To catch a falling robber

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• 作者：William B. Kinnersley^a ; ^{billk@uri.edu" class="auth_mail" title="E-mail the corresponding author} ; Paweł Prałat^b ; ¹ ; ^{pralat@ryerson.ca" class="auth_mail" title="E-mail the corresponding author} ; Douglas B. West^c ; ^d ; ² ; ^{west@math.uiuc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Cops-and-robber game ; Cop number ; Hypercube ; n-dimensional cube
• 刊名：Theoretical Computer Science
• 出版年：2016
• 出版时间：9 May 2016
• 年：2016
• 卷：627
• 期：Complete
• 页码：107-111
• 全文大小：225 K

文摘

We consider a Cops-and-Robber game played on the subsets of an n  -set. The robber starts at the full set; the cops start at the empty set. In each round, each cop moves up one level by gaining an element, and the robber moves down one level by discarding an element. The question is how many cops are needed to ensure catching the robber when the robber reaches the middle level. Alan Hill posed the problem and provided a lower bound of 16001614&_mathId=si1.gif&_user=111111111&_pii=S0304397516001614&_rdoc=1&_issn=03043975&md5=97edc1645723038ac2f994bdcc1397fc" title="Click to view the MathML source">2^n/2$2^{n/2}$ for even n   and 16001614&_mathId=si19.gif&_user=111111111&_pii=S0304397516001614&_rdoc=1&_issn=03043975&md5=8011c8a62226bbcaa86ee0d82d37cf23">16001614-si19.gif">$\left(\begin{array}{c}\hfill n\hfill \\ \hfill \lceil n/2\rceil \hfill \end{array}\right)2^{-\lfloor n/2\rfloor }$ (which is asymptotic to 16001614&_mathId=si3.gif&_user=111111111&_pii=S0304397516001614&_rdoc=1&_issn=03043975&md5=9ac74b5458afc33281d330f76789204e">16001614-si3.gif">$2^{\lceil n/2\rceil }/\sqrt{\pi n/2}$) for odd n  . Until now, no nontrivial upper bound was known. In this paper, we prove an upper bound that is within a factor of 16001614&_mathId=si4.gif&_user=111111111&_pii=S0304397516001614&_rdoc=1&_issn=03043975&md5=4b21e1f42068f674dfb943c081c43494" title="Click to view the MathML source">O(ln⁡n)$O(\ln n)$ of this lower bound.