1
Grooming Traffic to Maximize
Throughput
Charles J. Colbourn, Gaetano Quattrocchi, and Violet R. Syrotiuk
Abstract—Using a graph-theoretic formulation, a traffic grooming in a SONET ring network
may be interpreted as a decomposition G of
an undirected graph. In such a decomposition,
each subgraph specifies a set of primitive rings
assigned to the same wavelength. Bounding the
maximum throughput tp(c, n, ) of a c-grooming
G of an n-vertex graph with load d(G) ≤ is
addressed. Precise determinations of maximum
throughput for grooming ratios c = 2, 3, and 4
are given. These underlie substantially improved
bounds for larger grooming ratios.
Index Terms—traffic grooming; optical network; graph decompositions; throughput maximization
I. I NTRODUCTION : G ROOMING AND
G RAPH D ECOMPOSITION
A unidirectional SONET ring connects n nodes
in a circular fashion. Naming the nodes 0, . . . , n −
1, a connection is provided from node i to node
i+1 mod n for 0 ≤ i < n. Each connection supports
a number of independent wavelengths on which
communications can be carried, and each wavelength can be multiplexed to carry a number of
traffic streams. In addition, each wavelength has
a capacity, and the traffic streams assigned to a
specific wavelength on any one connection cannot
exceed the capacity of the wavelength.
Traffic requirements specify a set of traffic
streams; each traffic stream consists of a source
node, a destination node, and a required capacity.
The assignment of a stream from i to j with
capacity requirement p to a particular wavelength
λ consumes p of the available capacity on each of
the links {(k, k + 1) : i ≤ k < j} when i < j, or on
each of the links {(k mod n, k + 1 mod n) : j ≤ k <
n + i} when i > j.
Traffic grooming is the process of choosing an
assignment for each traffic stream, so that for
C. J. Colbourn and V. R. Syrotiuk are with Arizona State
University, and G. Quattrocchi is with the Università di
Catania.
every connection on every wavelength, there is
sufficient capacity to carry the union of the traffic
assigned [1]. A natural objective is to minimize
the number of wavelengths required in a grooming, but it is by no means the only one. When
a traffic stream from i to j is assigned to a
wavelength λ, although the traffic traverses intermediate nodes on the ring, it may optically bypass
these nodes. However, at nodes i and j, a conversion is needed to add or drop traffic; an add-drop
multiplexer (ADM) is needed. Equipping nodes
with ADMs incurs a substantial hardware cost,
often called the drop cost. By grooming traffic
streams with overlapping source and destination
nodes onto the same wavelength, the drop cost
can be reduced.
The minimization of drop cost has been a topic
of much study [1], [2], [3], [4], [5], [6]. Wang and
Gu [7] remark that prior work has assumed that
every node is capable of housing as many ADMs
as needed. Yet there are physical limitations on
the number of ADMs that can be housed and
on the number of ports that can be active at a
node. Hence minimizing the maximum number of
ADMs at a node is also of concern. Indeed when
this minimum number exceeds the limitations on
a node, instead one is concerned with maximizing
the throughput of the traffic streams that are
assigned. This leads to many competing objectives
for the utility of a traffic grooming.
The complexity of finding good traffic groomings has led to numerous general techniques (see
[8], for example), and to a focus on simplified
models of traffic requirements. Uniform traffic
requirements specify that all traffic streams with
nonzero traffic requirement have the same traffic requirement. Symmetric traffic requirements
specify that the traffic requirement from node i
to node j is the same as that from node j to
node i. All-to-all traffic requirements arise when
for every two distinct nodes i and j, the traffic
requirement from i to j is nonzero. The case
of all-to-all uniform traffic (which is necessarily
symmetric as well) has been studied extensively
[6], [9]. When traffic requirements are symmetric,
the traffic stream from node i to node j and that
from node j to node i form a complete circuit of
the ring, called a primitive ring.
We make the objectives for grooming in symmetric uniform SONET ring networks precise
by adopting a graph-theoretic formulation that
is explicitly developed in [2], [10]. First, it suffices to consider only groomings that assign both
traffic streams of a primitive ring to the same
wavelength, or leave both unassigned (see [7] for
a proof). Therefore rather than mapping traffic
streams to wavelengths, we map primitive rings
to wavelengths. The key observation is that the
primitive ring {(i, j), (j, i)} can be represented as
an unordered pair {i, j}. This enables us to interpret groomings as decompositions of undirected
graphs. In such a decomposition, a subgraph specifies a set of primitive rings to be assigned to the
same wavelength. (An extensive survey of graph
decompositions appears in [11].)
Specifically, we consider the problem of bounding the maximum throughput tp(c, n, ) of a cgrooming G of an n-vertex graph with load d(G) ≤
. In this setting, we determine precise values for
maximum throughput for small grooming ratios
c = 2, 3 and 4. These determinations for small
grooming ratios improve the bounds for larger
grooming ratios.
3)
4)
5)
6)
ec(G) is also referred to as the throughput
tp(G).
