On Searching a Table Consistent with Division Poset

Suppose $P_n=\{1,2,...,n\}$ is a partially ordered set with the partial order defined by divisibility, that is, for any two distinct elements $i,j\in P_n$ satisfying $i$ divides $j$, $i<_{P_n} j$. A table $A_n=\{a_i|i=1,2,...,n\}$ of distinct real numbers is said to be \emph{consistent} with $P_n$, provided for any two distinct elements $i,j\in \{1,2,...,n\}$ satisfying $i$ divides $j$, $a_i< a_j$. Given an real number $x$, we want to determine whether $x\in A_n$, by comparing $x$ with as few entries of $A_n$ as possible. In this paper we investigate the complexity $\tau(n)$, measured in the number of comparisons, of the above search problem. We present a $\frac{55n}{72}+O(\ln^2 n)$ search algorithm for $A_n$ and prove a lower bound $({3/4}+{17/2160})n+O(1)$ on $\tau(n)$ by using an adversary argument.Comment: 16 pages, no figure; same results, representation improved, add reference

Visualizing the impact of Covid-19 vaccine passports on pedestrian access to metro stations in Hong Kong

Pedestrian infrastructures in Hong Kong enable multilevel city life in a vertical metropolis plagued by land scarcity. Public spaces integrated into pedestrian networks play an indispensable role in neighbourhood accessibility. We visualize the impact of the Covid-19 vaccine passport (VP) restrictions on the use of public space on pedestrian accessibility to all 97 metro stations in Hong Kong. Pedestrians without a vaccine passport (PwoVP) need to walk significantly longer alternative routes. Specifically, VP-related access restrictions to indoor walkways have doubled the shortest travel time for PwoVP and a 50% reduction in accessibility of two-thirds of stations