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Dynamic Programming Tutorial

Dynamic Programming Tutorial. Elaine Chew QMUL: ELE021/ELED021/ELEM021 26 March 2012. Sources. Hillier, F. S. & G. J. Lieberman. Introduction to Operations Research. Chapter 11: Dynamic Programming, 7 th , 8 th , 9 th editions, McGraw-Hill.

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Dynamic Programming Tutorial

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  1. Dynamic Programming Tutorial Elaine Chew QMUL: ELE021/ELED021/ELEM021 26 March 2012

  2. Sources • Hillier, F. S. & G. J. Lieberman. Introduction to Operations Research. Chapter 11: Dynamic Programming, 7th, 8th, 9th editions, McGraw-Hill. • Dixon, S. & Widmer, G. (2005). MATCH: A Music Alignment Tool Chest. In Proceedings of the 6thIntl Conf on Music Information Retrieval, London, UK, 492-497. • Yang, A., E. Chew & A. Volk (2005). A Dynamic Programming Approach to Adaptive Tatum Assignment for Rhythm Transcription. In Proceedings of the 1st IEEE Intl Wkshop on Multimedia Information Processing and, Irvine, CA, 577-584.

  3. Sources • Hillier, F. S. & G. J. Lieberman. Introduction to Operations Research. Chapter 11: Dynamic Programming, 7th, 8th, 9th editions, McGraw-Hill. • Dixon, S. & Widmer, G. (2005). MATCH: A Music Alignment Tool Chest. In Proceedings of the 6thIntl Conf on Music Information Retrieval, London, UK, 492-497. • Yang, A., E. Chew & A. Volk (2005). A Dynamic Programming Approach to Adaptive Tatum Assignment for Rhythm Transcription. In Proceedings of the 1st IEEE Intl Wkshop on Multimedia Information Processing and, Irvine, CA, 577-584.

  4. Shortest Path Problem • V = set of vertices (or nodes) • E = set of edges (or arcs), including s (source) and t (sink) • C = [cij] = set of arc costs • X = indicator variable for whether arc ij is used • Linear Programming Problem Formulation: Min Σijcijxij s.t. Σixij= 1 Σjxij= 1 0 ≤ xij≤ 1

  5. Gringott’s Wizarding Bank Exit to Charing Cross Road Quality Quidditch Supplies Entrance to Wizarding World 7 C M R Q E O G 2 2 5 Ollivander’s 5 4 3 4 1 1 7 4 Magical Menagerie The Leaky Cauldron Hogwarts (Shortest Path) Example • Determine which route from entrance to exit has the shortest distance

  6. Algorithm • At iteration i • Objective: find the i-th nearest node to the origin • Input: i-1 nearest node to the origin, shortest path and distance • Candidates for i-th nearest node: nearest unsolved node to one solved • Calculation for i-th nearest node: for each new candidate, add new edge to previously found shortest path to thei-1-th nearest node. Among the new candidates, the node with shortest distance becomes the i-th nearest node.

  7. Implementation

  8. 7 C R Q E O G M 2 2 5 5 4 3 4 1 1 7 4 Hogwarts (Shortest Path) Example 0

  9. Implementation

  10. 7 M Q E O G C R 0 2 2 5 5 5 4 3 4 1 1 7 4 Hogwarts (Shortest Path) Example

  11. Implementation

  12. 7 C R Q E O G M 2 2 5 5 4 3 4 1 1 7 4 Hogwarts (Shortest Path) Example 0 5 6

  13. Implementation

  14. 7 C R Q E O G M 2 2 5 5 4 3 4 1 1 7 4 Hogwarts (Shortest Path) Example 0 5 9 6

  15. Implementation

  16. 7 G R Q E O M C 2 2 5 5 4 3 4 1 1 7 4 Hogwarts (Shortest Path) Example 0 5 9 10 6

  17. Implementation

  18. 7 G R Q E O M C 2 2 5 5 4 3 4 1 1 7 4 Hogwarts (Shortest Path) Example 11 0 5 9 10 6

  19. Implementation

  20. Implementation

  21. 7 O R Q E G M C 2 2 5 5 4 3 4 1 1 7 4 Hogwarts (Shortest Path) Example 11 0 5 9 13 10 6

  22. Sources • Hillier, F. S. & G. J. Lieberman. Introduction to Operations Research. Chapter 11: Dynamic Programming, 7th, 8th, 9th editions, McGraw-Hill. • Dixon, S. & Widmer, G. (2005). MATCH: A Music Alignment Tool Chest. In Proceedings of the 6thIntl Conf on Music Information Retrieval, London, UK, 492-497. • Yang, A., E. Chew & A. Volk (2005). A Dynamic Programming Approach to Adaptive Tatum Assignment for Rhythm Transcription. In Proceedings of the 1st IEEE Intl Wkshop on Multimedia Information Processing and, Irvine, CA, 577-584.

  23. Dynamic Time Warping

  24. Euclidean Distance

  25. Sources • Hillier, F. S. & G. J. Lieberman. Introduction to Operations Research. Chapter 11: Dynamic Programming, 7th, 8th, 9th editions, McGraw-Hill. • Dixon, S. & Widmer, G. (2005). MATCH: A Music Alignment Tool Chest. In Proceedings of the 6thIntl Conf on Music Information Retrieval, London, UK, 492-497. • Yang, A., E. Chew & A. Volk (2005). A Dynamic Programming Approach to Adaptive Tatum Assignment for Rhythm Transcription. In Proceedings of the 1st IEEE Intl Wkshop on Multimedia Information Processing and, Irvine, CA, 577-584.

  26. Tatum Segmentation • Tatum = smallest perceptual time unit in music

  27. 140 Coarse Manual Segmentation bepthq3 120 son08_3 100 Tatum 80 60 40 0 100 200 300 400 Beats Example Results

  28. Segmentation • Example of tatum selection • Input: • Note onset times, O(1,n) = {O1, O2,…, On} • Output: • Segmentation points, S = {S1, S2,…,Sm} • Optimal tatum in each segment

  29. Remainder Squared Error p o(1) … o(j) 1 … j j+1 p

  30. ERR(.) The error incurred by a tatum assignment, p, for onsets O(i+ 1,k) is given by the remainder squared error (RSE) function: where oj (= Oj+1 – Oj) is the inter onset interval (IOI) between onsets j and j+1.

  31. Optimal Segmentation • Define OPT(k) to be the best segmentation (cost) for a given set of onsets O(1,k): where ERR(.) returns the error incurred by the best tatum assignment for the set of onsets O(i+1,k). k i i+1 … 1 …

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