Jason J. Siwula, PE – Safety Engineer HSIP HORIZONTAL ALIGNMENT SIGNING.
SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity:...
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Transcript of SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity:...
![Page 1: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/1.jpg)
SMAWK
![Page 2: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/2.jpg)
REVISE
![Page 3: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/3.jpg)
Global alignment (Revise)
ag
a
g
c
a
t
c
agcagcaa 31
1
2
3
5
4 65 7 80
7
6
8
2
4
Alignment graph for S = aacgacga, T = ctacgaga
Complexity: O(n2)
V(i,j) = max {V(i-1,j-1) + (S[i], T[j]),V(i-1,j) + (S[i], -),V(i,j-1) + (-, T[j])
}
![Page 4: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/4.jpg)
DIST and OUT matrix (Revise)
O
g
a
gca
G0
20
1
2 3 4
13
4
55
I
DIST matrix OUT matrixI (input borders)
Block – sub-sequences “acg”, “ag”
0 1 2 3 4 5
I0 0 -1 -2 -3 △ △
I1 -1 -1 -2 -1 -3 △
I2 -2 0 0 1 -1 -3
I3 △ -2 -2 0 -2 -2
I4 △ △ -2 0 -1 -1
I5 △ △ △ -2 -1 0
0 1 2 3 4 5
1 0 -1 -2 - -
1 1 0 1 -1 -
1 3 3 4 2 0
-12 0 0 2 0 0
-13 -13 -1 1 0 0
-14 -14 -14 1 2 3
I0=1
I1=2
I2=3
I3=2
I4=1
I5=3
O0 O1 O2 O3 O4 O5
1 3 3 4 2 3
max col
![Page 5: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/5.jpg)
Compute O without explicit OUT
O
g
a
gca
G0
20
1
2 3 4
13
4
55
I
DIST matrix I (input borders)
Block – sub-sequences “acg”, “ag”
0 1 2 3 4 5
I0 0 -1 -2 -3 △ △
I1 -1 -1 -2 -1 -3 △
I2 -2 0 0 1 -1 -3
I3 △ -2 -2 0 -2 -2
I4 △ △ -2 0 -1 -1
I5 △ △ △ -2 -1 0
I0=1
I1=2
I2=3
I3=2
I4=1
I5=3
O0 O1 O2 O3 O4 O5
1 3 3 4 2 3
SMAWK
![Page 6: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/6.jpg)
• Aggarwal, Park and Schmidt observed that DIST and OUT matrices are Monge arrays.
• Definition: a matrix M[0…m,0…n] is totally monotone if either condition 1 or 2 below holds for all a,b=0…m; c,d=0…n; a<b and c<d1. Convex condition:
M[a,c]M[b,c]M[a,d]M[b,d].2. Concave condition:
M[a,c]M[b,c]M[a,d]M[b,d].
![Page 7: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/7.jpg)
SMAWK
• Aggarwal et. al. gave a recursive algorithm, called SMAWK, which can find
all row and column maxima of a totally monotone matrixby querying only O(n) elements of the matrix.
![Page 8: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/8.jpg)
Presentation Outline
• What is Monge arrays?– Monge Totally monotone
• Why DIST alignment matrix is Monge arrays?
• How to compute totally monotone arrays efficiently?– SMAWK
• Given a totally monotone arrays• Compute all columns maxima in O(n)
![Page 9: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/9.jpg)
MONGE AND TOTALLY MONOTONE PROPERTIES
![Page 10: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/10.jpg)
Monge
• A matrix M[0…m, 0…n] is Monge if either condition 1 or 2 below holds for all a,b=0…m; c,d=0…n; a<b and c<d 1. M[a, c] + M[b, d] M[a, d] + M[b, c]2. M[a, c] + M[b, d] M[a, d] + M[b, c]
c d z
a M[a,c] M[a,d] …
b M[b,c] M[b,d]x … …
![Page 11: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/11.jpg)
Totally monotone
• A matrix M[0…m, 0…n] is totally monotone if either condition 1 or 2 below holds for all a,b=0…m; c,d=0…n; a<b and c<d 1. Convex condition:
M[a,c]M[b,c] M[a,d]M[b,d]2. Concave condition:
M[a,c]M[b,c] M[a,d]M[b,d]• Monge Totally monotone
c d z
a M[a,c] M[a,d] …
b M[b,c] M[b,d]x … …
![Page 12: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/12.jpg)
Intuition
• Monge: Quadrangle inequality:
a
cb
d
xz
c d z
a M[a,c] M[a,d] …
b M[b,c] M[b,d]
x … …
M[a, c] + M[b, d] M[a, d] + M[b, c]
![Page 13: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/13.jpg)
History
• Computational Geometry• All nearest neighbor problem– Shamos and Hoey proved (n log n) in 1975
• All farthest neighbor problem– F.P.Reparata proved (n log n) in 1977
• All farthest neighbor problem in convex polygon– Lee and Preparata proved O(n) in 1978
![Page 14: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/14.jpg)
SMAWK
• Aggarwal et.al. proved O(n) for farthest in convex polygon in 1987
• Aggarwal et. al. gave a recursive algorithm, called SMAWK, which can find
all row and column maxima of a totally monotone matrixby querying only O(n) elements of the matrix.
