1. Dynamic Programming

2. Dynamic Programming
Dynamic Programming is a general algorithm design technique
It breaks up a problem into a series of overlapping sub-problems ( sub-problem whose results can be reused several times)
Main idea:
set up a recurrence relating a solution to a larger instance to solutions of some smaller instances
- solve smaller instances once
record solutions in a table
extract solution to the initial instance from that table

3. Discussed topics Assembly-line scheduling Partition of Data Sequence
Route Segmentation and Classification for GPS data
Longest Common Subsequence (LCS)

4. Assembly-line scheduling
Recursive equation:
where f1[j] and f2[j] are the accumulated time cost to reach station S1,j and S2,j
a1,j is the time cost for station S1,j
t2,j-1 is the time cost to change station from S2,j-1 to S1,j

5. Partition of Data Sequence
Partition of the sequence X into to K non-overlapping groups with given cost functions f(xi, xj) so that the total value of the cost function is minimal:

6. Partition of Data Sequence
G(k,n) is cost function for optimal partition of n points into
k non-overlapping groups:
Recursive equation:

7. Examples: Image Quantization
Input
Quantized
Here, cost function is:
Square Error
Quantize value

8. Examples: Polygonal Approximation
5004 points are approximated by 78 points.
Given approximated points M, errors are minimized.
Given error ε, number of approximated points are minimized.
Here, cost function is:

9. Examples: Route segmentation
Ski
Running and Jogging
Non-moving
Divide the routes into several segments by
speed consistency
Here, cost function is:
Speed variance
Time duration

10. Examples: Route classification
Determine the moving type only by speed will cause mis-classification.
Frequent Moving type dependency of 1st order HMM

11. Examples: Route classification (cont.)
1st order HMM, maximize:
mi : moving type of segment i
Xi : feature vector (e.g. speed)
Solve by dynamic programming similar with the Assembly-line scheduling problem

12. Longest Common Subsequence (LCS)
Find a maximum length common subsequence between two sequences.
For instance,
Sequence 1: president
Sequence 2: providence
Its LCS is priden.
president
providence

13. How to compute LCS? Let A=a1a2…am and B=b1b2…bn .
len(i, j): the length of an LCS between a1a2…ai and b1b2…bj
With proper initializations, len(i, j) can be computed as follows.

14. Running time and memory: O(mn) and O(mn).

16. Time Series Matching using Longest Common Subsequence
delta = time matching region (left & right)
epsilon = spatial matching region (up & down)
Recursive equation:

17. Spatial Data Matching using Longest Common Subsequence
epsilon = spatial matching distance
Recursive equation:

18. Conclusion Main idea: Divide into sub-problems
Design cost function and recursive function
Optimization (fill in the table)
Backtracking