Two-Channel Conflict-Free Square Grid Aggregation

Abstract

The conflict-free data aggregation problem in an arbitrary wireless network is NP-hard, both in the case of a limited number of frequencies (channels) and with an unlimited number of channels. However, on graphs with a particular structure, this problem sometimes becomes polynomially solvable. For example, when the network is a square grid (lattice), at each node of which there is a sensor, and the transmission range does not exceed 2, the problem is polynomially solvable. In this paper, we consider the problem of conflict-free data aggregation in a square grid, when network elements use two frequencies, and the transmission range is at least 2. It consists in finding an energy-efficient conflict-free (we will give later the definition of a conflict) schedule of minimum length for the transfer of aggregated data from all vertices of the lattice to the center node (base station).

Full text

Two-Channel Conflict-Free Square Grid
Aggregation?
Adil Erzin1,2[0000−0002−2183−523X]and Roman Plotnikov1[0000−0003−2038−5609]
1Sobolev Institute of Mathematics, SB RAS, Novosibirsk 630090, Russia
2Novosibirsk State University, Novosibirsk 630090, Russia
{adilerzin,prv}@math.nsc.ru
Submitted to Special Session 4 “Intractable Problems of Combinatorial Op-
timization, Computational Geometry, and Machine Learning: Algorithms and
Theoretical Bounds”
Abstract. The conflict-free data aggregation problem in an arbitrary
wireless network is NP-hard, both in the case of a limited number of fre-
quencies (channels) and with an unlimited number of channels. However,
on graphs with a particular structure, this problem sometimes becomes
polynomially solvable. For example, when the network is a square grid
(lattice), at each node of which there is a sensor, and the transmission
range does not exceed 2, the problem is polynomially solvable. In this pa-
per, we consider the problem of conflict-free data aggregation in a square
grid, when network elements use two frequencies, and the transmission
range is at least 2. It consists in finding an energy-efficient conflict-free
(we will give later the definition of a conflict) schedule of minimum length
for the transfer of aggregated data from all vertices of the lattice to the
center node (base station).
We find polynomially solvable cases, and also develop an efficient algo-
rithm that builds a schedule with a guaranteed accuracy estimate. For
example, when the transmission range is 2, the algorithm constructs ei-
ther an optimal schedule or a schedule whose length exceeds the optimal
latency by no more than 1. For a transmission range more than 2, an esti-
mate of the reduction in the length of the schedule is obtained compared
to the case when only one frequency is used.
Keywords: Multichannel aggregation ·Square grid ·Conflict-free schedul-
ing.
1 Introduction
Data transmission in wireless networks, such as sensor networks, is carried out us-
ing radio communications. During convergecasting, each network element trans-
mits a packet of aggregated data received from its children, as well as its data to
the parent vertex once during the entire aggregation session. This requirement
?The research is partly supported by the Russian Science Foundation (pro jects 18–
71–00084, section 3 and 19–71–10012, sections 1, 2, 3.1, 4)

