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Dynamic Programming Tutorial

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Dynamic Programming Tutorial

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Dynamic Programming Tutorial

Elaine Chew

QMUL: ELE021/ELED021/ELEM021

26 March 2012

- 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.

- 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.

- 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

Gringott’s

Wizarding Bank

Exit to

Charing Cross Road

Quality

Quidditch

Supplies

Entrance to Wizarding World

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Ollivander’s

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Magical Menagerie

The Leaky Cauldron

- Determine which route from entrance to exit has the shortest distance

- 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.

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- 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.

- Tatum = smallest perceptual time unit in music

140

Coarse Manual

Segmentation

bepthq3

120

son08_3

100

Tatum

80

60

40

0

100

200

300

400

Beats

- 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

p

o(1) … o(j)

1 … j j+1

p

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.

- 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 …