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Analysis of the contact graph routing algorithm: Bounding interplanetary paths

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Analysis of the contact graph routing algorithm:Bounding interplanetary paths
Edward Birrane
a,
n
, Scott Burleigh
b
, Niels Kasch
c
a
Johns Hopkins University, Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723, USA
b
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
c
University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA
a r t i c l e i n f o
Article history:
Received 2 February 2011Received in revised form30 January 2012Accepted 3 February 2012Available online 24 February 2012
Keywords:
Delay-tolerant networksContact graph routingSolar system internet
a b s t r a c t
Interplanetary communication networks comprise orbiters, deep-space relays, andstations on planetary surfaces. These networks must overcome node mobility, con-strained resources, and signiﬁcant propagation delays. Opportunities for wirelesscontact rely on calculating transmit and receive opportunities, but the Euclidean-distance diameter of these networks (measured in light-seconds and light-minutes)precludes node discovery and contact negotiation. Propagation delay may be largerthan the line-of-sight contact between nodes. For example, Mars and Earth orbiters maybe separated by up to 20.8 min of signal propagation time. Such spacecraft may
never
share line-of-sight, but may uni-directionally communicate if one orbiter knows theother’s future position. The Contact Graph Routing (CGR) approach is a family of algorithms presented to solve the messaging problem of interplanetary communica-tions. These algorithms exploit networks where nodes exhibit
deterministic
mobility.For CGR, mobility and bandwidth information is pre-conﬁgured throughout the net-work allowing nodes to construct transmit opportunities. Once constructed, routingalgorithms operate on this contact graph to build an efﬁcient path through the network.The interpretation of the contact graph, and the construction of a bounded approximatepath, is critically important for adoption in operational systems. Brute force approaches,while effective in small networks, are computationally expensive and will not scale.Methods of inferring cycles or other librations within the graph are difﬁcult to detectand will guide the practical implementation of any routing algorithm. This paperpresents a mathematical analysis of a multi-destination contact graph algorithm(MD-CGR), demonstrates that it is NP-complete, and proposes realistic constraints thatmake the problem solvable in polynomial time, as is the case with the srcinallyproposed CGR algorithm. An analysis of path construction to complement hop-by-hopforwarding is presented as the CGR-EB algorithm. Future work is proposed to handle thepresence of dynamic changes to the network, as produced by congestion, link disrup-tion, and errors in the contact graph. We conclude that pre-computation, and thus CGR style algorithms, is the only efﬁcient method of routing in a multi-node, multi-pathinterplanetary network and that algorithmic analysis is the key to its implementation inoperational systems.
&
2012 Elsevier Ltd. All rights reserved.
1. Overview
Interplanetary networks operate in the presence of transmission delays, sparse connectivity, and unavoidable
Contents lists available at SciVerse ScienceDirectjournal homepage: www.elsevier.com/locate/actaastro
Acta Astronautica
0094-5765/$-see front matter
&
2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.actaastro.2012.02.004
n
Corresponding author.