2 A. Erzin et al.
is dictated by the extreme power consumption of the transmission process and
entails the need to build a spanning aggregation tree (AT) with arcs directed to
the sink, which is called the base station (BS). The faster the aggregated data
reaches the BS, the better the schedule. In a TDMA scheduling, time is divided
in equal-length slots under assumptions that each slot is long enough to send or
receive one packet [4]. Minimizing time for the aggregated convergecast in this
case is equivalent to minimizing the number of time slots required for all packets
to reach the sink [17].
The solution of the problem includes two components: an AT, and a schedule,
which assigns a transmitting time slot for each node so that every node transmits
after all its children in the tree have, and potentially interfered links scheduled to
send in different time slots. The last condition means that the TDMA schedule
should be interference-free, i.e., no receiving node is within the interference range
of the other transmitting node. There are two types of interferences or collisions
in wireless networks: primary and secondary. A primary collision occurs when
more than one node transmits to the same destination. In tree-based aggregation,
it corresponds to the case when two or more children of the same parent node
send their packets in the same time slot. A secondary collision occurs when a
node overhears transmissions intended for another node. Links in the underlying
communication graph cause such kind of collision, but not in the aggregation
tree. If the links employ different frequencies (communication channels), then
this type of conflict does not occur.
The conflict-free data aggregation problem was proven to be NP-hard [3]
even if AT is known [5]. Therefore, almost all existing results in literature are
polynomial algorithms for finding approximate solutions when the network el-
ements use one channel [1,3, 10–14, 16–18] or several channels [2, 9, 15]. In [1]
presented a novel cross-layer approach for reducing the latency in disseminat-
ing aggregated data to the BS over multi-frequency radio links. Their approach
forms the aggregation tree to increase the simultaneity of transmissions and re-
duce buffering delay. Aggregation nodes picked, and time slots are allocated to
the individual sensors so that the most number of ready nodes can transmit their
data without delay. The use of different radio channels allows to avoid colliding
transmissions. Their approach is validated through simulation and outperforms
previously published schemes. [3] considered a min-length scheduling problem to
aggregate data in the BS. The authors study the problem with an equal trans-
mission range of all sensors. They assume that, in each time slot, data sent by
a sensor reaches exactly all sensors within its transmission range, and a sensor
receives data if it is the only data that reaches the sensor during this time slot.
They first prove that the problem is NP-hard even when all sensors have de-
ployed a grid, and data from all sensors are required to be aggregated in the
BS. A (∆−1)-approximation algorithm is designed, where ∆+ 1 equals the
maximum number of sensors within the transmission range of any sensor. The
authors also simulate the proposed algorithm and compare it with the existing
algorithm. The obtained results show that their algorithm has much better per-

Two-Channel Conflict-Free Square Grid Aggregation 3
formance in practice than the theoretically proved guarantee and outperforms
other algorithms.
In [12], the authors investigate the question: “How fast can information be
collected from a wireless sensor network organized as a tree?” To address this,
they explore and evaluate several different techniques using realistic simulation
models under the many-to-one communication paradigm known as a converge-
casting. First, min-time scheduling on a single frequency channel considered.
Next, they combine scheduling with transmission power control to mitigate the
effects of interference and show that while power control helps in reducing the
schedule length under a single frequency, scheduling transmissions using multi-
ple frequencies is more efficient. The authors gave lower bounds on the schedule
length without interference conflicts, and proposed algorithms that achieve these
bounds. They also evaluate the performance of various channel assignment meth-
ods and find empirically that for moderate size networks of about 100 nodes, the
use of multifrequency scheduling can suffice to eliminate most of the interfer-
ence. Then, the data collection rate no longer remains limited by interference
but by the topology of the routing tree. To this end, they construct degree-
constrained spanning trees and capacitated minimal spanning trees and show
significant improvement in scheduling performance over different deployment
densities. Lastly, they evaluate the impact of different interference and channel
models on the schedule length.
In [16], the authors consider the problem of aggregation convergecast schedul-
ing. The solution to aggregation convergecast satisfies the aggregation process,
expressed as precedence constraints, combined with the impact of the shared
wireless medium, expressed as resource constraints. Both sets of constraints in-
fluence the routing and scheduling. They propose an aggregation tree construc-
tion suitable for aggregation convergecast that is a synthesis of a tree tailored to
precedence constraints and another tree tailored to resource constraints. Addi-
tionally, they show that the scheduling component modeled as a mixed graph col-
oring problem. Specifically, the extended conflict graph introduced, and through
it, a mapping from aggregation convergecast to mixed graphs described. Bounds
for the graph coloring provided and a branch-and-bound strategy developed
from which the authors derive numerical results that allow comparison against
the current state-of-the-art heuristic.
[18] focuses on the latency of data aggregation. Since the problem is NP-
hard, many approximate algorithms have proposed to address this issue. Using
maximum independent set and first-fit algorithms, in this study a scheduling
algorithm, Peony-tree-based Data Aggregation (PDA), designed which has a la-
tency bound of 15R+∆−15, where Ris the network radius (measured in hops)
and ∆is the maximum node degree. They theoretically analyze the performance
of PDA based on different network models and further evaluate it through ex-
tensive simulations. Both the analytical and simulation results demonstrate the
advantages of PDA over the state-of-art algorithm, which has a latency bound
of 23R+∆−18.

