Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Extended Islands of Tractability for

Parsimony Haplotyping

Rudolf Fleischer1, Jiong Guo2, Rolf Niedermeier3, JohannesUhlmann3, Yihui Wang1, Mathias Weller3, and Xi Wu1

1Fudan University Shanghai, 2Universitat des Saarlandes,and 3Friedrich-Schiller-Universitat Jena

June 22, 2010

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

First, Some Biology...

A G T T A G C G A

A G T C A G C A A

approx. 0.1% of human nucleotide sites differ between individuals

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

First, Some Biology...

A G T T A G C G A

A G T C A G C A A

approx. 0.1% of human nucleotide sites differ between individuals

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

First, Some Biology...

A G T T A G C G A

A G T C A G C A A

the sequence of SNPs is called a haplotype

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

First, Some Biology...

T G

C A

the sequence of SNPs is called a haplotype

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

First, Some Biology...

humans are diploid ; 2 chromosome setshaplotype = SNP sequence in one chromosome setgenotype = SNP sequence in the combined chromosome sets

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

First, Some Biology...

humans are diploid ; 2 chromosome setshaplotype = SNP sequence in one chromosome setgenotype = SNP sequence in the combined chromosome sets

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Motivation

Haplotype Inference

goal: find relation between certain SNPs and genetic diseasesproblem: difficult (expensive) to sequence both haplotypesbut: easy (cheap) to sequence the genotype instead; idea: sequence genotype and computationally infer haplotypes

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Motivation

Haplotype Inference

goal: find relation between certain SNPs and genetic diseasesproblem: difficult (expensive) to sequence both haplotypesbut: easy (cheap) to sequence the genotype instead; idea: sequence genotype and computationally infer haplotypes

Problems

impossible to infer haplotypes of just 1 genotype; sequence and infer groups/populations

which explanation should be preferred if there are multiple?; parsimony (“Ockham’s razor”)

how to perform the actual computation fast?

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Motivation

Haplotype Inference

goal: find relation between certain SNPs and genetic diseasesproblem: difficult (expensive) to sequence both haplotypesbut: easy (cheap) to sequence the genotype instead; idea: sequence genotype and computationally infer haplotypes

Problems

impossible to infer haplotypes of just 1 genotype; sequence and infer groups/populations

which explanation should be preferred if there are multiple?; parsimony (“Ockham’s razor”)

how to perform the actual computation fast?

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Preliminary Definitions

Definition (Haplotype, Genotype, Resolve)

haplotype h = string over {0, 1}genotype g = string over {0, 1, 2}h1, h2 resolve g ⇔

for all i ∈ N, g [i ] = h1[i ] = h2[i ] or g [i ] = 2 and h1[i ] 6= h2[i ]multiset H ; res (H)multiset H resolves multiset G ⇔ G ⊆ res (H)

Example

haplotype1: 0 0 1 0 1 1 1 0 0 0 1haplotype2: 0 0 1 1 0 1 0 0 1 1 1genotype: 0 0 1 2 2 1 2 0 2 2 1

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Preliminary Definitions

Definition (Haplotype, Genotype, Resolve)

haplotype h = string over {0, 1}genotype g = string over {0, 1, 2}h1, h2 resolve g ⇔

for all i ∈ N, g [i ] = h1[i ] = h2[i ] or g [i ] = 2 and h1[i ] 6= h2[i ]multiset H ; res (H)multiset H resolves multiset G ⇔ G ⊆ res (H)

Example

haplotype1: 0 0 1 0 1 1 1 0 0 0 1haplotype2: 0 0 1 1 0 1 0 0 1 1 1genotype: 0 0 1 2 2 1 2 0 2 2 1

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Preliminary Definitions

Definition (Haplotype, Genotype, Resolve)

haplotype h = string over {0, 1}genotype g = string over {0, 1, 2}h1, h2 resolve g ⇔

for all i ∈ N, g [i ] = h1[i ] = h2[i ] or g [i ] = 2 and h1[i ] 6= h2[i ]multiset H ; res (H)multiset H resolves multiset G ⇔ G ⊆ res (H)

Example

haplotype1: 0 0 1 1 1 1 1 0 0 0 1haplotype2: 0 0 1 0 0 1 0 0 1 1 1genotype: 0 0 1 2 2 1 2 0 2 2 1

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Preliminary Definitions

Definition (Haplotype Graph)

Given H and G ⊆ res (H), define the haplotype graph of H and G :

|H| vertices (labeled by H)

|G | edges (labeled by G )

haplotypes of the endpoints of an edge resolve its genotype

Example

11011

01001 11111

10110 11100

1212011122

21221

12212

22222

21021

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Haplotype Inference by Parsimony

Definition (Haplotype Inference by Parsimony)

Input: multiset G of length-m genotypes, integer k ≥ 0Question: ∃ multiset H of k haplotypes that resolves G ?

