Distributed Algorithm Finding Regular Clustering
in Unit Disc Graphs
Krzysztof Krzywdziński
SOFSEM 2010
Definition of UDG
Definition
Unit Disc Graph (UDG)
◮
vertices - points on the plane.
◮
two vertices are connected by an edge iff they are at distance
at most 1.
UDG is natural model for:
◮
cellphone networks,
◮
wireless networks,
◮
sensor networks
◮
ad hoc networks
Distributed model of computing
◮
Each vertex in UDG knows its position on the plane (GPS).
◮
Local clocks of the vertices of UDG can be synchronized – we
perform computations in rounds
◮
In each round a vertex can send, receive messages from its
neighbors, and can perform some local computations.
◮
The time complexity of the algorithm – the number of rounds
needed to find a solution.
Regular clustering - idea
Goal- dividing graph into disjoint connected components of size in
some interval.
Regular clustering - result
Let λ be some chosen parameter.
(KK)
Division of UDG into disjoint connected subgraphs H1 , H2 , . . . , Hk
such that for all 1 ¬ i ¬ k inequality λ ¬ |V (Hi )| ¬ 5λ − 4 holds.
Algorithm works in O(poly (λ)) time.
Ruling set - definition
Definition
A subset V ′ ⊆ V (G ) we call a (d, b)-Ruling set if for any
vi , vj ∈ V ′ inequality dist(vi , vj ) ­ d holds and V ′ if b-dominating
set.
Regular clustering - result
(KK)
Boundary 5λ − 4 is the best of possible.
Thank you