4 A. Erzin et al.
In [9], the authors focus on designing a multi-channel minimum latency ag-
gregation scheduling protocol, named MC-MLAS, using a new joint approach for
tree construction, channel assignment, and transmission scheduling. To the best
knowledge of the authors, this is the first work in the literature that combines
orthogonal channels and partially overlapping channels to consider the total la-
tency involved in data aggregation. Extensive simulations verify the superiority
of MC-MLAS in WSNs.
In [15], the authors consider a problem of minimum length scheduling for the
conflict-free aggregation convergecast in wireless networks in a case when each
element of a network uses its frequency channel. This problem is equivalent to
the well-known NP-hard problem of telephone broadcasting since only the con-
flicts between the children of the same parent taken into account. They propose
a new integer programming formulation and compare it with the known one
by running the CPLEX software package. Based on the results of a numerical
experiment, they concluded that their formulation is preferable in practice to
solve the considered problem by CPLEX than the known one. The authors also
propose a novel heuristic algorithm, which based on a genetic algorithm and a
local search metaheuristic. The simulation results demonstrate the high quality
of the proposed algorithm compared to the best-known approaches.
However, if the network has a regular structure, for example, it is a lattice,
the problem is solved in polynomial time. Known that in a square lattice, in each
node of which information is located, the process of single-channel data aggrega-
tion is simple [8]. Moreover, in some cases, for example, when the transmission
range equals 1 [8] or 2 [6], one can build an optimal schedule. If the transmission
range is greater than 2, then one can find a solution close to the optimal [5, 7].
1.1 Our contribution
In this paper, for the first time, the problem of two-channel conflict-free aggrega-
tion in a square lattice is considered, when the transmission distance is not less
than 2. If the transmission distance is 1, then the problem does not differ from
single-channel aggregation and is solved completely [8]. We have developed and
analyzed an efficient algorithm that builds either an optimal or near-optimal so-
lution. We estimated the reduction in the length of the schedule compared with
the case when one channel is used.
The rest of the paper is organized as follows. Section 2 contains the state-
ment of the problem. The main results with the description and analysis of the
algorithm make up the contents of Section 3. Section 4 concludes the paper.
2 Problem formulation
We suppose that the network elements are positioned at the nodes of a square
grid of size (n+ 1) ×(m+ 1). For convenience, we will call the network elements
sensors, vertices, or nodes equivalently. A sink node (or BS) is located at the
point (0,0). At each time slot, any sensor except the sink node can either be