The grooming ratio gr(G) of decomposition
G is maxsi=1 |Ei |. A decomposition G of G =
(V, E) into subgraphs {G1 , . . . , Gs } is a cgrooming if gr(G) ≤ c.
The wavelength cost wc(G) is the number s
of subgraphs in the decomposition.
The drop cost dc(G) of a graph G is the
number of vertices of nonzero degree in G.
The drop cost dc(G) of
as graph decomposition
G = {G1 , . . . , Gs } is i=1 dc(Gi ).
The load d(G, v) on vertex v in decomposition G is the number of subgraphs in which
v has nonzero degree. The load d(G) of
decomposition G is maxv∈V d(G, v).
In optical networking, tp(G) determines the
number of pairs between which traffic can be
routed, and hence larger throughput is desired.
On the other hand, gr(G) determines the fraction
of a wavelength available (in the worst case) for
communication between two nodes; alternatively
it specifies a maximum for the number of such
communications that are groomed onto a single
wavelength. To maximize capacity, one wants to
minimize the grooming ratio. Once the number of
nodes n, the throughput m, and the grooming ratio c are selected, it is important to minimize each
of wc(G), dc(G), and d(G), over all c-groomings of
graphs on n vertices and m edges. Each determines an amount of hardware needed to employ
the grooming for wavelength assignment, and
their minimization arises in reducing hardware
costs.
Typically, the node count n, throughput m, and
grooming ratio c are specified as input, and we
wish to find a decomposition G with nc(G) = n,
tp(G) ≥ m, and gr(G) ≤ c that minimizes one (or
more) of wc(G), dc(G), and d(G). Let wc(c, n, m),
dc(c, n, m), and d(c, n, m) be the minima. The
minimum wc(c, n, m) is easily determined in general: It is m
c ; to see this, choose any n-vertex,
m-edge graph and partition its edge set into m
c classes each having at most c edges.
Minimizing drop cost is a much more challenging problem. A lower bound is obtained by
first determining the graph Mc with at most
c edges for which the ratio of the number of
vertices of nonzero degree in Mc to the number
of edges in Mc is minimized; call this minimum
ρc . Then dc(c, n, m) ≥ ρc m. The calculation of
II. P ROBLEM F ORMULATION AND
R ELATED W ORK
Let G = (V, E) be a finite simple graph. A
subgraph of G is a graph (V , E ) with the property that E ⊆ E and e ⊆ V for every e ∈ E .
The subgraph is spanning if V = V ; henceforth
we consider only spanning subgraphs. Subgraphs
G1 = (V, E1 ) and G2 = (V, E2 ) are disjoint if
E1 ∩ E2 = ∅. When subgraphs {Gi = (V, Ei ) : 1 ≤
Gs } is a (subi ≤ s} are pairwise disjoint, {G1 , . . . , graph) packing. When in addition, si=1 Ei = E,
G = {G1 , . . . , Gs } is a (subgraph) decomposition.
We introduce six fundamental parameters of decompositions as used in traffic grooming in optical
networks.
1) The node count nc(G) of decomposition G is
the number of vertices in G.
2) The edge count ec(G) of decomposition G is
the number of edges in G. The edge count
2
ρc is straightforward:
Choose the smallest ν for
which ν2 ≥ c, and compute ρc = min(ρc−1 , νc ).
(Hence ρ1 = 2, ρ2 = 32 , ρ3 = 1, ρ4 = 1, ρ5 = 45 ,
ρ6 = 23 , ρ7√= 23 , and so on.) We derive a lower
bound of √2c on ρc in general. Suppose to the
work, a connection must be set up on the same
wavelength channel if it traverses multiple hops
(single-hop traffic grooming), the transceivers in
a node are tunable to any wavelength on the
fiber, all data in a connection request must follow
the same route, and a node can multiplex or
demultiplex as many connections to a lightpath
as needed, as long as the aggregated traffic does
not exceed the capacity of the lightpath. An extension of this problem to include wavelength
conversion, i.e., allowing a connection to traverse
multiple lightpaths (multihop traffic grooming),
is straightforward. As expected, multihop traffic
grooming leads to higher throughput than the
single-hop case, for randomly generated traffic
matrices.
√
contrary that ρc < √2c . Then there must exist
a √simple graph with c ≤ c edges and less than
c√
2
vertices. On this number of vertices at most
c
√
√
c√
2 c√
( c2
c
2
− 1)/2 = (cc) − √c2c < c edges can arise,
so the graph does not exist. √This bound is not
sharp: for example, ρ7 = 23 > √27 .
In general, the simple lower bound on drop
cost obtained by using ρc is not sufficient. The
minimum drop cost
has been determined in the
case when m = n2 for small values of c as follows:
1) dc(2, n, n2 ) = 32 n2 , by Wan [6], for example.
2) Bermond
net al. [12] show
dc(3,
n,
⎧
2n) =
if n ≡ 1, 3 (mod 6)
⎪
⎪
n2
⎨
+
2
if
n ≡ 5 (mod 6)
n2
n
+
if
n ≡ 0, 2, 4, 6, 10 (mod 12)
⎪
⎪ n2 n 4
⎩
+
+
1
if
n
≡ 8 (mod 12).
