Related papers: Minimizing the Minimizers via Alphabet Reordering

We study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order, where the quantity to be maximized is given by a monotone submodular function on subsets of edges. This…

We prove that a uniformly random automaton with $n$ states on a 2-letter alphabet has a synchronizing word of length $O(n^{1/2}\log n)$ with high probability (w.h.p.). That is to say, w.h.p. there exists a word $\omega$ of such length, and…

For a word $S$, let $f(S)$ be the largest integer $m$ such that there are two disjoints identical (scattered) subwords of length $m$. Let $f(n, \Sigma) = \min \{f(S): S \text{is of length} n, \text{over alphabet} \Sigma \}$. Here, it is…

We present theoretical guarantees for an alternating minimization algorithm for the dictionary learning/sparse coding problem. The dictionary learning problem is to factorize vector samples $y^{1},y^{2},\ldots, y^{n}$ into an appropriate…

We show that several versions of Floyd and Rivest's algorithm Select for finding the $k$th smallest of $n$ elements require at most $n+\min\{k,n-k\}+o(n)$ comparisons on average and with high probability. This rectifies the analysis of…

We consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists…

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$.…

Word Break is a prototypical factorization problem in string processing: Given a word $w$ of length $N$ and a dictionary $\mathcal{D} = \{d_1, d_2, \ldots, d_{K}\}$ of $K$ strings, determine whether we can partition $w$ into words from…

The palindromic fingerprint of a string $S[1\ldots n]$ is the set $PF(S) = \{(i,j)~|~ S[i\ldots j] \textit{ is a maximal }\\ \textit{palindrome substring of } S\}$. In this work, we consider the problem of string reconstruction from a…

This paper defines multidimensional sequential optimization numbers and prove that the unsigned Stirling numbers of first kind are 1-dimensional sequential optimization numbers. This paper gives a recurrence formula and an upper bound of…

In this paper we study the problem of minimizing a submodular function $f : 2^V \rightarrow \mathbb{R}$ that is guaranteed to have a $k$-sparse minimizer. We give a deterministic algorithm that computes an additive $\epsilon$-approximate…

Consider a compact $M \subset \mathbb{R}^d$ and $r > 0$. A maximal distance minimizer problem is to find a connected compact set $\Sigma$ of the minimal length, such that \[ \max_{y \in M} dist (y, \Sigma) \leq r. \] The inverse problem is…

A string $S[1,n]$ is a power (or tandem repeat) of order $k$ and period $n/k$ if it can decomposed into $k$ consecutive equal-length blocks of letters. Powers and periods are fundamental to string processing, and algorithms for their…

We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our…

The Burrows-Wheeler Transform (BWT) is a string transformation technique widely used in areas such as bioinformatics and file compression. Many applications combine a run-length encoding (RLE) with the BWT in a way which preserves the…

Given a string $w$ over a finite alphabet $\Sigma$ and an integer $K$, can $w$ be partitioned into strings of length at most $K$, such that there are no \emph{collisions}? We refer to this question as the \emph{string partition} problem and…

We give an $\mathcal{O}(n \log n)$-time, $\mathcal{O}(n)$-space algorithm for factoring a string into the minimum number of palindromic substrings. That is, given a string $S [1..n]$, in $\mathcal{O}(n \log n)$ time our algorithm returns…

Sampling (evenly) the suffixes from the suffix array is an old idea trading the pattern search time for reduced index space. A few years ago Claude et al. showed an alphabet sampling scheme allowing for more efficient pattern searches…

Given a string $S$ of length $n$, the classic string indexing problem is to preprocess $S$ into a compact data structure that supports efficient subsequent pattern queries. In the \emph{deterministic} variant the goal is to solve the string…

We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the $1$-dimensional homology classes with $\mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been…