Two-Channel Conflict-Free Square Grid Aggregation 5
idle, send the data to another sensor within its transmission range, or receive the
data from another sensor within its transmission range. We assume that each
sensor has the same transmission distance d≥2 in L1metric. A sink node can
only receive the data at any time slot. Each data transmission is performed using
one of the available frequency channels (for short, further, they are referred to
as channels) and each sensor can use any channel for data transmission and
receiving. Besides, we suppose that the following conditions met:
–each vertex sends a message only once during the aggregation session (except
the sink which always can only receive messages);
–once a vertex sends a message, it can no longer be a destination of any
transmission;
–if some vertex sends a data packet by the channel c, then during the same
time slot none of the other vertices within a receiver’s interference range can
post a message by the channel c;
–a vertex cannot receive and transmit at the same time slot.
For simplicity, we assume that the interference range equals the transmission
range. We have precisely two available channels for data transmission: 0 and 1.
The problem consists in constructing the conflict-free min-length schedule of the
data aggregation from all the vertices to the BS.
3 Building a conflict-free schedule with different
transmission ranges
As mentioned before, the considered problem is NP-hard in the general case.
Therefore, our goal is to find some individual cases and propose polynomial
algorithms that construct either optimal solutions or solutions with guaranteed
estimations on the schedule length. In this section, we offer such algorithms and
estimates for the different values of the transmission range.
Let us introduce some notations that are used further.
Definition 1. Let the distance from the vertex ito the sink be the minimum
number of time slots that are necessary to transmit the data from ito the sink.
Definition 2. The most remote vertex (MRV) is such vertex, that the distance
from it to the sink is maximum among all vertices.
In square grid (n+ 1) ×(m+ 1), the distance from the node (x, y) to the sink
equals d(x+y)/de, where dis transmission range and daeis a smallest integer
not less than a. Obviously, the vertex (n, m) is MRV. There can be more than
one MRV, and the distance depends of the remainder of division of nand mby
d.
We will also refer to the distance from a vertex to the sink as its remoteness.
Let Dbe the remoteness of MRV. The following two propositions are obvious:
Proposition 1. The aggregation time cannot be less than D.

6 A. Erzin et al.
Proposition 2. If remoteness of at least two different vertices is R, then the
aggregation time cannot be less than R+ 1.
The validity of the last statement follows from the property that any vertex
during one time round can receive no more than one message.
For convenience, we call the row a set of sensors that are positioned at the
points with the same ordinates. All the presented below algorithms consist of two
stages: vertical and horizontal aggregation. During the vertical transmissions, all
sensors except ones in the row 0 transmit the data downwards or upwards. At the
end of the vertical stage, all data are aggregated in row 0. At the second stage,
the sensors in the row 0 transmit the data horizontally until the sink receives all
the data.
Further, we will consider separately two cases: when d= 2 and when d > 2.
3.1 The transmission range is 2
Suppose that d= 2. Let us prove the following
Lemma 1. In the square grid (n+ 1) ×(m+ 1), if the transmission range is
2 and the number of channels is 2, then the length of any feasible aggregation
schedule is not less than bn/2c+bm/2c+ 1, where bacis the largest integer not
exceeding a.
Proof. If at least one of two values, nand m, is odd, then the remoteness of
(n, m) is bn/2c+bm/2c+ 1, and, according to the Proposition 1, the aggregation
time cannot be less than bn/2c+bm/2c+ 1 in this case. If nand mare even
then there are three MRV whose remoteness is bn/2c+bm/2c. Therefore, in
this case, according to the Proposition 2, aggregation time cannot be less than
bn/2c+bm/2c+ 1.
Lemma 2. In the square grid (n+1)×(m+1), if nand mare odd, the transmis-
sion range is 2 and the number of channels is 2, then the length of any feasible
aggregation schedule is not less than bn/2c+bm/2c+ 2.
Proof. In this case, three MRV, (n−1, m), (n, m −1), and (n, m) are at the dis-
tance bn/2c+bm/2c+ 1 from the sink. Therefore, according to the Proposition
2, aggregation time cannot be less than bn/2c+bm/2c+ 2.
Let us describe the Algorithm 1 of aggregation scheduling in the square grid
(n+ 1) ×(m+ 1) when the transmission range is 2, and the number of channels
is 2. The pseudo-code of the vertical aggregation stage is given in the Algorithm
1. The examples of the proposed vertical aggregation algorithm are illustrated in
Fig. 1. As it follows from lines 3–8, during the first dm/2e − 1 time slots, at each
time slot, the highest two rows that did not transmit yet simultaneously transmit
the data downwards at a distance 2. Subroutine T ransmitRow(j, k, ch, t) which
is called in lines 5–6 assigns transmission of each sensor in j-th row vertically
to the corresponding sensors in k-th row at the time slot tusing the channel

