Path computation in multi-layer networks: Complexity and algorithms

0.82 MB·

by  Mohamed Lamine Lamali

Path computation in multi-layer networks:
Complexity and algorithms
Mohamed Lamine Lamali∗, Nasreddine Fergani∗, Johanne Cohen†, He´lia Pouyllau‡
∗Nokia Bell Labs. France.
†LRI, Univ. Paris-Sud, CNRS, Universite´ Paris-Saclay. France.
†Thales Research & Technology. France.
mohamed [email protected] [email protected] [email protected]
Abstract—Carrier-grade networks comprise several layers Dealing with protocol heterogeneity becomes increasingly
6 where different protocols coexist. Nowadays, most of these important nowadays. In addition to the IPv4/IPv6 migration,
1 networks have different control planes to manage routing on this heterogeneity appears in tunneling, some architectures
0 different layers, leading to a suboptimal use of the network (e.g., The Pseudo-Wire architecture [3] allows the emulation
2 resourcesandadditionaloperationalcosts.However,somerouters
– and thus the encapsulation – of lower layer protocols over
n areabletoencapsulate,decapsulateandconvertprotocolsandact Packet-Switched Networks), hybrid networks (e.g., National
a asaliaisonbetweentheselayers.Aunifiedcontrolplanewouldbe
Research and Education Networks – NRENs – which may
J usefultooptimizetheuseofthenetworkresourcesandautomate
the routing configurations. Software-Defined Networking (SDN) have optical and IP interconnection points), and last but not
1
based architectures, such as OpenFlow, offer a chance to design least,mostcarrier-gradenetworks,whichhaveseparatecontrol
2
suchacontrolplane.Oneofthemostimportantproblemstodeal planes for IP and Transport layers. In all these contexts, a

withinthisdesignisthepathcomputationprocess.Classicalpath unified control plane would be very useful for optimizing the
]
I computationalgorithmscannotresolvetheproblemastheydonot network resources and reduce operational and management
N takeintoaccountencapsulationsandconversionsofprotocols.In costs.
. thispaper,weproposealgorithmstosolvethisproblemandstudy
s several cases: Path computation without bandwidth constraint, OpenFlow is a chance to design such a control plane.
c
under bandwidth constraint and under other Quality of Service Some previous works [4], [5] present an OpenFlow-based
[
constraints. We study the complexity and the scalability of our architecture to achieve this challenge, but they only focus

2 algorithms and evaluate their performances on real topologies. on the convergence of packet and circuit networks. Other
v Theresultsshowthattheyoutperformthepreviousonesproposed works tackle the traffic engineering problem in SDNs but
6 in the literature. circumscribe it on a single layer [6] or in the IPv4/IPv6
8
Keywords—Multi-layer networks; Path computation; Protocol migrationcontext[7].However,animportantproblemtosolve
7
heterogeneity; Unified control plane. remainsthepathcomputationprocessinamulti-layercontext.
1
Taking into account the adaptation functions is not trivial and
0
. I. INTRODUCTION classical algorithms such as Dijkstra’s one [8] cannot achieve
1 the task as they do not handle these functions.
0 Carrier-grade networks generally encompass several layers
6 involving different technologies and protocols. To support Here, we design several algorithms to compute shortest
1 someservices,suchasaVirtualPrivateNetwork(VPN),apath paths dealing with protocol changes and adaptation functions.
: across network equipments must be identified and the equip-
v
i mentsbeconfiguredaccordingly.Understringentrequirements Our contributions:
X
ofQualityofService(QoS–e.g.,end-to-enddelay,geographic
1) We widely generalize the model and the polynomial
r zone avoidance, etc.), computing such a path within a single
a algorithms described in Lamali et al. [9], [10] to
layer is not always possible. Hence, one of the key challenges
perform path computation in multi-layer networks
is to determine the end-to-end path that uses the appropriate
(withoutbandwidthconstraint).Ourmodeltakesinto
adaptation functions over the protocols: The mapping from
account all possible types of protocol changes (en-
a protocol to another being realized through encapsulation
capsulation,conversion,etc.)andanyadditivemetric.
(e.g., Ethernet over IP/MPLS [1]), decapsulation (the reverse
Wedrasticallyimprovethealgorithmcomplexityand
operation) or conversion (e.g., IPv4 to IPv6 [2]) functions.
realize the first implementation, showing their effi-
Consequently, the path computation process should take into
ciency on two real topologies.
account the adaptation function capabilities of the network
2) For simulation purposes, we empirically study the
equipments in order to ensure path feasibility: If a protocol
distribution of adaptation functions over the network
is encapsulated in another one, it must be decapsulated (or
nodes and its impact on feasible path existence. We
unwrapped) further in the path. If several encapsulations are
exhibit a phase transition phenomenon, i.e., a gap
nested, the corresponding decapsulations must occur in the
where the probability of existence of a feasible path
right order. Here, the multi-layer context should be taken in
hugely increases.
a broad sense: Presence of several protocols and technologies
3) We prove that path computation in multi-layer net-
that can be nested, encapsulated, converted, etc.
works under bandwidth constraint is NP-complete
PreprintofthepaperacceptedforIEEEINFOCOM2016. even with two protocols and on symmetric graphs,
2
thus improving a result of Kuipers and Dijkstra [11]. This configuration leads to a decapsulation of an IPv6 packet
We also obtain results on the complexity of some fromanEthernetdatagram(bytheborderrouterofDomain2)
subproblems: It is polynomial on Directed Acyclic whereas at this stage the datagram encapsulates IPv4 packets.
Graphs (DAG) and the general problem is not ap-
This example depicts the constraints to comply with when
proximable.Weproposeanewheuristictoresolvethe
computing a multi-layer (and multi-domain, in this case)
problem and show its efficiency through simulations.
path: Being physically linked is not sufficient to establish
4) We propose the first algorithm to perform path
connectivity. Protocol continuity (by analogy with wavelength
computation in multi-layer networks under several
continuity in optical networks) must hold and the adaptation
QoS constraints by adapting the Self-Adaptive Mul-
functions should occur in the right order. Moreover, feasible
tiple Constraints Routing Algorithm (SAMCRA –
paths can involve loops and their subpaths are not necessary
Van Mieghem and Kuipers [12]) to the multi-layer
feasible [11], [15]. Nowadays, such paths are manually deter-
context. We study its scalability through simulations.
mined and configurations are operated and applied by scripts.
