{"title":"Robust Max Selection","authors":"Trung Dang, Zhiyi Huang","doi":"arxiv-2409.06014","DOIUrl":null,"url":null,"abstract":"We introduce a new model to study algorithm design under unreliable\ninformation, and apply this model for the problem of finding the uncorrupted\nmaximum element of a list containing $n$ elements, among which are $k$\ncorrupted elements. Under our model, algorithms can perform black-box\ncomparison queries between any pair of elements. However, queries regarding\ncorrupted elements may have arbitrary output. In particular, corrupted elements\ndo not need to behave as any consistent values, and may introduce cycles in the\nelements' ordering. This imposes new challenges for designing correct\nalgorithms under this setting. For example, one cannot simply output a single\nelement, as it is impossible to distinguish elements of a list containing one\ncorrupted and one uncorrupted element. To ensure correctness, algorithms under\nthis setting must output a set to make sure the uncorrupted maximum element is\nincluded. We first show that any algorithm must output a set of size at least $\\min\\{n,\n2k + 1\\}$ to ensure that the uncorrupted maximum is contained in the output\nset. Restricted to algorithms whose output size is exactly $\\min\\{n, 2k + 1\\}$,\nfor deterministic algorithms, we show matching upper and lower bounds of\n$\\Theta(nk)$ comparison queries to produce a set of elements that contains the\nuncorrupted maximum. On the randomized side, we propose a 2-stage algorithm\nthat, with high probability, uses $O(n + k \\operatorname{polylog} k)$\ncomparison queries to find such a set, almost matching the $\\Omega(n)$ queries\nnecessary for any randomized algorithm to obtain a constant probability of\nbeing correct.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Data Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce a new model to study algorithm design under unreliable
information, and apply this model for the problem of finding the uncorrupted
maximum element of a list containing $n$ elements, among which are $k$
corrupted elements. Under our model, algorithms can perform black-box
comparison queries between any pair of elements. However, queries regarding
corrupted elements may have arbitrary output. In particular, corrupted elements
do not need to behave as any consistent values, and may introduce cycles in the
elements' ordering. This imposes new challenges for designing correct
algorithms under this setting. For example, one cannot simply output a single
element, as it is impossible to distinguish elements of a list containing one
corrupted and one uncorrupted element. To ensure correctness, algorithms under
this setting must output a set to make sure the uncorrupted maximum element is
included. We first show that any algorithm must output a set of size at least $\min\{n,
2k + 1\}$ to ensure that the uncorrupted maximum is contained in the output
set. Restricted to algorithms whose output size is exactly $\min\{n, 2k + 1\}$,
for deterministic algorithms, we show matching upper and lower bounds of
$\Theta(nk)$ comparison queries to produce a set of elements that contains the
uncorrupted maximum. On the randomized side, we propose a 2-stage algorithm
that, with high probability, uses $O(n + k \operatorname{polylog} k)$
comparison queries to find such a set, almost matching the $\Omega(n)$ queries
necessary for any randomized algorithm to obtain a constant probability of
being correct.