Mursalin Habib

Nick Fischer, Mursalin Habib

We revisit the Binary Closest String problem, which asks, given a set of binary strings $X \subseteq \{0, 1\}^n$, to compute a string minimizing the maximum Hamming distance to $X$. A long line of work has focused on parameterized algorithms with respect to the optimal distance $d$, yielding a sequence of improvements from $O^*(d^d)$ through $O^*(16^d)$, $O^*(9.513^d)$, $O^*(8^d)$, $O^*(6.731^d)$ to the current best-known running time of $O^*(5^d)$ [Chen, Ma, Wang; Algorithmica '16]. We present a faster randomized algorithm running in time $O^*(4^d)$. Our result matches a recent fine-grained lower bound [Abboud, Fischer, Goldenberg, Karthik C.S., Safier; ESA '23], and is therefore conditionally optimal. As an extra benefit, our algorithm is remarkably simple.

Sudatta Bhattacharya, Sanjana Dey, Elazar Goldenberg et al.

A function $φ:\{0,1\}^n \to \{0,1\}^N$ is called an isometric embedding of the $n$-dimensional Hamming metric space to the $N$-dimensional edit metric space if, for all $x,y\in\{0,1\}^n$, the Hamming distance between $x$ and $y$ is equal to the edit distance between $φ(x)$ and $φ(y)$. The rate of such an embedding is defined as the ratio $n/N$. It is well known in the literature how to construct isometric embeddings with rate $Ω(1/\log n)$. However, achieving even near-isometric embeddings with positive constant rate has remained elusive until now. In this paper, we present an isometric embedding with rate $1/8$ by discovering connections to synchronization strings, which were studied in the context of insertion-deletion codes (Haeupler-Shahrasbi [JACM'21]). At a technical level, we introduce a framework for obtaining high-rate isometric embeddings using a novel object called misaligners. As an immediate consequence of our constant-rate isometric embedding, we improve known conditional lower bounds for various optimization problems in the edit metric, now with optimal dependence on the dimension. We complement our results by showing that no isometric embedding $φ:\{0,1\}^n \to \{0,1\}^N$ can have rate greater than $15/32$ for all positive integers $n$. En route to proving this upper bound, we uncover fundamental structural properties necessary for every Hamming-to-edit isometric embedding. We also prove similar upper and lower bounds for embeddings over larger alphabets. Finally, we consider embeddings $φ:Σ_{\mathrm{in}}^n \to Σ_{\mathrm{out}}^N$ between different input and output alphabets, where the rate is given by $\frac{n\log|Σ_{\mathrm{in}}|}{N\log|Σ_{\mathrm{out}}|}$. In this setting, we show that the rate can be made arbitrarily close to $1$.