The paper is organized as follows: Section II describes
the problem of path computation in multi-layer networks and B. Related work
recalls the related work; Section III formalizes the problem
The initial works dealing with protocol and technology
and describes our model of multi-layer network; Section IV
heterogeneity circumscribed the problem at the optical layer.
proposes algorithms to perform path computation without
For instance, Chlamtac et al. [16] described a model and
bandwidth constraint and shows their efficiency through sim-
algorithms to compute a path under wavelength continuity
ulations, it also studies the phase transition phenomenon in
constraints. Zhu et al. [17] addressed the same problem in
multi-layernetworks;SectionVstudiesthecomplexityofpath
WDMmeshnetworkstacklingtrafficgroomingissues.In[18],
computationunderbandwidthconstraintandproposesheuristic
GongandJabbariprovidedanalgorithmtocomputeanoptimal
solutions to tackle the problem; Section VI proposes the first
pathunderconstraintsonseverallayers:wavelengthcontinuity,
algorithmcomputingpathsunderadditiveQoSconstraintsand
label continuity, etc.
studies its scalability; finally, Section VII concludes the paper.
However, the models of these past works are not adapted
II. PATHCOMPUTATIONINMULTI-LAYERNETWORKS to the problem of nested encapsulation and decapsulation
capabilities for which a kind of stack mechanism is needed.
A. Connectivity in multi-layer networks In[19],Dijkstraetal.addressedthisissueinthecontextofthe
ITU-T G.805 recommendations on adaptation functions. They
We aim to present the different concepts of path com-
stressed the lack of solutions on path computation. Kuipers
putation in multi-layer networks through an example. While
and Dijkstra [11] demonstrated that the problem of path
thisexamplerelatestomulti-domainmulti-layernetworks,the
computation with encapsulation and decapsulation capabilities
underlying problem of path computation is the same as in
is NP-complete under bandwidth constraint. They proposed a
a single domain network1. Figure 1 (inspired by the Inter-
Breadth-FirstSearch(BFS)algorithmthatexploresallpossible
Provider Reference Model [14]) depicts a network involving
paths until finding a feasible one. In [9], [10], Lamali et al.
multiple domains and adaptation function capabilities of net-
demonstrated that the problem is polynomial if the bandwidth
work equipments: A company owning a Local Area Network
constraintisrelaxed.Theirapproachwastomodelthenetwork
(LAN) wishes the Virtual Machines (VMs) of a data-center
asaPush-DownAutomatonandtouseautomataandlanguage
to be within the same routing domain (for instance through
theory tools to compute a shortest feasible path, but only
a Layer 2 VPN or a Generic Routing Encapsulation tunnel).
considering the number of hops or adaptation functions. More
Hence, the switches of the LAN and the VMs of the data-
recently, Iqbal et al. [20] underlined the need of path com-
center must communicate through Ethernet datagrams and a
putation algorithms in NRENs. They proposed a new matrix-
path has to be determined across the Domains 1 and 2.
basedmodelformulti-layernetworksandalgorithmsbasedon
On Figure 1, Domains 1 and 2 use IPv6/MPLS-TE tech- k-shortest paths and LOOK-AHEAD methods. However, the
nology and are linked by equipments providing Ethernet modeldealswithtechnologies2 insteadofprotocols.Thus,the
encapsulation and decapsulation. The Provider Edge (PE) of nested protocols are not transparent to the nodes. Moreover,
Domain 1 is linked to the Customer Edge (CE) of the data- the proposed exact algorithm is exponential and can compute
center. The adaptation capabilities of each node are shown only loopless feasible paths.
above it. An example of feasible path would cross the PE
of Domain 1 converting IPv4 packets into IPv6 ones. Then C. Proposed approach
it would apply the encapsulation and decapsulation of the
Our goal is to study the path computation problem in
border routers of Domains 1 and 2 respectively and the PE
a multi-layer context and to propose efficient algorithms to
of Domain 2 would apply a conversion of IPv6 packets into
resolve it. To this end, we focus on three cases: Path compu-
IPv4 ones. The protocol stacks of the packets at each stage
tation without bandwidth constraint (by adapting the language
are illustrated at the bottom of Figure 1. As an example of
theoretic approach of Lamali et al. [10]), under bandwidth
unfeasible path, a direct Ethernet connection between the CE
constraint(byusinggraphtransformationinordertoovercome
of the data-center and the border router of Domain 1 appears.
the problem complexity) and under several QoS constraints.
1Thealgorithmspresentedinthispapercanbeappliedinasingle-domain The simulations showing the efficiency of our algorithms
or a multi-domain context. For the latter, however, a mechanism for sharing
thenetworkinformation(suchasthetopology)isneeded.Thiscanbedone 2A technology is an exhaustive description of the protocol stack at some
throughaPCEforexample[13]. node,e.g.,IPoverEthernetoverATM.
3
Fig.1. Carrier-gradenetworkcomprisingseveraldomainsanddifferentlayers.
follow a methodology based on the probabilistic distribution B. Path feasibility
of the adaptation functions over the nodes.
Let (S,D) be a pair of nodes in G corresponding to the
III. MODELANDPROBLEMFORMALIZATION source and the destination of the path to be computed. We
consider a path from S to D as a sequence of nodes and
Thissectiondescribesamathematicalmodelofmulti-layer adaptation functions Sf U f U f ...U f D where each U ,
0 1 1 2 2 n n i
networks and formalizes the notion of path feasibility. i = 1,...,n, is a node and each f is an adaptation function
i
(f being fictitious). A path is feasible if:
A. Multi-layer network model 0
Notation convention. In order to avoid confusion, lowercase 1) ThesequenceSU1U2...UnDisapathinG =(V,E)
letters denote protocols (e.g., a,b,c,x,y) or functions (e.g., and each fi ∈F(Ui);
f,h,(cid:96)). Capital letters denote nodes and links (e.g., U,V,E). 2) Each encapsulated protocol is decapsulated before
Finally, calligraphic letters denote sets (e.g., G,V,E). reaching D according to its encapsulation order and
protocol continuity must hold (i.e., if the sequence
We consider a multi-layer network as a 4-tuple N =
contains a function f s.t. f = (a → b), a,b ∈ A,
i i
(G,A,F,h) where: then f = (b → a(cid:48)) or f = (b → ba(cid:48)) or
i+1 i+1
• G =(V,E) is a directed graph modeling the network fi+1 =(a(cid:48) →a(cid:48)b), a(cid:48) ∈A).
topology. The set of nodes V models the routers of
Actually, the protocol sequences of feasible paths can be
the network. The set of edges E models the physical
characterized as a well-parenthesized language [10].
links between the routers.