![Page 15: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/15.jpg)
DIST AND OUT MATRICES
![Page 16: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/16.jpg)
• Assumption– row and column maxima of a
totally monotone matrixcan be computed in O(n)
• Why DIST and OUT matrices of the alignment problem is totally monotone?
![Page 17: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/17.jpg)
DIST and OUT matrix (Revise)
O
g
a
gca
G0
20
1
2 3 4
13
4
55
I
DIST matrix OUT matrixI (input borders)
Block – sub-sequences “acg”, “ag”
0 1 2 3 4 5
I0 0 -1 -2 -3 △ △
I1 -1 -1 -2 -1 -3 △
I2 -2 0 0 1 -1 -3
I3 △ -2 -2 0 -2 -2
I4 △ △ -2 0 -1 -1
I5 △ △ △ -2 -1 0
0 1 2 3 4 5
1 0 -1 -2 - -
1 1 0 1 -1 -
1 3 3 4 2 0
-12 0 0 2 0 0
-13 -13 -1 1 0 0
-14 -14 -14 1 2 3
I0=1
I1=2
I2=3
I3=2
I4=1
I5=3
O0 O1 O2 O3 O4 O5
1 3 3 4 2 3
max col
![Page 18: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/18.jpg)
Compute O without explicit OUT
O
g
a
gca
G0
20
1
2 3 4
13
4
55
I
DIST matrix I (input borders)
Block – sub-sequences “acg”, “ag”
0 1 2 3 4 5
I0 0 -1 -2 -3 △ △
I1 -1 -1 -2 -1 -3 △
I2 -2 0 0 1 -1 -3
I3 △ -2 -2 0 -2 -2
I4 △ △ -2 0 -1 -1
I5 △ △ △ -2 -1 0
I0=1
I1=2
I2=3
I3=2
I4=1
I5=3
O0 O1 O2 O3 O4 O5
1 3 3 4 2 3
SMAWK
![Page 19: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/19.jpg)
DIST is Monge
O
g
a
gca
G0
20
1
2 3 4
13
4
55
I
![Page 20: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/20.jpg)
DIST is Monge array
• Monge• M[a, c] + M[b, d] M[a, d] + M[b, c]
• Totally monotone by Concave condition:• M[a,c]M[b,c] M[a,d]M[b,d]
![Page 21: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/21.jpg)
Comment on this approach
• Advantages– Easy to parallelize– Easy to combine
• Disadvantages– Need to compute/keep more information
![Page 22: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/22.jpg)
Applications
• Parallel sequence alignment– O(log m log n) time – Using O(m n / log m) processors (CREW PRAM)
• Best non-overlapping alignment score– O(n2 log2 n) time
• Tandem approximate repeat– O(n2 log n) time
• Common Substring Alignment
![Page 23: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/23.jpg)
SMAWK
![Page 24: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/24.jpg)
0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 1512 21 35 48 65 76 85 105 123 1303 13 26 35 51 58 67 86 100 1044 10 20 28 42 48 56 75 86 885 20 29 33 44 49 55 73 82 806 13 21 24 35 39 44 59 65 597 19 25 28 38 42 44 57 61 528 35 37 40 48 48 49 62 62 499 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 2911 58 56 54 55 47 41 50 47 2912 66 64 61 61 51 44 52 45 2413 82 76 72 70 56 49 55 46 2314 99 91 83 80 63 56 59 46 2015 124 116 107 100 80 71 72 58 2816 133 125 113 106 86 75 74 59 2517 156 146 131 120 97 84 80 65 3118 178 164 146 135 110 96 92 73 39
[a b][c d]
Find all column mimimas of the following totally monotone arrays
b < d a < cb = d a c
![Page 25: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/25.jpg)
0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 1512 21 35 48 65 76 85 105 123 1303 13 26 35 51 58 67 86 100 1044 10 20 28 42 48 56 75 86 885 20 29 33 44 49 55 73 82 806 13 21 24 35 39 44 59 65 597 19 25 28 38 42 44 57 61 528 35 37 40 48 48 49 62 62 499 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 2911 58 56 54 55 47 41 50 47 2912 66 64 61 61 51 44 52 45 2413 82 76 72 70 56 49 55 46 2314 99 91 83 80 63 56 59 46 2015 124 116 107 100 80 71 72 58 2816 133 125 113 106 86 75 74 59 2517 156 146 131 120 97 84 80 65 3118 178 164 146 135 110 96 92 73 39
[a b][c d]
a > c b > da = c b d
Find all column mimimas of the following totally monotone arrays
b < d a < cb = d a c
![Page 26: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/26.jpg)
