12-DP Tutorial Chew - EECS€¦ · Chew:&Dynamic&Programming&Tutorial& 3/26/12...

Chew: Dynamic Programming Tutorial 3/26/12

QMUL: ELE021/ELED021/ELEM021 1

Dynamic Programming Tutorial

Elaine Chew

QMUL: ELE021/ELED021/ELEM021

26 March 2012

Sources

•  Hillier, F. S. & G. J. Lieberman. IntroducMon to OperaMons Research. Chapter 11: Dynamic Programming, 7th, 8th, 9th ediMons, McGraw-‐Hill.

•  Dixon, S. & Widmer, G. (2005). MATCH: A Music Alignment Tool Chest. In Proceedings of the 6th Intl Conf on Music InformaMon Retrieval, London, UK, 492-‐497.

•  Yang, A., E. Chew & A. Volk (2005). A Dynamic Programming Approach to AdapMve Tatum Assignment for Rhythm TranscripMon. In Proceedings of the 1st IEEE Intl Wkshop on MulMmedia InformaMon Processing and, Irvine, CA, 577-‐584.



Sources




Shortest Path Problem

•  V = set of verMces (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 FormulaMon:

Min Σij cijxij s.t. Σi xij = 1 Σj xij = 1 0 ≤ xij ≤ 1



Hogwarts (Shortest Path) Example

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

7

4

2 2

5

1 4

4

5

1 7

G

O E Q

R

M C

3

Gringoi’s Wizarding

Bank

Entrance to Wizarding World

Quality Quidditch Supplies Ollivander’s

Exit to Charing Cross

Road

The Leaky Cauldron

Magical Menagerie

Algorithm

•  At iteraMon i – ObjecMve: 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

– CalculaMon for i-‐th nearest node: for each new candidate, add new edge to previously found shortest path to the i-‐1-‐th nearest node. Among the new candidates, the node with shortest distance becomes the i-‐th nearest node.



ImplementaMon Itera&on, i Solved nodes

connected to unsolved

Closest connected unsolved node

Total distance involved

i-‐th nearest node

Minimum distance

Last connec&on


7

4

2 2

5

1 4

4

5

1 7

G

O E Q

R

M C

3

0








Minimum distance

Last connec&on

1 R Q 5 Q 5 Q


7

4

2 2

5

1 4

4

5

1 7

G

O E Q

R

M C

3

5

0








Minimum distance

Last connec&on

1 R Q 5 Q 5 R

2 Q R

C C

6 7

C 6 Q


7

4

2 2

5

1 4

4

5

1 7

G

O E Q

R

M C

3

5

0

6








Minimum distance

Last connec&on

1 R Q 5 Q 5 R

2 Q R

C C

6 7

C 6 Q

3 Q C

O O

9 9

O 9 Q C


7

4

2 2

5

1 4

4

5

1 7

G

O E Q

R

M C

3

5

0

6

9








Minimum distance

Last connec&on

1 R Q 5 Q 5 R

2 Q R

C C

6 7

C 6 Q

3 Q C

O O

9 9

O 9 Q C

4 Q C O

G M M

12 10 10

M 10 C O


7

4

2 2

5

1 4

4

5

1 7

G

O E Q

R

M C

3

5

0

6

9

10








Minimum distance

Last connec&on

1 R Q 5 Q 5 R

2 Q R

C C

6 7

C 6 Q

3 Q C

O O

9 9

O 9 Q C

4 Q C O

G M M

12 10 10

M 10 C O

5 Q O M

G G E

12 11 14

G 11 O


7

4

2 2

5

1 4

4

5

1 7

G

O E Q

R

M C

3

5

0

6

9

10

11








Minimum distance

Last connec&on

1 R Q 5 Q 5 R

2 Q R

C C

6 7

C 6 Q

3 Q C

O O

9 9

O 9 Q C

4 Q C O

G M M

12 10 10

M 10 C O

5 Q O M

G G E

12 11 14

G 11 O






Minimum distance

Last connec&on

5 Q O M

G G E

12 11 14

G 11 O

6 O M G

E E E

14 14 13

E 13 G




7

4

2 2

5

1 4

4

5

1 7

G

O E Q

R

M C

3

5

0

6

9

10

11

13

Sources






Dynamic Time Warping

Euclidean Distance



(i,j)



Sources




Tatum SegmentaMon

•  Tatum = smallest perceptual Mme unit in music



Example Results

40

60

80

100

120

140

0 100 200 300 400 Beats

Tatum

Coarse Manual Segmentation bepthq3 son08_3

SegmentaMon

•  Example of tatum selecMon •  Input:

– Note onset Mmes, O(1,n) = {O1, O2,…, On}

•  Output: – SegmentaMon points, S = {S1, S2,…,Sm} – OpMmal tatum in each segment



Remainder Squared Error

o(1) … o(j)

p

1 … j j+1

p

ERR(.)

The error incurred by a tatum assignment, p, for onsets O(i+ 1,k) is given by the remainder squared error (RSE) funcMon:

where oj (= Oj+1 – Oj) is the inter onset interval (IOI) between onsets j and j+1.

€

e(p,i + 1,k) =ojp−ojp

+12

⎢

⎣ ⎢

⎥

⎦ ⎥

⎛

⎝ ⎜

⎞

⎠ ⎟

j =1

k − i −1

∑2

,

€

ERR(O(i + 1,k)) =pmin e(p,i + 1,k).



OpMmal SegmentaMon

•  Define OPT(k) to be the best segmentaMon (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). €

OPT(k) =1≤ i ≤kmin {OPT(i)+ ERR(O(i + 1,k))},

k i i+1 … 1 …

Date post:	27-Apr-2020
Category:	Documents
Upload:	others
View:	15 times
Download:	0 times

12-DP Tutorial Chew - EECS€¦ · Chew:&Dynamic&Programming&Tutorial& 3/26/12...

Documents