MotivationSegmental duplications (SDs) or low-copy repeats, are segments of DNA > 1 Kbp with high sequence identity that are copied to other regions of the genome. SDs are among the most important sources of evolution, a common cause of genomic structural variation and several are associated with diseases of genomic origin including schizophrenia and autism. Despite their functional importance, SDs present one of the major hurdles for de novo genome assembly due to the ambiguity they cause in building and traversing both state-of-the-art overlap-layout-consensus and de Bruijn graphs. This causes SD regions to be misassembled, collapsed into a unique representation, or completely missing from assembled reference genomes for various organisms. In turn, this missing or incorrect information limits our ability to fully understand the evolution and the architecture of the genomes. Despite the essential need to accurately characterize SDs in assemblies, there has been only one tool that was developed for this purpose, called Whole-Genome Assembly Comparison (WGAC); its primary goal is SD detection. WGAC is comprised of several steps that employ different tools and custom scripts, which makes this strategy difficult and time consuming to use. Thus there is still a need for algorithms to characterize within-assembly SDs quickly, accurately, and in a user friendly manner.ResultsHere we introduce SEgmental Duplication Evaluation Framework (SEDEF) to rapidly detect SDs through sophisticated filtering strategies based on Jaccard similarity and local chaining. We show that SEDEF accurately detects SDs while maintaining substantial speed up over WGAC that translates into practical run times of minutes instead of weeks. Notably, our algorithm captures up to 25% ‘pairwise error’ between segments, whereas previous studies focused on only 10%, allowing us to more deeply track the evolutionary history of the genome.Availability and implementationSEDEF is available at https://github.com/vpc-ccg/sedef.

Haplotype reconstruction of distant genetic variants remains an unsolved problem due to the short-read length of common sequencing data. Here, we introduce HapTree-X, a probabilistic framework that utilizes latent long-range information to reconstruct unspecified haplotypes in diploid and polyploid organisms. It introduces the observation that differential allele-specific expression can link genetic variants from the same physical chromosome, thus even enabling using reads that cover only individual variants. We demonstrate HapTree-X’s feasibility on in-house sequenced Genome in a Bottle RNA-seq and various whole exome, genome, and 10X Genomics datasets. HapTree-X produces more complete phases (up to 25%), even in clinically important genes, and phases more variants than other methods while maintaining similar or higher accuracy and being up to 10×  faster than other tools. The advantage of HapTree-X’s ability to use multiple lines of evidence, as well as to phase polyploid genomes in a single integrative framework, substantially grows as the amount of diverse data increases.

We analyze the computational complexity of several new variants of edge-matching puzzles. First we analyze inequality (instead of equality) constraints between adjacent tiles, proving the problem NP-complete for strict inequalities but polynomial-time solvable for nonstrict inequalities. Second we analyze three types of triangular edge matching, of which one is polynomial-time solvable and the other two are NP-complete; all three are #P-complete. Third we analyze the case where no target shape is specified and we merely want to place the (square) tiles so that edges match exactly; this problem is NP-complete. Fourth we consider four 2-player games based on 1×n edge matching, all four of which are PSPACE-complete. Most of our NP-hardness reductions are parsimonious, newly proving #P and ASP-completeness for, e.g., 1 × n edge matching. Along the way, we prove #P-and ASP-completeness of planar 3-regular directed Hamiltonicity; we provide linear-time algorithms to find antidirected and forbidden-transition Eulerian paths; and we characterize the complexity of new partizan variants of the Geography game on graphs.

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When can n given numbers be combined using arithmetic operators from a given subset of {+, −, ×, ÷} to obtain a given target number? We study three variations of this problem of Arithmetic Expression Construction: when the expression (1) is unconstrained; (2) has a specified pattern of parentheses and operators (and only the numbers need to be assigned to blanks); or (3) must match a specified ordering of the numbers (but the operators and parenthesization are free). For each of these variants, and many of the subsets of {+, −, ×, ÷}, we prove the problem NP-complete, sometimes in the weak sense and sometimes in the strong sense. Most of these proofs make use of a rational function framework which proves equivalence of these problems for values in rational functions with values in positive integers.