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Space Efficient Alignment
Algorithms and Affine Gap
Penalties
Dr. Nancy Warter-Perez
Outline
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Algorithm complexity
Complexity of dynamic programming
alignment algorithms
Memory efficient algorithms
Hirschberg’s Divide and Conquer algorithm
Affine gap penalty
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Algorithm Complexity
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Indicates the space and time (computational) efficiency
of a program
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Generally written in Big-O notation
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O represents the complexity (order)
n represents the size of the data set
Examples
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Space complexity refers to how much memory is required to
execute the algorithm
Time complexity refers to how long it will take to execute
(compute) the algorithm
O(n) – “order n”, linear complexity
O(n2) – “order n squared”, quadratic complexity
Constants and lower orders ignored
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O(2n) = O(n) and O(n2 + n + 1) = O(n2)
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Complexity of Dynamic Programming
Algorithms for Global/Local Alignment
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Time complexity – O(m*n)
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For each cell in the score matrix, perform 3
operations
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Space complexity – O(m*n)
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Compute Up, Left, and Diagonal scores
O(3*m*n) = O(m*n)
Size of scoring matrix = m*n
Size of trace back matrix = m*n
O(2*m*n) = O(m*n)
Where, m and n are the lengths of the
sequences being aligned.
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Since m  n, O(n2 ) – quadratic complexity!
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Memory Requirements
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For a sequence of 200-500 amino acids
or nucleotides
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O(n2) = 5002 = 250,000
If store each score as a 32-bit value = 4
bytes, it requires 1,000,000 bytes to
represent the scoring matrix!
If store each trace back symbol as a
character (8-bit value), it requires 250,000
bytes to represent the trace back matrix
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Simple Improvement for
Scoring Matrix
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In reality, the space complexity of the
scoring matrix is only linear, i.e.,
O(2*min(m,n)) = O(min(m,n))
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O(min(m,n))  O(n) for sequences of
comparable lengths
2,000 bytes (instead of 1 million)
But, trace back still quadratic space
complexity
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Hirschberg’s “Divide and Conquer”
Space Efficient Algorithm
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Compute the score matrix(s)
between the source (0,0) and
(n, m/2). Save m/2 column of
s. Compute the reverse score
matrix (sreverse) between the
sink (n, m) and (0,m/2). Save
the m/2 column of sreverse.
Find middle (i, m/2) satisfies
max 0 in {s(i, m/2) + sreverse(ni, m/2)}
Recursively partition problem
into 2 subproblems
Source
(0,0)
i
m/2
m
m/2
m
middle
n
(0,0)
(0,0)
(n,m)
Sink
m/2
m
n
(0,0)
(n,m)
m
middle
middle
middle
n
(0,0)
n
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(n,m)
m
n
(0,0)
(n,m)
n
(n,m)
m
(n,m)
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Pseudo Code of SpaceEfficient Alignment Algorithm
Path (source, sink)
If source and sink are in consecutive columns
output the longest path from the source to the
sink
Else
middle middle vertex between source and sink
Path (source, middle)
Path (middle, sink)
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Complexity of Space-Efficient
Alignment Algorithm
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Time complexity
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Equal to the sum of the areas of the rectangles
Area + ½ Area + ¼ Area + …  2*Area
where, Area = n*m
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Space complexity
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O(2n*m) = O(n*m)
Quadratic time/computation complexity (same as before)
Need to save a column of s and sreverse for each computation (but can
discard after computing middle)
O(min(n,m)) – if m < n, switch the sequences (or save a row of s
and sreverse instead)
Linear space complexity!!
Reference:
http://www.csse.monash.edu.au/~lloyd/tildeAlgDS/Dynamic/Hirsch/
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Gap Penalties
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Gap penalties account for the introduction of
a gap - on the evolutionary model, an
insertion or deletion mutation - in both
nucleotide and protein sequences, and
therefore the penalty values should be
proportional to the expected rate of such
mutations.
http://en.wikipedia.org/wiki/Sequence_alignment#Assessment_of_significance
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Source: http://www.apl.jhu.edu/~przytyck/Lect03_2005.pdf
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Project Verification - Use EMBOSS
Pairwise Alignment Tool
http://www.ebi.ac.uk/Tools/emboss/align/index.html
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Project Verification – LALIGN
http://www.ch.embnet.org/software/LALIGN_form.html
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