Approximately Sampling Elements with Fixed Rank in Graded Posets

Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call $\textit{balanced bias}$. A noteworthy application of our method is sampling restricted classes of integer partitions of $n$. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of $n$ from general restricted classes. Several observations allow us to improve the efficiency of this chain to require $O(n^{1/2}\log(n))$ space, and for unrestricted integer partitions, expected $O(n^{9/4})$ time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.Comment: 23 pages, 12 figure

Genomic insights into fine-scale recombination variation in adaptively diverging threespine stickleback fish (Gasterosteus aculeatus)

Meiotic recombination is one of the major molecular mechanisms generating genetic diversity and influencing genome evolution. By shuffling allelic combinations, it can directly influence the patterns and efficacy of natural selection. Studies in various organisms have shown that the rate and placement of recombination varies substantially within the genome, among individuals, between sexes and among different species. It is hypothesized that this variation plays an important role in genome evolution. In this PhD thesis, I investigated the extent and molecular basis of recombination variation in adaptively diverging threespine stickleback fish (Gasterosteus aculeatus) to further understand its evolutionary implications. I used both ChIP-sequencing and whole genome sequencing of pedigrees to empirically identify and quantify double strand breaks (DSBs) and meiotic crossovers (COs). Whole genome sequencing of large nuclear families was performed to identify meiotic crossovers in 36 individuals of diverging marine and freshwater ecotypes and their hybrids. This produced the first genome-wide high-resolution sex-specific and ecotype-specific map of contemporary recombination events in sticklebacks. The results show striking differences in crossover number and placement between sexes. Females recombine nearly 1.76 times more than males and their COs are distributed all over the chromosome while male COs predominantly occur near the chromosomal periphery. When compared among ecotypes a significant reduction in overall recombination rate was observed in hybrid females compared to pure forms. Even though the known loci underlying marine-freshwater adaptive divergence tend to fall in regions of low recombination, considerable female recombination is observed in the regions between adaptive loci. This suggests that the sexual dimorphism in recombination phenotype may have important evolutionary implications. At the fine-scale, COs and male DSBs are nonrandomly distributed involving ‘semi-hot’ hotspots and coldspots of recombination. I report a significant association of male DSBs and COs with functionally active open chromatin regions like gene promoters, whereas female COs did not show an association more than expected by chance. However, a considerable number of COs and DSBs away from any of the tested open chromatin marks suggests possibility of additional novel mechanisms of recombination regulation in sticklebacks. In addition, we developed a novel method for constructing individualized recombination maps from pooled gamete DNA using linked read sequencing technology by 10X Genomics®. We tested the method by contrasting recombination profiles of gametic and somatic tissue from a hybrid mouse and stickleback fish. Our pipeline faithfully detects previously described recombination hotspots in mice at high resolution and identify many novel hotspots across the genome in both species and thereby demonstrate the efficiency of the novel method. This method could be employed for large scale QTL mapping studies to further understand the genetic basis of recombination variation reported in this thesis. By bridging the gap between natural populations and lab organisms with large clutch sizes and tractable genetic tools, this work shows the utility of the stickleback system and provides important groundwork for further studies of heterochiasmy and divergence in recombination during adaptation to differing environments

The White-Box Adversarial Data Stream Model

We study streaming algorithms in the white-box adversarial model, where the stream is chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the $L_1$-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our $L_1$-heavy hitters algorithm even further and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in a stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for {\it randomized} algorithms robust to a white-box adversary. In particular, our results show that for all $p\ge 0$, there exists a constant $C_p>1$ such that any $C_p$-approximation algorithm for $F_p$ moment estimation in insertion-only streams with a white-box adversary requires $\Omega(n)$ space for a universe of size $n$. Similarly, there is a constant $C>1$ such that any $C$-approximation algorithm in an insertion-only stream for matrix rank requires $\Omega(n)$ space with a white-box adversary. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. (Abstract shortened to meet arXiv limits.)Comment: PODS 202

Adaptive Path Planning for Depth Constrained Bathymetric Mapping with an Autonomous Surface Vessel

This paper describes the design, implementation and testing of a suite of algorithms to enable depth constrained autonomous bathymetric (underwater topography) mapping by an Autonomous Surface Vessel (ASV). Given a target depth and a bounding polygon, the ASV will find and follow the intersection of the bounding polygon and the depth contour as modeled online with a Gaussian Process (GP). This intersection, once mapped, will then be used as a boundary within which a path will be planned for coverage to build a map of the Bathymetry. Methods for sequential updates to GP's are described allowing online fitting, prediction and hyper-parameter optimisation on a small embedded PC. New algorithms are introduced for the partitioning of convex polygons to allow efficient path planning for coverage. These algorithms are tested both in simulation and in the field with a small twin hull differential thrust vessel built for the task.Comment: 21 pages, 9 Figures, 1 Table. Submitted to The Journal of Field Robotic