E-mail addresses:
Edward.Birrane@jhuapl.edu (E. Birrane),scott.c.burleigh@jpl.nasa.gov (S. Burleigh),nkasch1@umbc.edu (N. Kasch).Acta Astronautica 75 (2012) 108–119
node mobility, all of which present signiﬁcant and uniquerouting challenges. These networks comprise orbiters,deep-space relays, and stations on planetary surfacescommunicating via radio frequencies. As radio wavespropagate at approximately the speed of light, existingdistance measurements based on the speed of light alsoserve as measures of propagation delays within the net-work. For example, Earth and Mars vary between 4.5 and20.8 light-minutes apart at any given time, which meansthat nodes on or near these planetary bodies requireapproximately 4.5 to 20.8 min for one-way signal propa-gation while communicating.The orbital dynamics in these networks present com-plex geometries that must be understood when determin-ing opportunities for wireless contact. Spacecraft orbitcelestial bodies in ways not synchronous with the rota-tion of these bodies about their axes, or the orbit of thesebodies about the sun. Envisioned interplanetary networksare constrained to our solar system, which removes theneed to correct for the motion of the sun through thegalaxy. Even so, this complex mobility model results inrelatively few opportunities for line-of-sight contact. Thesigniﬁcant propagation delays encountered at interplane-tary distances further reduce contact opportunities. It ispossible, for example, that spacecraft orbiting differentcelestial bodies experience periods of line-of-sight shorterthan the propagation delay between them. Such a systemis illustrated in Fig. 1.In this ﬁgure, if spacecraft A was to transmit to space-craft B based on existing line of sight, by the time thesignal traversed the distance between them, spacecraft Awould be in a different place. Therefore, both musttransmit based on where the receiving spacecraft will beat some point in the future. This ﬁgure captures anothersigniﬁcant constraint on space wireless communication:directional transmission. Deep-space spacecraft are typi-cally low-powered devices whose transmitters can dom-inate instantaneous component power requirements.Persistent, omni-directional transmission is not an optionin most power budgets, which means that the timing of when transmitters are powered, as well as how dishes onthe platform are pointed, presents an additional logisticchallenge. For example, an orbiter performing a sciencemission may or may not be able to slew so that aparticular transmitter or receiver dish is appropriatelypositioned to partake in a relay circuit.The combination of propagation delays, node mobility,transceiver power cycling, and directional transmissionprovide signiﬁcant obstacles to the automatic discoveryand synchronization of the network topology. To date,there is no practical automation mechanism that per-forms a global synchronization function. Regardless, pack-etized, multi-path, addressed data networks must providea reliable mechanism for constructing routes betweendata sources and sinks. Deterministic mechanisms forroute computation rely on the ability to construct, at eachnode, a reasonable approximation of the network topol-ogy. Any useful subset of the network topology mustcontain the source and sink of the message and sufﬁcientintermediate nodes to represent a path.The use of
pre-conﬁgured
topology informationexploits the observation that mobility in space networksis both regular and deterministic. While the motion of nodes and celestial bodies remains complex, this motioncan be modeled far into the future. Similarly, spacecraftpower load cycles, trajectory corrections, and other man-euvers are planned in advanced. Unlike ad-hoc mobilenetworks, the rate of node addition to the networks is of sufﬁciently low frequency as to allow for conﬁgurationand convergence on new topologies in operationallyuseful timescales.
2. Motivation
Packetized data requires mechanisms to discoverroutes through the network. These mechanisms mustoperate over distances whose propagation delays pre-clude bandwidth negotiation and topological conversionbased on discovery. Several routing algorithms used forDelay-Tolerant Networks (DTNs) have been proposed inthe literature [1–4]. However, these algorithms requireinformation exchange that likely does not scale in net-work size or distance. The Contact Graph Routing (CGR)algorithm, proposed by Burleigh et al. [5], presents aheuristic-based approach to route computation whichexploits regularity in space asset mobility to calculate aglobal network topology and its evolution over time.The Bundle Protocol [6] is a packetized, overlay, store-and-forward protocol proposed for the exchange of datain interplanetary networks. The protocol data unit (PDU)for this protocol is the ‘‘Bundle’’, which contains a header,a payload, and zero or more extensions that govern theprocessing of the bundle at various nodes.This paper is motivated by the need for a generalizedformulation of the routing problem within high delay net-works utilizing the BP, and to present that formulation as amechanism for evaluating new routing approaches. Such aformulation codiﬁes the need to optimize transient connec-tivity without reliance on state measurement and synchro-nization. Complexity analysis of this formulation shows thatthere is no practical optimal solution and implementingnetworks must build approximations based on networkingassumptions. The paper is further motivated by the desire to
Fig. 1.
Bidirectional communications between multi-body orbitingsystems across great distances is difﬁcult to optimize.
E. Birrane et al. / Acta Astronautica 75 (2012) 108–119
109
consider the proposed CGR algorithm in the context of thisformulation, whether this algorithm supports identiﬁedapproximation strategies, and how CGR may be applied tocurrent and envisioned space networks.
