“Traffic Grooming in Path, Star, and Tree Networks:
Complexity, Bounds, and Algorithms”
Shu Huang, Rudra Dutta, George N. Rouskas
All-optical networks
In an all-optical network, optical fibers are used to provide much
more bandwidth than legacy copper wire.
• In the simplest case, a single wavelength in the fiber is used to
carry information (point-to-point link of a given bandwidth).
• Using Wavelength Division Multiplexing (WDM), we can exploit
different wavelengths in the same fiber, each one utilizing
bandwidths comparable to that which the original fiber was
providing.
2
Lightpaths
Ideally, we would like to connect each source-destination pair with
paths using the same wavelength on each physical link (lightpaths).
However, this is not always feasible due to limitations on the number
of wavelengths that can be used, and hardware constraints at the
network nodes.
3
Architecture of a WDM wavelength routed network
• Point-to-point optical fiber interconnections (physical topology).
• Limited availability of wavelengths.
• Wavelength router setup to forward traffic from source to
destination nodes using the same wavelength (lightpaths-logical
topology).
• Routing of traffic over a single lightpath (“cheap”), or a sequence
of more than one lightpaths (“costly”).
4
Virtual Topology Design breakdown
1. Topology: Determine the virtual topology to be imposed on the
physical topology, that is determine the lightpaths in terms of
their source and destination nodes.
2. Lightpath Routing: Determine the physical links which each
lightpath consists of, that is route the lightpaths over the
physical topology.
3. Wavelength Assignment: Determine the wavelength each
lightpath uses, that is assign a wavelength to each lightpath in
the virtual topology so that wavelength restrictions are obeyed
for each physical link.
4. Traffic Routing: Route packet traffic between source and
destination nodes over the virtual topology obtained.
5
Traffic streams - grooming
Each lightpath has a high bandwidth which may not be fully utilized
by single users. Lightpaths must be viewed as transport channels in
the backbone network, in which traffic from multiple user
applications is multiplexed.
The pattern of multiplexing traffic onto lightpaths affects the
efficiency of optical forwarding of information through wavelength
routers, since all information on an entire lightpath will need to
undergo electro-optic conversion and electronic routing at an
intermediate node if even one lower speed traffic stream from that
lightpath has to be terminated at the intermediate node.
6
Traffic Grooming - problem setting
Given:
• a physical topology,
• C: the grooming factor,
• W : the number of wavelength that each physical link can
support,
• T = [tij ]: the demand matrix (we allow tij > C), and
• F : the goal.
Is it possible to form a valid logical topology and route all traffic in T
over the lightpaths of the logical topology so that the total electronic
switching cost is ≤ F ?
Cost model: every time a lightpath terminates, one unit of cost is
incurred for each traffic unit that has to be electronically switched.
7
Bifurcation of traffic
• In general, a traffic component tij may be split into integer
subcomponents which may follow different routes from source to
destination.
• Bifurcation of traffic may or may not be allowed.
• If bifurcation is not allowed:
– for any source-destination pair such that tij ≤ C we require
that all tij traffic units be carried on the same sequence of
lightpaths.
j k
t
– if tij > C, we allow the traffic demand to be split into Cij
subcomponents of magnitude C and at most one
subcomponent of magnitude < C. The no-bifurcation
requirement applies to each subcomponent independently.
8
Complexity of Traffic Grooming on paths
• Traffic Grooming on unidirectional path networks without
bifurcation of traffic is NP-complete.
• We give a reduction from the Subset Sum problem (given a set of
positive integers S and a goal sum B, is there a subset of S
whose elements add up to B?).
Suppose that we are given the set S = {s1 , s2 , . . . , sn } and goal B.
P
Let B1 = max {B, si − B}. Obviously, the Subset Sum problem
can be posed equivalently with goal B or B1 .
We restrict the Subset Sum instance to a non-trivial one, where:
X
0 < B, B1 <
si
9
We construct an instance of Traffic Grooming on a path with
P
N = 2n + 1 nodes, W = 2 and C = 1 + si .
C+1
s1
C+1
C+1
B1+1
s2
si
sn
B1+1
C+1
C+1
C+1
C-B1
The traffic pattern forces the logical topology shown above.
10
C+1
s1
C+1
s2
C+1
si
B1+1
sn
B1+1
C+1
C+1
C+1
C-B1
Before taking into account the si -streams, the only two-hop lightpath
has a free capacity of B1 traffic units.
