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In Section 3, we formalise the problem of Data Ag-
gregation Scheduling (DAS) in a WSN with multiple
sinks. Then, in Section 4, we show a lower bound for
solving a variant of DAS called weak DAS. Finally,
in Section 5, we discuss future work and conclude the
paper.
2 RELATED WORK
Communication protocols that have been developed
for WSNs with multiple sinks can be found in (Mot-
tola and Picco, 2011; Kawano and Miyazaki, 2008;
Bo and Li, 2011; Thulasiraman et al., 2007; Tuy-
suz Erman and Havinga, 2010; Hui Zhou and Xu,
2012). A data collection protocol that tries to decrease
the number of redundant transmissions has been pro-
posed in (Mottola and Picco, 2011). This protocol
uses information about the neighbourhood nodes to
reduce transmissions while collecting data from many
nodes to many sinks.
In (Thulasiraman et al., 2007), the authors propose
an algorithm that builds two node-disjoint paths from
every node to two different sinks. If one path fails, the
other is used to route the data. In (Tuysuz Erman and
Havinga, 2010), the authors propose a routing pro-
tocol with hexagon-based architecture. Nodes in the
network are grouped into hexagons according to their
locations. A routing protocol proposed in (Hui Zhou
and Xu, 2012), is based on trees. In this protocol, dif-
ferent trees rooted at different sinks are used to for-
ward data.
The schemes that have been proposed in (Kawano
and Miyazaki, 2008; Bo and Li, 2011) have more rel-
evance to our work. In (Kawano and Miyazaki, 2008),
the authors propose two algorithms: an algorithm that
builds shortest path trees rooted at each root and a
scheduling algorithm that exploits a graph colouring
algorithm to allow nodes to forward their messages to
their closest sink without message collisions. The au-
thors of (Bo and Li, 2011), propose two data aggre-
gation scheduling algorithms for multiple-sink sen-
sor networks. The first algorithm is Voronoi-based
scheduling where the sensing area is divided into re-
gions forming k forests, one forest for each sink. Then
the algorithm assigns slots to nodes. The second al-
gorithm is Independent scheduling which differs from
the first one in forest construction. However, in both
of these algorithms different portions of sensor nodes
send their data to a single different sink, i.e., many-
to-one communication, whereas we consider the case
where many nodes send their data to many sinks.
3 PROBLEM FORMULATION
We present the following definitions that we will use
in this paper.
Definition 1 (Schedule) A schedule S : V → 2
N
is a
function that maps a node to a set of time slots.
Definition 2 (DAS-label) Given a network G =
(V, E), a sink ∆, a schedule S and a path γ = n ·
m.. . ∆, we say that n is DAS-labeled under S on γ
for ∆ if ∃t ∈ S (n) · ∃t
0
∈ S (m) : t
0
> t.
We call the node m on γ the ∆-parent of n and γ
the DAS-path for n.
Definition 3 (Strong and Weak schedule) Given a
network G = (V, E), a sink ∆ ∈ V and a schedule S,
S is said to be a strong DAS schedule for ∆ for a node
n ∈ V iff ∀ path γ
i
= n · m
i
. .. ∆, n is DAS-labeled un-
der S on γ
i
for ∆. S is a weak DAS schedule for ∆ for
n if ∃ path γ = n · m
i
. .. ∆ such that n is DAS-labeled
under S on γ for ∆.
A schedule S is strong DAS (resp. weak DAS) for
G iff ∀n ∈ V , S is strong DAS schedule (resp. weak
DAS schedule) for ∆ for n.
A strong schedule, which is impossible to de-
velop (Jhumka, 2010), in essence, is resilient to prob-
lems that occur in the network such as radio links not
working or node crashes during deployment. On the
other hand, a weak schedule is not resilient and, any
problem happening, will entail that a message from
node n to m will be lost.
Given a network with n sinks ∆
1
, ∆
2
,··· , ∆
n
we
wish to develop a weak schedule for all sinks. There
are different ways to achieve this. In general, to de-
velop a weak schedule, several works have adopted
the approach whereby a tree is first constructed,
rooted at the sink, and then slots assigned along the
branches to satisfy the data aggregation constraints.
A trivial solution is to construct n trees, each rooted
at a sink, and then to assign slots to nodes along the
trees. This means that nodes can have n slots, i.e.,
meaning that nodes may have to do n transmissions
for the same message. Thus, to reduce the number of
transmissions, we want to reduce the number of slots
for nodes to transmit in.
3.1 DAS Scheduling
We model our problem as follows:
We capture slots assignment with a set of decision
variables.
t
S
n
=
1 t ∈ S(n)
0 otherwise
A set value assignment to these variables represent
a possible schedule. The number of slots used, which
A Lower Bound on the Number of Nodes with Multiple Slots in Wireless Sensor Networks with Multiple Sinks
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