3. The generalized routing problem
3.1. Overview
The CGR algorithm presumes the ability to pre-com-pute network topology. Additionally, CGR assumes topo-logical changes occur less frequently than the time tosynchronize conﬁgurations. In cases of planned topologi-cal changes, conﬁgurations may certainly be propagatedin advance of application. In the case of unplannedtopological changes this approach relies on a networkmanagement function to update the conﬁgurations of affected nodes. This work assumes that unplanned topo-logical changes are infrequent and that node failures areindependent. We consider contact-graph routing to be a
family
of algorithms that operate on a conﬁgured graph of contacts within a system that, otherwise, cannot infer thetopology based on node discovery and state aggregation.Routing algorithms compute paths through the net-work that result in message delivery within requiredtimeframes. The actions applied to constructed paths varybased on speciﬁc algorithms. Some algorithms strictlyadhere to a path unless it collapses in transit. Others use asubset of the path before re-computing a new path toreduce reliance on the correctness of the perceived net-work topology. All approaches require the construction of comparable paths through the network, with path com-parison based on minimizing a cost metric to maximize anetwork metric. When considering the complexity analy-sis of this process, the hardest problem is determining anoptimal path through the network. Therefore, routinganalysis focuses on the path computation algorithm.The format of data sent through the network mayaffect the routing complexity. BP bundles, like mostpacketized data protocols, provide an information sourceand sink. Bundles also support a self-extension mechan-ism: the extension block. Within the space domain thereare several anticipated extensions to the bundle protocolthat place requirements on the routing subsystem.Speciﬁcally, the Bundle Security Protocol (BSP) extends BPto include authentication, integrity and conﬁdentialityciphersuites [7]. The integrity and conﬁdentiality mechan-isms within the BSP propose
security sources
and
securitydestinations
separate from the
bundle source
and
bundledestination
. These nodes represent required waypoints incalculating paths and determining optimal proximate nodes.Considering the network proposed in Fig. 2, even if a moreefﬁcient path exists between a particular bundle source anddestination, deﬁned waypoints may need to be visited toensurethatencryptionappliedseparatefromabundlesourceis decrypted before reaching the bundle destination. The useof multiple security destinations is a practical considerationwhen traversing networks comprising different administra-tive domains.Bundle security extensions are not the only example of data mandating subsets of the routing path. BP addressesbundles based upon their ‘‘End Point Identiﬁer’’ (EID).EIDs are unique within any BP overlay network. Themapping between EIDs and nodes in the network ismany-to-many. A node may be associated with severalEIDs, and one EID may be associated with multiple nodes.An EID used by multiple nodes approximates the func-tionality of a multi-cast address. The decision to engineermulti-cast EIDs into a particular space network is animportant engineering concern. Key managementamongst a group of nodes in a security domain presentsan example of a desirable multi-cast capability in a high-delay network. Consider the distribution of group-wisesymmetric keys encrypted by a larger group asymmetrickey. The destination for the encrypted symmetric keycould be an EID shared by all nodes in the group.We deﬁne a multi-destination contact graph routingmechanism (MD-CGR) in a directed, weighted graph
G
¼
(
V
,
E
,
T
,
c
), where
V
: The set of vertices (nodes) in the network.
E
: The contacts in the network representing the ability tosend data between nodes over time. The notation
e
(
i
,
j
)represents the edge between nodes
i
,
j
A
V
.
T
: A subset of the vertices in the network that must beincluded in an optimal path. This includes, at a mini-mum, the bundle source and destination.
c
: A cost function mapping
E
)
R
, associating a realnumber cost to traversing a particular contact in thenetwork.
c
(
i
,
j
) represents the cost of a particular edgebetween nodes
i
,
j
A
V
.MD-CGR is the optimization problem of ﬁnding amulti-destination route,
R
, with least cost and coveringall waypoints (
T
C
R
). A route is an ordered set of verticesand the cost of the route is the sum of the cost of theedges in the route. The decision problem is whether such
Fig. 2.
