2009 | OriginalPaper | Chapter
Euler Tour Lock-In Problem in the Rotor-Router Model
I Choose Pointers and You Choose Port Numbers
Authors : Evangelos Bampas, Leszek Gąsieniec, Nicolas Hanusse, David Ilcinkas, Ralf Klasing, Adrian Kosowski
Published in: Distributed Computing
Publisher: Springer Berlin Heidelberg
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The
rotor-router model
, also called the
Propp machine
, was first considered as a deterministic alternative to the random walk. It is known that the route in an undirected graph
G
= (
V
,
E
), where |
V
| =
n
and |
E
| =
m
, adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in
G
by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the
lock-in problem
. In recent work [11] Yanovski et al. proved that independently of the initial configuration of the rotor-router mechanism in
G
the agent locks-in in time bounded by 2
mD
, where
D
is the diameter of
G
.
In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. The case study is performed in the form of a game between a player
$\cal P$
intending to lock-in the agent in an Euler tour as quickly as possible and its adversary
$\cal A$
with the counter objective. First, we observe that in certain (easy) cases the lock-in can be achieved in time
O
(
m
). On the other hand we show that if adversary
$\cal A$
is solely responsible for the assignment of ports and pointers, the lock-in time Ω(
m
·
D
) can be enforced in any graph with
m
edges and diameter
D
. Furthermore, we show that if
$\cal A$
provides its own port numbering after the initial setup of pointers by
$\cal P$
, the complexity of the lock-in problem is bounded by
O
(
m
· min {log
m
,
D
}). We also propose a class of graphs in which the lock-in requires time Ω(
m
·log
m
). In the remaining two cases we show that the lock-in requires time Ω(
m
·
D
) in graphs with the worst-case topology. In addition, however, we present non-trivial classes of graphs with a large diameter in which the lock-in time is
O
(
m
).