Can You Solve Closest String Faster than Exhaustive Search?

• URL: http://arxiv.org/abs/2305.16878v2
• Date: Mon, 29 May 2023 07:24:43 GMT
• Title: Can You Solve Closest String Faster than Exhaustive Search?
• Authors: Amir Abboud, Nick Fischer, Elazar Goldenberg, Karthik C. S., and Ron Safier
• Abstract summary: We study the fundamental problem of finding the best string to represent a given set. In this paper, we investigate whether the Closest String problem admits algorithms that are faster than the trivial exhaustive search algorithm.
• Score: 4.963483138661772
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the fundamental problem of finding the best string to represent a given set, in the form of the Closest String problem: Given a set $X \subseteq \Sigma^d$ of $n$ strings, find the string $x^*$ minimizing the radius of the smallest Hamming ball around $x^*$ that encloses all the strings in $X$. In this paper, we investigate whether the Closest String problem admits algorithms that are faster than the trivial exhaustive search algorithm. We obtain the following results for the two natural versions of the problem: $\bullet$ In the continuous Closest String problem, the goal is to find the solution string $x^*$ anywhere in $\Sigma^d$. For binary strings, the exhaustive search algorithm runs in time $O(2^d poly(nd))$ and we prove that it cannot be improved to time $O(2^{(1-\epsilon) d} poly(nd))$, for any $\epsilon > 0$, unless the Strong Exponential Time Hypothesis fails. $\bullet$ In the discrete Closest String problem, $x^*$ is required to be in the input set $X$. While this problem is clearly in polynomial time, its fine-grained complexity has been pinpointed to be quadratic time $n^{2 \pm o(1)}$ whenever the dimension is $\omega(\log n) < d < n^{o(1)}$. We complement this known hardness result with new algorithms, proving essentially that whenever $d$ falls out of this hard range, the discrete Closest String problem can be solved faster than exhaustive search. In the small-$d$ regime, our algorithm is based on a novel application of the inclusion-exclusion principle. Interestingly, all of our results apply (and some are even stronger) to the natural dual of the Closest String problem, called the Remotest String problem, where the task is to find a string maximizing the Hamming distance to all the strings in $X$.

Related papers

• Quantum Algorithm for the Multiple String Matching Problem [0.0]
  We consider a sequence of $m$ strings, denoted by $S$, which we refer to as a dictionary. The objective is to identify all instances of strings from the dictionary within the text. We propose a quantum algorithm with $O(n+sqrtmLlog nlog n)$ query complexity and $O(n+sqrtmLlog n)=O*(n+sqrtmL)$ time complexity.
  arXiv  Detail & Related papers  (2024-11-22T10:50:43Z)
• Near-Optimal Quantum Algorithm for Finding the Longest Common Substring between Run-Length Encoded Strings [0.8057006406834466]
  We give a near-optimal quantum algorithm for the longest common (LCS) problem between two run-length encoded (RLE) strings. Our algorithm costs $tildemathcalO(n2/3/d1/6-o(1)cdotmathrmpolylog(tilden))$ time, while the query lower bound for the problem is $tildeOmega(n2/3/d1/6)$.
  arXiv  Detail & Related papers  (2024-10-21T15:52:08Z)
• Quantum Algorithms for the Shortest Common Superstring and Text Assembling Problems [11.048346250166073]
  We consider two versions of the Text Assembling problem. We are given a sequence of strings $s1,dots,sn$ of total length $L$ that is a dictionary, and a string $t$ of length $m$ that is texts. For both problems, we suggest new quantum algorithms that work better than their classical counterparts.
  arXiv  Detail & Related papers  (2023-06-18T14:16:49Z)
• Corralling a Larger Band of Bandits: A Case Study on Switching Regret for Linear Bandits [99.86860277006318]
  We consider the problem of combining and learning over a set of adversarial algorithms with the goal of adaptively tracking the best one on the fly. The CORRAL of Agarwal et al. achieves this goal with a regret overhead of order $widetildeO(sqrtd S T)$ where $M$ is the number of base algorithms and $T$ is the time horizon. Motivated by this issue, we propose a new recipe to corral a larger band of bandit algorithms whose regret overhead has only emphlogarithmic dependence on $M$ as long
  arXiv  Detail & Related papers  (2022-02-12T21:55:44Z)
• Logarithmic Regret from Sublinear Hints [76.87432703516942]
  We show that an algorithm can obtain $O(log T)$ regret with just $O(sqrtT)$ hints under a natural query model. We also show that $o(sqrtT)$ hints cannot guarantee better than $Omega(sqrtT)$ regret.
  arXiv  Detail & Related papers  (2021-11-09T16:50:18Z)
• Active Sampling for Linear Regression Beyond the $\ell_2$ Norm [70.49273459706546]
  We study active sampling algorithms for linear regression, which aim to query only a small number of entries of a target vector. We show that this dependence on $d$ is optimal, up to logarithmic factors. We also provide the first total sensitivity upper bound $O(dmax1,p/2log2 n)$ for loss functions with at most degree $p$ growth.
  arXiv  Detail & Related papers  (2021-11-09T00:20:01Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)
• Classical and Quantum Algorithms for Constructing Text from Dictionary Problem [0.0]
  We study algorithms for solving the problem of constructing a text from a dictionary (sequence of small strings) The problem has an application in bioinformatics and has a connection with the Sequence assembly method for reconstructing a long DNA sequence from small fragments.
  arXiv  Detail & Related papers  (2020-05-28T22:44:01Z)
• Agnostic Q-learning with Function Approximation in Deterministic Systems: Tight Bounds on Approximation Error and Sample Complexity [94.37110094442136]
  We study the problem of agnostic $Q$-learning with function approximation in deterministic systems. We show that if $delta = Oleft(rho/sqrtdim_Eright)$, then one can find the optimal policy using $Oleft(dim_Eright)$.
  arXiv  Detail & Related papers  (2020-02-17T18:41:49Z)
• Quantum Algorithms for the Most Frequently String Search, Intersection of Two String Sequences and Sorting of Strings Problems [0.0]
  We study algorithms for solving three problems on strings. The first one is the Most Frequently String Search Problem. The second is searching intersection of two sequences of strings. The third problem is sorting of $n$ strings of length $k$.
  arXiv  Detail & Related papers  (2020-01-07T07:22:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.