Bundle extensions create waypoints in the network.
E. Birrane et al. / Acta Astronautica 75 (2012) 108–119
110
a route,
R
, exists with cost of
k
. Formally, the decisionproblem is deﬁned asMD-CGR
¼
{
/
G
,
k
S
: graph
G
¼
(
V
,
E
,
T
,
c
) has a forwardingstrategy that will reach destination nodes
V
0
with totalcost
k
}.
3.2. NP-completeness
We claim that MD-CGR is NP-complete by construct-ing a polynomial-time reduction from the Graphic SteinerTree Problem (GSTP) [8], a known NP-complete problem.GSTP accepts a graph,
G
¼
(
V
,
E
,
T
,
c
) with
T
C
V
and costfunction
c
:
E
)
R
on the edges of
G
where
c
(
i
,
j
) is the costof traversing the edge between vertices
i
and
j
with
i
,
j
A
V
.The optimizing algorithm produces a Steiner tree throughall vertices
T
with minimum cost. The cost of a Steinertree is the sum of the cost of its edges.Given a graph
G
¼
(
V
,
E
,
V
0
,
c
), a traversal of edges con-necting
V
0
can be performed in
O
(
EV
) while accumulatingthe cost of traversing the edges to compare the total costof the route against
k
. Therefore, veriﬁcation of thedecision problem can be accomplished in polynomial timeand thus belongs to NP. We demonstrate GSTP
r
p
MD-CGR, thus showing that MD-CGR is NP-Hard and, com-bined with membership in NP, is NP-complete. Thereduction is trivial. Both GSTP and MD-CGR take the samegraph (Fig. 3a) as input. The optimal Steiner tree (Fig. 3b)
produced by GSTP is the multi-destination route pro-duced by MD-CGR.We prove this by contradiction. Let us suppose thatMD-CGR constructs a multi-terminal route through
G
with total cost
j
o
k
, containing at least every vertex in
T
and with no edge weight
o
¼
0. This implies that theroute must use either fewer edges, or less costly edgesthan those included in the Steiner tree through vertices
T
in
G
. However, connecting all vertices of
T
with fewer orless costly edges implies that a more efﬁcient Steiner Treeexists, which contradicts the optimal Steiner tree pro-duced by GSTP.It should be noted that while this is a relatively simplereduction, it demonstrates the inherent nature of thecomplexity of the multiple-destination problem for CGR.Were we to further complicate the problem by attemptingto provide an optimal,
ordered
traversal of
G
through
T
thecomplexity would increase, although likely stay in theclass NP-complete.
3.3. Approximations
Like most practical instantiations of Steiner Tree pro-blems, several polynomial-time approximations exist,with the most efﬁcient being an
1.5 approximation[9,14]. However, the existing MD-CGR formulation pro-
duces optimal results when the following simplifyingassumptions are made: single destinations, no fragmenta-tion, and properly selected cost functions.
3.4. Single destinations
A single bundle efﬁciently forwarded to multipledestinations is trivially converted to multiple bundleseach with a single destination. The resulting networkutilization cost is signiﬁcantly higher, but likely wellwithin margin for space networks in the foreseeablefuture.Security, while an important part of any operationalinternetwork, may initially be implemented only at theendpoints of data transmission, where messages areencrypted at the bundle origin and decrypted at thebundle destination. In cases where security sources andsecurity destinations are necessary at intermediate nodes,a concatenation approach may be taken, where thebundle is simply routed to the given security destinationas if it was the bundle destination and then routed to thebundle destination afterwards, as shown in Fig. 4. Thisgreedy strategy is sub-optimal for security destinationsnot already on the optimal route but likely sufﬁcient forsmall numbers of security destinations.This ﬁgure presents two types of paths through anetwork: (1) trusted sections of the network for whichpaths must be computed, and (2) untrusted sections of the network for which paths must be computed. Trustednetwork sections impose no waypoints beyond the
Fig. 3.
A least-cost Steiner Tree represents a least-cost multi-cast route.
Fig. 4.