• A = {a,b,c,...} is the set of protocols available in
the network, but not necessarily at each router. IV. PATHCOMPUTATIONWITHOUTBANDWIDTH
CONSTRAINT
• For each node U ∈ V, F(U) is the set of adaptation
functions available on node U. These functions are: This section proposes a polynomial algorithm to resolve
the path computation problem without bandwidth constraint
◦ Conversion: A protocol a is converted into a
and evaluates it through simulations.
protocol b without any change of the possible
underlying protocols. This function is denoted
by (a → b). E.g., Wavelength conversion on A. A polynomial algorithm for path computation
the optical layer, IPv4 to IPv6, etc.
Lamali et al. [10] proposed a language theoretic approach
◦ Passive function: A protocol a is left as it
to compute a shortest feasible path (involving encapsulations
is. It is a classical retransmission without any
anddecapsulationsofprotocols)inamulti-layernetwork.The
protocol change and can be considered as
metricconsideredwasthenumberofhopsorofencapsulations
a special case of protocol conversion where
in the path. The approach comprises the following steps:
a=b. Thus it is denoted by (a→a).
◦ Encapsulation:Aprotocolaisencapsulatedin
1) Consider the set of protocols as an alphabet and
a protocol b. It is denoted by (a→ab).
convert the multi-layer network into a Push-Down
◦ Decapsulation: A protocol a is decapsulated
Automaton (PDA);
from a protocol b. It is denoted by (a→ab).
2) If the considered metric is the number of encapsu-
• h : V ×F ×V → (cid:60) is the weight function. The lations, transform the automaton in order to bypass
+
value h(U,f,V) (where U,V ∈V and f ∈F(U) ) is passive transitions;
the cost of using the link (U,V) with the adaptation 3) ConvertthePDAtoaContext-FreeGrammar(CFG);
function f on U. Hence, function h allows represent- 4) Compute the shortest word generated by the CFG. It
ing any additive metric either associated only to the is the protocol sequence of a shortest path;
links or to both links and adaptation functions. 5) Compute a shortest path from this sequence.
4
We made several improvements to these algorithms:
• The PDA building is modified in order to support
protocol conversion by adding a new transition type;
• The PDA transitions are weighted in order to reflect
theweightfunction.Thus,ouralgorithmcomputesthe
shortestpathaccordingtoanyadditivemetric(instead
of just the number of hops or encapsulations);
• The PDA transformation is no longer useful thanks
to the weight function: Simply put h(U,f,V) = 1
(where U,V ∈V and f ∈F(U)) for all triples where
f isanencapsulation,andh(U,f,V)=0forallother
triples. It is also possible to set different weights to
eachtypeofencapsulationandminimizethepathcost
according to these weights; Fig.2. Probabilityofexistenceofafeasiblepath(andaloopintheshortest
one)accordingtotheprobabilityofexistenceofanadaptationfunction.
• The conversion of the PDA into a CFG is adapted:
As in [10], each transition from the PDA is converted
into a production rule set in the CFG according to • Topology T2 corresponds to the network of Exodus
a method described in [21]. However, the transition as in 2002. It has 79 nodes and 294 directed links.
weights are assigned to the corresponding production
Since these topologies are not layered, the adaptation
rules;
functions are randomly allocated to the nodes. For an alpha-
• Step 4 is different: Since the production rules are bet A, there are 3|A|2 possible adaptation functions (for each
weighted, the goal is no longer to compute the short- ordered pair of protocols: a conversion, an encapsulation and
est word but the word having the minimum weight a decapsulation). For each node U, each of these adaptation
derivation tree. This is done thanks to Knuth’s algo- functions is available on U with probability p. The source
rithm described in [22]. This word corresponds to the and the destination nodes are the diameter extremities, which
protocol sequence of a shortest path to compute; corresponds to 5 (resp. 10) hops for Topology T1 (resp. T2).
• Thealgorithmcomputingthepathmatchingtheproto- 2)Phase transition in path feasibility: Depending on the
col sequence is modified in order to take into account network topology and the adaptation function distribution,
the weights. thereisnotalwaysafeasiblepath.Itisinterestingtoknowthe
probabilityofafeasiblepathexistenceaccordingtoprobability
Due to the lack of space, we cannot detail our improved pinordertosetappropriateparametersforthesimulations.In
algorithm. The interested reader can find it (together with its caseofpathexistence,knowingtheprobabilitythattheshortest
correctness proof and complexity study) in Appendix A. one involves loops allows comparing the different algorithms
(some of them allow loops and others do not). To compute
Additionally to these improvements, the algorithm com-
this probability, we performed 200 runs for each value of p
plexityisdrasticallydecreased.In[10],Step4hasacomplex-
ity of O(|A|8×|V|7) in the worst case, which is the highest and counted the number of times there was a feasible path.
complexity in the whole process. Implementing Knuth’s al- Figure 2 shows the evolution of feasible path existence
gorithm with Fibonacci heaps gives an O(|Q|log|Q|+|R|) probability according to p and the proportion of shortest paths
complexity, where |Q| is the number of nonterminals in the that involve loops. Not surprisingly, the probability of feasible
CFG and |R| is the number of production rules [23]. Since path existence grows according to p. On both topologies, the
|Q| = O(|A|3 ×|V|2) and |R| = O(|A|5 ×|V|2 ×|E|) (see probability of path existence reaches 50% when p = 0.22
Appendix A), the complexity of the whole process is: and follows a phase transition phenomenon. For example, in
O(cid:0)|A|5×|V|2×|E|(cid:1) the interval p ∈ [0.10,0.38], the probability of path existence
in Topology T1 grows from 5% to 90%. This interval is the
This is a significant improvement compared to the complexity most suitable to perform simulations. The phase transition
O(|A|8×|V|7) in [10]. phenomenon also holds with more than 2 protocols. The more
the number of protocols is high, the more the phase transition
B. Simulations is shifted to the left. If there are few feasible paths (for small
p), the probability that the shortest ones involve loops is high.
Weimplementedouralgorithm(calledPDA)andcompared
However, this probability quickly decreases. For example, for
it to a classical BFS approach.
p > 18%, the proportion of shortest paths involving loops is
1)Networks used for the simulations and methodology: less than 20% in Topology T1. The trend of this proportion
Largemulti-layertopologiesaregenerallynotavailable.Some is not clear in T2, however it is less than 21% if p > 0.22.
publiconesastheInternet2network[24]arenotlargeenough The phase transition phenomenon can be seen in [20]. But the
to show the scaling of our algorithm. Thus we performed results consider only loopless paths and the distribution deals
simulations on two topologies described in [25]: with technologies rather than adaptation functions.
