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Packing internally disjoint Steiner paths of data center networks
Authors
Jou-Ming Chang
Rong-Xia Hao
Jaeun Lee
Wen-Han Zhu
Publication date
24 January 2024
Publisher
View
on
arXiv
Abstract
Let
S
β
V
(
G
)
S\subseteq V(G)
S
β
V
(
G
)
and
Ο
G
(
S
)
\pi_{G}(S)
Ο
G
β
(
S
)
denote the maximum number
t
t
t
of edge-disjoint paths
P
1
,
P
2
,
β¦
,
P
t
P_{1},P_{2},\ldots,P_{t}
P
1
β
,
P
2
β
,
β¦
,
P
t
β
in a graph
G
G
G
such that
V
(
P
i
)
β©
V
(
P
j
)
=
S
V(P_{i})\cap V(P_{j})=S
V
(
P
i
β
)
β©
V
(
P
j
β
)
=
S
for any
i
,
j
β
{
1
,
2
,
β¦
,
t
}
i,j\in\{1,2,\ldots,t\}
i
,
j
β
{
1
,
2
,
β¦
,
t
}
and
i
β
j
i\neq j
i
ξ
=
j
. If
S
=
V
(
G
)
S=V(G)
S
=
V
(
G
)
, then
Ο
G
(
S
)
\pi_{G}(S)
Ο
G
β
(
S
)
is the maximum number of edge-disjoint spanning paths in
G
G
G
. It is proved [Graphs Combin., 37 (2021) 2521-2533] that deciding whether
Ο
G
(
S
)
β₯
r
\pi_G(S)\geq r
Ο
G
β
(
S
)
β₯
r
is NP-complete for a given
S
β
V
(
G
)
S\subseteq V(G)
S
β
V
(
G
)
. For an integer
r
r
r
with
2
β€
r
β€
n
2\leq r\leq n
2
β€
r
β€
n
, the
r
r
r
-path connectivity of a graph
G
G
G
is defined as
Ο
r
(
G
)
=
\pi_{r}(G)=
Ο
r
β
(
G
)
=
min
{
Ο
G
(
S
)
β£
S
β
V
(
G
)
\{\pi_{G}(S)|S\subseteq V(G)
{
Ο
G
β
(
S
)
β£
S
β
V
(
G
)
and
β£
S
β£
=
r
}
|S|=r\}
β£
S
β£
=
r
}
, which is a generalization of tree connectivity. In this paper, we study the
3
3
3
-path connectivity of the
k
k
k
-dimensional data center network with
n
n
n
-port switches
D
k
,
n
D_{k,n}
D
k
,
n
β
which has significate role in the cloud computing, and prove that
Ο
3
(
D
k
,
n
)
=
β
2
n
+
3
k
4
β
\pi_{3}(D_{k,n})=\lfloor\frac{2n+3k}{4}\rfloor
Ο
3
β
(
D
k
,
n
β
)
=
β
4
2
n
+
3
k
β
β
with
k
β₯
1
k\geq 1
k
β₯
1
and
n
β₯
6
n\geq 6
n
β₯
6
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Last time updated on 22/08/2024