0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 1512 21 35 48 65 76 85 105 123 1303 13 26 35 51 58 67 86 100 1044 10 20 28 42 48 56 75 86 885 20 29 33 44 49 55 73 82 806 13 21 24 35 39 44 59 65 597 19 25 28 38 42 44 57 61 528 35 37 40 48 48 49 62 62 499 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 2911 58 56 54 55 47 41 50 47 2912 66 64 61 61 51 44 52 45 2413 82 76 72 70 56 49 55 46 2314 99 91 83 80 63 56 59 46 2015 124 116 107 100 80 71 72 58 2816 133 125 113 106 86 75 74 59 2517 156 146 131 120 97 84 80 65 3118 178 164 146 135 110 96 92 73 39
[a b][c d]
a > c b > da = c b d
b < d a < cb = d a c
Observation 1
![Page 27: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/27.jpg)
0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 1512 21 35 48 65 76 85 105 123 1303 13 26 35 51 58 67 86 100 1044 10 20 28 42 48 56 75 86 885 20 29 33 44 49 55 73 82 806 13 21 24 35 39 44 59 65 597 19 25 28 38 42 44 57 61 528 35 37 40 48 48 49 62 62 499 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 2911 58 56 54 55 47 41 50 47 2912 66 64 61 61 51 44 52 45 2413 82 76 72 70 56 49 55 46 2314 99 91 83 80 63 56 59 46 2015 124 116 107 100 80 71 72 58 2816 133 125 113 106 86 75 74 59 2517 156 146 131 120 97 84 80 65 3118 178 164 146 135 110 96 92 73 39
[a b][c d]
a > c b > da = c b d
Observation 2
b < d a < cb = d a c
![Page 28: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/28.jpg)
0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 151
2 21 35 48 65 76 85 105 123 130
3 13 26 35 51 58 67 86 100 104
4 10 20 28 42 48 56 75 86 88
5 20 29 33 44 49 55 73 82 80
6 13 21 24 35 39 44 59 65 59
7 19 25 28 38 42 44 57 61 52
8 35 37 40 48 48 49 62 62 49
9 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 29
11 58 56 54 55 47 41 50 47 29
12 66 64 61 61 51 44 52 45 24
13 82 76 72 70 56 49 55 46 23
14 99 91 83 80 63 56 59 46 20
15 124 116 107 100 80 71 72 58 28
16 133 125 113 106 86 75 74 59 25
17 156 146 131 120 97 84 80 65 31
18 178 164 146 135 110 96 92 73 39
[a b][c d]
a > c b > da = c b d
• SMAWK is a recursive algorithm of 2 steps– REDUCE– INTERPOLATE
b < d a < cb = d a c
![Page 29: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/29.jpg)
0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 151
2 21 35 48 65 76 85 105 123 130
3 13 26 35 51 58 67 86 100 104
4 10 20 28 42 48 56 75 86 88
5 20 29 33 44 49 55 73 82 80
6 13 21 24 35 39 44 59 65 59
7 19 25 28 38 42 44 57 61 52
8 35 37 40 48 48 49 62 62 49
9 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 29
11 58 56 54 55 47 41 50 47 29
12 66 64 61 61 51 44 52 45 24
13 82 76 72 70 56 49 55 46 23
14 99 91 83 80 63 56 59 46 20
15 124 116 107 100 80 71 72 58 28
16 133 125 113 106 86 75 74 59 25
17 156 146 131 120 97 84 80 65 31
18 178 164 146 135 110 96 92 73 39
[a b][c d]
a > c b > da = c b d
• SMAWK is a recursive algorithm of 2 steps– REDUCE– INTERPOLATE
• REDUCE removes rows• INTERPOLATE removes
half of the columns
b < d a < cb = d a c
![Page 30: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/30.jpg)
0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 1512 21 35 48 65 76 85 105 123 1303 13 26 35 51 58 67 86 100 1044 10 20 28 42 48 56 75 86 885 20 29 33 44 49 55 73 82 806 13 21 24 35 39 44 59 65 597 19 25 28 38 42 44 57 61 528 35 37 40 48 48 49 62 62 499 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 2911 58 56 54 55 47 41 50 47 2912 66 64 61 61 51 44 52 45 2413 82 76 72 70 56 49 55 46 2314 99 91 83 80 63 56 59 46 2015 124 116 107 100 80 71 72 58 2816 133 125 113 106 86 75 74 59 2517 156 146 131 120 97 84 80 65 3118 178 164 146 135 110 96 92 73 39
REDUCE
![Page 31: SMAWK. REVISE Global alignment (Revise) Alignment graph for S = aacgacga, T = ctacgaga Complexity: O(n 2 ) V(i,j) = max { V(i-1,j-1) + (S[i], T[j]),](https://reader030.fdocuments.net/reader030/viewer/2022032803/56649e2b5503460f94b191e5/html5/thumbnails/31.jpg)
0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 1512 21 35 48 65 76 85 105 123 1303 13 26 35 51 58 67 86 100 1044 10 20 28 42 48 56 75 86 885 20 29 33 44 49 55 73 82 806 13 21 24 35 39 44 59 65 597 19 25 28 38 42 44 57 61 528 35 37 40 48 48 49 62 62 499 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 2911 58 56 54 55 47 41 50 47 2912 66 64 61 61 51 44 52 45 2413 82 76 72 70 56 49 55 46 2314 99 91 83 80 63 56 59 46 2015 124 116 107 100 80 71 72 58 2816 133 125 113 106 86 75 74 59 2517 156 146 131 120 97 84 80 65 3118 178 164 146 135 110 96 92 73 39