2 4
3) dc(4, n, n2 ) = n2 if n = 2, 4 [2], [9].
4) Bermond
show
net al. [13]
1 n
)
=
4
·
n,
dc(5,
2
5 2 +
⎧
0 if n ≡ 0, 1 (mod 5) and n = 5
⎪
⎪
⎪
⎪
1
if n = 5
⎪
⎪
⎨
2 if n ≡ 2, 4 (mod 5) and n = 7
3 if n = 7
⎪
⎪
⎪
⎪
3 if n ≡ 3 (mod 5) and n = 8
⎪
⎪
⎩
4 if n = 8.
Similar results have been established for c = 6
[14], c = 7 [15], c = 8 [16], and partial results
for c = 9 [17]. Generalizations to two different
time periods, each supporting different all–to–all
traffic requirements, have been treated for c ∈
{2, 3} [18], [10] and for c = 4 [19].
A natural question is to determine the maximum throughput given a node count, grooming
ratio, and load [7]. Hence in this paper we are
primarily concerned with determining or bounding tp(c, n, ), the maximum throughput of a cgrooming G of an n-vertex graph having d(G) ≤ .
Among the first to consider the objective of
maximizing throughput are Zhu and Mukherjee
[20]. An integer linear program (ILP) is set up to
maximize the total successfully routed connection
requests under the following assumptions: that
the network is a single-fiber irregular mesh net-
The results from the solution to the ILP show
that the end-to-end aggregate traffic between the
same (s, d) pair tends to be groomed on to the
same lightpath. Based on this observation, Zhu
and Mukherjee [20] also propose two heuristic
algorithms, each with two stages. In the first
stage, the heuristic establishes lightpaths to satisfy aggregate end-to-end connection requests. If
there are enough resources in the network, every connection request is successfully multiplexed
onto a single-hop lightpath to minimize traffic
delay. In the second stage, the connection requests blocked in the first stage are carried on the
spare capacity of the established lightpaths. The
difference between the Maximizing Single-Hop
Traffic (MST) heuristic and the Maximizing Resource Ultilization (MRU) heuristic is that MST
chooses the (s, d) pair with the highest aggregate
uncarried traffic requests, while MRU chooses
the highest connection resource utilization which
simply averages the aggregate uncarried traffic
over the hop distance between s and d. The MRU
heuristic achieves higher throughput if tunable
transceivers are used while the MST heuristic
achieves higher throughput if fixed transceivers
are used.
Srinivas and Siva Ram Murthy [21] observe
that the MST and MRU heuristics do not re-route
or modify lightpaths once they are established. A
new heuristic algorithm is proposed for the static
multihop traffic grooming problem without any
constraint on the number of transceivers. After
the lightpaths are established, it checks k of the
possible paths between an (s, d) pair to see if
less utilized lightpaths can be deleted and their
3
traffic routed on to other existing lightpaths. This
can lead to better grooming of the connections
and better utilization of the channels, though
sometimes at higher cost. Yao and Ramamurthy
[22] propose rerouting schemes for dynamic traffic grooming.
ber of traffic demands accommodated subject to
each node being equipped with a limited number
of ADMs. More precisely, a graph partitioning approach is used. The set R is represented
by a traffic graph G = (V, E), an undirected
simple graph where V is the set of vertices in
UPSR and e = (x, y) ∈ E iff the primitive ring
{(x, y), (y, x)} ∈ R.
The maximum throughput traffic grooming
problem is formulated as a maximum connectivity c-edge partitioning problem on the traffic
graph: given an undirected simple graph G =
(V, E), grooming ratio given by the integer c ≤ |E|,
and the number of ADMs available at each node
given by integer (load), find subgraphs Gi =
(V, Ei ) of pairwise disjoint subsets of E such that
|Ei | ≤ c for each Gi ∈ G and d(G,
v) ≤ d(G) for
each v ∈ V . The objective is max Ei ∈G |Ei | which
is equivalent to maximizing throughput tp(G).
They prove that maximum throughput traffic
grooming problem is NP-hard, and propose a
(c + 1)-approximation algorithm where c is the
grooming ratio. They also study the all-to-all
traffic pattern for the maximum throughput traffic grooming problem and propose an algorithm
demands for a
the accommodates at least nc
2
UPSR with n nodes. Since any optimal algorithm
√
accommodates at most nc
their algorithm is
2
√
approximately 2 factor away from optimal.
Wang and Gu use a graph partitioning approach whereas we use a graph decomposition approach. Their graph represents what is available
to be groomed; they may leave some edges behind
and so they generate a graph packing. In our
formulation, the graph represents what has been
groomed. We select the graph, then decompose
it, getting a graph decomposition. Throughout we
use the vernacular of graph decompositions.