Two-Channel Conflict-Free Square Grid Aggregation 7
ch. Here one row uses channel 0, another one - channel 1, and therefore there
are no conflicts in such data transmission. After that, only one or two rows left
that did not transmit (except the row 0, which does not transmit at this stage):
one in a case when mis odd and two – if it is even. Then, according to lines
9–19, at each time slot, the highest row, that did not transmit yet, transmits
the data downwards at a distance 1 using two channels. It is performed without
conflicts because vertices with different parity of abscissa use different channels.
If mis odd, then the data transmission that is described in lines 9–19 requires
one time slot. Subroutine T ransmitS ensor((i, j),(l, k), ch, t) which is called in
line 19 assigns transmission of sensor (i, j) to sensor (l, k) at the time slot tusing
the channel ch. If mis even, it requires two time slots. Overall, the described
vertical aggregation takes bm/2c+ 1 time slots.
Algorithm 1 Transmission range equals 2. Vertical aggregation
1: t←0;
2: j←m;
3: while j > 2do
4: t←t+ 1;
5: T ransmitRow(j, j −2,0, t);
6: T ransmitRow(j−1, j −3,1, t);
7: j←j−2;
8: end while
9: while j > 0do
10: t←t+ 1;
11: for all i∈ {0, ..., n}do
12: ch ←0;
13: if iis odd then
14: ch ←1;
15: end if
16: T ransmitS ensor((i, j),(i, j −1), ch, t);
17: i←i−1;
18: end for
19: end while
Remark 1. The time complexity of vertical aggregation is O(m). During one time
round, all the vertices of the two highest layers send. After they have transferred,
it is enough to remember only the number of the highest layer, whose vertex has
not transmitted yet. Therefore, space (additional memory) needed to implement
vertical aggregation is O(1).
The pseudo-code for the horizontal aggregation stage is presented in Algo-
rithm 2. The examples that show how horizontal aggregation is performed are
presented in Fig. 2. According to lines 3–8, during the first dn/2e − 1 time slots,
at each time slot, the most remote two vertices that did not transmit yet simul-
taneously transmit the data to the left at a distance 2 using different channels.

8 A. Erzin et al.
Algorithm 2 Transmission range equals 2. Horizontal aggregation
1: t←0;
2: i←n;
3: while i > 2do
4: t←t+ 1;
5: T ransmitS ensor((i, 0),(i−2,0),0, t);
6: T ransmitS ensor((i−1,0),(i−3,0),1, t);
7: i←i−2;
8: end while
9: while i > 0do
10: t←t+ 1;
11: T ransmitS ensor((i, 0),(i−1,0),0, t);
12: i←i−1;
13: end while
After that, as it follows from lines 9–13, at each time slot, the most remote ver-
tex that did not transmit yet transmits the data to the left at a distance 1 until
such vertex exists. The last subroutine requires one time slot if nis odd, and two
time slots if it is even. Eventually, the horizontal aggregation takes bn/2c+ 1
time slots.
Remark 2. The time complexity of horizontal aggregation is O(n), and the ad-
ditional space required to implement the procedure is limited to O(1). Conse-
quently, the complexity of constructing a two-channel conflict-free aggregation
schedule in the (n+ 1) ×(m+ 1) grid with a transmission distance of 2 is
O(m+n), with additional memory space O(1). Although each vertex of the
grid must store its state, the total memory for storing incoming and current
information is O(mn).
t=1 t=2 t=3 t=4-5 t=4 t=5
m = 8 (even) m = 9 (odd)
Fig. 1. Example of vertical data aggregation when transmission distance equals 2.

