Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · cs/0612121

Power Assignment Problems in Wireless Communication

Stefan Funke·Soeren Laue·Zvi Lotker·Rouven Naujoks
View original source ↗Read source document ↗Export metadata ↓

Abstract

A fundamental class of problems in wireless communication is concerned with the assignment of suitable transmission powers to wireless devices/stations such that the resulting communication graph satisfies certain desired properties and the overall energy consumed is minimized. Many concrete communication tasks in a wireless network like broadcast, multicast, point-to-point routing, creation of a communication backbone, etc. can be regarded as such a power assignment problem. This paper considers several problems of that kind; for example one problem studied before in \cite{Carrots, Bilo} aims to select and assign powers to $k$ of the stations such that all other stations are within reach of at least one of the selected stations. We improve the running time for obtaining a $(1+ε)$-approximate solution for this problem from $n^{((α/ε)^{O(d)})}$ as reported by Bilo et al. (\cite{Bilo}) to $O(n+ {(\frac{k^{2d+1}}{ε^d})}^{\min{\{2k, (α/ε)^{O(d)} \}}})$ that is, we obtain a running time that is \emph{linear} in the network size. Further results include a constant approximation algorithm for the TSP problem under squared (non-metric!) edge costs, which can be employed to implement a novel data aggregation protocol, as well as efficient schemes to perform $k$-hop multicasts.

Explore related subjects

cs.CGcs.ARcs.NI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stefan Funke, Soeren Laue, Zvi Lotker, Rouven Naujoks. 2006-12-22. Power Assignment Problems in Wireless Communication. https://arxiv.org/abs/cs/0612121

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Tight Fréchet bounds for $λ$-low density curves

The Fréchet distance is a well-studied similarity measure between curves. We computing the Fréchet distance between $λ$-low-density curves, the most general of realistic curve assumptions, where every ball of radius $r$ intersects at most $λ$ edges of length at least $r$. Previous algorithms either assumed constant $λ$ or had no tight dependence on $λ$. For two $n$-vertex $λ$-low-density curves in $\mathbb{R}^d$, we give a $(1+\varepsilon)$-approximation algorithm for the continuous and discrete Fréchet distance running in $ \tilde{O}\!\left(\frac{λ^{2/d}n^{2-2/d}}{\varepsilon^2}\right) $ time. Our key insight is a tight property of simplifying $λ$-low density curves: the simplification of any $n$-vertex $λ$-low-density curve is $O(λ^{1/d}n^{1-1/d})$-low-density. We show this is tight, and this provides the structural property under simplification that was previously known for $c$-packed curves. We provide matching lower bounds for $n$ and $λ$: assuming the Orthogonal Vectors Hypothesis, for every $δ>0$, we rule out algorithms with running time $O\!\left( \left( \frac{λ^{2/d}n^{2-2/d}} {\varepsilon^{2-4/d}} \right)^{1-δ} \right). $ We extend our techniques to the map matching problem, where we also give tight bounds.

cs.CG↗
arXiv2026

Perfect Rectangular Tilings with Two Colors

We study a finite tiling problem, where tiles are unit squares whose four edges are colored with one of two colors. We ask whether a given rectangle admits a perfect rectangular tiling: every cell of the rectangle is occupied by one tile, neighboring edge colors match, and exactly $n_i$ tiles of type $i$ are used, where rotations of the tiles are allowed. Our problem is related to classical Wang tilings, more general finite tile-placement problems, and edge placement puzzles. But in our problem, the multiplicities of the tile types are part of the input and the tile alphabet is fixed and extremely small; thus the complexity of the problem arises from the interaction between the rectangle dimensions and the prescribed tile multiplicities. We provide a comprehensive study of the perfect rectangular tiling problem. For this we consider all classes of subsets of the six possible tile types for two-colored edges, and we characterize for each class whether multiplicities either always allow a perfect rectangular tiling or whether their existence can be decided efficiently.

cs.CG↗
arXiv2026

A constant-factor approximation of the Gromov-Hausdorff distance in the plane

We give the first polynomial-time constant-factor approximation of the Gromov-Hausdorff distance d_GH between finite point sets in the Euclidean plane; in fixed Euclidean dimension such an approximation was previously known only on the line (Majhi, Vitter and Wenk, 2024). Global alignment cannot succeed: the classical dimension drop defeats alignment by isometries, a multiplicity gap defeats alignment by bijections, and a reflection barrier defeats sorting under any single global reflection pattern. The algorithm is therefore local. Guessing the images of one diameter pair pins every point's longitudinal coordinate to within O(d_GH). Heights are read in windows whose height spread is at most a fixed multiple of their length, where a chain argument makes every compatible match local in the plane. One reflection sign per window is then chosen by 2-SAT; at the right frame and guess, any solution yields a correspondence of distortion O(d_GH). For the bijective relative of d_GH, half the least additive distortion over bijections, the same scheme reduces the planar problem to a matching question that we leave open.

cs.CG↗