Correia and Medeiros [23] use traffic grooming
for protection. Their idea is to use the available
bandwidth of working lightpaths (called vitrual
links) for backup, improving bandwidth utilization compared with traditional lightpath protection. Their objective
maximize total network
is to
where V is
throughput max s,d∈V y∈Y y · φs,d
y
the set of nodes, y is the granularities of the
connection requests (e.g., y = {3, 12, 48, 192}, and
is the number of OC-y connection requests
φs,d
y
from node s to node d established successfully,
all subject to many constraints. The throughput
analyzed for a network with G = (V, E), |V | = 6,
|E| = 8 for traffic matrices for OC-y, y ∈ Y
client connection requests have random numbers
uniformly distributed between 0-16 for OC-3, 0-8
for OC-12, 0-4 for OC-48, and 0-2 for OC-192. The
traffic was allowed to take multiple hops and only
single link failure scenarios were considered. The
conclusion is that traditional lightpath protection
requires more wavelengths compared to using
traffic grooming for protection.
Prathombutr et al. [24] consider a similar formulation of the traffic grooming problem in WDM
optical mesh networks as [23], and a similar
experimental set up. Their objective is to satisfy
multiple objectives: to maximize traffic throughput, to minimize the number of transceivers (or
lightpaths), and to minimize the average propagation delay. In order to simultaneously optimize these competing objectives they use a multiobjective evolutionary algorithm (MOEA), and
show how to apply it to the traffic grooming problem. They illustrate the effectiveness of a MOEA
in the traffic grooming problem by comparing
their results with the maximizing single-hop traffic (MST) and the maximizing resource utilization
(MRU) heuristic from [20]. In all experiments, the
MOEA yielded higher throughput than the two
other heuristics after 5000 generations.
III. M AXIMIZING T HROUGHPUT
FOR S PECIFIED L OAD
Wang and Gu [7] examine the problem of determining tp(c, n, ). Consider a grooming of an nvertex graph G with grooming ratio c and load
. Evidently its drop cost is at most n. What
is the largest number of edges, tp(c, n, ), that
G can have? It can be no larger than n
ρc . To
obtain a bound in the form given by Wang and
Gu [7], use the bound on ρc to establish that
the
√ maximum number of edges cannot exceed
n
√ c . Naturally using a better upper bound on
2
Perhaps the closest work to that presented here
is that of Wang and Gu [7]. They focus on unidirectional path-switched ring (UPSR) networks
with a set R of unitary duplex traffic demands.
The objective considered is to maximize the num4
the quantity ρc leads to a better upper bound on
tp(c, n, ). For example, taking ρ7 = 23 establishes
√
7n
√
.
that tp(7, n, ) ≤ 3n
2 , which improves upon
2
Lower bounds on the maximum throughput are
harder to produce. We first prove a general result
when the grooming ratio is ‘small’ relative to the
number of nodes. To do this, we use a powerful
theorem of Dukes and Ling [25]:
Theorem 3.1: Let G be a simple graph with
n vertices, m edges, and degree sequence
(d1 , . . . , dn ). Let γ be the greatest common divisor
of {d1 , . . . , dn }. Then there exists an NG such that
there is a decomposition of the complete graph of
order N ≥ NG into subgraphs each isomorphic to
G in which the subgraphs can be partitioned into
N
parallel classes each containing
N n vertex-disjoint
subgraphs, if and only if 2 ≡ 0 (mod m) and
N − 1 ≡ 0 (mod γ).
Theorem 3.2: For c fixed
and n sufficiently
large, tp(c, n, ) ≥ min( n2 , n
ρc − κc ) for some constant κc depending only on c.
Proof: Choose a simple graph G with c ≤ c
edges having c ρc vertices; such a graph exists
as a consequence of the definition of ρc . Let γ be
the greatest common divisor of its vertex degrees.
least integer n ≥ n for
When n ≥
NG , choose the
n
which 2 ≡ 0 (mod c ) and n − 1 ≡ 0 (mod γ).
By Theorem 3.1, there is a partition of the edges
(n )
c
parallel classes, each
Kn into 2n = (n −1)ρ
2
c
the correct growth as a function of n, it does not
provide groomings for small values of n. We establish a general result that explicitly constructs a
grooming, and does so for larger grooming ratios:
Lemma 3.3: Let n, c and be positive integers.
Let 1 ≤ c < c be a positive integer and g = cc .
n
Let n = g . Then tp(c, n, ) ≥ g 2 tp(c , n , ).
Proof: Consider any grooming on n vertices
with grooming ratio c and load . Replace every
vertex v by a set S(v) of g vertices. Then for each
subgraph S with edge set E of the grooming on
n nodes, define a subgraph with edges {{a, b} :
a ∈ S(v), b ∈ S(w), {v, w} ∈ E}. The resulting
subgraphs a grooming on gn ≤ n nodes with
grooming ratio g 2 c ≤ c. Adjoin n − gn nodes to
form the required grooming.
