1. Approximating Points by A Piecewise Linear Function: I
Danny Z. Chen and Haitao Wang
Computer Science and Engineering
University of Notre Dame
Indiana, USA

2. Outline
Problem definitions
Motivations and previous work
Our results

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

4. Two Problem Versions min-#: min-ε: Given: An error toleranceε≥ 0
Goal: f of minimized size, with e(p,f) ≤ε
min-ε:
Given: k>0
Goal: f of minimized error e(p,f), with size ≤k

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

8. Problems in 3-D Extend SF and WSF to 3-D (SF3 and WSF3)
The function f is represented by a set of rectangular faces parallel to the x-y plane and each face approximates a subset of points
Vertical distance measured by z-values

9. Problem Summary SF, PF, Weighted version: WSF, WPF
3D version: SF3, WSF3
min-# and min-ε

10. Motivations
Query optimizations and histogram constructions in database management systems
Regression analysis

11. Previous Work (2D problems)
min-#
min-ε SF
O(n) [Diaz-Banez ,Mesa,01]
O(n) [Fournier,Vigneron,08]
WSF
O(n) [Karras,Sacharidis,Mamoulis 07]
O(nlogn+k2log6n)[Guha,Shim, 07]
O(nlog4n) [Fournier,Vigneron,08] PF
O(n) [O’Rourke,81]
WPF

12. Our Results min-ε O(nlogn+k2log6n) [Guha,Shim,07] O(nlogn+k2log4n) WSF
O(nlog4n) [Fournier,Vigneron,08]
O(nlog2n) PF
O(n+k2log3loglogn)
O(nlogn)
WPF
O(nlogn+k2log5n)
O(nlog3n)

13. 3D Problems No previous result is found Our results: min-# min-ε SF3
NP-hard
2-Approx
1.5-nonapprox
WSF3

14. Technical Contributions
Two algorithm frameworks
A general formulation of path partition problem [Fournier,Vigneron,08]
Parametric search
Data structures for major components:
Vertical hull width query (PF)
2-D sublist LP query (WSF)
3-D sublist LP query (WPF)
A tiling modeling for 3-D problems

15. Where to Use Data Structures?
wij: The minimum error to approximate point subset Pij={pi,…,pj} by one segment
Need data structures to support queries on wij, 1≤i≤j≤n
wij pi pj

16. Where to Use Data Structures?
wij: The minimum error to approximate point subset Pij={pi,…,pj} by one segment
Need data structures to support queries on wij, 1≤i≤j≤n
wij pi pj

17. Vertical Hull Width Queries (PF)
Given: A point set P
Query q(i,j): The maximum vertical distance on the convex hull of Pij
Times: (n,lognloglogn), (nloglogn, logn)
Use compact interval tree
Vertical hull width

18. Main Idea For each query q(i,j)
Obtain the convex hull of Pij (Guibas, Hershberger, Snoeyink, 91)
Beginning from the root, at each node, in O(1) time, find the edges of the upper hull and lower hull spanning the node, and then determine which way to go

19. 2-D Sublist LP Queries (WSF)
Given: A halfplane set H={hi | 1≤i≤n}
Query q(i,j): The lowest point in the common intersection of Hij={hi, …, hj}
common intersection
lowest point

20. 2-D Sublist LP Queries (cont.)
Previously best-known result: (nlogn,log4n)
Our solution: (nlogn, log2n)
Our techniques: Fractional cascading and binary search on sorted arrays

21. An (nlogn, log3n) Solution
Preprocessing:
Build a complete binary tree T
The i-th leaf stores hi
Each internal node v stores the common intersection chain (Cv) of halfplanes stored in the subtree rooted at v
O(nlogn) time

22. Common intersection chain Cv of all halfplanes from lv to rv
Complete Binary Tree T
Common intersection chain Cv of all halfplanes from lv to rv v lv rv
stores halfplane hi

23. Query Algorithm q(i,j) Suppose p* is the sought lowest point
Find the LCA w of i and j
Find O(logn) internal nodes by following the two paths from w to i and to j
p* is the lowest point of the chains stored in O(logn) internal nodes

24. Query Algorithm q(i,j) (cont.)
p* is the lowest point in the common intersection of the O(log n) chains w
LCA of i and j i j

25. Query Algorithm q(i,j) (cont.)
Each chain is represented by a sorted array
Totally O(logn) arrays
Given a vertical line L, for each chain, spend O(log n) time computing its intersection with L
Need O(log2n) time to determine whether p* is in the left side of L or right side
p* can be located in O(log3n) time p* L

26. An (nlogn, log2n) Solution
Observation: The O(logn) chains for each query are organized along two ancestor-descendant paths in the tree T
By using fractional cascading
Given L, its intersections with all O(logn) chains can be obtained in O(logn) time
O(nlogn) preprocessing time
Each query: O(log2n) time

27. 3-D Sublist LP Queries (WPF)
Given: A halfspace set H={hi | 1≤i≤n}
Query q(i,j): The lowest point in the common intersection of Hij={hi, …, hj}
No previous solution has been found
Our solution: (nlogn, log3n)

28. Common intersection polyhedron Cv of all halfspaces from lv to rv
Complete Binary Tree T
Common intersection polyhedron Cv of all halfspaces from lv to rv v lv rv
stores halfspace hi

29. A projection of Cv to the XY plane
Data Structure for Cv
To support an efficient query, use Kirkpatrick’s hierarchical planar point location data structure to store Cv
A projection of Cv to the XY plane

30. Preprocessing Algorithm
Built the tree T in a bottom-up manner
Use Chazelle’s linear time convex polyhedra common intersection algorithm to construct Cv at an internal node v from its two children
Preprocessing time: O(nlogn)

31. Query Algorithm q(i,j)
p* is the lowest point in the common intersection of the polyhedra stored there w
LCA of i and j i j

32. 3D Problems NP-hardness proof: Based on reduction from planar-3SAT
Approximation algorithms: Use the solution of the hierarchical binary tiling problem (HBT) which can be solved optimally

33. Thank You
Questions?