Efficient Haplotype Inference From Pedigrees With Missing Data Using Linear Systems With Disjoint-Set Data Structures

Abstract

We study the haplotype inference problem from pedigree data under the zero recombination assumption, which is well supported by real data for tightly linked markers (i.e., single nucleotide polymorphisms (SNPs)) over a relatively large chromosome segment. We solve the problem in a rigorous mathematical manner by formulating genotype constraints as a linear system of inheritance variables. We then utilize disjoint-set structures to encode connectivity information among individuals, to detect constraints from genotypes, and to check consistency of constraints. On a tree pedigree without missing data, our algorithm can output a general solution as well as the number of total specific solutions in a nearly linear time O(mn · α(n)), where m is the number of loci, n is the number of individuals and α is the inverse Ackermann function4, which is a further improvement over existing ones3 , 8 , 12 , 15. We also extend the idea to looped pedigrees and pedigrees with missing data by considering existing (partial) constraints on inheritance variables. The algorithm has been implemented in C++ and will be incorporated into our PedPhase package8. Experimental results show that it can correctly identify all 0-recombinant solutions with great efficiency. Comparisons with other two popular algorithms show that the proposed algorithm achieves 10 to 10 5 -fold improvements over a variety of parameter settings. The experimental study also provides empirical evidences on the complexity bounds suggested by theoretical analysis.

An Efficient Algorithm for Haplotype Inference on Pedigrees with a Small Number of Recombinants

“…The heuristic algorithm in [25] runs for days on a PC even for medium-sized datasets. Hence, recent work on MRHC has been focused on the special case where the number of recombinants is zero, the so called zero-recombinant haplotype configuration (ZRHC) problem [5,16,20,21,23,28,30,31]. In particular, Li and Jiang [16] formulated ZRHC as a system of linear equations over the field F (2) and devised an O(m 3 n 3 ) time algorithm using Gaussian elimination, where m is the number of loci and n is the size of the input pedigree.…”

“…Xiao et al [30,31] improved the running time to O(mn 2 + n 3 log 2 n log log n) by using a compact system of linear equations, taking advantage of some special properties of a pedigree graph, and the low-stretch spanning tree technique. The recent results in [5,20,21] presented linear (i.e. O(mn)) time algorithms for ZRHC on tree pedigrees using different techniques.…”

“…We will refer to this parameterized problem as the k-recombinant haplotype configuration (k-RHC) problem. Although ZRHC (or 0-RHC) seems easy to solve [5,20,21,30,31], the general k-RHC problem remains very hard to tackle. Observe that, we could obtain a trivial algorithm for k-RHC with time complexity O((mn) k (mn 2 + n 3 log 2 n log log n)) by using the algorithm in [30,31] for ZRHC and exhaustively enumerating the possible locations of the k recombinants.…”

“…This is because the k-RHC instance can be easily transformed into a ZRHC instance once the recombinant locations are known. Similarly, one could obtain a trivial algorithm for k-RHC on tree pedigrees with time complexity O((mn) k+1 ) by using the linear time algorithms in [5,20,21] for tree ZRHC. We note in passing that the dynamic programming algorithm in [6] solves MRHC on tree pedigrees in O(nm 3k+1 2 m ) time when each parent-child pair is allowed to have at most k recombinants.…”

“…For each instance generated by the probabilistic model, we try to identify small areas of the pedigree where a recombinant might have occurred by comparing the linear (equality) constraints in the ILP. Once the locations of all k recombinants are determined (or enumerated), the instance is transformed to a tree ZRHC instance and solved in linear time by using one of the algorithms in [5,20,21].…”