In order to apply Lemma 3.3, we require groomings with small grooming ratio. Even examples
with grooming ratio 1 are useful here. The determination of tp(1, n, ) is easy: It is n
2 when
0 ≤ ≤ n − 1. When n is even, simply take onefactors from a one-factorization of the complete
graph. When n is odd, form a decomposition of
the complete graph into Hamilton cycles; include
2 of these completely, and when is odd choose
alternate edges from one further cycle, to add
n2 further edges. Using the exact solution for
tp(1, n, ) in Lemma 3.3, we obtain the lower
bound established in [7, Theorem 7]; this applies
only when < √nc :
√
n
√ 2 √c n c
≈
.
tp(c, n, ) ≥ ( c)
2
2
c ρc
containing cn ρc vertex-disjoint copies of G which
collectively include ρnc edges. To obtain the desired
grooming, we include of these parallel classes
and then delete n − n vertices. The number of
edges
deleted
in the process need not exceed
ρc −1
.
To
show that this is bounded by
n −n+c
ρc
a constant independent of n, we need only show
that n − n is bounded by a constant independent
of n. Let δ = n −n. Write n = αc +β for 0 ≤ β < c
and write n − 1 = χγ + ν for 0 ≤ ν < γ. Then
it suffices to choose the smallest nonnegative δ
for which δ ≡ −ν (mod γ) and δ ≡ −β (mod c ).
By the Chinese remainder theorem, there is a
solution for δ that is smaller than γc . Because
1 ≤ γ < c ≤ c, this depends only on c, not on n.
There is much room for improvement.
Choose
an integer x ≥ 2 and let c = x2 ; hence ρc =
2
n is sufficiently large, tp(c , n, ) ≥
x−1 . When
n (x−1)n
− κc ) by Theorem 3.2. Apply
min( 2 ,
2
Lemma 3.3 to obtain
⎛
⎞
√n (x
−
1)
⎜
⎟
c/c − κ c ⎟
tp(c, n, ) ≥ ( c/c )2 ⎜
⎝
⎠
2
√ √
n c x − 1
√
,
≈
2x
when 2ρc ≤ √ n − 1. As x increases, the
As n → ∞ for fixed c, then, the lower bound and
the upper bound on tp(c, n, ) differ by a constant
depending only on c. There are two principal
limitations to this result. First, it applies only
when c is fixed. Secondly, while it informs us of
c/c lower bound that results more closely matches
the upper
bound. More precisely, as x → ∞ the
value (x − 1)/x approaches 1, so as nx → ∞,
(2)
the lower bound approaches the upper bound.
5
n ≡ 0, 1 (mod 3), and to adjoin an edge that is
vertex-disjoint from each of these when n ≡ 2
(mod 3). This handles the subcase when = 3t+1.
For = 3t + 2, C can be partitioned into P3 s
and edges by Lemma 4.1, which necessarily add
2 to the load. When n ≡ 0 (mod 4), the remaining
edges have the form {{i, i + 2t + 1 mod n} : 0 ≤
i < n} ∪ {{i, i + 2t + 2 mod n} : 0 ≤ i < n2 }. These
remaining edges form a graph B that is 3-regular,
and contains the graph C. To treat the subcase
when = 3t+1, proceed as in the case when n ≡ 3
(mod 4). To treat the subcase when = 3t + 3, B
can be partitioned into P3 s and edges by Lemma
4.1, which necessarily add 2 or 3 to each vertex
load because B is 3-regular.
It remains to treat the most involved situation,
when = 3t + 2 and n = 4t + 4 ≡ 0 (mod 4). Form
the n P3 s {[2i+2t+2, 2i+1, 2i+2t+4] : 0 ≤ i < n2 }.
Then every vertex with odd label has load 1, and
every vertex with even label has load 2, in this
set containing n edges in total. For 0 ≤ i < t+1
3 ,
replace the three paths {[6i + 2j + 2t + 2, 6i + 2j +
1, 6i+ 2j + 2t + 4], [6i+ 2j, 6i+ 2j + 2t + 3, 6i+ 2j + 2] :
0 ≤ j < 3}, by the eight paths {[6i+3j +2t+2, 6i+
3j + 1, 6i + 3j + 2t + 3], [6i + 3j + 2t + 3, 6i + 3j +
2, 6i+3j +2t+4], [6i+3j +1, 6i+3j +2t+4, 6i+3j +
3], [6i+3j +2, 6i+3j +2t+5, 6i+3j +3] : 0 ≤ j < 2}.
If t ≡ 0 (mod 3) add a subgraph containing the
single edge {4t+3, 2t+1}, and when t ≡ 1 (mod 3)
add edges {4t + 1, 2t − 1} and {4t + 3, 2t + 1}. In the
process, to the 4t + 4 edges, we have added 4t+4
3
edges when t ≡ 2 (mod 3), 4t+3
edges when t ≡ 0
3
(mod 3), and 4t+2
edges when t ≡ 1 (mod 3). In
3
each case, the number of edges totals 2t(4t + 4) +
2(3t+2)(4t+4)
4t + 4 + 4t+4
, as required.
