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The circular median problem in the Double-Cut-and-Join (DCJ) distance asks to find, for three given genomes, a fourth circular genome that minimizes the sum of the mutual distances with the three other ones. This problem has been shown to…
The sequence reconstruction problem involves a model where a sequence is transmitted over several identical channels. This model investigates the minimum number of channels required for the unique reconstruction of the transmitted sequence.…
This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and…
We propose a combinatorial and graph-theoretic theory of dropout by modeling training as a random walk over a high-dimensional graph of binary subnetworks. Each node represents a masked version of the network, and dropout induces stochastic…
The human genome contains repetitive DNA at different level of sequence length, number and dispersion. Highly repetitive DNA is particularly rich in homo-- and di--nucleotide repeats, while middle repetitive DNA is rich of families of…
Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex.…
The fragmentation processes of exchangeable partitions have already been studied by several authors. In this paper, we examine rather fragmentation of exchangeable compositions, that means partitions of $\mathbb{N}$ where the order of the…
We present a minimal computational model, which allows very fast, on-the-fly construction of three dimensional haploid interphase genomes from single cell Hi-C contact maps using the HOOMD-blue molecular dynamics package on graphics…
This paper introduces a new family of reconstruction codes which is motivated by applications in DNA data storage and sequencing. In such applications, DNA strands are sequenced by reading some subset of their substrings. While previous…
The trace reconstruction problem studies the number of noisy samples needed to recover an unknown string $\boldsymbol{x}\in\{0,1\}^n$ with high probability, where the samples are independently obtained by passing $\boldsymbol{x}$ through a…
A "genome structure" is a labeled directed graph with vertices of degree 1 or 2. A set of operations over such graphs is fixed, and each of the operations has a certain cost, a strictly positive number. The transformation problem consists…
In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. (*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a…
Motivated by applications to DNA storage, we study reconstruction and list-reconstruction schemes for integer vectors that suffer from limited-magnitude errors. We characterize the asymptotic size of the intersection of error balls in…
Breakage-Fusion-Bridge (BFB) is a mechanism of genomic instability characterized by the joining and subsequent tearing apart of sister chromatids. When this process is repeated during multiple rounds of cell division, it leads to patterns…
Motivated by empirical observations of algebraic duplicated sequence length distributions in a broad range of natural genomes, we analytically formulate and solve a class of simple discrete duplication/substitution models that generate…
We prove an interesting fact describing the location of the roots of the generating polynomials of the numbers of derangements of length $n$, counted by their number of cycles. We then use this result to prove that if $k$ is the number of…
Conformation-dependent design of polymer sequences can be considered as a tool to control macromolecular self-assembly. We consider the monomer unit sequences created via the modification of polymers in a homogeneous melt in accordance with…
This article discusses the problem of estimation of parameters in finite mixtures when the mixture components are assumed to be symmetric and to come from the same location family. We refer to these mixtures as semi-parametric because no…
We prove a general theorem on cutoffs for symmetric exclusion and interchange processes on finite graphs $G_N=(V_N,E_N)$, under the assumption that either the graphs converge geometrically and spectrally to a compact metric measure space,…
We study high order random walks in high dimensional expanders; namely, in complexes which are local spectral expanders. Recent works have studied the spectrum of high order walks and deduced fast mixing. However, the spectral gap of high…