• Topology T1 is a simplified version of Time Warner 3)Simulation results: Our algorithm is compared to a
network. It has 41 nodes and 296 directed links. classical BFS which explores all possible paths until reaching
5
be crossed more than twice. The (optimization) problem of
computing the shortest path in a multi-layer network under
bandwidth constraint is defined as follows:
(cid:88)
min h(P)= h(U,f,V)
(U,f,V)∈P
 P is a feasible path between S and D
(1)

s.t.
q (E)
 minE∈P nbb(E) ≥qbmin
where nb(E) is the number of times a link E is crossed by
path P, q (E) is the bandwidth capacity of E and qmin is the
b b
bandwidth constraint.
Fig. 3. Comparison of processing time of PDA algorithm and BFS on
B. Path computation complexity under bandwidth constraint
TopologiesT1andT2.
The bandwidth constraint impacts the complexity of feasi-
ble path computation. In a single-layer network, computing a
thedestination.Duringtheexplorationprocess,alldominated3
path under bandwidth constraint is trivial: It suffices to prune
pathsaredeleted.BFScanbeseenasaversionofthealgorithm
all the links without enough bandwidth. This is no longer
in [11] where the bandwidth constraint is relaxed. The first
possibleinamulti-layernetwork.Infact,thedecisionproblem
results showed that BFS algorithm is extremely slow even
is NP-complete as shown by Kuipers and Dijkstra [11]. But
for small values of p (processing time of the order of several
this proof does not work on symmetric directed graphs4.
hours). It was impossible to perform a comparison with our
However, most communication networks are symmetric. We
algorithm. Due to this tremendous running time, we fixed a
show that the decision version of the problem remains NP-
maximum length to the explored paths by BFS algorithm. If a
complete even with two protocols and in a symmetric graph.
path exceeds 10 hops (resp. 14 hops) on Topology T1 (resp.
Consider the following problem:
T2), it is deleted and no more considered. We performed 100
runs for each value of p and averaged the processing time. Problem (1’). Given a multi-layer network N = (G =
Figure3showstheprocessingtimeofPDAalgorithmandBFS (V,E),A,F,h), a function assigning to each link E ∈ E an
algorithmonTopologiesT1andT2accordingtothevaluesof available bandwidth q (E), a bandwidth constraint qmin and
b b
p. For small values of p (< 0.22 for T1 and < 0.04 for T2) a pair S and D of nodes in V. Is there a feasible path from S
BFS algorithm is faster than PDA. However, the processing to D satisfying the bandwidth constraint?
time of BFS explodes. We cannot put it on Figure 3 because
Proposition 1: Problem(1’)isNP-completewithtwopro-
it would be unreadable. For example, the processing time of
tocols even if G =(V,E) is a symmetric directed graph.
BFS algorithm on Topology T2 for p = 0.24 is more than
14 minutes, while that of PDA algorithm is 10 seconds. On Proof:Clearly,theproblemisinNP.Thus,weonlydetail
Topology T1, for p = 0.38, the processing time of BFS the proof of NP-hardness.
algorithmismorethan7minutes,whilethatofPDAalgorithm
is 7 seconds. These results show that our algorithm clearly FirstconsidertheproblemoffindingaHamiltonianpathin
outperforms the BFS approach. asymmetricdirectedgraphbetweentwonodesS(cid:48) andD(cid:48).Call
this problem SYM-HAM. SYM-HAM is NP-complete (for a
detailed proof, see Appendix B).
V. ADDRESSINGBANDWIDTHCONSTRAINT
Now we provide a polynomial reduction from SYM-HAM
This section studies the complexity of path computation
toProblem(1’)restrictedtosymmetricdirectedgraphandtwo
underbandwidthconstraintandproposesheuristicsolutionsto
protocols. Given an instance of SYM-HAM, i.e., a symmetric
resolve the problem.
directed graph H = (V(cid:48),E(cid:48)) and a pair of nodes (S(cid:48),D(cid:48)),
we build an instance of Problem (1’), i.e., a network N =
A. Problem formalization
(G,A,F,h) and a pair of nodes (S,D) as following:
For Traffic Engineering purposes, a feasible path may Step 1: Splitting the nodes. For each node U(cid:48) ∈ V(cid:48), four
be constrained by a minimal bandwidth. But it is possible
nodes U ,U ,U and U are created in G. Links (U ,U )
that feasible paths in a multi-layer network involve loops 1 2 3 4 i i+1
and (U ,U ) are created for i = 1,...,3. For each link
(i.e., involving the same link several times but using different (U(cid:48),V(cid:48))i+∈1 Ei(cid:48), a link (U ,V ) is created in G. This step is
protocols).Itimpliesthatthebandwidthconstraintisnolonger 1 1
illustrated on Figure 4.
prunable: Even if the links with not enough bandwidth are
deleted by topology filtering prior to path computation, other Step 2: Adding a tail. G = (V,E) is augmented by a set
links can have enough bandwidth if they are selected once but C = {C ,...,C } of nodes (n = |V(cid:48)|), where C = S
0 n+1 0
not if more. For example, if a link has a capacity of 10Gbps is the source node. There are a link (C ,C ) and a link
i i+1
and the bandwidth constraint is 5Gbps, then this link cannot (C ,C ) for i = 0,...,n. Moreover, there is also a link
i+1 i
3In this context, a path dominates another one if they have the same 4Asymmetricdirectedgraphisagraphwherealink(U,V)existsifand
extremitiesandthesameprotocolstack,andthefirstpathisshorter. onlyifthereverselink(V,U)exists.
6
Node Adaptationfunctions Finally, node X decapsulates protocol a from protocol b and
node D receives protocol a as emitted by S. Thus, P is
Ci, i=1...n (a→aa)
a feasible path, and each link is crossed at most once, the
Cn+1 (a→ab) bandwidth constraint is satisfied.
U1s.t.U(cid:48)∈V(cid:48) (b→bb),(a→ab)
Conversely, we show that from any feasible path P sat-
U2s.t.U(cid:48)∈V(cid:48) (b→bb),(a→a) isfying the bandwidth constraint in N, one can extract a
U3s.t.U(cid:48)∈V(cid:48) (a→ab),(a→a) Hamiltonian path between S(cid:48) and D(cid:48) in H. A feasible path
U4s.t.U(cid:48)∈V(cid:48) (a→aa) mustcrossallnodesU4 s.t.U(cid:48) ∈V(cid:48) inordertodecapsulateall
occurrences of protocol a encapsulated when crossing the tail.
