Improved approximation algorithms for single-tiered relay placement

Abstract

We consider the problem of Single-Tiered Relay Placement with Basestations, which takes as input a set $S$ of sensors and a set $B$ of basestations described as points in a normed space $(M,d)$, and real numbers $0< r\le R$. The objective is to place a minimum cardinality set $Q$ of wireless relay nodes that connects $S$ and $B$ according to the following rules. The sensors in $S$ can communicate within distance $r$, relay nodes in $Q$ can communicate within distance $R$, and basestations are considered to have an infinite broadcast range. Together the sets $S, B$, and $Q$ induce an undirected graph $G=(V,E)$ defined as follows: $V=S\cup B\cup Q$ and $E=\{uv|u,v\in B\}\cup \{uv|u\in Q$ and $v\in Q\cup B$ and $d(u,v)\le R\} \cup \{uv|u\in S$ and $v\in S\cup Q\cup B$ and $d(u,v)\le r\}$. Then $Q$ connects $S$ and $B$ when this induced graph is connected. In the case of the two-dimensional Euclidean plane, we get a $(1+\ln 6+\epsilon )<2.8$-approximation algorithm, improving the previous best ratio of 3.11. Let $\varDelta$ be the maximum number of points on a unit ball with pairwise distance strictly bigger than 1. Under certain assumptions, we have a $\left( 1+\ln (\varDelta +1)+\epsilon \right)$-approximation algorithm. When biconnectivity is required, we show that a variant of our previously proposed algorithm has approximation ratio of $\varDelta + 2$. In the case of the two-dimensional Euclidean plane, our ratio of 7 improves our previous bound of 16.