Two-Channel Conflict-Free Square Grid Aggregation 9
t=1
t=2
t=3
t=4
t=1
t=2
t=3
t=4
t=5
t=5
n = 8 (even) n = 9 (odd)
Fig. 2. Example of horizontal data aggregation when transmission distance equals 2.
As a result, the entire aggregation process takes bm/2c+bn/2c+2 time slots.
Due to the Lemma 2, a solution that is constructed by the proposed algorithm
is optimal if nand mare odd. According to the Lemma 1, in all other cases, the
length of a constructed schedule does not exceed the minimum schedule length
by more than 1. As it is proved in [6], in a case when the only channel is used
the minimum length of an aggregation schedule is bn/2c+bm/2c+ 3. Therefore,
the usage of two channels allows decreasing the length of a schedule by one.
3.2 The transmission range is at least 3
In this section we propose the algorithm that constructs a schedule of two-
channels aggregation on a square grid (n+ 1) ×(m+ 1) when dis arbitrary
not less than 3. Assume that m=Md +rv, rv< d and n=Nd +rh, rh< d.
As well as the algorithm described in previous section, this algorithm consists
of two stages: vertical and horizontal aggregation. We will describe each stage
separately.
Vertical aggregation. For convenience purposes, let us colorize all the vertices.
Initially, let each vertex (i, j), i, j ≤mbe colored in red if m−j≡0 (mod d) and
let it be colored in blue otherwise. After the moment when a vertex transmits
the data, it becomes grey. When all the vertices of the same row colored in the
same color, we will also assign the corresponding color to the row. Initially there
are bm/dc+ 1 red rows and m− bm/dcblue rows.
The pseudo-code has given in Algorithm 3. At first, as it stated in lines 2–14,
a pair of top blue rows iteratively transmit the data downstairs at a distance
dby different channels during one time slot. These transmissions are repeated
until all of (M−1)(d−1) top blue rows send their data. Meanwhile, the top red
row transmits downwards at a distance das soon as the number of blue rows
within 2drows below becomes not more than one, otherwise, the two top blue
rows would transmit at the same time slot, and this would generate a conflict.

10 A. Erzin et al.
After every (M−1)(d−1) top blue rows transmitted, in lines 15–19, the top red
rows sequentially transmit downwards at a distance duntil the moment when
only two red rows remain. Note that after that exactly d+rv+ 1 non-grey rows
left. Then, in lines 20–29, the top rvrows transmit downstairs during the next
drv/2etime slots. After that, in lines 30–41, the rows between row 0 and row d
transmit the data. Here the data is transmitted at less distance than d. It is easy
to see that to transmit all the data of some row to another row at distance d−k
without conflicts by two channels d(k+ 1)/2etime slots are required. Note that
for this, we slightly changed a signature of the method T ransmitRow to denote
such data transmission. Finally, only two non-grey rows remain: row 0 and row
d. At the last time slot, row dtransmits at distance ddownwards. Overall, the
described vertical aggregation uses
d(M−1)(d−1)/2e+drv/2e+ 2
b(d−1)/2c+1
X
i=2
di/2e+xdd(d/2 + 1)/2e+ 2
time slots, where xd= 1 if dis even and 0 otherwise. An example of vertical
aggregation is presented in Fig. 3.
t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10-11 t=12
Fig. 3. Example of two-channel vertical data aggregation when d= 4.
Horizontal aggregation. The horizontal aggregation performed on row 0 of
the grid when all the data aggregated to this row. At first, we assume that
n=Nd and propose the algorithm for this case. For simplicity, in this subsection
we refer to sensor (i, 0) as sensor i,i= 0, . . . , n. Suppose that each sensor is