P
Suppose that there is a T ⊆ S such that t∈T t = B1 . Then:
X
X
OPT = (n − 1) · B1 + n ·
si − B1 = n ·
si − B1
11
C+1
s1
C+1
s2
C+1
si
0
B1+1
sn
B1+1
C+1
C+1
C+1
C-B1
P
Otherwise, let T ⊆ S such that t∈T 0 t = A < B1 and A is
maximum. Then:
X
X
X
OPT = (n − 1) · A + n ·
si − A = n ·
si − A > n ·
si − B1
12
Complexity of Traffic Grooming on paths
Therefore, if we set as goal of the Traffic Grooming instance the
P
number F = n · si − B1 , we have a reduction from Subset Sum.
Note that:
• The reduction above demonstrates that Traffic Grooming on
unidirectional path networks is NP-complete even when a logical
topology is provided.
• Therefore, Traffic Grooming is inherently more difficult than
Routing and Wavelength Assignment.
• The only feasible routing of traffic in the instance we created is
P
0
the optimal one. So, we are free to set F = n · si + C − B1 ,
the maximum possible electronic switching cost. Therefore,
deciding whether a given logical topology admits of any feasible
traffic routing at all is also NP-complete.
13
More complexity results
• Traffic Grooming on unidirectional path networks with
bifurcation of traffic is NP-complete.
• Traffic Grooming on unidirectional path networks with
bifurcation of traffic is NP-complete, even when a candidate
logical topology is provided.
• Traffic Grooming on bidirectional path networks (with or
without bifurcation of traffic) is NP-complete.
• Traffic Grooming on bidirectional ring networks (with or without
bifurcation of traffic) is NP-complete, even when every node has
full wavelength conversion capability.
• There is no FPTAS, nor any constant-factor approximation
algorithm for the minimization version of Traffic Grooming on
unidirectional paths with bifurcation of traffic, unless P = NP.
14
Complexity of Traffic Grooming on stars
In a star network with N + 1 nodes:
• There is a single hub node which is connected
to every other node by a physical link.
• The N nodes other than the hub are numbered
from 1 to N , and the hub is numbered 0.
1
N
2
0
• Each physical link consists of a fiber in each
direction.
In an optimal solution, no node except for the hub switches traffic,
either electronically or optically. So we have only single-hop
lightpaths between the hub and a non-hub node, and two-hop
lightpaths between non-hub nodes.
Deciding which of the traffic components between non-hub nodes are
optically switched at the hub fixes both the logical topology and the
traffic routing.
15
...
Complexity of Traffic Grooming on stars
• Traffic Grooming on star networks is NP-complete.
• We will see a reduction from the Knapsack problem.
Let U be the given set of n elements, w1 , . . . , wn their weights,
v1 , . . . , vn their values, B the target weight and K the target value.
The question is whether there exists a binary vector x = (x1 , . . . , xn )
P
P
such that
xi wi ≤ B and
xi vi ≥ K.
We construct a Traffic Grooming instance on a star with n + 2
non-hub nodes with W = n, C = maxi (wi + vi ) + 1 and optical
P
switching goal Q = K + (C − wi − vi ).
On a star with N non-hub nodes, minimizing the total electronic switching
F is equivalent to maximizing the total optical switching Q, where:
Q=
X
1≤i,j≤N
16
tij − F
1
i
2
C-wi
n+1
0
C-wi-vi
n+2
n+1
i
(n-2)C+wi
Ȉwi-B
n+2
0
n
• Each component t0,i fills (n − 2) wavelengths of the link (0, i)
completely, and takes up wi traffic units of another wavelength.
So, at most one of tn+1,i , tn+2,i can be optically routed.
• Suppose that the Knapsack instance is a yes-instance, and let
x = (x1 , . . . , xn ) be a suitable binary vector. Consider the
candidate solution to the Traffic Grooming instance where tn+1,i
is optically routed iff xi = 1.
17
i
C-wi
n+1
C-wi-vi
(n-2)C+wi
Ȉwi-B
n+2
0
P
Vector x satisfies
xi wi ≤ B. Transforming this constraint into
Traffic Grooming instance terms, we get:
X
X
X
xi w i ≤ B ⇔
xi (C − tn+1,i ) ≤
(C − tn+1,i ) − tn+1,0
X
X ⇔
xi tn+1,i + tn+1,0 ≤ n −
xi C
i.e., the single-hop lightpaths from node (n + 1) to the hub can carry
the hub traffic, as well as all traffic components which have not been
given a two-hop lightpath.