Single-destination routing constructs an end-to-end route as aseries of shorter routes.
E. Birrane et al. / Acta Astronautica 75 (2012) 108–119
111
message destination. Untrusted network segments musthave deﬁned entrance and exit nodes that are the securitysource and destinations associated with some integrityand/or conﬁdentiality ciphersuites.Engineering space networks are likely to imposesecurity waypoints only at administrative boundaries,such as between assets controlled by different spaceagencies. These boundaries are anticipated to be few innumber and, therefore, not likely to overburden the net-work if sub-optimal routes are computed.
3.5. No fragmentation
Assuming that sufﬁcient contacts exist to transmitbundles without fragmentation is a simplifying assump-tion. Currently, when a bundle must be fragmented, eachbundle fragment is individually routed by the routingalgorithm. It has yet to be shown whether this approachperturbs the optimality of message transmission givencontact capacity, especially when the fragmentation sizemay be driven by differing capacities from differingcontacts in the contact graph.
3.6. Cost functions
The cost of traversing a path is assumed to be con-sistent. All things being equal, if a particular path wasdeemed ‘‘not desirable’’ previously it should be deemed‘‘not desirable’’ for all future path evaluations for thesame bundle along the same route. This is especiallyimportant in cases where node loss causes a restart of the routing process at an intermediate node. Cost func-tions that present a consistent cost for bundle traversal(such as elapsed time) exhibit this property whereas costfunctions whose cost can change over time (distance fromdestination, % capacity utilized) can result in loops in thecalculated path.As previously mentioned, there is no guarantee that anend-to-end path will ever exist for a random mobilenetwork. It is assumed safe to posit that an engineeredspace network will exhibit sufﬁcient determinism andcoverage to provide paths. It is further assumed that acost function will take as input any bundle characteristicthat would preclude ﬁnding a reasonable path. For exam-ple, engineered networks are engineered with particularconcepts for operational data ﬂow. Data ﬂows that are notappropriate given the network design may or may not beaccommodated by the network. Bundles with exceedinglyshort expiration times, in this example, may not betransmitted through the network even if an end-to-endpath exists. Any cost function used in path computationshould understand this
before
the bundle is queued forany transmission so as to reduce the unnecessary use of bandwidth for messages that cannot be delivered in time.
4. The CGR algorithm
4.1. Overview
CGR exploits prior information regarding the change of the network topology over time in accordance with theassumptions put forward in the prior section. Thisapproach forms the basis for the encompassing MD-CGR routing generalization. This section presents the CGR algorithm itself, published as an experimental Internetdraft through the DTNRG [10]. CGR has been implemen-ted within the open-source ‘‘Interplanetary Overlay Net-work (ION)’’ source code distribution [11] available viaOpenChannel [12] and ﬂight-tested using the DeepImpact Network Experiments (DINET) [13].CGR accepts a series of node contacts and associateddistance ranges. From this input it constructs an overall
contact plan
that identiﬁes the perceived network topol-ogy. This topology is consolidated into a weighted, direc-ted
contact graph
over which message paths may beevaluated. CGR analyzes this contact graph and producesa set of next-hop nodes that would be part of anyplausible route through the network. When a message issent over the next hop, CGR is run again at the down-stream node. The processing performed by CGR is illu-strated in Fig. 5.Inputs to the contact plan comprise information relat-ing to when data may be exchanged between nodes(contacts) and the distance between those nodes (ranges).Calculating data exchange opportunities is a non-trivialendeavor in resource-constrained RF systems. Pointing,error rates, transmission power, frequency matching, andadministrative policy must be considered while determin-ing when platforms may attempt to exchange data. Sincepropagation time eclipses contact duration, the timesassociated with contacts may be very different based ontransmission or reception.Since precise contact calculation is network and plat-form speciﬁc, the CGR algorithm assumes that this infor-mation can be condensed into a single set of uniform
Fig. 5.
The CGR algorithm has three processing steps.
E. Birrane et al. / Acta Astronautica 75 (2012) 108–119
112

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