X (a→ab)
TABLEI. THEADAPTATIONFUNCTIONSAVAILABLEONTHENODESIN Thus, it involves Sequence (2) for all U(cid:48) ∈ V(cid:48). By removing
THEPOLYNOMIALREDUCTION. thetailpartandthenodesX andDfromP andreplacingeach
occurrenceofSequence(2)bythecorrespondingnodeU(cid:48),the
resulting path starts from S(cid:48) and crosses all the nodes in H
before reaching D(cid:48). The only problem is the possibility that
from C to S (the first node resulting from the splitting of
S(cid:48)) andnc+o1nverse1ly. Figure 5 shows this construction. Finally, there are other sequences than Sequence (2) in the remaining
path. There are two possible cases:
two nodes X and D are added, as well as the four links
(D ,X),(X,D ),(X,D) and (D,X) (recall that D is the
1 1 1 • An incomplete Sequence (2) where U is not reached
first node resulting from the splitting of D(cid:48), see Step 1). 4
(e.g., U fU f(cid:48)U f(cid:48)(cid:48)U f(cid:48)(cid:48)(cid:48)U ): This cannot happen
1 2 3 2 1
Step 3: Allocating the adaptation functions and available because such a sequence forbids to reach U4 later,
bandwidth. All the links have available bandwidth 1. The and thus one encapsulated occurrence of protocol
bandwidth constraint is set to 1. Thus, any feasible path must a is never decapsulated and P cannot be feasible.
crossalinkatmostonce.Thereisnopossibleloop.Lettheset Such a sequence cannot occur after an occurrence of
of protocols be A={a,b}. Node S emits packets of protocol Sequence (2) on the same nodes because if a node Ui
a. For i = 1...,n, each node C in the tail can encapsulate (i = 2,3) is reached in a Sequence (2) it cannot be
i
protocol a in itself. Node C can only encapsulate a in b. reached again due to the bandwidth constraint.
n+1
For each node U(cid:48) ∈V(cid:48), node U can encapsulate any protocol
1 • A sequence U fV f(cid:48)W : Let P be a feasible path
in b. Node U can either decapsulate protocol b from itself or 1 1 1
2 from S to D containing a sequence U fV f(cid:48)W
passively transmit protocol a. Node U can either decapsulate 1 1 1
3 (where U and W may be the same node). These
protocol a from b or passively transmit protocol a. Node U 1 1
4 three nodes can only encapsulate protocol a or b in
is able to decapsulate protocol a from itself. Finally, node X
protocolb.Thus,aftercrossingsuchasequence,there
can decapsulate protocol a from b. Table I summarizes the
are three occurrences of protocol b on the top of the
allocation of the adaptation functions.
protocol stack. However, in network N, there is no
Now, we prove that there is a Hamiltonian path from S(cid:48) to possible sequence of nodes and adaptation functions
D(cid:48) in H if and only if there is a feasible path from S to D in abletodecapsulateprotocolbthreeconsecutivetimes.
N that satisfies the bandwidth constraint. First, assuming that Thus, P is not feasible.
there is a Hamiltonian path from S(cid:48) to D(cid:48) in H, we construct
a feasible path P in N as follows: Starting from S in N, P Thus, if a feasible path exists, then it contains only one
crosses the tail and each C (i = 1...n) adds an occurrence occurrence of Sequence (2) for each node U(cid:48) ∈V(cid:48). Replacing
i
of protocol a in the stack of encapsulated protocols. Then each Sequence (2) by the corresponding node in V(cid:48) induces a
crossing C adds b as current protocol. Thus, at the end Hamiltonian path in H. This concludes the proof.
n+1
of the tail, there are n+1 encapsulated protocols a (the one
Unfortunately, the previous negative result implies:
emitted by S and n occurrences added in the tail) and the
current protocol is b. Following the same node order as in the Corollary 1: Problem(1)isnotapproximable(unlessP=
Hamiltonian path, replace each occurrence of a node U(cid:48) ∈V(cid:48) NP).
(includingS(cid:48)andD(cid:48))intheHamiltonianpathbythesequence:
Proof: Since the existence of a feasible path (indepen-
U (b→bb)U (b→bb)U (a→ab)U (a→aa)U (a→a) dently of its cost) is NP-complete to decide, any polynomial
1 2 3 4 3
U (a→a)U (a→ab) approximation algorithm would imply P=NP.
2 1
(2)
On the other hand, the problem is tractable on some
particular topologies:
Thus, at node U an encapsulation of protocol b occurs, at
1
U protocol b is decapsulated, at U it is decapsulated again,
2 3 Corollary 2: Problems (1) and (1’) are polynomial if the
and at U protocol a is decapsulated. Path P then crosses
4 graph G =(V,E) is a Directed Acyclic Graph (DAG).
passively nodes U and U , and finally encapsulates protocol
3 2
b at U . Thus, at each time the path crosses a Sequence (2), Proof:TheNP-completenessofProblem(1’)resultsfrom
1
thenoneoccurrenceofprotocolaisremovedfromtheprotocol the fact that the bandwidth constraint is not prunable when
stack. Crossing all U s.t. U(cid:48) ∈ H removes all encapsulated feasible paths involve loops. In a DAG, every link is involved
4
occurrences of protocol a except the first one. When the path at most once in a feasible path due to the absence of cycles.
leaves D to reach node X, the current protocol is b and Thusthebandwidthconstraintisprunableandtheproblemcan
1
there is a last occurrence of protocol a which is encapsulated. be resolved using the method described in Section IV.
7
Fig.4. ReductionfromSYM-HAMtofeasiblepathunderbandwidthconstraint(nodesplitting).
Fig.5. ReductionfromSYM-HAMtofeasiblepathunderbandwidthconstraint(graphtransformation).
C. DAG Heuristic
AsseeninSectionIV-B2,shortestfeasiblepathsinvolving
loops are infrequent (for p>20%). Combining this fact with
Corollary2suggestsaheuristictocomputefeasiblepathunder
bandwidth constraint: Convert the network into a DAG and
performthePDAalgorithmtocomputeashortestfeasiblepath.
DAG Conversion. The network is converted into a DAG in
the following way:
1) Set the number 0 to node S and |V|−1 to node D
(recall that S and D are the extremities of the graph
diameter);
2) Perform a BFS algorithm starting from node S and
number the nodes in the visit order. The nodes
Fig.6. ProbabilityoffeasiblepathexistencebeforeandafterDAGconversion
at the same distance from S are visited randomly,
onTopologiesT1andT2.
thus performing several times this heuristic does not
always give the same node numbering and the same
DAG; The simulation conditions (parameters, topology, number of
3) Delete all the links that start at a node and end at a runs, etc.) are the same as in Section IV-B3. The bandwidth
node with a smaller number. capacityofthelinksisrandomlyanduniformlyselectedinthe
set {1,2,...,10}. The bandwidth constraint is set to 2.