18
i
C-wi
n+1
C-wi-vi
(n-2)C+wi
Ȉwi-B
n+2
0
P
Additionally, vector x satisfies
xi vi ≥ K. Transforming into Traffic
Grooming instance terms, we get:
X
X
X
xi vi ≥ K ⇔
xi (tn+1,i − tn+2,i ) ≥ Q −
tn+2,i
X
⇔
(xi tn+1,i + xi tn+2,i ) ≥ Q
i.e., the total amount of optical switching at the hub is at least equal
to the optical switching goal.
Therefore, the Traffic Grooming instance is a yes-instance.
19
i
C-wi
n+1
C-wi-vi
(n-2)C+wi
Ȉwi-B
n+2
0
Conversely, suppose that the Knapsack instance is a no-instance.
Then any binary vector x will violate at least one of the Knapsack
satisfiability criteria. This means that for any subset of
{tn+1,i : 1 ≤ i ≤ n} that is chosen to be routed on two-hop lightpaths,
the resulting candidate solution to the Traffic Grooming instance will
either be infeasible, or not achieve the optical switching goal. So, the
Traffic Grooming instance that we created is a no-instance.
20
Complexity of Traffic Grooming on trees
It is known that the Routing and Wavelength Assignment
subproblem is NP-complete. Therefore, Traffic Grooming on trees
must also be NP-complete.
However, the reduction we just saw establishes that Traffic Grooming
on tree networks is NP-complete, even when every interior node has
full wavelength conversion capability.
• Indeed, the tree can be restricted to the star, and the preceding proof
goes through even if wavelength conversion is available at the hub
node of the star.
21
RWA on trees with 2-hop lightpaths
• The RWA problem on tree networks is NP-complete, even for
logical topologies in which no lightpath is longer than three
physical hops.
• We will demonstrate an exact polynomial-time algorithm for
RWA on trees where each lightpath is either single-hop or 2-hop.
This algorithm will be used to develop a heuristic for the general
RWA problem on trees, based on star decomposition.
22
RWA on trees with 2-hop lightpaths
The algorithm uses exactly L colors, where L is the maximum
lightpath loading on any directed link in the tree.
If a physical link is traversed by k twohop lightpaths which have been optimally
colored, then there are L−k unused colors
with which we have to color at most L − k
single-hop lightpaths on this link.
Therefore, we are free to ignore single-hop lightpaths and concentrate
on optimally coloring only the 2-hop lightpaths.
23
RWA on trees with 2-hop lightpaths
e2
e1
e3
e1
e2
e3
e5
e4
e5
e1
e1
e2
e2
e3
e3
e4
e4
e5
e5
e4
∆ colors suffice for coloring this bipartite graph, so
L colors suffice for coloring the paths in the original tree.
24
Bounds for path networks
• Consider a logical topology in which no node switches traffic
optically (completely opaque topology). Clearly, this solution
incurs the maximum possible electronic switching cost and is a
trivial upper bound on the cost of the solution produced by any
Traffic Grooming algorithm.
• Now, consider a logical topology in which the distance between
adjacent opaque nodes is no more than 2. The logical topology
now consists of single-hop and 2-hop lightpaths, which are
determined by optimally solving the traffic grooming problem for
all path segments between opaque nodes.
25
A greedy heuristic for path networks with bifurcation
Reduction of a traffic matrix T = [tij ]:
• We reduce T so that all elements are less than the capacity C of
t a single wavelength. For each tsd ≥ C, we assign Csd lightpaths
from s to d, which are filled completely. At the same time, we set
tsd to the leftover traffic and reduce the number of available
wavelengths on the physical links we used by the number of
lightpaths we created.
• This procedure does not preclude us from reaching an optimal
solution, since breaking such a lightpath costs at least C, while
the benefit we can get by having that wavelength available for
grooming traffic cannot exceed C.
26
A greedy heuristic for path networks with bifurcation
1. Reduce the traffic matrix T .
2. Consider the logical topology obtained by assigning a lightpath
to each non-zero traffic component of the reduced matrix T
(completely transparent topology). If this is feasible, stop.