3 =
3
Although this technique easily yields asymptotic results, the reliance on Theorem 3.2 prevents application for specific (small) values of n.
Application to all numbers of vertices requires
that we determine tp(c, n, ) more precisely for
‘small’ values of c and all n.
IV. M AXIMUM T HROUGHPUT FOR
G ROOMING R ATIO c = 2
We represent a path containing the two edges
{x, y} and {y, z} by [x, y, z]. We require a preliminary result.
Lemma 4.1: [26], [27] Let G be a connected
graph. Then the edges of G can be partitioned into
copies of P3 when the number of edges is even,
and into a single edge together with copies of P3
when the number of edges is odd.
Theorem 4.2: For
≥ 2 and 0 ≤ ≤ 3(n−1)
,
4
n2n
n
tp(2, n, ) = min( 2 , 3 ).
Proof: Necessity is immediate from the fact
that ρ2 = 32 . We must establish sufficiency. Write
n = 4t+4, n = 4t+1, n = 4t+2, n = 4t+3, with t an
integer. When t = 0 the proof is trivial, so assume
that t ≥ 1. In each case we take the vertex set
to be Zn = {0, . . . , n − 1}. Write s = min(t, 3 ).
Choose pairs {{ai0 , ai1 } : i = 1, . . . , s} of positive
integers with | ∪si=1 {ai0 , ai1 }| = 2s; aij ≤ t for 1 ≤
i ≤ s and j ∈ {0, 1}; ensure that {a10 , a11 } = {1, 2}.
When s ≥ 2, start with a set of P3 s: {[j + ai0 , j, j +
ai1 ] : 0 ≤ j < n, 2 ≤ i ≤ s}, arithmetic being done
modulo n. This yields (2s−2)n edges with load 3s−
3 at each vertex. Let Gj = [j +1, j, j +2] for 0 ≤ j <
n, arithmetic modulo n. Order these subgraphs
(H0 , . . . , Hn−1 ) where Hi = G3i mod n when n ≡
1, 2 (mod 3); when n ≡ 0 (mod 3) set Hi = G3i ,
Hi+n/3 = G3i+1 , and Hi+2n/3 = G3i+2 for 0 ≤ i <
n
n
3 . Include the first 3 subgraphs and when n ≡
2 (mod 3), include the single edge {n − 2, n − 1} as
well. This completes all cases with s ≤ 3t.
It remains to treat cases with > 3t. We always
start by including P, the set of P3 s defined above
covering 2tn edges with load 3t. When n ≡ 1
(mod 4), no cases remain. When n ≡ 2 (mod 4),
the n/2 edges that remain are pairwise disjoint,
so adding each edge as a subgraph in the grooming increases the load by 1 while covering all
edges. When n ≡ 3 (mod 4), the remaining edges
have the form {{i, i + 2t + 1 mod n} : 0 ≤ i < n}.
Because 2t + 1 and 4t + 3 are relatively prime,
these edges form a single n-cycle C. It is then
easy to pack C with n3 vertex-disjoint P3 s when
V. M AXIMUM T HROUGHPUT FOR
G ROOMING R ATIO c = 3
A triangle or triple is a subgraph {x, y, z} containing three edges {{x, y}, {x, z}, {y, z}}.
Theorem 5.1: For n ≥ 2,
⎧
n
if ≤ n−1
⎨
2 and n ≡ 0 (mod 3)
(n −1) if ≤ n−1
tp(3, n, ) =
2 and n ≡ 0 (mod 3)
⎩
n
otherwise
2
Proof: The presence of at most n edges follows from the fact that ρ3 = 1, and equality occurs
only when the number n of edges is a multiple
of 3. Now we treat sufficiency. Let
6
⎧
⎪
⎪
⎨
n(n−2)
n2 2
n(n−2)
n2 −
2 −4
if n ≡ 0, 2 (mod 6)
if n ≡ 1, 3 (mod 6)
.
mn =
⎪
1 if n ≡ 4 (mod 6)
⎪
⎩
if n ≡ 5 (mod 6)
Then there exists a set of m3n edge-disjoint
triangles on n nodes (see [28], for example); this
is a partial triple system. Hence there is a partial
triple system on n nodes with t triples whenever
0 ≤ t ≤ m3n . Andersen, Hilton, and Mendelsohn
[29] establish that whenever a partial triple system on t triples and n nodes exists, there is one
in which every two nodes appear in numbers
of triples that differ by at most one; it is baln
anced. So when ≤ n−1
2 , set t = 3 when
n
n ≡ 0, 1 (mod 3) and t = 3 + 1 when n ≡ 2
(mod 3). Form a balanced partial triple system
on t triples. When n ≡ 2 (mod 3), exactly one
node x occurs in + 1 triples while all other nodes
occur in . Select any triple containing x, and
remove x from the triple to leave a single edge
on the remaining two nodes. This realizes the
maximum throughput whenever n ≤ mn . There
remains one case with ≤ n−1
2 , namely when
n ≡ 5 (mod 6) and = (n − 1)/2. In this case, a
maximum partial triple system leaves four edges
uncovered, and they form a 4-cycle. Taking all
triples as subgraphs of the grooming, along with
a subgraph containing any three edges of the 4cycle, establishes the result.
