Related papers: Substring Complexity in Sublinear Space
The recently developed matrix based Renyi's entropy enables measurement of information in data simply using the eigenspectrum of symmetric positive semi definite (PSD) matrices in reproducing kernel Hilbert space, without estimation of the…
In grammar-based compression a string is represented by a context-free grammar, also called a straight-line program (SLP), that generates only that string. We refine a recent balancing result stating that one can transform an SLP of size…
For an undirected tree with $n$ edges labelled by single letters, we consider its substrings, which are labels of the simple paths between pairs of nodes. We prove that there are $O(n^{1.5})$ different palindromic substrings. This solves an…
The Stochastic Gravitational Wave Background (SGWB) from cosmic superstrings offers one of the few known possibilities to test String Theory within current experimental reach. However, in order to be compatible with the existing…
Nishimoto and Tabei [CPM, 2021] proposed r-enum, an algorithm to enumerate various characteristic substrings, including maximal repeats, in a string $T$ of length $n$ in $O(r)$ words of compressed working space, where $r \le n$ is the…
We investigate symbolic sequences and in particular information carriers as e.g. books and DNA--strings. First the higher order Shannon entropies are calculated, a characteristic root law is detected. Then the algorithmic entropy is…
We present a new on-line algorithm for computing the Lempel-Ziv factorization of a string that runs in $O(N\log N)$ time and uses only $O(N\log\sigma)$ bits of working space, where $N$ is the length of the string and $\sigma$ is the size of…
We describe an alternative method (to compression) that combines several theoretical and experimental results to numerically approximate the algorithmic (Kolmogorov-Chaitin) complexity of all $\sum_{n=1}^82^n$ bit strings up to 8 bits long,…
It is not obvious how to extend Shannon's original information entropy to higher dimensions, and many different approaches have been tried. We replace the English text symbol sequence originally used to illustrate the theory by a discrete,…
The Subset Sum problem asks whether a given set of $n$ positive integers contains a subset of elements that sum up to a given target $t$. It is an outstanding open question whether the $O^*(2^{n/2})$-time algorithm for Subset Sum by…
This paper considers the problem of specifying a simple approximating density function for a given data set (x_1,...,x_n). Simplicity is measured by the number of modes but several different definitions of approximation are introduced. The…
Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say $\tilde{O}(n^{1+\delta})$ bits. There is an inherent tradeoff between the…
A regular expression specifies a set of strings formed by single characters combined with concatenation, union, and Kleene star operators. Given a regular expression $R$ and a string $Q$, the regular expression matching problem is to decide…
We consider the problem of minimizing the number of matrix-vector queries needed for accurate trace estimation in the dynamic setting where our underlying matrix is changing slowly, such as during an optimization process. Specifically, for…
We investigate adaptive sublinear algorithms for detecting monotone patterns in an array. Given fixed $2 \leq k \in \mathbb{N}$ and $\varepsilon > 0$, consider the problem of finding a length-$k$ increasing subsequence in an array $f \colon…
Shannon entropy is widely used to measure the complexity of DNA sequences but suffers from saturation effects that limit its discriminative power for long uniform segments. We introduce a novel metric, the entropy rank ratio R, which…
The $r$-index represented a breakthrough in compressed indexing of repetitive text collections, outperforming its alternatives by orders of magnitude in query time. Its space usage, $O(r)$ where $r$ is the number of runs in the…
In the trace reconstruction problem, one seeks to reconstruct a binary string $s$ from a collection of traces, each of which is obtained by passing $s$ through a deletion channel. It is known that $\exp(\tilde O(n^{1/5}))$ traces suffice to…
Compressed Counting (CC), based on maximally skewed stable random projections, was recently proposed for estimating the p-th frequency moments of data streams. The case p->1 is extremely useful for estimating Shannon entropy of data…
We solve an open problem related to an optimal encoding of a straight line program (SLP), a canonical form of grammar compression deriving a single string deterministically. We show that an information-theoretic lower bound for representing…