The DAG heuristic is as follows:
1)Comparison of the feasibility ratio: Converting the
1) Convert the network into a DAG; network topology into a DAG deletes some feasible paths
2) Prune the links without enough bandwidth; in the original network. We measure how much feasible
3) Perform the PDA algorithm of Section IV. paths are lost by comparing the probability of feasible path
existence before and after the DAG conversion according
D. Simulations to the probability of existence of adaptation functions (p).
Figure 6 shows that the probability of feasible path existence
WestudytheefficiencyoftheDAGheuristic(calledDAG- is shifted to the right after the DAG conversion. The ratio
PDA) and compare it with the algorithm of Kuipers and ProbabilityoffeasiblepathexistenceinTi (i = 1,2) is clearly de-
Dijkstra [11]. The latter is an exact (and thus exponential) ProbabilityoffeasiblepathexistenceinDAGTi
creasing and is less than 50% if p>0.34, which is important
algorithm that performs a BFS and explores all the paths that
but balanced by the improvement of the processing time.
are not dominated and that satisfy the bandwidth constraint.
AsinSectionIV-B3,theBFSalgorithmisslow.Thus,wealso 2)Comparison of the processing time: Figure 7 shows the
compare our algorithm with DAG-BFS algorithm, where the processingtimeofDAG-PDA,DAG-BFSandBFSalgorithms
network is converted into a DAG before performing the BFS. onbothtopologiesaccordingtotheprobabilityofexistenceof
8
protocol and passive transitions. Thus the decision version
associated to Problem 3 is also NP-complete.
C. ML-SAMCRA
As computing a multi-layer path under QoS constraints
is NP-complete, any algorithm able to solve this problem is
exponential in the worst case (unless P = NP). We propose
to adapt the Self-Adaptive Multiple Constraints Routing Al-
gorithm (SAMCRA) to the multi-layer context in order to
compute a shortest feasible path under QoS constraints.
SAMCRA is an exact QoS routing algorithm proposed
by Van Mieghem and Kuipers [12]. It computes the shortest
Fig. 7. Comparison of the processing time of DAG-PDA, DAG-BFS and
pathunderseveral(additive)QoSconstraintsbutitignoresthe
BFSalgorithmsonTopologiesT1andT2.
feasibilityconstraintasdefinedinourpaper.SAMCRAhasan
exponentialworstcasecomplexity,butitexhibitsareasonable
an adaptation function. BFS algorithm is slow even for small processing time in practice.
valuesofp.Forp<0.3(resp.0.4)onTopologyT1(resp.T2),
1)ThemainconceptsofSAMCRA: TheideaofSAMCRA
DAG-BFS is faster than DAG-PDA. Beyond these values, the
is to maintain a path list from the source node S to all other
processing time of DAG-BFS explodes. For example, for p=
nodes until reaching the destination node D. It progressively
0.5,theprocessingtimeofDAG-BFSismorethan35minutes
removesthepathsthatdonotcomplywiththeQoSconstraints.
on Topology T1 and more than 53 minutes on Topology T2,
The main concepts of SAMCRA are:
while that of DAG-PDA is 3.8 seconds on T1 and 24 seconds
on T2. These results show that the DAG-PDA algorithm is • Non-linear path length: In SAMCRA, the path length
clearly faster when there is a significant number of adaptation isdefinedasanon-linearfunctionoftheQoSparame-
functions, but the exponential DAG-BFS algorithm is faster if ters of each link. It reduces the solution space to scan
there are few of them (for small values of p). but the algorithm can apply with any metric. Hence,
it is not a strict requirement.
VI. PATHCOMPUTATIONUNDERQOSCONSTRAINTS
• The k-shortest path algorithm: The k-shortest path
A. Multi-constrained feasible path algorithm maintains the list of the paths that are not
(yet) removed from the path list.
Let N be a multi-layer network. Each link E = (U,V)
is associated to a set of m additive QoS metrics q(E) = • Non-dominance: A multi-constrained path P dom-
qt(hqbe1(E(pE)a.)c,kT.eh.te-.ls,oeqsmsa,d(eEdtic)t.i)veinmeatdrdicitsiocnantobeittsheavdaeillaayb,lelogbaarnidthwmidothf i(cid:80)QnoaESte∈spPa(cid:48)raqanim(oEetht)eerr)(.i.eAp.a,tphiafthPPP(cid:48)isiisfbento∀tenir-,d(cid:80)tohmaEni∈nPaPtqe(cid:48)id(fEoifr)tehaec≤rhe
Let qmin be the bandwidth constraint and qmax = is no path which dominates it. The concept of non-
(qmax,qmbax...,qmax) be a vector of QoS constraints, the dominance induces a partial order over the paths. It
1 2 m
problem of computing a shortest feasible path under these avoids the exploration of several paths thus substan-
constraints is formalized as: tially reducing the average complexity of SAMCRA.
(cid:88)
min h(P)= h(U,f,V) The path length definition is not impacted by the multi-layer
contextandusingalinearpathlengthfunctionisnotforbidden.
(U,f,V)∈P
 P is a feasible path between S and D Thek-shortestpathalgorithmisnotimpactedeither.However,
 q (E) (3) tfheaesciboinlicteypctoonfstdroaminitnaanndcetomtuasktebientroedaecficnoeudnttopomsseiebtlethleooppast.h
s.t. min b ≥qmin
E∈P nb(E) b 2)Extension of the non-dominance definition: A multi-
 (cid:80)E∈P(qi(E)×nb(E))≤qimax, i=1...m lsataycekrpaattthheisdcehsatriancatteiorinzendodbey.iTtshnuosdienstbhuetaallgsoorbityhmitsppartohtolcisotl,
each path should be stored with its protocol stack at its final
B. Complexity of multi-constrained feasible path computation node. A multi-layer path can involve the same link several
times. Before checking if this path complies with some QoS
The problem of QoS multi-constrained path computation
parameters, the parameters of each link should be multiplied
(on a single layer) is well studied. It is well-known that the
by the number of times this link is involved in the path. The
decision version associated to this problem is NP-complete,
bandwidthconstraintisnotprunableinmulti-layercontext,the
even with 2 additive and/or multiplicative constraints [26].
new non-dominance definition should take it into account.
Van Mieghem and Kuipers [12] gave an exponential time
algorithm but showed that the instances that really require A path P dominates a path P(cid:48) if the four following
an exponential computation time are infrequent. The classical conditions are satisfied:
multi-constrained path problem is a particular case of Prob-
lem 3, corresponding to the case where there is only one • minE∈P nqbbP(E(E)) ≥minE∈P(cid:48) nqbPb((cid:48)E(E))
9
constraint and under additive QoS constraints. For the first
case,wewidelygeneralizedpolynomialalgorithmsinthestate
oftheartanddecreasedtheircomplexity.Throughsimulations,
we showed that they outperform previous approach in the
literature. For the second case, we obtained several time
complexity results and proposed efficient heuristics. Finally,
we designed the first algorithm to resolve the third case. In
futureworks,weplantodesignheuristicstodealwithadditive
QoS metrics, as the exact approach seems to be not scalable.
