All-to-All Broadcast in WDM Linear Array with 3-length extension
K. MANOHARAN1, M. SABRIGIRIRAJ2
1Department of ECE, SNS College of Technology, Coimbatore, 641 035, Tamilnadu, INDIA
2 Department of ECE, SVS College of Engineering, Coimbatore, 642 109, Tamilnadu, INDIA
Abstract: - All-to-all broadcast communication, distributing messages from each node to every other node, is a
dense communication pattern and finds numerous applications in advanced computing and communication
networks from the control plane to datacenters. In this article, a linear array is extended by directly linking all
nodes which are separated by two intermediate nodes with additional fibers and this network is referred as
linear array with 3-length extension. The wavelength allotment methods are proposed to realize all-to-all
broadcast over WDM optical linear array with 3-length extension under multiple unicast routing model and the
wavelength number needed atmost to establish all-to-all broadcast is determined. The wavelength number
needed atmost to establish all-to-all broadcast in a linear array with 3-length extension is reduced by a
minimum of 61% and a maximum of 66% when compared to a basic linear array. Similarly, the wavelength
number needed atmost to establish all-to-all broadcast is reduced by a minimum of 24% and a maximum of
33% when compared to linear array with 2-length extension.
Key-Words: - All-to-All Broadcast, WDM Optical Network, Linear Array, Wavelength Assignment, RWA,
Modified Linear Array
Received: June 20, 2021. Revised: March 15, 2022. Accepted: April 17, 2022. Published: May 6, 2022.
1 Introduction
Wavelength Division Multiplexing (WDM)
technology working over optical networks is proven
to be a successful to cater the tremendous bandwidth
demand of emerging high performance and
computing applications. A WDM optical network
employs numerous optical nodes and nodes are
interconnected using optical fibers in some fashion.
WDM technology permits the passage for multiple
wavelength optical signals through the same fiber
and thus provides abundant bandwidth. Each optical
node employs required optical sources (Ex: laser
diodes) at the transmitter section to modulate the
input electrical signals and required optical detectors
(Ex: photo diodes) at the receiver section to
demodulate the received signal and extract the input
signal that was fed at the transmitter. Though the
same fiber can be used for signal transmission in
both forward and reverse directions, it is normally
assumed that each optical link is a set of two fibers,
with one fiber dedicated to forward transmission
and another one for reverse transmission. An optical
connection (lightpath) (m, n) corresponds to the
establishment of an optical path for transfer of a
packet from source m to destination n on a unique
wavelength. In the absence of wavelength
converters at the intermediate optical nodes, each
lightpath needs to be on the same wavelength from
source to destination.
All-to-all broadcast communication, distributing
messages from each node to every other node, is a
dense communication pattern and finds numerous
applications from network control plane to
datacenters [1-3]. In general, all-to-all broadcast is
employed for numerous applications in advanced
computing and communication systems which
employ WDM optical networks comprising
hundreds of optical nodes at the backbone and
involving huge number of operating wavelengths [4-
12]. Wavelength need to be assigned for various
lightpaths in such a way that no two lightpaths are
established using the same wavelength, if they share
any common link along entire route. Wavelengths
being a scarce and costly resource, its use need to be
restricted to reduce the complexity and cost of the
network [13]. Linear array topology, due to its small
node degree and regularity finds application in
interconnection networks [14-19]. However,
research on identifying new topologies with better
properties for interconnection networks is an
interesting and challenging area [20-21].
Researches have also been carried out in
modifying/extending, few of the popular topologies
namely linear array [22] and ring [20-21, 23]
towards reducing the hop count, wavelength
requirements and network survivability. On the
same way, in this paper, linear array topology is
extended in another fashion and all-to-all broadcast
communication is investigated for the same and the
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reduction in wavelength requirements is identified.
In addition, the results obtained in this paper will be
useful to analyse practical long-haul backbone
networks, as such networks may be partitioned into
multiple extended linear array and/or linear array
networks. Section 2 gives an overview of the basics
needed to understand the investigation done in this
paper. Wavelength number required atmost to
establish all-to-all broadcast and its associated link
load is obtained in section 3. Then, section 4
discusses about the significance of the results got
from this paper. Finally, section 5 completes the
paper highlighting future research avenues.
