27,540 research outputs found
Two Lower Bounds for Shortest Double-Base Number System
In this letter, we derive two lower bounds for the number of terms in a double-base number system (DBNS), when the digit set is {1}. For a positive integer n, we show that the number of terms obtained from the greedy algorithm proposed by Dimitrov, Imbert, and Mishra [1] is . Also, we show that the number of terms in the shortest double-base chain is Θ(log n).LETTE
Remarks on Category-Based Routing in Social Networks
It is well known that individuals can route messages on short paths through
social networks, given only simple information about the target and using only
local knowledge about the topology. Sociologists conjecture that people find
routes greedily by passing the message to an acquaintance that has more in
common with the target than themselves, e.g. if a dentist in Saarbr\"ucken
wants to send a message to a specific lawyer in Munich, he may forward it to
someone who is a lawyer and/or lives in Munich. Modelling this setting,
Eppstein et al. introduced the notion of category-based routing. The goal is to
assign a set of categories to each node of a graph such that greedy routing is
possible. By proving bounds on the number of categories a node has to be in we
can argue about the plausibility of the underlying sociological model. In this
paper we substantially improve the upper bounds introduced by Eppstein et al.
and prove new lower bounds.Comment: 21 page
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table
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