Research Article
Characterization of the Congestion Lemma on
Layout Computation
Jia-Bao Liu ,
1
Arul Jeya Shalini ,
2
Micheal Arockiaraj,
3
and J. Nancy Delaila
3
1
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
2
Department of Mathematics, Women’s Christian College, Chennai 600006, India
3
Department of Mathematics, Loyola College, Chennai 600034, India
Correspondence should be addressed to Arul Jeya Shalini; aruljeyashalini@gmail.com
Received 8 June 2021; Accepted 15 October 2021; Published 27 October 2021
Academic Editor: Firdous Shah
Copyright © 2021 Jia-Bao Liu et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An embedding of a guest network GN into a host network HN is to find a suitable bijective function between the vertices of the
guest and the host such that each link of GN is stretched to a path in HN. e layout measure is attained by counting the length of
paths in HN corresponding to the links in GN and with a complexity of finding the best possible function overall graph
embedding. is measure can be computed by summing the minimum congestions on each link of HN, called the congestion
lemma. In the current study, we discuss and characterize the congestion lemma by considering the regularity and optimality of the
guest network. e exact values of the layout are generally hard to find and were known for very restricted combinations of guest
and host networks. In this series, we derive the correct layout measures of circulant networks by embedding them into the path-
and cycle-of-complete graphs.
1. Introduction
Nowadays, there is an emerging demand for high-per-
formance concurrent functional in different fields which
can be successfully achieved through parallel processing
techniques. e core of a parallel processing system is the
interconnected network by which the system processors
areconnected.Oneoftheimportantchallengesinparallel
processing techniques is how to allocate the subprocesses
to the processors within the system in such a way that the
total communication cost is minimized. is issue in
parallel processing can be reduced to a graph embedding
problem [1, 2]. For this purpose, the network topology is
formulated as a simple graph, in which the vertex set
denotes the system processors and the edge set denotes
the links connecting them.
In this paper, the collection of vertices and edges of a simple
graph network GN are, respectively, represented by V(GN)
and E(GN). A graph embedding of a guest network GN into a
host network HN is a kind of vertex and edge labeling denoted
bya1 − 1 and onto mapping μ: V(GN) ⟶ V(HN) together
with 1 − 1 mapping R: E(GN) ⟶ R(HN) such that R(e) is
a μ(x) to μ(y) path in HN, where e �(x, y) and R(HN)
contains the collection of routes or paths in HN [2, 3]. e
congestion of an edge s of HN is measured by counting the
routes in R(e) { }
e∈E(GN)
such that s is in the route R(e) and
denoted by EC
μ
(s). In other words,
EC
μ
(s)�| e ∈ E(GN): s ∈ E(R(e)) { }|. e layout/wire length
[4, 5] of GN by embedding μ in HN is defined as
L
μ
(GN, HN)�
e∈E(GN)
|E(R(e))| �
s∈E(HN)
EC
μ
(s).
(1)
Let D be any subset of E(HN). If we represent
EC
μ
(D)�
e∈D
EC
μ
(e), then L
μ
(GN, HN)�
p
i�1
EC
f
(F
i
),
where E(HN)� F
1
,F
2
, ... ,F
p
is a partition. For λ ≥ 1,
construct a set based on the edges of HN such that each
edge in HN is duplicated λ-times.Suchasetisdenotedby
E
λ
(HN). en,
L
μ
(GN, HN)�
1
λ
s∈E(HN)
EC
μ
(s).
(2)
Furthermore, if E
λ
(HN)� D
1
,D
2
, ... ,D
m
, then
L
μ
(GN, HN)�(1/λ)
m
i�1
EC
μ
(D
i
). e correct layout of
GN by embedding in HN is measured by
Hindawi
Journal of Mathematics
Volume 2021, Article ID 2984703, 5 pages
https://doi.org/10.1155/2021/2984703