1. Representing a Functional Curve by Curves with Fewer Peaks
Danny Z. Chen Chao Wang Haitao Wang
Computer Science and Engineering
University of Notre Dame
Indiana, USA

2. General Problem Definition
Input: a piecewise linear functional curve
Output: one or more curves to approximate the input curve
The output curves are structurally simpler

3. An Example
The new curve is structually simpler

4. General Problem Definition (cont.)
The optimization criteria
Approximation error
Output size
All curves discussed in our problems are non-negative (required by application)

5. Formal Problem Definitions
Input: A nonnegative piecewise linear functional curve F=(f1,f2,…,fn) f2 fn fi f1

6. Peaks and Valleys
a peak
a valley

7. Approximation Error: L∞
Givenε>0, F can be approximated by another curve G with approximation error є if and only if max|fi-gi|≤є, i.e., G lies in the area between F +εand F –ε
F +ε G F
F -ε

8. Three Problems Uphill-downhill pair representation (UDPR)
Unimodal representation (UR)
Fewer-peak representation (FPR)

9. Uphill-downhill Pair Representation (UDPR)
Find two non-negative piecewise linear curves, one non-decreasing (uphill) and one non-increasing (downhill), to use their sum to approximate F

10. An Example for UDPR F
uphill curve
downhill curve

11. Two Problem Versions of UDPR
Feasibility version: Givenε>0, determine whether there is a feasible UDPR solution for F with approximation errorε
min-ε: Compute the minimum errorε* such that there is a feasible UDPR solution with an approximation error ε*

12. Unimodal Representation (UR)
Find a set of non-negative unimodal (single-peak) curves to use their sum to approximate F

13. An Example for UR F

14. Two Problem Versions of UR
min-k: Givenε>0, find a minimum number of unimodal curves with approximation errorε
min-ε: Given k>0, find at most k unimodal curves that minimizes the approximation error

15. Fewer-peak Representation (FPR)
Compute a non-negative piecewise linear curve with fewer peaks to approximate F F

16. Two Problem Versions of FPR
min-k: Givenε>0, find a curve with the minimum number of peaks
min-ε: Given k>0, find a curve with at most k peaks that minimizes the approximation error

17. Problem Summary Uphill-downhill pair representation (UDPR)
Feasibility version
min-εversion
Unimodal representation (UR)
min- k version
Fewer-peak representation (FPR)
min-k version

18. Motivations
A dose decomposition problem in intensity-modulated radiation therapy (IMRT)
A modern cancer treatment technique to deliver a prescribed dose to a target tumor
The prescribed dose to be delivered is F (nonnegative)
A unimodal dose function can be delivered smoothly
Approximate F by a set of unimodel functions (UR)
To solve UR, we need to solve UDPR
FPR can be solved in a similar way as UDPR

19. Related Work [Chun, Sadakane, Tokuyama 2006]
gave an O(n) time algorithm to approximate a piecewise linear curve by a single unimodal curve under L2 error metric
O(n2(k+log n)) time algorithm for computing a curve with k peaks to approximate a piecewise linear curve, under the Lp error metric
[Stout 2008] studied the problem of approximating points by a unimodal step function, O(nlog n), O(n) and O(n) time algorithms for L1, L2 and L∞ error metrics

20. Our Results UDPR UR FPR Feasibility version: O(n) time
min-εversion: O(n) time UR
min-k version: O(n) time
min-εversion: O(nlogn) time
FPR

21. Our Approaches
Geometric observations on the topological structure of the problem
Interesting pruning and search techniques, and data structures

22. The Characteristic Curve
Given F and є, define its characteristic curve R(F,є)=(r1,…,rn),
r1 = f1+є
ri = fi-є if ri-1<fi-є, fi+є if ri-1>fi-є and ri-1 otherwise
F+ε
F-ε

23. UDPR Feasibility Version
Givenε>0, determine whether there exist
An uphill (non-decreasing) curve Y=(y1,…,yn)
A downhill (non-increasing) curve Z=(z1,…,zn)
Y and Z are non-negative
Y+Z=(y1+z1,…, yn+zn) approximates F with errorε
If yes, compute the two curves Y and Z

24. UDPR Feasibility Version (cont.)
If the nonnegativeness is not required, the problem is easy even for є=0. Compute Y and Z as follows: F Y Z
y-coordinate is 0

25. I(F) and D(F) F I(F) D(F) I(F): The increasing curve of F
D(F): The decreasing curve of F

26. Algorithm for UDPR Feasibility
I(R)
D(R)
f1+ є – the sum of the decreasing parts in R(F, є)

27. UDPR min- є Problem
Compute the minimum approximation errorε* such that there is a feasible UDPR solution with errorε*

28. Algorithm for UDPR min- є
R(F, є)
F+ ε
F- ε
y=0
D(R)

29. The Changing of y-coordinate of the Lowest Point in D(R) as є Increasesε ε*

30. Determine the Critical Errors
For every critical error, a peak “touches” a valley
ε* must be in S={|fi-fj|/2 | for any i, j}; the size of S is O(n2)
A simple O(nlog n) solution without producing S explicitly

31. Demo …

32. Determine the Critical Errors (cont.)
The number of critical errors is at most n
All critical errors can be computed in O(n) time
UDPR min-εis solvable in O(n) time

33. The Problem UR
Find a set of non-negative unimodal (single-peak) curves to use their sum to approximate F

34. An Observation
Let Hi with 1≤i≤k be k unimodal curves and the index of the peak in Hi be pi; suppose 1≤p1≤…≤pk ≤n and H=∑Hi, then
H[1…p1] is uphill
H[pi-1…pi] has a UDPR solution with error zero
H[pk…n] is downhill

35. p1 p2 p3 p4 p5

36. The Inverse is also True
Given a curve H defined on [1…n], if
1≤p1≤…≤pk≤n
H[1…p1] is uphill
H[pi-1…pi] has a UDPR solution with error zero
H[pk…n] is downhill
Then, we can find k unimodal curves Hi, 0<i<k+1, such that H = ∑Hi

37. Solving UR
min-k
Model the problem UR as a series of UDPR problems
Use our UDPR algorithms as subroutines
A greedy algorithm, O(n) time
min-є
Determine a error set of O(n4) size containing є*
O(n+mlog m) time, m is the number of peaks in F
Use pruning and search techniques and data structures

38. Conclusions UDPR UR FPR Feasibility version: O(n) time
min-εversion: O(n) time UR
min-k version: O(n) time
min-εversion: O(n+mlogm) time
FPR

39. Thank you