The problem of efficient generation of random topologies
being widely open, it would be interesting to analytically
study the phase transition phenomenon in order to generate
topologies having a suitable number of feasible paths.
Fig.8. ProcessingtimeofML-SAMCRATopologiesT1andT2.
REFERENCES
(cid:80) (cid:80)
• q (E)×nb (E)≤ q (E)×nb (E)
E∈P i P E∈P(cid:48) i P(cid:48)
∀i=1,...,m [1] L. Martini, E. Rosen, N. El-Aawar, and G. Heron, “RFC4448 - En-
capsulationMethodsforTransportofEthernetoverMPLSNetworks,”
• P and P(cid:48) have the same final node; 2008.
[2] F. Baker, X. Li, C. Bao, and K. Yin, “RFC6144 - Framework for
• P and P(cid:48) have the same protocol stack at this node. IPv4/IPv6Translation,”2011.
[3] S. Bryant and P. Pate, “RFC3985 - Pseudo Wire Emulation Edge-to-
Where nb (E) (resp. nb (E)) is the number of times the
P P(cid:48) Edge(PWE3)Architecture,”2005.
link E is involved in path P (resp. P(cid:48)). According to this
[4] S.Das,G.Parulkar,N.McKeown,P.Singh,D.Getachew,andL.Ong,
new definition of non-dominance, ML-SAMCRA explores all
“Packet and circuit network convergence with openflow,” in Optical
the possible paths until reaching the destination node with FiberCommunicationConference. OpticalSocietyofAmerica,2010.
satisfactoryQoSparameters.Alongtheexploration,itremoves [5] L. Liu, D. Zhang, T. Tsuritani, R. Vilalta, R. Casellas, L. Hong,
all paths that are dominated or not feasible. I. Morita, H. Guo, J. Wu, R. Mart´ınez et al., “Field trial of an
openflow-based unified control plane for multilayer multigranularity
opticalswitchingnetworks,”LightwaveTechnology,Journalof,vol.31,
D. Simulations
no.4,pp.506–514,2013.
We know study the efficiency of ML-SAMCRA through [6] S.Agarwal,M.S.Kodialam,andT.V.Lakshman,“Trafficengineering
simulations and check if it is as scalable in a multi-layer insoftwaredefinednetworks,”inProceedingsoftheIEEEINFOCOM
2013,Turin,Italy,April14-19,2013,2013,pp.2211–2219.
contextasSAMCRAinasinglelayercontext.Figure8shows
[7] S.Li,Y.Shao,S.Ma,N.Xue,S.Li,D.Hu,andZ.Zhu,“Flexibletraffic
the processing time of ML-SAMCRA on Topologies T1 and
engineering:Whenopenflowmeetsmulti-protocolip-forwarding,”IEEE
T2 according to the probability of existence of an adaptation
CommunicationsLetters,vol.18,no.10,pp.1699–1702,2014.
function(p).Theresultsshowthatforp>0.08(resp.0.10)on
[8] E. W. Dijkstra, “A note on two problems in connexion with graphs.”
Topology T1 (resp. T2) the processing time explodes (more NumerischeMathematik,vol.1,pp.269–271,1959.
than 1 minutes). Clearly, ML-SAMCRA does not scale above
[9] M.L.Lamali,H.Pouyllau,andD.Barth,“Pathcomputationinmulti-
these values. There are two reasons: layermulti-domainnetworks,”inNetworking(1),2012,pp.421–433.
[10] ——,“Pathcomputationinmulti-layermulti-domainnetworks:Alan-
1) The paths are less comparable in term of the new
guagetheoreticapproach,”ComputerCommunications,vol.36,no.5,
non-dominancedefinition:Theyshouldhavethesame pp.589–599,2013.
protocolstack.Astherearelessdominatedpaths,the [11] F.A.KuipersandF.Dijkstra,“Pathselectioninmulti-layernetworks,”
algorithm complexity increases; ComputerCommunications,2009.
2) Taking into account loops increases the number and [12] P. V. Mieghem and F. A. Kuipers, “Concepts of exact QoS routing
the length of the paths, which also increases the algorithms,”IEEE/ACMTrans.Netw.,vol.12,no.5,pp.851–864,2004.
algorithm complexity. [13] A. Farrel, J. Vasseur, and J. Ash, “RFC4655 - A Path Computation
Element(PCE)-BasedArchitecture,”2006.
So, path computation under QoS constraints in multi-layer [14] M. Bocci and S. Bryant, “RFC5659 - An Architecture for Multi-
networksismorecomplexthaninsinglelayernetworks.Thus, SegmentPseudowireEmulationEdge-to-Edge,”2009.
exact algorithms are suitable only for small instances. [15] F.Dijkstra,J.V.derHam,P.Grosso,andC.deLaat,“Apathfinding
implementation for multi-layer networks,” Future Generation Comp.
Syst.,vol.25,no.2,pp.142–146,2009.
VII. CONCLUSION
[16] I.Chlamtac,A.Farago´,andT.Zhang,“Lightpath(Wavelength)Rout-
Most of carrier-grade networks manage their different lay- ing in Large WDM Networks,” IEEE Journal on Selected Areas in
ers thanks to separate control planes. Designing a unified con- Communications,vol.14,no.5,pp.909–913,1996.
trol plane would allow the network resources to be optimized [17] H. Zhu, H. Zang, K. Zhu, and B. Mukherjee, “A novel generic graph
model for traffic grooming in heterogeneous WDM mesh networks,”
and the operational management costs to be reduced. One key
IEEE/ACMTrans.Netw.,vol.11,no.2,pp.285–299,2003.
problemtoaddressispathcomputationtakingintoaccountthe
[18] S. Gong and B. Jabbari, “Optimal and Efficient End-to-End Path
protocolheterogeneityandthemulti-layercontextdealingwith
ComputationinMulti-LayerNetworks,”inICC,2008,pp.5767–5771.
encapsulation,conversionanddecapsulationofprotocols.This
[19] F. Dijkstra, B. Andree, K. Koymans, J. van der Ham, P. Grosso, and
papertacklesthisissuebypartitioningitintothreecases:Path
C. de Laat, “A multi-layer network model based on ITU-T G.805,”
computation without bandwidth constraint, under bandwidth Comput.Netw.,2008.