2 Preliminaries
Fig.1 shows a 16-node linear array network with 3-
length extension. A linear array is extended by
additionally linking two nodes which are separated
by two intermediate nodes as in [22] and similar to
that done for a ring network [20-21, 23]. This
network is referred as linear array with 3-length
extension. Here, in addition to immediate
neighbouring node, every node is additionally
linked to one another node on its right if feasible
and one additional node on its left, if feasible. As a
result, data can move from node to node
and nodeto nodeif such nodes exist. This
arrangement provides additional paths and reduces
the hop count for various lightpaths and also
wavelength number required to establish various
lightpaths.
Fig. 1: A 16-node linear array with 3-length
extension
The following concepts are essential to prove the
main results of this work.
Definition 1: An optical connection (lightpath) (m,
n) corresponds to the establishment of an optical
path under a prescribed routing method for transfer
of a packet from source m to destination n on a
unique wavelength.
Definition 2: An length connection is one, whose
difference between the destination node index
and the source node index is. For example,
in Fig. 1, the length of the connection that connects
node 1 to node 4 is 3.
Definition 3: “A link that directly joins two
adjacent nodes whose difference of index equals to
unity is said to be a ‘shorter link’. For example, in
Fig. 1, the link that joins node 1 with node 2 is said
to be a shorter link.” [22]
Definition 4: “A link that directly joins two nodes
whose difference of index equals to 3 is said to be a
‘longer link’. For example, Fig. 1, the link that joins
node 3 with node 6 is said to be a longer link.” [22]
Definition 5: “A connection that selects the longest
link among all the available links at the source node
and at each of the intermediate nodes to reach the
destination node is said to follow ‘longest link first
routing’. For example, in Fig. 1, under longest link
first algorithm, a connection from node 1 to node 8
selects the links joining the nodes 1 with 4, then
nodes 4 with 7 and nodes 7 with 8.” [22]
Definition 6: For a connection if, then
the connection is termed as rightward connection.
Else, if, then the connection is termed as
leftward connection.
Example: An example of wavelength allotment for
a 16-node linear array with 3-length extension under
longest link first algorithm to establish all-to-all
broadcast is described below.
Consider the 16-node linear array with 3-length
extension shown in Fig. 1. As rightward connections
and leftward connections do not share any fiber
because they use different sets of fibers, they can be
assigned same set of wavelengths. Hence, only all
connections going in rightward direction of all-to-all
broadcast are listed below:
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The above list of connections is partitioned into
multiple groups. All connections inside each group
are nonoverlpping with each other and are allotted a
common wavelength and is given below:
Thus, 23 wavelengths are needed atmost for a 16-
node linear array with 3-length extension to
establish all-to-all broadcast.
3 Main Results
Let the total number of nodes in a linear array with
3-length extension be represented by N.
Let be positive integers.
Let the wavelength number wanted atmost under
longest link first routing to establish all-to-all
broadcast be represented by Ws.
Lemma 1a: Let, where is a positive
integer. Then for each, such that
the
wavelength number wanted atmost to establish all
connections of length, in a node linear array
with 3-length extension is if (mod 3) = 0,
if
(mod 3 ) = 1,
if( mod6) = 5 and
if ( mod
6) = 2.
Proof: First, all connections of length in a
network with nodes are listed as follows:
where
is such that Then, the number of
connections of length is . Then the
listed connections are partitioned into two or more
groups such that no two connections in any group
overlap with each other. It is shown below:
Case i)mod 3 = 0
.
.
.
for
and if where
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By assigning a unique wavelength to the
connections of every group, wavelengths are
wanted to establish all j length connections.
Case ii)mod 3 = 1
.
.
.
for
and if where
By assigning a unique wavelength to the
connections of every group,
wavelengths are
wanted to establish all length connections.
Case iii)mod 3 = 2
.
.
.
for
and if where
By assigning a unique wavelength to the
connections of every group
if (mod 6) = 5 and
if (mod 6) = 2 wavelengths are wanted to
establish all length connections.
The following lemmas 1b through 1f, can be proved
similar to lemma 1a.
