[PPT] - Locality sensitive hashing for the edit distance Guillaume Marc PowerPoint Presentation

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Locality sensitive hashing for the edit distance

Guillaume Marc ¸ais, Dan DeBlasio, Prashant Pandey, Carl Kingsford

Carnegie Mellon University

SLIDE 2

Overlap computation

Compute overlaps between reads (HMAP)
Instance of “Nearest Neighbor Problem”

for edit distance

Use multiple hash tables
Need meaningful hash collisions

Reads Overlap?

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SLIDE 3

Overlap computation

Compute overlaps between reads (HMAP)
Instance of “Nearest Neighbor Problem”

for edit distance

Use multiple hash tables
Need meaningful hash collisions

Reads Overlap?

1

SLIDE 4

Overlap computation

Compute overlaps between reads (HMAP)
Instance of “Nearest Neighbor Problem”

for edit distance

Use multiple hash tables
Need meaningful hash collisions

Reads Overlap? Hash Tables

1

SLIDE 5

Overlap computation

Compute overlaps between reads (HMAP)
Instance of “Nearest Neighbor Problem”

for edit distance

Use multiple hash tables
Need meaningful hash collisions

Reads Overlap? Hash Tables

1

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Locality Sensitive Hashing

Pick h at random from H: Locality sensitive hash family Family H of hash functions where similar elements are more likely to have the same value than distant elements.

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Locality Sensitive Hashing

Pick h at random from H: Locality sensitive hash family Family H of hash functions where similar elements are more likely to have the same value than distant elements.

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Locality Sensitive Hashing

The family H is sensitive for distance D if there exists d1 < d2, p1 > p2 such that for all x, y ∈ U D(x, y) ≤ d1 = ⇒ Pr

h∈H[h(x) = h(y)] ≥ p1

D(x, y) ≥ d2 = ⇒ Pr

h∈H[h(x) = h(y)] ≤ p2

Low distance ⇐

⇒ High collisions

High distance ⇐

⇒ Low collisions Locality sensitive hash family Family H of hash functions where similar elements are more likely to have the same value than distant elements.

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SLIDE 9

LSH for the edit distance How to design an LSH for edit distance?

LSH for Jaccard distance (minHash) used as proxy
Jaccard distance is significantly different than edit distance

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LSH for the edit distance How to design an LSH for edit distance?

LSH for Jaccard distance (minHash) used as proxy
Jaccard distance is significantly different than edit distance

3

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LSH for the edit distance How to design an LSH for edit distance?

LSH for Jaccard distance (minHash) used as proxy
Jaccard distance is significantly different than edit distance

3

SLIDE 12

Jaccard distance

Jaccard distance between sets A, B: J(A, B) = 1 − |A ∩ B| |A ∪ B|

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Jaccard distance

Jaccard distance between sets A, B: J(A, B) = 1 − |A ∩ B| |A ∪ B| Jaccard between sequences x, y: Jaccard distance of their k-mer sets J(x, y) = J(K(x), K(y))

Low D(x, y) =

⇒ Low J(x, y)

High D(x, y)
=

⇒ High J(x, y)

4

SLIDE 14

Jaccard ignores k-mer repetition

x =

n−k

AAAAAAAAAAAAAAA

k

CCCCC y = AAAAA

k

CCCCCCCCCCCCCCC

n−k

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Jaccard ignores k-mer repetition

x =

n−k

AAAAAAAAAAAAAAA

k

CCCCC → {AAAAA, AAAAC, AAACC, AACCC, ACCCC, CCCCC} y = AAAAA

k

CCCCCCCCCCCCCCC

n−k

→ {AAAAA, AAAAC, AAACC, AACCC, ACCCC, CCCCC}

5

SLIDE 16

Jaccard ignores k-mer repetition

x =

n−k

AAAAAAAAAAAAAAA

k

CCCCC → {AAAAA, AAAAC, AAACC, AACCC, ACCCC, CCCCC} y = AAAAA

k

CCCCCCCCCCCCCCC

n−k

→ {AAAAA, AAAAC, AAACC, AACCC, ACCCC, CCCCC} Jaccard distance J(x, y) = 0 Edit distance D(x, y) ≥ 1 − 2k

n

Identical k-mer content and high edit distance

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SLIDE 17

Weighted Jaccard handles repetitions

x =

n−k

AAAAAAAAAAAAAAA

k

CCCCC →

(AAAAA,1),(AAAAA,2),...,(AAAAA,11)

(AAAAC,1),(AAACC,1),(AACCC,1),(ACCCC,1),(CCCCC,1)

y = AAAAA

k

CCCCCCCCCCCCCCC

n−k

→

(AAAAA,1),(AAAAC,1),(AAACC,1),(AACCC,1),(ACCCC,1),

(CCCCC,1),(CCCCC,2),...,(CCCCC,11)

Weighted Jaccard Jw(x, y) = 1 − k+2

n

Edit distance D(x, y) ≥ 1 − 2k

n

Weighted Jaccard = Jaccard for multi-sets

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Jaccard and weighted Jaccard ignore relative order

x = CCCCACCAACACAAAACCC y = AAAACACAACCCCACCAAA

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Jaccard and weighted Jaccard ignore relative order

x = CCCCACCAACACAAAACCC → AAAA,AAAC,AACA,AACC,ACAA,ACAC,ACCA,ACCC

CAAA,CAAC,CACA,CACC,CCAA,CCAC,CCCA,CCCC

y = AAAACACAACCCCACCAAA

→ AAAA,AAAC,AACA,AACC,ACAA,ACAC,ACCA,ACCC

CAAA,CAAC,CACA,CACC,CCAA,CCAC,CCCA,CCCC

x, y: de Bruijn sequences,

contain all 16 possible 4-mers once

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Jaccard and weighted Jaccard ignore relative order

x = CCCCACCAACACAAAACCC → AAAA,AAAC,AACA,AACC,ACAA,ACAC,ACCA,ACCC

CAAA,CAAC,CACA,CACC,CCAA,CCAC,CCCA,CCCC

y = AAAACACAACCCCACCAAA

→ AAAA,AAAC,AACA,AACC,ACAA,ACAC,ACCA,ACCC

CAAA,CAAC,CACA,CACC,CCAA,CCAC,CCCA,CCCC

x, y: de Bruijn sequences,

contain all 16 possible 4-mers once J(x, y) = Jw(x, y) = 0 D(x, y) = 0.63

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SLIDE 21

Jaccard is different from edit distance

Unlike edit distance, Jaccard is insensitive to:

1. k-mer repetitions
2. relative positions of k-mers

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SLIDE 22

OMH: Order Min Hash

minHash is an LSH for Jaccard
OMH is a refinement of minHash
OMH is sensitive to
repeated k-mers
relative order of k-mers

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minHash & OMH sketches

x = AGTTGAGCGGAAGGTG, k = 2

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minHash & OMH sketches

x = AGTTGAGCGGAAGGTG, k = 2

AG GT TT TG GA GC CG GG AA AG GT TT TG GA GC CG GG AA AC AT CA CT GT TA TC

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SLIDE 25

minHash & OMH sketches

x = AGTTGAGCGGAAGGTG, k = 2, m = 6

AG GT TT TG GA GC CG GG AA AG GT TT TG GA CG GG AA

2

AG GT TT TG GA GC CG GG AA

3

AG GT TT TG GA GC CG GG AA

4

AG GT TT TG GA GC CG GG AA

5

AG GT TT TG GA GC CG GG AA

6 1

10

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minHash & OMH sketches

x = AGTTGAGCGGAAGGTG, k = 2, m = 6

AG GT TT TG GA GC CG GG AA AG GT TT TG GA CG GG AA

2

AG GT TT TG GA GC CG GG AA

3

AG GT TT TG GA GC CG GG AA

4

AG GT TT TG GA GC CG GG AA

5

AG GT TT TG GA GC CG GG AA

6 1

10

SLIDE 27

minHash & OMH sketches

x = AGTTGAGCGGAAGGTG, k = 2, m = 6

AG GT TT TG GA GC CG GG AA AG GT TT TG GA CG GG AA

2

AG GT TT TG GA GC CG GG AA

3

AG GT TT TG GA GC CG GG AA

4

AG GT TT TG GA GC CG GG AA

5

AG GT TT TG GA GC CG GG AA

6 1

AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG,11 AG, 0 GT, 1 TG,14 GA, 9 GG,12

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SLIDE 28

minHash & OMH sketches

x = AGTTGAGCGGAAGGTG, k = 2, m = 6

AG GT TT TG GA GC CG GG AA AG GT TT TG GA CG GG AA

2

AG GT TT TG GA GC CG GG AA

3

AG GT TT TG GA GC CG GG AA

4

AG GT TT TG GA GC CG GG AA

5

AG GT TT TG GA GC CG GG AA

6 1

AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG,11 AG, 0 GT, 1 TG,14 GA, 9 GG,12

2

AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG,11 AG, 0 GT, 1 TG,14 GA, 9 GG,12

3

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

4

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12 AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

5

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

6 1

10

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minHash & OMH sketches

x = AGTTGAGCGGAAGGTG, k = 2, m = 6

AG GT TT TG GA GC CG GG AA AG GT TT TG GA CG GG AA

2

AG GT TT TG GA GC CG GG AA

3

AG GT TT TG GA GC CG GG AA

4

AG GT TT TG GA GC CG GG AA

5

AG GT TT TG GA GC CG GG AA

6 1

AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG,11 AG, 0 GT, 1 TG,14 GA, 9 GG,12

2

AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG,11 AG, 0 GT, 1 TG,14 GA, 9 GG,12

3

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

4

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12 AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

5

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

6 1

10

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minHash & OMH sketches

x = AGTTGAGCGGAAGGTG, k = 2, m = 6, ℓ = 2

AG GT TT TG GA GC CG GG AA AG GT TT TG GA CG GG AA

2

AG GT TT TG GA GC CG GG AA

3

AG GT TT TG GA GC CG GG AA

4

AG GT TT TG GA GC CG GG AA

5

AG GT TT TG GA GC CG GG AA

6 1

AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG,11 AG, 0 GT, 1 TG,14 GA, 9 GG,12

2

AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG,11 AG, 0 GT, 1 TG,14 GA, 9 GG,12

3

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

4

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12 AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

5

AG,11 AG, 5 GT,13 TT, 2 TG, 3 GA, 4 GC, 6 CG, 7 GG, 8 AA,10 AG, 0 GT, 1 TG,14 GA, 9 GG,12

6 1

GA AG GG GC AG TG

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SLIDE 31

OMH is a LSH for edit distance

Theorem: OMH is a LSH for edit distance There exists (d1, d2, p1, p2) such that OMH is sensitive for the edit distance.

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Conclusion

OMH:
improvement on minHash
easy to compute
locality sensitive for edit distance
LSH for other alignment scores?
Smallest “gap” achievable?

12

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Conclusion

OMH:
improvement on minHash
easy to compute
locality sensitive for edit distance
LSH for other alignment scores?
Smallest “gap” achievable?

12

SLIDE 34

Conclusion

OMH:
improvement on minHash
easy to compute
locality sensitive for edit distance
LSH for other alignment scores?
Smallest “gap” achievable?

12

SLIDE 35

GBMF4554