10
[20] F. Iqbal, J. van der Ham, and F. Kuipers, “Technology-aware multi- • If (a → ab) ∈ EN(U) then a ∈ In(U) and b ∈
domain multi-layer routing,” Computer Communications, vol. 62, pp. Out(U)
85–96,2015.
[21] J.E.Hopcroft,R.Motwani,andJ.D.Ullman,“Introductiontoautomata • If (a→ab) ∈ DE(U) then b ∈ In(U) and a ∈
theory,languages,andcomputation.” Addison-Wesley,2006. Out(U)
[22] D.E.Knuth,“AGeneralizationofDijkstra’sAlgorithm,”Inf.Process.
Lett.,vol.6,no.1,pp.1–5,1977. Obviously, several paths can have the same trace. The set
[23] M.L.FredmanandR.E.Tarjan,“FibonacciHeapsandTheirUsesin oftracesofthefeasiblepathsinanetworkN isacontext-free
Improved Network Optimization Algorithms,” J. ACM, vol. 34, no. 3, language but it is not regular as the encapsulations and decap-
pp.596–615,Jul.1987.
sulationsshouldbebalanced.Infact,itisawell-parenthesized
[24] R. Summerhill, “The new internet2 network,” in 6th GLIF Meet- language, and thus requires a stack to be recognized and
ing, 2006, available at http://www.internet2.edu/products-services/
computed. PDAs are the classical tools to recognize context-
advanced-networking.
free languages. Using weighted PDAs allows associating a
[25] R. Mahajan, N. Spring, D. Wetherall, and T. Anderson, “Inferring
link weights using end-to-end measurements,” in Proceedings of the weight to each link and adaptation function in order to model
2nd ACM SIGCOMM Workshop on Internet measurment. ACM, any additive metric.
2002, pp. 231–236. [Online]. Available: http://research.cs.washington.
edu/networking/rocketfuel/
B. Definition of WPDA
[26] Z.WangandJ.Crowcroft,“Quality-of-ServiceRoutingforSupporting
MultimediaApplications,”IEEEJournalonSelectedAreasinCommu-
A weighted PDA (WPDA) is a 8-tuple PDA =
nications,vol.14,no.7,pp.1228–1234,1996.
(S,Σ,Γ,δ,Q ,Z ,S ,ω)whereS isthesetofstates,Σisthe
[27] I. Petre and A. Salomaa, “Algebraic Systems and Pushdown Au- 0 0 F
tomata,”inHandbookofWeightedAutomata,M.Droste,W.Kuich,and input alphabet, Γ is the stack symbol set (i.e., stack alphabet)
H. Vogler, Eds. Springer Publishing Company, Incorporated., 2009, notnecessarilydifferentfromΣ,δ isthesetoftransitions,Q0
ch.7,pp.257–290. is the initial state, Z is the initial stack symbol, S is the set
0 F
of final (accepting) states and ω is a weight function over the
APPENDIXA set of transitions (i.e., ω : δ →(cid:60)+).
POLYNOMIALALGORITHMSFORPATHCOMPUTATIONIN
A transition t ∈ δ is denoted by t = (Q ,(cid:104)x,α,β(cid:105),Q ),
MULTI-LAYERNETWORKS i j
where Q is the state of PDA before the transition, Q is the
i j
The sequence of protocols involved in a feasible multi- stateafterthetransition,x∈Σ∪{(cid:15)}isaninputsymbol,α∈Γ
layer path is a context-free language. Based on this fact, is the symbol which is popped from the top of the stack, and
Lamali et al. [10] used automata and language theory tools β ∈Γ∗ is the symbol sequence which is pushed on the top of
to compute the shortest feasible path in hops or in adaptation the stack.
functions. We improve their algorithm in order to compute
Remark. WPDAs are more often formalized as 6-tuples
the shortest path according to any additive metric. We also
PDA = (S,Γ,M,q ,Z ,S ) where M, called the Push-
substantially reduce its complexity. 0 0 F
Down transition matrix, is a matrix over a semiring of formal
power series. The input alphabet Σ, the transitions set δ
A. Theoretical language aspects of multi-layer paths and the weight function ω are expressed by a single entity
M ∈ ((R(cid:104)(cid:104)Σ∗(cid:105)(cid:105))S×S)Γ∗×Γ∗, where R(cid:104)(cid:104)Σ∗(cid:105)(cid:105) denotes the
Considering a path P = Sf U f U f ...U f D, let
0 1 1 2 2 n n collection of all power series from Σ∗ into a semiring R.
H = f ...f denotes the sequence of adaptation functions
P 1 n For simplification purposes, we opted for defining a WPDA
along P. Let define as an alphabet the set A = {a | a ∈ A}
as a classical PDA with a weight function over the transition
and the set A={a|a∈A}.
set. For the theoretical foundations of WPDAs, the interested
T =x ...x is the sequence of protocols used along reader can refer to [27].
P 1 n+1
path P. It is called the trace of P. For each x :
i
C. From the graph to the WPDA
• x = a and x = b, b or b means that U converts
i i+1 i
protocol a into b ( a,b,b,b∈A∪A∪A) Algorithm 1 converts a multi-layer network N with a
specified pair of nodes (S,D) into a WPDA PDA =
• x = a and x = b, b or b means that U
i i+1 i (S,Σ,Γ,δ,Q ,Z ,S =Q ,ω).
encapsulates protocol a in b 0 0 F F
Computing a feasible path requires to know the current
• x = a and x = b, b or b means that U
i i+1 i protocol and the last encapsulated one (in order to know if a
decapsulates protocol b from a.
decapsulation can be performed). Thus Algorithm 1 creates a
Here, some additional definitions are needed. The set stateUx foreachnodeU andeachprotocolx∈In(U).Being
of protocol conversions available on node U is denoted by in state Ux indicates that the current protocol is x. The last
CO(U). The set of encapsulations available on node U is encapsulated protocol is the one on the top of the stack.
denoted by EN(U) and the set of decapsulations available
The conversion functions (x → y) between node U
on node U is denoted by CO(U).
and node V are turned into transitions (U ,(cid:104)x,α,α(cid:105),V )
x y
In(U) (resp. Out(U)) is the set of protocols that node U in the WPDA. The encapsulation functions (x → xy) are
can receive (resp. send). More formally: converted into pushes of x on the stack (Ux,(cid:104)x,α,xα(cid:105),Vy)
and the decapsulation functions into pops of x from the stack
• If(a→b)∈CO(U)thena∈In(U)andb∈Out(U) (U ,(cid:104)y,x,∅(cid:105),V ).
y x