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In the evolution of a genome, the gene sequence is sometimes rearranged, for example by transposition of two adjacent gene blocks. In biocombinatorics, one tries to reconstruct these rearrangement incidents from the resulting permutation.…
Motivation: Genome rearrangement plays an important role in evolutionary biology and has profound impacts on phenotype in organisms ranging from microbes to humans. The mechanisms for genome rearrangement events remain unclear. Lots of…
Questions in computational molecular biology generate various discrete optimization problems, such as DNA sequence alignment and RNA secondary structure prediction. However, the optimal solutions are fundamentally dependent on the…
Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. Next,…
Given two genomes with duplicate genes, \textsc{Zero Exemplar Distance} is the problem of deciding whether the two genomes can be reduced to the same genome without duplicate genes by deleting all but one copy of each gene in each genome.…
The genome-wide recombination rate ($RR$) of a species is often described by one parameter, the ratio between total genetic map length ($G$) and physical map length ($P$), measured in centiMorgans per Megabase (cM/Mb). The value of this…
It is widely believed that the folding of the chromosome in the nucleus has a major effect on genetic expression. For example co-regulated genes in several species have been shown to colocalize in space despite being far away on the DNA…
We analyze models of genome evolution based on both restricted and unrestricted double-cut-and-join (DCJ) operations. We compare the number of operations along the evolutionary trajectory to the DCJ distance of the genome from its ancestor…
Understanding the dynamics of genome rearrangements is a major issue of phylogenetics. Phylogenetics is the study of species evolution. A major goal of the field is to establish evolutionary relationships within groups of species, in order…
We define the (random) $k$-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon except now a node must be cut $k$ times before it is destroyed. The…
The study of genetic map linearization leads to a combinatorial hard problem, called the {\em minimum breakpoint linearization} (MBL) problem. It is aimed at finding a linearization of a partial order which attains the minimum breakpoint…
The variation in DNA copy number carries information on the modalities of genome evolution and misregulation of DNA replication in cancer cells; its study can be helpful to localize tumor suppressor genes, distinguish different populations…
In this work, we consider a restricted case of the well studied Sorting by Block Interchanges problem. We put an upper bound k on the length of the blocks (substrings) to be interchanged at each step. We call the problem Sorting by k-Block…
The wide array of currently available genomes display a wonderful diversity in size, composition and structure with many more to come thanks to several global biodiversity genomics initiatives starting in recent years. However, sequencing…
Modellers of large scale genome rearrangement events, in which segments of DNA are inverted, moved, swapped, or even inserted or deleted, have found a natural syntax in the language of permutations. Despite this, there has been a wide range…
We study complexity of rearrangement problems in the generalized breakpoint model and settle several open questions. The model was introduced by Tannier et al. (2009) who showed that the median problem is solvable in polynomial time in the…
Tree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, R\'acz, and Rashtchian used combinatorial…
The Single Cut or Join (SCJ) operation on genomes, generalizing chromosome evolution by fusions and fissions, is the computationally simplest known model of genome rearrangement. While most genome rearrangement problems are already hard…
The incipient infinite cluster appearing at the bond percolation threshold can be decomposed into singly-connected ``links'' and multiply-connected ``blobs.'' Here we decompose blobs into objects known in graph theory as 3-blocks. A 3-block…
The seriation problem seeks to reorder a set of elements given pairwise similarity information, so that elements with higher similarity are closer in the resulting sequence. When a global ordering consistent with the similarity information…