest point

24. Computing w(i,j,q) for ORWSF (cont.)
Compute the q points with highest lower end and the q points with lowest upper end
O(q) time
Determine the value of w(i,j,q)
O(qlogq) time

25. Computing w(i,j,q) for ORPF/ORWPF
Total time: O((nq1.5)1+δ) time
3-D segment dragging queries:
A convex polyhedron G in 3-D
Each query is specified by a line segment e outside G and a direction perpendicular to e,
find the first point (if any) on G that is hit by e if we move e along the direction
Our result: With O(n) preprocessing, answer each query in O(logn) time e G

26. Thank you