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Lemma 1b: Let, where is a
positive integer. Then for each, such that
wavelength number wanted atmost to establish
all connections of length , in a node linear array
with 3-length extension is if (mod 3) = 0,
if
( mod 3) = 1,
if (mod 6) = 5 and
if ( mod
6) = 2.
Lemma 1c: Let , where is a positive
integer. Then for each, such that
wavelength number needed atmost to establish all
connections of length, in a N node linear array
with 3-length extension is if (mod 3) = 0,
if(
mod 3) = 1,
if (mod 6) = 5 and
if ( mod 6)
= 2 under longest link first routing.
Lemma 1d: Let, where is a
positive integer. Then for each, suchthat
wavelength number wanted atmost to establish
connections of length , in a node linear array
with 3-length extension isif ( mod 3) = 0,
if
(mod 3) = 1,
if (mod 6) = 5 and
if ( mod
6) = 2.
Lemma 1e: Let , where is a positive
integer. Then for each, such that
wavelength number wanted atmost to establish
connections of length, in a node linear array
with 3-length extension isif ( mod 3) = 0,
if
(mod 3 ) = 1,
if (mod 6 ) = 5 and
if ( mod
6 ) = 2.
Lemma 1f: Let, where is a
positive integer. Then for each, such that
wavelength number wanted atmost to establish
all connections of length , in a N node linear array
with 3-length extension is if(mod 3) = 0,
if
mod 3 ) = 1,
if ( mod 6 ) = 5 and
if( mod
6) = 2.
Lemma 2: Let be a positive integer, then for =1
and = 3 one wavelength is sufficient to establish
all connections of length =1 and = 3 in a node
linear array with 3-length extension.
Proof: It is easy to observe that all connections of
length 1 namely
use only shorter link. Similarly, all connections of
length 3, namely
all use only longer links because of
longest link first algorithm. It is also observed that
all connections of length 1 uses only one shorter
link and connections of length 3 uses only one
longer link and hence they do not share any
common link. So, one wavelength is sufficient to
route all connections of length 1 and 3.
Lemma 3: Let be a positive integer, then for
andtwo wavelengths are sufficient to
establish all connections of length and
in a node linear array with 3-length extension.
Proof: It is easy to observe that all connections of
length 2 namely
use two shorter links alone for all
connections.Similarly all connections of length 6,
namely
all uses two longer links alone for all
connection in longest link first algorithm and hence
they do not share any common link. So, two
wavelengths are sufficient to route all connections
of length 2 and 6.
Lemma 4a: Let, where is a positive
integer. Then for each , such that
wavelength number wanted atmost to establish
connections of length
, in a node linear array
with 3-length extension is
if
,
if
,
if
.
Proof: First, all connections of length
in a
network with nodes are listed as follows,
where is such that
Then,
the number of connections of length
is
.
Then the listed connections are partitioned into two
or more groups such that no two connections in any
group overlap with each other. It is shown below:
Case i)
mod 3 = 0
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.
.
.
where
, for each sub collection, a
unique wavelength can be associated. The number
of connections of length
is
.
Therefore, atleast
wavelengths are
needed to establish all connections of length
.
Case ii)
mod 3 = 1
.
.
.
where
, for each sub collection, a
unique wavelength can be associated. The number
of connections of length
is
.
Therefore, atleast
wavelengths are
needed to establish all connections of length
.
Case iii)
mod 3 = 2
.
.
.
where
, for each sub collection, a
unique wavelength can be associated. The total
number of sub such of connections of length
is
. Therefore, atleast
wavelengths are needed to establish all connections
of length
The following lemmas 4b through 4f can be proved
similar to lemma 4a.
Lemma 4b: Let, where is a
positive integer. Then for each , such that
wavelength number wanted atmost to establish
connections of length
, in a node linear
array with 3-length extension is
if
,
if
and
if
Lemma 4c: Let , where is a positive
integer. Then for each, such that
wavelength number wanted atmost to establish
connections of length
, in a node linear array
with 3-length extension is
if
,
,
and
.
Lemma 4d: Let, where is a positive
integer. Then for each , such that
wavelength number wanted atmost to establish
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connections of length
, in a node linear
array with 3-length extension is
if
,
if
,
if
.
