Fast Elimination of Redundant Linear Equations and Reconstruction of Recombination-free Mendelian Inheritance on a Pedigree

1
Fast Elimination of Redundant Linear Equations
and Reconstruction of Recombination-free
Mendelian Inheritance on a Pedigree

• Authors
• Lan Liu Tao Jiang, Univ.
  California, Riverside
• Jing Xiao, Lirong Xia, Tsinghua
  Univ. , China

2
Outline

• Introduction and problem definition
• A new system of linear equations for ZRHC
• An O(mn3) time algorithm for ZRHC
• An improved algorithm for ZRHC
• Conclusion

3
Pedigree

• An example British Royal Family

4
Biological Background

• Mendelian Law one haplotype comes from the
  father and the other comes from the mother.

• Basic concepts

maternal
paternal
11 22 homozygous
12 heterozgyous
12
21
5
Notations and Recombinant
6
Haplotype Configuration Reconstruction

• Haplotypes useful, but expensive to obtain
• Genotypes not so informative, but cheaper to
  obtain
• In biological application, genotypes instead of
  haplotypes are collected.

• How to reconstruct haplotype from genotype?
• recombination-free assumption

7
The ZRHC problem

• Problem definition
• Given a pedigree and the genotype
  information for each member, find a
  recombination-free haplotype configuration for
  each member that obeys the Mendelian law of
  inheritance.

8
Previous Work

• Li and Jiang introduced a system of linear
  equations over F2 and presented an
  time algorithm for ZRHC LJ03 , where m is loci
  and n is members in pedigree.
• Several attempts have been made recently, but the
  authors failed to prove the correctness of their
  algorithms in all cases, especially when the
  input pedigree has mating loops CZ04 LCL06.
• Recently, Chan et al. proposed a linear-time
  algorithm in CCC06, which only works for
  pedigree without mating loops.

9
Related work

• Methods based on fast matrix multiplication
  algorithms could achieve an asymptotic speed of
  O(k2.376) on k equations with k unknowns
• The Lanczos and conjugate gradient algorithms are
  only heuristics GV96.
• The Wiedeman algorithm has expected quadratic
  running time W86

10
Our Result

• We present a much faster algorithm for ZRHC with
  running time .

O(n)
redundancy elimination
O(n)
transformation
Axb
Axb
Axb

• O(n log2n log log n)

11
Outline

• Introduction and problem definition
• A new system of linear equations for ZRHC
• An O(mn3) time algorithm for ZRHC
• An improved algorithm for ZRHC
• Conclusion

Axb
12
The New Linear System

• n, m
• m loci n members in pedigree

13
The New Linear System
pj1,21 pj1,30
14
The Linear System

• O(mn) equations on O(mn) unknowns.
• Given a homozygous locus i on a member j (with a
  child j1), pji and pj1i are pre-determined.

15
Pedigree Graph

• A pedigree with genotype

16
Locus Graph

• Locus graph Gi

Gi (V, Ei), where Ei (k,j) k is a parent of
j, wki1
12
22
11
1
3
2
6
4
7
5
12
12
11
12
8
12
9
Zero-weight
22

(a) Genotype info
Example Locus graph for the 3rd locus
17
Outline

• Introduction and problem definition
• A new system of linear equations for ZRHC
• An O(mn3) time algorithm for ZRHC
• An improved algorithm for ZRHC
• Conclusion

O(n)
transformation
Axb
Axb
O(mn)
18
An Observation

• For any cycle or any path in a locus graph
  connecting two pre-determined vertices, the
  summation of h-variables along the path is a
  constant.

We can use paths to denote constraints!
19
Examples of Linear Constraints
?
1
0
1
3
2
1
1
0
6
4
7
5
1
h6,8
8
0
h8,9
1
9
(a) 1st locus graph h6,8 h8,9 1
20
Linear Constraints

• Obviously, the linear constraints are necessary.
  We can also show that these constraints are
  sufficient.
• Moreover, we can upper bound constraints in
  each locus graph as O(n), while the trivial
  analysis gives an upper bound O(n2).
• Total constraints O(mn).

21
The ZRHC-PHASE algorithm
Algorithm ZRHC_PHASE input a pedigree G(V,E)
and genotype gj output a general solution of
pj begin Step 1. Preprocessing Step 2.
Linear constraint generation on h-variables
Step 3. Solve h-variables by Gaussian
Elimination Step 4. Solve the p-variables by
propagation from pre-determined p-variables to
others. end
22
Outline

• Introduction and problem definition
• A new system of linear equations for ZRHC
• An O(mn3) time algorithm for ZRHC
• An improved algorithm for ZRHC
• Conclusion

O(n)
redundancy elimination
O(n)
transformation
Axb
Axb
Axb
O(mn)
O(n log2n log log n)
23
Redundant Equation Elimination

• An observation

j0
j1

• Given a cycle , assume
  that there are constraints among each pair of
  vertices.
• Originally, there are O(k2) constraints. Notice
  that they are not independent.
• However, we can replace the original constraints
  by an equivalent set of constraints with size
  O(k).

j2
jk