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Dynamic Programming

Dynamic Programming. 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:

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Dynamic Programming

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  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-overlappinggroups with given cost functions f(xi, xj) sothat the totalvalue 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 5004points are approximated by 78 points. Givenapproximatedpoints M, errorsareminimized. Givenerrorε, number of approximatedpointsareminimized. Here, cost function is:

  9. Examples: Route segmentation Ski Running and Jogging Non-moving Divide the routes into severalsegmentsby speedconsistency Here, cost function is: Speedvariance 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…aiand b1b2…bj With proper initializations, len(i, j) can be computed as follows.

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

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

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

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

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