It remains to treat cases with > n−1
2 . When
n is even, the edges left uncovered by a maximum
partial triple system form a set of vertex-disjoint
edges, together with zero or one vertex-disjoint
stars K1,3 [28]. Taking each as a subgraph in a
grooming, along with all triples of the maximum
partial triple system, yields load n−1
2 + 1 and
handles all edges of the complete graph.
and taking a 3-cycles and b 4-cycles, all disjoint,
because n can be written in this form except when
n ∈ {2, 5}. When n = 2,selecting
the unique edge
gives throughput 1 = 22 . When n = 5, a single
4-cycle or kite gives throughput 4.
Now suppose that ≥ 2. Form a graph G =
(V, E) on n2 vertices that is -regular when
n2 ≡ 0 (mod 2), and has one vertex y of degree
+ 1 and all others of degree when n2 ≡ 1
(mod 2). If a vertex of degree + 1 is present,
ensure that it belongs to a triangle {y, a, b}. When
n is even, replace each vertex v ∈ V by two
vertices v0 and v1 ; whenever {v, w} ∈ E, form the
4-cycle (v0 , w0 , v1 , w1 ). When n ≡ 2 (mod 4), replace the 4-cycles (y0 , a0 , y1 , a1 ) and (y0 , b0 , y1 , b1 )
by two 3-cycles {y0 , a0 , a1 } and {y1 , b0 , b1 }. This
set of cycles has load , and contains n edges.
On the other hand, when n is odd, select one
distinguished vertex z ∈ V , with z ∈ {y, a, b}
when n2 is odd; replace each vertex v ∈ V \ {z}
by two vertices v0 and v1 ; whenever {v, w} ∈ E
and z ∈ {v, w}, form the 4-cycle (v0 , w0 , v1 , w1 ).
Finally when {v, z} ∈ E, form the 3-cycle (triple)
{v0 , z, v1 }. When n2 ≡ 0 (mod 2), this set of 3cycles and 4-cycles has load and contains n
edges. When n2 ≡ 1 (mod 2), to complete the
construction, replace the 4-cycles (y0 , a0 , y1 , a1 )
and (y0 , b0 , y1 , b1 ) by two 3-cycles {y0 , a0 , b0 } and
{y1 , a1 , b1 }.
VII. C ONCLUSION
Using combinatorial techniques, we have determined the maximum throughput for given
node count and load for grooming ratio c ≤ 4.
These complete results improve upon the bounds
for maximum throughput for arbitrary grooming ratios. Indeed, similar (but not necessarily
complete) results for larger c can be used to
strengthen these bounds as needed. Although
these bounds provide accurate statements of the
amount of traffic that can be routed for a specified load, they provide no easy method to specify
which traffic will in fact be routed. Nevertheless, the extension to certain cases in which the
traffic is not all–to–all is amenable to similar
techniques.
VI. M AXIMUM T HROUGHPUT FOR
G ROOMING R ATIO c = 4
A 4-cycle (w, x, y, z) is a subgraph containing four edges {{w, x}, {x, y}, {y, z}, {w, z}}. A kite
[x, y, z : w] is a subgraph containing four edges
{{x, y}, {x, z}, {y, z}, {x, w}}.
=
Theorem
≥
2, tp(4, n, )
6.1: For n
min(n, n2 ) except when n = 5 and = 1,
for which tp(4, 5, 1) = 4.
Proof: Necessity follows from the fact that
ρ4 = 1. Now we treat sufficiency. It suffices to
treat cases with ≤ n−1
2 by Theorem 5.1. The
cases with = 1 are treated by writing 3a+4b = n
A CKNOWLEDGEMENTS
Thanks to Alan Ling and Dianhua Wu for helpful discussions.
7
R EFERENCES
[21] N. Srinivas and C. S. R. Murthy, “Throughput maximization in traffic grooming in WDM mesh networks,”
Journal of High Speed Networks, vol. 13, pp. 139–154,
2004.
[22] W. Yao and B. Ramamurthy, “Rerouting schemes for
dynamic traffic grooming in optical WDM networks,”
Computer Networks, vol. 52, pp. 1891–1904, 2008.
[23] N. S. C. Correia and M. C. R. Medeiros, “Traffic grooming
applied to network protection: Throughput and grooming
port cost analysis,” in Proceedings of the 7th International Conference on Transparent Optical Networks (ICTON’05), 2005, pp. 389–393.
[24] P. Prathombutr, J. Stach, and E. K. Park, “An algorithm for traffic grooming in wdm optical mesh networks
with multiple objectives,” Telecommunications Systems,
vol. 28, no. 3,4, pp. 369–386, 2005.
[25] P. Dukes and A. C. H. Ling, “Asymptotic existence of
resolvable graph designs,” Canad. Math. Bull., vol. 50,
no. 4, pp. 504–518, 2007.