Lemma 4e: Let, where is a positive
integer. Then for each, such that
wavelength number wanted atmost to establish
connections of length
, in a node linear array
with 3-length extension is
if
,
,
if
and
if
.
Lemma 4f: Let, where is a positive
integer. Then for each , such that
wavelength number wanted atmost to establish
connections of length
, in a node linear
array with 3-length extension is
if
,
if
,
if
and
if
.
Theorem 1: If, then
.
Proof: By Lemma 1a, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively,
By Lemma 4a, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in the first
term, second term and third term respectively,
By combining (1) and (2),
Theorem 2: If, then
.
Proof: By Lemma 1b, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
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Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By Lemma 4b, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By combining (3) and (4),
Theorem 3: If, then
.
Proof: By Lemma 1c, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
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By Lemma 4c, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By combining (5) and (6),
Theorem 4: If, then
.
Proof: By Lemma 1d, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By Lemma 4d, the wavelength number wanted to
establish all connections of length
, where
is
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Substitute
in the first
term, second term and third term respectively in the
above expression,
By combining (7) and (8),
Theorem 5: If, then
.
Proof: By Lemma 1e, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By Lemma 4e, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in
the first term, second term and third term
respectively in the above expression,
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By combining (9) and (10),
Theorem 6: If, then
.
Proof: By Lemma 1f, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By Lemma 4f, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in
the first term, second term and third term
respectively,
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By combining (11) and (12),
Theorem 7: If, then
.
Proof: By Lemma 1a, Lemma 2 and Lemma3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in the first term, second term, third term and
fourth term respectively in the above expression,
By Lemma 4a, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in the first
term, second term and third term respectively in the
above expression,
By combining (13) and (14),
Theorem 8: If, then
.
Proof: By Lemma 1b, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
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Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By Lemma 4b, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By combining (15) and (16),
Theorem 9: If, then
.
Proof: By Lemma 1c, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
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By Lemma 4c, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in
the first term, second term, third term and fourth
term respectively,
By combining (17) and (18),
Theorem 10: If, then
Proof: By Lemma 1d, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By Lemma 4d, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in the first
term, second term and third term respectively,
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By combining (19) and (20),
Theorem 11: If, then
Proof: By Lemma 1e, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively in the above expression,
By Lemma 4e, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
in
the first term, second term, third term and fourth
term respectively,
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By combining (21) and (22),
Theorem 12: If,then
Proof: By Lemma 1f, Lemma 2 and Lemma 3, the
wavelength number wanted to establish all
connections of length less than or equal to
is
Substitute
in
the first term, second term, third term and fourth
term respectively,
By Lemma 4f, the wavelength number wanted to
establish all connections of length
, where
is
Substitute
andin the
first term, second term, third term and fourth term
respectivelyin the above expression,
By combining (23) and (24),
Now, the link load of the linear array with 3-
length extension for all-to-all broadcast is derived.
The link load of a network is the maximum number
of lightpaths that share a common link. First,
consider an arbitrary shorter link,joining the
nodes indexed and.It is easy to observe
that number of connections share the link
to transmit the message to other nodes. Hence, the
total number of connections that share any arbitrary
shorter link isNext, consider the link load of
an arbitrary longer link. For, let
represent the longer link joining the nodes indexed
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and.It is easy to observe that
number of nodes before node shares the link to
transmit the message to number of
nodes after node. Therefore, the number of
connections sharing any link is
. Hence, the link load of a longer link
is
. As the link load of
longer link is higher than that of shorter link, so the
link load of a longer link is the link load of the
network. Hence, the link load of the network is
given by,
4 Results and Discussion
Table 1 shows the wavelength requirement for all-
to-all broadcast in a linear array with 3-length
extension and its associated link load. Table 2
shows the wavelength number wanted atmost to
establish all-to-all broadcast and the associated link
load for certain values of node number N in a linear
array with 3-length extension. It can be noted that
the value of link load is slightly greater than the
wavelength number. Hence, the results derived in
the previous section are either optimum or near
optimum. The use of wavelength converter may
reduce wavelength requirement to the minimum
value but wavelength converter is very expensive.