REDUCE
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REDUCE
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REDUCE
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REDUCE
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REDUCE
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REDUCE
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REDUCE
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REDUCE
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INTERPOLATE
Remove all odd indexed colums
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INTERPOLATE
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RECURSIVE
Find all row minima
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4 10 20 286 24 35 397 42
10 35 33 44 43 2911 2912 2413 2314 2016 25
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0 1 2 3 4 5 6 7 8 9
1 25 42 57 78 90 103 123 142 1512 21 35 48 65 76 85 105 123 1303 13 26 35 51 58 67 86 100 1044 10 20 28 42 48 56 75 86 885 20 29 33 44 49 55 73 82 806 13 21 24 35 39 44 59 65 597 19 25 28 38 42 44 57 61 528 35 37 40 48 48 49 62 62 499 37 36 37 42 39 39 51 50 37
10 41 39 37 42 35 33 44 43 2911 58 56 54 55 47 41 50 47 2912 66 64 61 61 51 44 52 45 2413 82 76 72 70 56 49 55 46 2314 99 91 83 80 63 56 59 46 2015 124 116 107 100 80 71 72 58 2816 133 125 113 106 86 75 74 59 2517 156 146 131 120 97 84 80 65 3118 178 164 146 135 110 96 92 73 39
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APPROXIMATE TANDEM REPEATApplication of DIST and SMAWK
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Tandem repeat
• IRQI QLWLR QIWIR LRQL
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Social City
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Observation
• Approximate tandem repeat– With the Mid-point c
– Alignments• start at column c• end at row c
c
c
0 n
n
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• 4 cases– Cross column n/2– Cross row n/2– In side sub-triangle
[0,n/2]– In side sub-triangle
[n/2,n]
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Algorithm
1. Find all repeats that cross– row n/2– column n/2
2. Recursively solve the – sub-array
[0..n/2, 0..n/2]– sub-array
[n/2..n, n/2..n]
c10n/2c2
c1
c2
c3
c3
n/2
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Cross column n/2
• Combine– Best path from column c
to (k,n/2)– Best path from (k,n/2) to
row c
c
c
0 n
n
n/2
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Cross column n/2
• Sub-problems:– DIST_col(c,n/2)[i,j]
– DIST_row(c,n/2)[i,j]
c10n/2c2
c1
c2
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Cross column n/2
• DIST_col(c,n/2)[i,j] : O(n3) words
• Encode in array of binary trees • Using O(n2 log n) words • B[j,c] is a binary tree • B[j,c](i) is a leaf of the tree • Read an entry of DIST_col(c,n/2)[i,j] in O(log n)
c10n/2c2
c1
c2
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Algorithm1. Find all repeats O(n2 logn)
– cross row n/2– column n/2
1. Recursively solve the – sub-array
[0..n/2, 0..n/2]– sub-array
[n/2..n, n/2..n]
c10n/2c2
c1
c2
c3
c3
n/2
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References
• Aggarwal, A. and Park, J. Notes on Searching in Multidimensional Monotone Arrays. IEEE
• Jeanette P. Schmidt. All highest scoring paths in weighted grid graphs and their application to finding all approximate repeats in strings. SIAM.
• Lawrence L. Larmore. The SMAWK Algorithm. UNLV.• Apostolico, A. and Atallah, M.J. and Larmore, L.L. and
McFaddin, S.. Efficient Parallel Algorithms for String Editing and Related Problems. SIAM J. Comput.
• Landau, G.M. and Ziv-Ukelson, M. On the Common Substring Alignment Problem. J. of Algorithms