Two-Channel Conflict-Free Square Grid Aggregation 11
colored in blue or red color: sensor iis colored in red if i≡0 (mod d), and it
is colored in blue otherwise. As well as it was previously, the sensor becomes
grey as soon as it transmits the data. With such colorizing, the blue sensors are
divided by the red sensors into groups of d−1 sensors. We classify these groups
into 3 types: A,B, and C. The group {i, . . . , i +d−2}has type A (B or C) if
b(n−i)/dc ≡0 (1 or 2) (mod 3). For example, the group {n−d+ 1, . . . , n −1}has
type A. The idea of the algorithm is that in each time slot, sensors of groups of
the same type perform the same transmissions. For this reason, we enumerate
the sensors of each group from 1 to d-1.
The pseudo-code of the proposed procedure is given in Algorithm 4. Here
we use the method T ransmitS ensor whose signature differs from the similar
method used above. For instance, T ransmitS ensor(1,0, A, 0, t) means that by
channel 0 at time slot tsensor n−d+ 1 transmits the data to sensor n−d,
sensor n−4d+ 1 transmits to sensor n−4d, and so on. As noted in lines 6–
16, during first b(d−2)/2ctime slots, only sensors in groups of types A and
B transmit the data, while sensors in groups of type C remain in the idle state
because transmission from any of them would lead to a conflict. After that, in
each group of types A and B, only two blue sensors remain if dis odd and one –
if it is even. The last blue sensors of groups A and B transmit the data in lines
18–30. And, when it is possible, some sensors of groups of type C transmit the
data at the same time. In lines 33–39, the last blue sensors of groups C transmit
the data during the next b(d−2)/2ctime slots. In total, all blue sensors transmit
the data during the first d−1 time slots of the aggregation session. In lines 40–44,
red sensors sequentially transmit the data from right to left. Note that the first
red sensor transmits simultaneously with the last data transmission in groups C.
After that, during the next N−1 time slots, other red sensors transmit the data.
Overall, the horizontal aggregation takes N+d−2 time slots. As an illustration,
two examples are presented in Fig. 4 and Fig. 5. In first one dis odd (5) and in
second it is even (6).
t=1
t=2
t=3
t=4
t=5
t=6
t=7
t=8
A B C A B
Fig. 4. Example of two-channel horizontal data aggregation when d= 5.

12 A. Erzin et al.
t=1
t=2
t=3
t=4
t=5
t=6
t=7
t=8
A B C A
Fig. 5. Example of two-channel horizontal data aggregation when d= 6.
It is easy to observe that if nis not a multiple of d, then the horizontal
aggregation time increases by 1. Indeed, the sensors 1, . . . , n −dbn/dc − 1 can
transmit the data during the first d−1 time slots in any case, and it is required a
time slot more to aggregate the data from the red sensors. Eventually, horizontal
aggregation takes dn/de+d−2 time slots.
In total the proposed procedure constructs a schedule of length
d(bm/dc− 1)(d−1)/2e+dn/de+drv/2e+2
b(d−1)/2c+1
X
i=2
di/2e+xdd(d/2+ 1)/2e+d.
Remark 3. Note that the time complexity of the described algorithm has the
same order as the length of the schedule O(n+m), and the additional mem-
ory space required for its implementation is O(1). However, the length of the
input data determined by the size of the grid; therefore, when implementing the
algorithm, O(nm) memory space is used.
We compared the schedule length obtained by this algorithm with the best-
known approach for the one-channel aggregation, which proposed in [7]. The
results presented in Table 1. In most cases, the two-channel aggregation schedule
constructed by the proposed algorithm has less length than the length of the one-
channel aggregation schedule.
Table 1. Comparison of schedule length for aggregation by 1 and 2 channels
d11 ×11 26 ×26 51 ×51 101 ×101
2 ch 1 ch 2 ch 1 ch 2 ch 1 ch 2 ch 1 ch
3 12 15 22 25 38 43 72 75
4 14 19 24 25 39 39 61 61
5 15 20 24 26 36 36 56 56
7 21 35 29 41 43 43 59 59
10 30 49 40 61 52 57 67 67

