{"title":"A Nearly Tight Lower Bound for the d-Dimensional Cow-Path Problem","authors":"N. Bansal, John Kuszmaul, William Kuszmaul","doi":"10.48550/arXiv.2209.08427","DOIUrl":null,"url":null,"abstract":"In the $d$-dimensional cow-path problem, a cow living in $\\mathbb{R}^d$ must locate a $(d - 1)$-dimensional hyperplane $H$ whose location is unknown. The only way that the cow can find $H$ is to roam $\\mathbb{R}^d$ until it intersects $\\mathcal{H}$. If the cow travels a total distance $s$ to locate a hyperplane $H$ whose distance from the origin was $r \\ge 1$, then the cow is said to achieve competitive ratio $s / r$. It is a classic result that, in $\\mathbb{R}^2$, the optimal (deterministic) competitive ratio is $9$. In $\\mathbb{R}^3$, the optimal competitive ratio is known to be at most $\\approx 13.811$. But in higher dimensions, the asymptotic relationship between $d$ and the optimal competitive ratio remains an open question. The best upper and lower bounds, due to Antoniadis et al., are $O(d^{3/2})$ and $\\Omega(d)$, leaving a gap of roughly $\\sqrt{d}$. In this note, we achieve a stronger lower bound of $\\tilde{\\Omega}(d^{3/2})$.","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inf. Process. Lett.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2209.08427","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
In the $d$-dimensional cow-path problem, a cow living in $\mathbb{R}^d$ must locate a $(d - 1)$-dimensional hyperplane $H$ whose location is unknown. The only way that the cow can find $H$ is to roam $\mathbb{R}^d$ until it intersects $\mathcal{H}$. If the cow travels a total distance $s$ to locate a hyperplane $H$ whose distance from the origin was $r \ge 1$, then the cow is said to achieve competitive ratio $s / r$. It is a classic result that, in $\mathbb{R}^2$, the optimal (deterministic) competitive ratio is $9$. In $\mathbb{R}^3$, the optimal competitive ratio is known to be at most $\approx 13.811$. But in higher dimensions, the asymptotic relationship between $d$ and the optimal competitive ratio remains an open question. The best upper and lower bounds, due to Antoniadis et al., are $O(d^{3/2})$ and $\Omega(d)$, leaving a gap of roughly $\sqrt{d}$. In this note, we achieve a stronger lower bound of $\tilde{\Omega}(d^{3/2})$.