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Revisiting $O(n \log \log n)$ chaining for anchored edit distance

arXiv:2606.03929v1 Announce Type: new Abstract: Colinear chaining is a classical heuristic for sequence alignment: it enables scalable genome comparison and is a main component of many state-of-the-art read mappers based on seed-chain-extend. The earliest $O(n \log \log n)$ time algorithms by Eppstein et al. (J. ACM, 1992) chained $n$ fragments between two sequences $T$ and $Q$ while minimizing a gap cost based on the diagonal distance $\Delta_{\text{diag}}$ between consecutive fragments.

arXiv CS 7d ago

Complementary Time-Space Tradeoff for Self-Stabilizing Leader Election: Polynomial States Meet Sublinear Time

arXiv:2505.23649v3 Announce Type: replace Abstract: We study the self-stabilizing leader election (SS-LE) problem in the population protocol model, assuming exact knowledge of the population size $n$. Burman, Chen, Chen, Doty, Nowak, Severson, and Xu [BCC+21a] (PODC) showed that this problem can be solved in $O(n)$ expected time with $O(n)$ states. Recently, G\k{a}sieniec, Grodzicki, and Stachowiak [GGS25] (PODC) proved that $n+O(\log n)$ states suffice to achieve $O(n \log n)$ time both in...

arXiv CS 8d ago

Exact Sampling of Permutations with a Fixed Longest Increasing Subsequence

arXiv:2606.02263v1 Announce Type: new Abstract: We study exact uniform sampling of permutations of length $n$ whose longest increasing subsequence (LIS) has prescribed length $k$. For $k \in \Theta(n)$, we give a direct rejection sampler whose expected running time is $O(n\log\log n)$ in the word-RAM model. The sampler uses an expanded proposal space consisting of permutations together with a specified increasing subsequence, and accepts exactly those proposals whose specified subsequence is...

arXiv CS 8d ago

Infinite sequences with optimal diaphony, periodic $L_2$-discrepancy, and beyond

arXiv:2606.05482v1 Announce Type: new Abstract: We investigate the periodic $L_2$-discrepancy of infinite sequences $\S_d$ in $[0,1)^d$ and its analytic counterpart, the diaphony. We prove that infinite order-2 digital sequences over $\mathbb{F}_2$ attain the optimal order $L_{2,N}^{{\rm per}}(\S_d) \le C_d (\log N)^{d/2}/N$ for all $N \in \mathbb{N}\setminus \{1\}$, matching known lower bounds for infinitely many $N \in \mathbb{N}$.

arXiv CS 5d ago

Palindrome complexity versus factor complexity

arXiv:2606.08127v1 Announce Type: cross Abstract: Let ${\bf x} = (a_i)_{i \geq 0}$ be an infinite word over a finite alphabet $\Sigma$. Let $\rho (n)$ be the factor complexity function for $\bf x$ and ${\rm Pal}(n)$ be the palindrome complexity function for $\bf x$. We give a new relationship between these two quantities; namely, if $\bf x$ is not ultimately periodic, then $$ \lim_{n \rightarrow \infty} {{ {\rm Pal} (n) \log ({\rm Pal} (n) + 1)} \over {\rho (n)}} = 0.

arXiv CS 1d ago

Empirical Approximation of $L_p$ Norms

arXiv:2606.00347v1 Announce Type: cross Abstract: We study empirical $L_p$ moments of a random vector $\pmb\varphi$ based on its i.i.d.\ copies $\pmb\varphi^1,\ldots,\pmb\varphi^m$, that is, $\frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p$. Our main result is a new estimate for the expected uniform deviation \[ \mathbb{E}\sup_{y\in D}\biggl| \frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p -\mathbb{E}|\langle \pmb\varphi,y\rangle|^p \biggr| \] over an arbitrary index set $D$....

arXiv CS 8d ago

Parallel Metric Skiplists and Nearest Neighbor Search

Announce Type: new Abstract: The metric skip-list is a data structure designed for efficient nearest and $k$-nearest neighbor search in metric spaces. For many real-world datasets with reasonable distributions - specifically, those with a constant expansion rate - it supports $\tilde{O}(n)$ construction time and $O(k\log n)$ query time, where $n$ is the input size and $k$ is the number of nearest neighbors in queries. Notably, unlike alternative approaches, it does not require a bounded...

arXiv CS 7d ago

Geometric Routing in Geometric Inhomogeneous Random Graphs

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arXiv CS 7d ago

When and why randomised exploration works (in linear bandits)

arXiv:2502.08870v2 Announce Type: replace Abstract: We provide an approach for the analysis of randomised exploration algorithms like Thompson sampling that does not rely on forced optimism or posterior inflation. With this, we demonstrate that in the $d$-dimensional linear bandit setting, when the action space is smooth and strongly convex, randomised exploration algorithms enjoy an $n$-step regret bound of the order $O(d\sqrt{n} \log(n))$. Notably, this shows for the first time that there...

arXiv CS 6d ago

Incremental BPE Tokenization

arXiv:2605.30813v1 Announce Type: new Abstract: We propose a novel algorithm for incremental Byte Pair Encoding (BPE) tokenization. The algorithm processes each input byte in worst-case $\mathcal{O}(\log^2 t)$ time, leading to an overall complexity of $\mathcal{O}(n \log^2 t)$, where $n$ is the input length and $t$ is the maximum token length. The algorithm incrementally maintains BPE tokenization results for every prefix of the input text, implementing the standard BPE merge procedure...

arXiv CS 9d ago