Two-Channel Conflict-Free Square Grid Aggregation 13
4 Conclusion
In this paper, we considered the problem of two-channel conflict-free dater aggre-
gation in a square lattice and proposed an efficient algorithm that yields a better
solution than the convergecasting using one frequency. However, with increas-
ing transmission distance, the advantages of two-channel aggregation disappear.
This drawback is associated with insufficient consideration of the specifics of
two-channel aggregation. We plan to fix this shortcoming in the future.
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Two-Channel Conflict-Free Square Grid Aggregation 15
Algorithm 3 Arbitrary transmission range. Vertical aggregation
1: t←0;
2: while The number of blue rows exceeds d−1 + rvdo
3: t←t+ 1;
4: j0←the maximum number of red row;
5: if There are not more than one blue row between j0-th and (j0−2d)-th rows
then
6: T ransmitRow(j0, j0−d, 1, t);
7: end if
8: j1←the maximum number of blue row;
9: T ransmitRow(j1, j1−d, 0, t);
10: j2←the maximum number of blue row less than j1;
11: if j2> d +rvthen
12: T ransmitRow(j2, j2−d, 1, t);
13: else
14: j←the maximum number of red row;
15: T ransmitRow(j, j −d, 1, t);
16: end if
17: end while
18: while The number of red rows exceeds 2 do
19: t←t+ 1;
20: j←the maximum number of red row;
21: T ransmitRow(j, j −d, 0, t);
22: end while
23: j←d+rv
24: while j > d do
25: t←t+ 1;
26: T ransmitRow(j, j −d, 0, t);
27: j←j−1
28: if j > d then
29: T ransmitRow(j, j −d, 1, t);
30: j←j−1
31: end if
32: end while
33: for all k= 1, . . . b(d−1)/2cdo
34: tδ← d(k+ 1)/2e;
35: T ransmitRow(k , d, {0,1},[t+ 1, t +tδ]);
36: t←t+tδ;
37: T ransmitRow(d−k , 0,{0,1},[t+ 1, t +tδ]);
38: t←t+tδ;
39: end for
40: if dis even then
41: tδ← d(d/2 + 1)/2e;
42: T ransmitRow(d/2,0,{0,1},[t+ 1, t +tδ]);
43: t←t+tδ;
44: end if
45: T ransmitRow(d, 0,0,[t+ 1, t +tδ]);
46: t←t+tδ;

16 A. Erzin et al.
Algorithm 4 Arbitrary transmission range. Horizontal aggregation
1: t←0;
2: sA0←d−2;
3: sA1←d−3;
4: sB0←2;
5: sB1←3;
6: for all i= 1,...,b(d−2)/2cdo
7: t←t+ 1;
8: T ransmitS ensor(sA0, d, A, 0, t);
9: sA0←sA0−2;
10: T ransmitS ensor(sA1, d −1, A, 1, t);
11: sA1←sA1−2;
12: T ransmitS ensor(sB0,0, B, 0, t);
13: sB0←sB0+ 2;
14: T ransmitS ensor(sB1,1, B, 1, t);
15: sB1←sB1+ 2;
16: end for
17: t←t+ 1;
18: if dis odd then
19: T ransmitS ensor(1, d, A, 0, t);
20: T ransmitS ensor(d−1,0, A, 1, t);
21: T ransmitS ensor(d−1,0, B, 0, t);
22: t←t+ 1;
23: T ransmitS ensor(1, d, B, 1, t);
24: T ransmitS ensor(d−2, d, C, 0, t);
25: T ransmitS ensor(d−1,0, C, 1, t);
26: else
27: T ransmitS ensor(d−1, d+ 1, A, 0, t) except sensor n−1 that transmits to sensor
n;
28: T ransmitS ensor(1,0, B, 0, t);
29: T ransmitS ensor(d−1,0, C, 1, t);
30: end if
31: sC0←1;
32: sC1←2;
33: for all i= 1,...,b(d−2)/2cdo
34: t←t+ 1;
35: T ransmitS ensor(sCO ,0, C, 0, t);
36: sCO ←sC O + 2;
37: T ransmitS ensor(sC1, d, C, 1, t);
38: sC1←sC1+ 2;
39: end for
40: T ransmitM ostRemoteRedSensor (t);
41: for all i= 1,...,N −1do
42: t←t+ 1;
43: T ransmitM ostRemoteRedSensor (t);
44: end for