Manuscript on bioRxiv
We also have a Poster
LEAP: A Generalization of
the Landau-Vishkin Algorithm
with Custom Gap Penalties
Hongyi Xin1, Jeremie Kim1, Sunny Nahar1,4, Can Alkan3 and Onur Mutlu1,2
1Carnegie
Mellon University
2ETH Zürich
3Bilkent University
4Google
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Approximate String Matching Problem (ASM)
• A fundamental problem in Bioinformatics
• DNA and protein sequence mapping and comparisons
• Computes a similarity score between two strings
• Differences include mismatches as well as indels
• Common scoring schemes
• Edit-distance (mismatch and indels have same penalty score)
• Affine gapping (gap opening is more penalized than gap extension)
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The Landau-Vishkin (1989) Algorithm
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• Dynamic-programming algorithm for editdistance
• Diagonally oriented
• Find out the furthest nodes reachable at e
edit-distance
• Use nodes at e to compute the furthest nodes
at e+1 edit-distance
• Procedure (3 steps)
• Pick the furthest starting position at e (initiation)
• Traverse diagonally until hitting a mismatch
(elongation)
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• Notify neighbor diagonals with new potential
furthest starting positions at e+1 (termination)
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• Furthest node reachable at e
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Benefit of the Landau-Vishkin Algorithm
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• Compared against Ukkonen’s banded algorithm
• Less bookkeeping
• Only stores starting positions and elongation length
• Compute fewer nodes
• Only the start and end positions in the diagonal
• Less compute per node
• Elongation only checks for the next mismatch
• Termination updates the furthest node for next edit
iteration
• Only one termination per edit per diagonal
• Less work! (although still being O(k*m) )
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Limitations
• Landau-Vishkin was proposed for edit-distance
• Correctness for custom gap penalties has not been proven
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Our Contribution
• Prove that Landau-Vishkin also works for custom gap penalties
• Including affine-gapping
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Problem Setup
• Convert the process of computing the DP tableA into
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• Nodes in the table  vertices
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• Horizontal, vertical and diagonal transitionsA  1edges
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• Mismatches, indels  weights on the edges
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• Goal: find a path from start to
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Problem Statement of LEAP
• Toad swims in a swimming pool with hurdles in it
swim
stride
leap
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Three Theorems
• Delaying a leap in the path until the next hurdle does not add cost to
the path
• There exists an optimal path where the toad only leaps before a
hurdle
• Landau-Vishkin algorithm finds such optimal path
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Pseudo-Proof 1
• Blue path has the same cost as the red path
• No change in leaps, no additional strides
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Pseudo-Proof 2
• Must exist an optimal path where leaps are right before hurdles
• Iteratively use theorem 1
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Pseudo-Proof 3
• Can be proved using induction
• Energy cost monotonically increases along the path
• Check out our paper/poster for further details!
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Further Optimizations
• Elongation finds the next mismatch
• Elongation can be sped up using the de Bruijn sequence technique
• Think of it as a hashing function
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No need to iteratively
check for letter matches. Use bit-parallel operations instead!
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* De Bruijn sequence
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lookup
XOR
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Results
• Two implementations
• LEAP (without De Bruijn
sequence optimization)
• LEAP-BV (with optimization)
• Compared to 3 state-of-theart implementations
• Myer’s bit vector  SeqAn
• NW-SIMD
• Canonical LV (equivalent to
LEAP for Levenshtein dist.)
Takeaway: LEAP-BV attains as much as 7.4x speedup against Myers’ bit-vector and
32x speedup against NW-SIMD
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Conclude
• We prove that the Landau-Vishkin algorithm can be extended to
support custom gap penalties
• We further optimized the Landau-Vishkin method
• We achieved up to 7.4x speed up over the state-of-the-art
implementation
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Special
thanks to:
Acknowledgement
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Q&A
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