3. Pick a non-zero component tij of the reduced matrix T .
4. Determine a sequence of lightpaths (l1 , . . . , lr ) over which to
route tij .
5. For each lk in the previous sequence, if s and d are its endpoints
add an amount of traffic equal to tij to the component tsd .
6. Set tij to zero.
7. Repeat from step 1.
27
A greedy heuristic for path networks with bifurcation
The performance of the previous algorithm depends on:
• how we pick a traffic component in step 3, and
• how we determine the sequence (l1 , . . . , lr ) over which this
component is routed in step 4.
A possible way to pick tij in step 3 is to pick the traffic component
that has to travel the smallest distance, breaking ties in favor of the
smallest component.
To determine the sequence of lightpaths (l1 , . . . , lk ) in step 4, we
designate in advance either the odd-numbered nodes or the
even-numbered nodes as opaque. Initially, the traffic component tij
that was picked is routed on a direct lightpath from i to j. We break
this lightpath at the opaque endpoint of the most congested physical
link that it uses.
28
A greedy heuristic for path networks with bifurcation
Fixed lightpaths (by
matrix reduction)
Candidate completely
transparent topology
tij
Step 3 of the algorithm indicates that the traffic component tij picked
in step 2 should be split into two traffic components, as follows:
29
A greedy heuristic for path networks with bifurcation
In another variation of the algorithm, the traffic component picked in
step 2 is broken both at the opaque endpoint of its most congested
link, and at the next opaque node in the path.
3
• The running time of both versions is O N .
• Both versions are guaranteed to do no worse than the logical
topology with alternating opaque and transparent nodes.
• The two versions have similar performance in practice, with the
first version performing slightly better due to the fact that it
tends to use less lightpaths than the second one, for the same
traffic matrix.
30
Bounds for star networks
1
As we have mentioned, deciding which traffic components are
electronically switched at the
hub and which optically bypass
it fixes both the logical topology
and the routing of traffic.
N
2
0
...
If this solution does not violate any wavelength or traffic constraints,
then it is a feasible solution.
Therefore, the search space of the problem is quite large: in order to
find the optimal solution, a brute-force algorithm would have to
evaluate a total of 2N (N −1) candidate feasible solutions, for a star
with N non-hub nodes.
31
Representing solutions
We represent a proposed solution to a given problem instance by a
mask matrix M = [mij ], 1 ≤ i, j ≤ N , i 6= j, where mij ∈ {O, E}.
• mij = O means that the traffic component tij is optically
switched at the hub in this solution.
• mij = E means that the traffic component tij is electronically
switched at the hub in this solution.
It is also convenient to consider partial solutions to a given problem
instance, i.e. solutions in which not every traffic component has been
routed yet. For this purpose, we introduce a third possible value for
mij :
• mij = U means that it is not specified whether the traffic
component tij should be electronically switched or optically
routed at the hub.
32
Generating the search tree
• We seek an efficient way to generate the search tree, so that we
can easily prune invalid and suboptimal solutions.
• Partial solutions allow us to prune away whole subtrees of the
search tree, when the fixed parts of a solution are certain to yield
invalid or suboptimal solutions.
• The root of the tree is a mask matrix in which all elements have
the value U.
• Level i + 1 of the tree is generated by picking a single traffic
component tsd and generating two children for each vertex of
level i, one with msd = O and one with msd = E.
33
U U U
U
U U
U U
U
U U U
E U U
U
O U E
U U
U U
U
U
U U
U U U
E
O
E O
O O
O O
O
O O O
E
O
O O O
U
U
O
O O E
U
U
U
U U U
U E U
34
U
U E U
E U
U U
E U
U U
O E E
E U
U U
U
U E U
E O E
U
E U
U U
U U U
O O
O O
O E E
E U
U U
E O
U
U E U
E O E
U
U U
U
O O E
U
E U
U U
U E U
U
Generating the search tree
• Let Πt = (t1 , t2 , t3 , . . .) denote the ordering of the traffic
components that is used to generate the levels of the tree.
• In a complete search, the resulting tree will always be a full
binary tree of depth N (N − 1), whose leaves will be all possible
complete mask matrices. However:
– Pruning will allow us to avoid generating all the leaves, even
in an exhaustive search.