Table 3 shows the wavelength number wanted
atmost to establish all-to-all broadcast for certain
values of node number N in a linear array with 3-
length extension, 2-length extension and a basic
linear array. The results show that the wavelength
number wanted atmost to establish all-to-all
broadcast in a linear array with 3-length extension is
reduced by a minimum of 61% and a maximum of
66% when compared to a basic linear array.
Similarly, the wavelength number needed atmost to
establish all-to-all broadcast is reduced by a
minimum of 24% and a maximum of 33% when
compared to linear array with 2-length extension.
Table 1. Wavelength number wanted atmost to
establish all-to-all broadcast along with its link load
in a linear array with 3-length extension.
Network
Topology
with N
nodes
Wavelength number
wanted atmost to
establish all-to-all
broadcast
Link Load
Linear
array
with 3-
length
extension
m is a positive
integer
for
Table 2. Wavelength number wanted atmost to
establish all-to-all broadcast along with its link load
for certain values of node number N in a linear array
with 3-length extension
Node
number
N
Wavelength
number
wanted
atmost
Link
load
Difference
between
wavelength
number and link
load
12
13
12
1
13
16
14
2
14
18
16
2
15
19
18
1
16
23
21
2
17
26
24
2
18
28
27
1
19
33
30
3
20
36
33
3
21
38
36
2
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22
44
40
4
23
48
44
4
24
50
48
2
25
56
52
4
26
60
56
4
27
62
60
2
28
69
65
4
29
74
70
4
30
77
75
2
31
85
80
5
32
90
85
5
33
93
90
3
34
102
96
6
35
108
102
6
36
111
108
3
37
120
114
6
38
126
120
6
39
129
126
3
40
139
133
6
41
146
140
6
42
150
147
3
43
161
154
7
44
168
161
7
45
172
168
4
46
184
176
8
47
192
184
8
48
196
192
4
50
216
208
8
60
305
300
5
70
420
408
12
80
546
533
13
90
682
675
7
100
849
833
16
Table 3. Comparison of wavelength number wanted
atmost to establish all-to-all broadcast for certain
values of node number in a linear array, linear
array with 2-length extension and linear array with
3-length extension.
Node
number
N
Linear
array
[14]
Linear
array with
2-length
extension
[22]
Linear array
with 3-
length
extension
12
36
18
13
13
42
21
16
14
49
24
18
15
56
28
19
16
64
32
23
17
72
36
26
18
81
40
28
19
90
45
33
20
100
50
36
21
110
55
38
22
121
60
44
23
132
66
48
24
144
72
50
27
182
91
62
30
225
112
77
33
272
136
93
36
324
162
111
39
380
190
129
42
441
220
150
45
506
253
172
48
576
288
196
51
650
325
220
60
900
450
305
70
1225
612
420
80
1600
800
546
90
2025
1012
682
100
2500
1250
849
125
3906
1953
1322
150
5625
2812
1887
175
7656
3828
2581
200
10000
5000
3366
300
22500
11250
7525
500
62500
31250
20916
700
122500
61250
40949
1000
250000
125000
83499
5 Conclusion and Future Work
In this work, the wavelength number wanted atmost
to establish all-to-all broadcast in a WDM optical
linear array with 3-length extension is determined
and the wavelength allotment technique is given.
The wavelength number needed is nearly equal to
the link load and so the results are near to optimum
or optimum. The wavelength number wanted
atmost to establish all-to-all broadcast in a linear
array with 3-length extension is reduced by a
minimum of 61% and a maximum of 66% when
compared to a basic linear array. Similarly, the
wavelength number wanted atmost to establish all-
to-all broadcast is reduced by a minimum of 24%
and a maximum of 33% when compared to linear
array with 2-length extension. Wavelength number
requirement needs to be investigated with still
higher order extensions, to judge the rate of
reduction in wavelength number requirements with
increasing extension and is a challenging issue.
Also, deriving a generalized expression for
wavelength number requirement in a linear
array/ring network with k-length extension (k is any
positive integer and k < N where N is the total
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.9
K. Manoharan, M. Sabrigiriraj
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Volume 21, 2022
number of nodes in the network) is another
interesting and challenging future work. Examining
the effects of physical layer impairments and
network survivability on routing and wavelength
assignment are other research issues with these
extended networks.
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problem to the final findings and solution.
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