# Sparse Hop Spanners for Unit Disk Graphs

@article{Dumitrescu2020SparseHS, title={Sparse Hop Spanners for Unit Disk Graphs}, author={Adrian Dumitrescu and Anirban Ghosh and Csaba D. T'oth}, journal={ArXiv}, year={2020}, volume={abs/2002.07840} }

A unit disk graph $G$ on a given set of points $P$ in the plane is a geometric graph where an edge exists between two points $p,q \in P$ if and only if $|pq| \leq 1$. A subgraph $G'$ of $G$ is a $k$-hop spanner if and only if for every edge $pq\in G$, the topological shortest path between $p,q$ in $G'$ has at most $k$ edges. We obtain the following results for unit disk graphs.
(i) Every $n$-vertex unit disk graph has a $5$-hop spanner with at most $5.5n$ edges. We analyze the family of…

## 3 Citations

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## References

SHOWING 1-10 OF 52 REFERENCES

Plane Hop Spanners for Unit Disk Graphs: Simpler and Better

- Mathematics, Computer ScienceComput. Geom.
- 2020

This paper presents a simple algorithm that constructs a plane hop spanner for the unit disk graph, a widely employed model for the study of wireless networks, and establishes several results on the plane geometry.

On Geometric Spanners of Euclidean and Unit Disk Graphs

- Mathematics, Computer ScienceSTACS
- 2008

A very simple linear time algorithm is obtained that, given a unit disk graph embedded in the plane, constructs a geometric spanner with the above stretch factor and degree bound, and also containing an EMST as a subgraph.

Lower Bounds on the Dilation of Plane Spanners

- Computer Science, MathematicsInt. J. Comput. Geom. Appl.
- 2016

(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners.
(II) For…

A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs

- Mathematics
- 2007

Let G = (V,E) be an undirected weighted graph on |V | = n vertices and |E| = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V,ES), ES ⊆ E, such that the distance between any pair…

Simple Algorithms for Spanners in Weighted Graphs

- Mathematics, Computer ScienceEncyclopedia of Algorithms
- 2016

The objective is to design an algorithm that, for any weighted graph on n vertices, computes a .2k 1/-spanner of O.n1C1=k/ size, and the most ambitious aim would be to achieve the linear time complexity.

Generating low-degree 2-spanners

- Mathematics, Computer ScienceSODA '94
- 1994

It is shown that the problem of finding a 2-spanner in a given graph is at least as hard to approximate as set cover, and a randomized approximation algorithm is provided with approximation ratio of $\tilde O(\Delta^{1/4})$.

Planar Hop Spanners for Unit Disk Graphs

- Computer ScienceALGOSENSORS
- 2010

This paper presents an algorithm that, given a set X of terminals in the plane, constructs a planar hop spanner with constant stretch factor for the Unit Disk Graph defined by X, and improves on previous constructions in the sense that it ensures the planarity of the whole spanner.

Plane Hop Spanners for Unit Disk Graphs

- Mathematics, Computer ScienceWADS
- 2019

The unit disk graph (UDG) is a widely employed model for the study of wireless networks and a hop spanner is a spanning subgraph H such that for every edge in the UDG the topological shortest path between p and q in H has a constant number of edges.

Efficient construction of low weighted bounded degree planar spanner

- Mathematics, Computer ScienceInt. J. Comput. Geom. Appl.
- 2004

This work gives an O(nlogn)-time centralized algorithm that constructs a planar t-spanner for V, for such that the degree of each node is bounded from above by , where 0<α<π/2 is an adjustable parameter.

Efficient Construction of Low Weight Bounded Degree Planar Spanner

- Mathematics, Computer ScienceCOCOON
- 2003

This method can be converted to a localized algorithm where the total number of messages sent by all nodes is at most O(n) (under broadcasting communication model) and the constants are all worst case constants due to the proofs.