“迹线重建的下限”勘误表

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2022-08-01 DOI:10.1214/22-aap1827
N. Holden, R. Lyons
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引用次数: 0

摘要

引理3.1断言,对于所有足够大的n,Eyn[Z(yn)]−Exn[Z(xn)]=(n−1/2)和Eyn[Z(yn)]>Exn[Z(xn。这是一个正确的证明。结果背后的直觉是,具有我们所考虑的类型的缺陷的字符串,即01的字符串中的10,可能比没有缺陷的字符串在跟踪中导致更多的11。由于与xn中的缺陷相比,yn中的缺陷向右移动,因此yn的缺陷比xn的缺陷更容易“落入”迹线的测试窗口{2np+1,…,2np+√npq}。更准确地说,概率的差异是n−1/2阶。在下面的证明中,我们使这种直觉变得严格。证明。我们在整个证明中假设k∈{2np+1,…,2np+√npq}。设E(m,k)表示输入字符串中的位m被复制到轨迹中的位置k的事件。首先要注意
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Erratum to “Lower bounds for trace reconstruction”
Lemma 3.1 asserts that Eyn[Z(̃yn)] − Exn[Z(̃xn)] = (n−1/2) and Eyn[Z(̃yn)] > Exn[Z(̃xn)] for all sufficiently large n. Our proof was not correct: As Benjamin Gunby and Xiaoyu He pointed out to us, we missed four terms in the computation of equation (3.3). Those terms contribute a negative amount, so the proof is more delicate. Here is a correct proof. The intuition behind the result is that a string with a defect of the type we consider, namely, a 10 in a string of 01’s, is likely to cause more 11’s in the trace than a string without the defect. Since the defect in yn is shifted to the right as compared to the defect in xn, the defect of yn is slightly more likely to “fall into” the test window { 2np + 1 , . . . , 2np + √npq } of the trace than is the defect of xn. More precisely, the difference in probability is of order n−1/2. In the proof below, we make this intuition rigorous. PROOF. We assume throughout the proof that k ∈ { 2np + 1 , . . . , 2np + √npq }. Let E(m,k) denote the event that bit m in the input string is copied to position k in the trace. First observe that
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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