– We may wish to stop before generating the full tree and
extract information in the form of bounds.
• The search is characterized by Πt , since changing the order in
which we consider the traffic components may change the bounds
we get, or the amount of pruning possible.
35
Pruning for invalidity
In any valid mask matrix, the following must hold for each non-hub
node i:
• The available capacity on the link (i, 0) must be able to
accommodate the traffic demand originating from node i which
has not been optically routed.
• Similarly for traffic having node i as destination.
As a result, whenever a new mask matrix is created by changing the
status of some msd from U to O, some traffic components may be
forced to be electronically switched at the hub.
36
Pruning for suboptimality
• Check whether all the unassigned traffic components of a nearly
complete partial mask matrix can be optically routed without
violating the feasibility conditions.
• Let M , M 0 be two partial mask matrices for which no children
have been generated yet. If the lower bound on the value of the
best solution that can be obtained by M 0 is greater than the
upper bound on the value of the best solution that can be
obtained by M , then we can safely prune away the whole subtree
rooted at M 0 .
37
Bounds for star networks
Consider a search tree fully generated up to level i. Let Li be the set
of nodes at level i, and let v be one of them.
• Replacing all unassigned elements of node v by E yields a feasible
solution of cost ψ (v), an upper bound on the cost of the optimal
solution in the subtree rooted at v. The tightest upper bound
that we can derive from level i of the tree is:
Ψi = min ψ (v)
v∈Li
• Similarly, replacing all unassigned elements of node v by O yields
a lower bound φ (v), and the tightest lower bound from level i is:
Φi = max φ (v)
v∈Li
38
Bounds for star networks
• It turns out that these bounds converge most quickly when Πt is
an ordering of the traffic components in descending order. Then
the following holds for each level i:
X
i
Ψi − Φ i ≤ 1 −
tsd
N (N − 1)
• To obtain an upper bound from a node v, use the following
greedy algorithm: consider the traffic components in descending
order; if a component is unassigned in node v, and if it can be
routed optically without violating the feasibility conditions,
assign a 2-hop lightpath to it.
• Thus another sequence of upper bounds is defined:
(g)
Ψi
= min ψ (g) (v)
v∈Li
39
Bounds for tree networks
Decomposed star network for a tree node p:
• A star network with the same traffic pattern as the one “seen” by
node p in the tree.
• p is the hub of the star.
• The neighbors of p in the tree are the non-hub nodes of the star.
t1
t1
t4
t4
t2+t3
t2
t3
40
Bounds for tree networks
Let φ (p) be the electronic switching cost on node p that would be
obtained by solving the decomposed star network for node p
optimally.
• φ (p) is a lower bound on the electronic switching node p can
perform under any feasible logical topology and traffic routing.
Let ψ (p) be the amount of electronic switching node p performs in a
completely opaque logical topology.
• ψ (p) is an upper bound on the electronic switching node p can
perform under any feasible logical topology and traffic routing.
P
• Ψ = ψ (p) is an upper bound on the cost of an optimal
solution in the tree network.
41
A heuristic based on star decomposition
• For any two adjacent nodes p and q in the tree, the optimal
electronic switching costs φ (p) and φ (q) may not be
simultaneously achievable.
• We seek a logical topology in which at least one opaque node is
interposed between any two transparent nodes of the tree.
• To find the best star decomposition, we seek a maximum weight
independent set for the tree network where the weight of each
node p is ψ (p) − φ (p).
• The resulting logical topology has at most 2-hop lightpaths,
therefore the RWA subproblem can be solved optimally.
42
A greedy heuristic for tree networks
1. Consider traffic components in descending order:
• Assign an end-to-end lightpath, if possible.
• If no wavelength is available on the whole path, form a
lightpath up to the intermediate node for which there is an
available wavelength; try the rest of the path in a similar
manner.
• Otherwise, abandon the current traffic component.
2. Route the abandoned components over single-hop lightpaths.
43
References
[1] Huang, Dutta, and Rouskas: “Traffic Grooming in Path, Star, and
Tree Networks: Complexity, Bounds, and Algorithms.”
[2] Dutta and Rouskas: “A Survey of Virtual Topology Design
Algorithms for Wavelength Routed Optical Networks.”
[3] Dutta and Rouskas: “On Optimal Traffic Grooming in WDM
Rings.”
[4] ?
44