Example

genotypes haplotype graph haplotypes111221212012212210212122122222

11011

01001 11111

10110 11100

12120

11122

21221

12212

22222

21021

0100110110110111110011111

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Previous Work

Complexity

NP-hard [Halldorsson et al., DMTCS ’03]

APX-hard [Lancia et al., INFORMS ’04]

many special cases in P (e.g. [Lancia & Rizzi, ORL ’06])

Algorithms

Factor-2d−1-Approximation [Lancia & Rizzi, ORL ’06]

O(2|G |·d) Branch&Bound [Wang & Xu, Bioinformatics ’03]

O(k2k2· m) Algorithm [Sharan et al., TCBB ’06]

k := #haplotypes, m := genotype length, d := max # 2’s in a genotype

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Our Contribution

k4k · poly (|G |,m) time algorithm

polynomial-time solvable special case:“Induced Haplotype Inference”

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Inference Graph

Definition (Inference Graph)

inference graph Γ of G = an order-k graph with edgesconsistently labeled by the genotypes in G

Example

12120

11122

21221

12212

22222

21021

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

If Only We Had an Inference Graph...

Observation (non-bipartite components of Γ)

C = cycle in Γ; for all i , |{g ∈ C | g [i ] = 2}| is even

non-bipartite components of Γ ; O(|Γ| · m) time

Example

12120

11122

21221

12212

22222

21021

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

If Only We Had an Inference Graph...

Observation (bipartite components of Γ)

all g [i ] = 2 for some i ⇒ choose arbitrarily

bipartite components of Γ ; O(|Γ| · m) time

Example

1212011122

21221

22222

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

If Only We Had an Inference Graph...

Observation

(G , k) yes-instance with solution H⇒ ∃ Γ extendable (O(|Γ| ·m) time) to a haplotype graph of H and G

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

If Only We Had an Inference Graph...

Observation

(G , k) yes-instance with solution H⇒ ∃ Γ extendable (O(|Γ| ·m) time) to a haplotype graph of H and G

algorithmic idea: guess Γ

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

If Only We Had an Inference Graph...

Observation

(G , k) yes-instance with solution H⇒ ∃ Γ extendable (O(|Γ| ·m) time) to a haplotype graph of H and G

algorithmic idea: guess Γbetter idea: guess a “spanning” subgraph of Γ

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Algorithm Outline

Algorithm

1 guess “spanning” subgraph of Γ

2 infer the haplotype multiset H ; O(k · m) time

3 check whether H resolves G ; O(k2 · m) time

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Algorithm Outline

Algorithm

1 guess “spanning” subgraph of Γ1 guess a size-k genotype subset of G ; O(k2k) possibilities2 for these genotypes, guess 2 (of k) vertices ; O(k2k ) possibilities

2 infer the haplotype multiset H ; O(k · m) time

3 check whether H resolves G ; O(k2 · m) time

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Algorithm Outline

Algorithm

1 guess “spanning” subgraph of Γ1 guess a size-k genotype subset of G ; O(k2k) possibilities2 for these genotypes, guess 2 (of k) vertices ; O(k2k ) possibilities

2 infer the haplotype multiset H ; O(k · m) time

3 check whether H resolves G ; O(k2 · m) time

Theorem

Haplotype Inference by Parsimony can be solved in O(k4k+2 · m)time.

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Example

11122

21221

12212

22222

21021

22020

01022

21221

12222

01001

11011

10110 11100

11111

10010 01000

0101111101

12120

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Example

11122

21221

12212

22222

21021

22020

01022

21221

12222

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Example

11122

21221

12212

22222

21021

22020

01022

21221

12222

?10?1

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Example

11122

21221

12212

22222

21021

22020

01022

21221

12222

?10?1

11011

1??1? 111??

111?1

1?0?0 010?0

010?111??1

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Example

11122

21221

12212

22222

21021

22020

01022

21221

12222

01001

11011

10110 11100

11111

100?0 010?0

010?1111?1

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Example

11122

21221

12212

22222

21021

22020

01022

21221

12222

01001

11011

10110 11100

11111

10010 01000

0101111101

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Induced Haplotype Inference

recall: H resolves G ⇔ G ⊆ res (H)what if G = res (H)? ; haplotype graph is a clique

Definition

G = res (H) ; H induces G

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Induced Haplotype Inference

recall: H resolves G ⇔ G ⊆ res (H)what if G = res (H)? ; haplotype graph is a clique

Definition

G = res (H) ; H induces G

Definition (Induced Haplotype Inference by Parsimony)

Input: multiset G of length-m genotypesQuestion: ∃ multiset H of haplotypes that induces G ?