[26] D. de Werra, “Equitable colorations of graphs,” RAIRO
R-3, pp. 3–8, 1971.
[27] E. B. Yavorskii, “Representations of directed graphs and
ψ-transformations,” Theoretical and Applied Questions of
Differential Equations and Algebra, pp. 247–250, 1978,
a. N. Sharkovskii (Editor).
[28] C. J. Colbourn and A. Rosa, Triple systems. Oxford and
New York: Oxford University Press, 1999.
[29] L. D. Andersen, A. J. W. Hilton, and E. Mendelsohn,
“Embedding partial Steiner triple systems,” Proc. London Math. Soc. (3), vol. 41, no. 3, pp. 557–576, 1980.
[1] E. Modiano and P. Lin, “Traffic grooming in WDM networks,” IEEE Communications Magazine, vol. 39, no. 7,
pp. 124–129, 2001.
[2] J.-C. Bermond and D. Coudert, “Traffic grooming in
unidirectional WDM ring networks using design theory,”
in IEEE Conf. Commun. (ICC’03), IEEE, Ed., vol. 2, Los
Alamitos, CA, 2003, pp. 1402–1406.
[3] J.-C. Bermond, D. Coudert, and X. Muñoz, “Traffic
grooming in unidirectional WDM ring networks: the
all-to-all unitary case,” in ONDM’03: 7th IFIP Working
Conference on Optical Network Design and Modelling,
2003, pp. 1135–1153.
[4] R. Berry and E. Modiano, “Reducing electronic multiplexing costs in SONET/WDM rings with dynamically
changing traffic,” IEEE J. Selected Areas Comm., vol. 18,
pp. 1961–1971, 2000.
[5] R. Dutta and N. Rouskas, “Traffic grooming in WDM
networks: Past and future,” IEEE Network, vol. 16, no. 6,
pp. 46–56, 2002.
[6] P. J. Wan, Multichannel optical networks. Norwell, MA:
Kluwer Academic Publishers, 2000.
[7] Y. Wang and Q.-P. Gu, “Maximum throughput traffic
grooming in optical networks,” Journal of Optical Networking, vol. 7, pp. 895–904, 2008.
[8] O. Goldschmidt, D. Hochbaum, A. Levin, and E. Olinick,
“The SONET edge-partition problem,” Networks, vol. 41,
pp. 13–23, 2003.
[9] J. Hu, “Optimal traffic grooming for wavelength-divisionmultiplexing rings with all-to-all uniform traffic,” OSA
Journal of Optical Networks, vol. 1, no. 1, pp. 32–42,
2002.
[10] C. J. Colbourn, G. Quattrocchi, and V. R. Syrotiuk,
“Grooming for two-period optical networks,” Networks,
vol. 58, pp. 307–324, 2008.
[11] D. Bryant, P. Adams, and M. Buchanan, “A survey on
the existence of G-designs,” J. Combinatorial Designs,
vol. 16, pp. 373–410, 2008.
[12] J.-C. Bermond and S. Ceroi, “Minimizing SONET ADMs
in unidirectional WDM rings with grooming ratio 3,”
Networks, vol. 41, pp. 83–86, 2003.
[13] J.-C. Bermond, C. J. Colbourn, A. C. H. Ling, and M. L.
Yu, “Grooming in unidirectional rings: K4 − e designs,”
Discrete Mathematics, vol. 284, pp. 67–72, 2004.
[14] J.-C. Bermond, C. J. Colbourn, D. Coudert, G. Ge, A. C. H.
Ling, and X. Muñoz, “Traffic grooming in unidirectional
WDM rings with grooming ratio c = 6,” SIAM J. Discrete
Mathematics, vol. 19, pp. 523–542, 2005.
[15] C. J. Colbourn, H. L. Fu, G. Ge, A. C. H. Ling, and H. C.
Lu, “Minimizing SONET ADMs in unidirectional WDM
rings with grooming ratio 7,” SIAM J. Discrete Math.,
vol. 23, pp. 109–122, 2008.
[16] C. J. Colbourn, G. Ge, and A. C. H. Ling, “Optimal
grooming with grooming ratio eight,” Discrete Applied
Math., vol. 157, pp. 2763–2772, 2009.
[17] ——, “Optimal grooming with grooming ratio nine,” submitted for publication, 2010.
[18] C. J. Colbourn, G. Quattrocchi, and V. R. Syrotiuk,
“Lower bounds for graph decompositions via linear programming duality,” Networks, vol. 58, pp. 299–306, 2008.
[19] J.-C. Bermond, C. J. Colbourn, L. Gionfriddo, G. Quattrocchi, and I. Sau, “Drop cost and wavelength optimal
two-period grooming with ratio 4,” SIAM J. Discrete
Mathematics, to appear.
[20] K. Zhu and B. Mukherjee, “Traffic grooming in an optical
WDM mesh network,” IEEE Journal on Selected Areas
in Communications, vol. 20, no. 1, pp. 122–133, 2002.
8