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Observations & Algorithm

Observation

G can be partitioned “nicely” into G0, G1, and G2.

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Observations & Algorithm

Observation

G can be partitioned “nicely” into G0, G1, and G2.

Observation

G2 6= ∅ but G0 = ∅ or G1 = ∅ ; poly

|G0| = |G1| = 1 ; poly (although we may get 2 solutions)

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Observations & Algorithm

Observation

G can be partitioned “nicely” into G0, G1, and G2.

Observation

G2 6= ∅ but G0 = ∅ or G1 = ∅ ; poly

|G0| = |G1| = 1 ; poly (although we may get 2 solutions)

Strategy: Divide & Conquer

divide-stepbase casesmerge-step

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Observations & Algorithm

Observation

G can be partitioned “nicely” into G0, G1, and G2.

Observation

G2 6= ∅ but G0 = ∅ or G1 = ∅ ; poly

|G0| = |G1| = 1 ; poly (although we may get 2 solutions)

Strategy: Divide & Conquer

divide-step ; OKbase casesmerge-step

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Observations & Algorithm

Observation

G can be partitioned “nicely” into G0, G1, and G2.

Observation

G2 6= ∅ but G0 = ∅ or G1 = ∅ ; poly

|G0| = |G1| = 1 ; poly (although we may get 2 solutions)

Strategy: Divide & Conquer

divide-step ; OKbase cases ; OKmerge-step

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Observations & Algorithm

Observation

G can be partitioned “nicely” into G0, G1, and G2.

Observation

G2 6= ∅ but G0 = ∅ or G1 = ∅ ; poly

|G0| = |G1| = 1 ; poly (although we may get 2 solutions)

Strategy: Divide & Conquer

divide-step ; OKbase cases ; OKmerge-step ; problem!

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Observations & Algorithm

Observation

G can be partitioned “nicely” into G0, G1, and G2.

Observation

G2 6= ∅ but G0 = ∅ or G1 = ∅ ; poly

|G0| = |G1| = 1 ; poly (although we may get 2 solutions)

Strategy: Divide & Conquer

divide-step ; OKbase cases ; OKmerge-step ; problem!

need to find a way to compute H1 for given H0 and G2

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Extending a Subclique Solution

Observation

Let H0 induce G0 and let g be a genotype in G2 with the smallestnumber of 2’s. ; All h ∈ H0 that are consistent with g areequal.

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Extending a Subclique Solution

Observation

Let H0 induce G0 and let g be a genotype in G2 with the smallestnumber of 2’s. ; All h ∈ H0 that are consistent with g areequal.

Proof Idea

H0H1

g = 2120 . . .h′ = 1100 . . . h = 0110 . . .

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Extending a Subclique Solution

Observation

Let H0 induce G0 and let g be a genotype in G2 with the smallestnumber of 2’s. ; All h ∈ H0 that are consistent with g areequal.

Proof Idea

H0H1

g′

g = 2120 . . .h′ = 1100 . . . h = 0110 . . .

h′′ =?1?0 . . .

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Concluding The Induced Problem

Divide & Conquer like algorithm

divide steps O(|G | · k)base solutions O(|G | · m)

extends (merges) O(|G | · k · m)

all in all O(|G | · k · m)

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Conclusion

what we saw. . .

Induced Haplotyping in O(|G | · k · m) time

2O(k2 log k)-time algorithm improved to 2O(k log k)

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Conclusion

what we saw. . .

Induced Haplotyping in O(|G | · k · m) time

2O(k2 log k)-time algorithm improved to 2O(k log k)

also in the paper

results also hold for sets instead of multisets

results basically also hold for constrained variant

data reduction (O(2k · k2)-bit kernel)

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Conclusion

what we saw. . .

Induced Haplotyping in O(|G | · k · m) time

2O(k2 log k)-time algorithm improved to 2O(k log k)

also in the paper

results also hold for sets instead of multisets

results basically also hold for constrained variant

data reduction (O(2k · k2)-bit kernel)

future work

find polynomial kernel (or prove nonexistence)

distance from triviality measures

find 2O(k) time algorithm

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Introduction Improved FPT algorithm Induced Haplotyping Conclusion

Thank you

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