# Inherently Safe Backup Routing with BGP

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4 (6 5 2), (6 5 2) (5 2), (5 2) 6 5 ( 2) ( 2), ( 5 2) ( 2) ( 2), ( 5 2) 0 2 ( 6 5 2), ( 6 5 2) (0 2), (0 2), (0 5 2) (2) 2 2 Fig.. AS graph with - and peer-peer relationships, with permissible paths to node 2 We also permit paths O AE\$a<` O where OCA is either an empty path ; or an uphill path, <` is a peer-peer edge, and O is either an empty path ; or a downhill path. For example, * (\W is a permissible path. However, we do not permit paths that have steps since this would violate the restriction that peers only transit traffic between their respective s. Formally, a peer-peer edge <` is a step in path O A =B<` O if either the last edge of O A is a peer-peer or - edge, or the first edge in O is a - edge. For example, the path,(9vfwfy is not permissible because it has two steps: V` W and ^. Likewise, the path 5X is not permissible because it has two steps: and " X. Fourth, we assume that paths are preferable to peer and paths. For example, node ( would rank the path ([WU^ higher than peer path ('*U^. This is consistent with the commercial relationships between ASes. An AS has a financial incentive to limit the amount of traffic that traverses peer and links, and may receive payment based on the amount of traffic that travels via links. We do not impose any restriction on the ranking of different paths, or of different peer and paths. In summary, we consider a stable paths problem with the following four properties: (acyclic - digraph) The directed graph induced by the - relationships is acyclic. (no-valley) If =< > C< A < D _6 c and A 6 ) +L( "*,+=< for some f F, then A *E(<g!g"*,+-\$<,K for all!. (no-step) If =< > < A < D 69c and A 6`"*,+=< for some F, then A 6 *E(<g!!"*,+=< for %Y F and < A 6 ) +L( "*,+=< K for!. (prefer-) If 0 6 ) +L(B"*E+\$. and <:6%`"*,+-\$. *E(<g!!"*,+=, then =905 O A I and O. =Y< ]O - for all paths O A These assumptions result in the following theorem from [0], expressed in S terminology: Theorem II.: Any S with an acyclic - digraph and the no-valley, no-step, and prefer- properties is inherently safe. The theorem is a special case of Theorem IV., presented in the next section. Fig.. Node providing backup service for node 2 III. INHERENTLY SAFE ROUTING WITH STES eer-peer backup relationships violate the restrictions introduced in Section II-B. After introducing the notion of a backup path, we explain why backup paths should always have a lower ranking than other paths. Then, we explore policies for ranking between backup paths and show that preferring backup paths over peer and backup paths can result in an unsafe system. Finally, we propose a policy that prefers paths with fewer steps over paths with more steps, with preference for paths among paths with the same number of steps. A. Backup aths With Steps Normally, the selection and export of routes is constrained by the - and peer-peer relationships. Backup relationships introduce additional paths that provide connectivity in the event of a failure and change how nodes select among paths to the same destination. Consider the example in Figure after removing the edge (^ W. The new solution to the stable paths problem may select different paths for some node pairs. For example, deleting edge (^ W^ would remove the path,(uwx, forcing node 5 to use the other path,(hv^ to reach node 2. However, node ( would have to use a peer path,( *5 to reach node 0, since the path ( W5^ is no longer available. If ( W and V X both fail, then nodes, 5, and 6 do not have any permissible path to node, even though the graph is still connected. Ensuring that these nodes have a path to node requires relaxing the restrictions on the set of permitted paths. For example, relaxing the no-step restriction would make (5V[WUX into a permissible paths with a step involving edge V W^. Enlarging the set of permissible paths has important implications on whether the system remains inherently safe. On the one extreme, every path could be permissible. However, paths with valleys violate the basic notion of hierarchical relationships between nodes. Instead, we define a slightly weaker notion of reachability where the set of permitted paths can include paths that have steps. Figure shows an example where node provides backup service for node 2. This backup relationship allows paths like ]?.W^]O and? V O to ensure that node 2 can reach destinations in the rest of the Internet. In addition, path like \$V_ X]O and \$W_ X]O ensure that the rest of the Internet can reach node 2 and its downstream s. These paths fall into the three categories shown in Figure 5. In Figure 5(a), node v has a backup relationship with node u to allow backup paths =0a <` O, corresponding to the path \$W X O

5 w v u v w u w u v (a) rovider, peer peer (b) eer peer, (c) eer peer, peer peer Fig. 5. Allowing backup paths with a step 2 in Figure. 0 Fig. 6. Backup paths should be global In Figure 5(b), node w has a backup relationship with node u to allow backup paths =0J< ]O through s node <, corresponding to the path ]?.W^]O. In Figure 5(c), node w has a backup relationship with node u to include backup paths \$0J <` O, corresponding to both paths?.v O and =V X]O. B. Lower Ranking for aths With Steps A backup path should have global significance. That is, if a path O is a backup path, then \$ <` O is also a backup path. A backup path should not be used unless all primary paths are unavailable. More formally, if path O A has no steps and path O has one or more steps, then \$O A I O. Having a lower ranking for backup paths is important for the safety of the system. In the example in Figure 6, nodes 2,, and each have a relationship with node and peer-peer relationships with each other. Node also has a peer-peer relationship with 0, which is a of 2. Limiting the consideration to primary, valley-free paths, nodes 2,, and would each select a path that uses the edge from 2 to 0, and node would select a direct path. Now, suppose that the set of permissible paths is extended to include paths with steps, such as Z and W5 ^, following the approach in Figure 5(a). Nodes 2,, and should still prefer the primary path that uses the edge from 2 to 0. If the edge between 0 and 2 fails, nodes 2,, and select paths to 0 that include node. Each of these paths has a step. For example, node 2 should not prefer the path [W/ ^ over path Z^ just because the subpath ][W9, does not involve a step. Otherwise, the example in Figure 6 could devolve to the BAD GADGET scenario in Figure Fig. 7. Unsafe system when backup paths are preferred over peer backup paths C. Ranking Among aths With Steps In addition to preferring primary paths over backup paths, each node needs to rank amongst the backup paths. The simplest policy ranks the backup paths based on the path length, favoring shorter backup paths over longer backup paths. This approach has the desirable property of minimizing the length of backup paths. In addition, always selecting the shortest backup path ensures that the system is inherently safe. Informally, ranking primary paths over backup paths reduces the problem to two subproblems related to primary paths and backup paths, respectively. The prefer- requirement in Section II-B addresses the subproblem concerning primary paths, and the shortest-path requirement addresses the subproblem concerning backup paths. However, the shortest-path policy is very restrictive, since it does not incorporate information about the commercial relationships between nodes or the number of steps in the backup paths. The shortest-path policy would prefer a path with two steps over a longer path with one step. Extending the prefer- policy to backup paths is an appealing first approach to allowing a more liberal ranking of backup paths. In this scheme, a node would assign a higher rank to a backup path through a over a backup path through a peer or. However, this policy can result in an unsafe system. Consider the example in Figure 7, where 5

6 Fig. 8. Unsafe system when backup paths are preferred over peer backup paths each node prefers backup paths over peer and backup paths. Also, suppose that each node ranks among paths (or among peer and paths) based on the length of the path. Nodes, 2,, and do not have any stepfree paths to node 0. The prefer- property would result in node preferring a path via node and node preferring a path via node 2. Node prefers the shorter path via node over the path via node 2. In addition, node 2 prefers the shorter path via node over the peer path via node 8. This system is unsafe. Similar examples can be shown for preferring peers over s, preferring s over peers, and for preferring s over peers. Figure 8 shows a an example of an unsafe system where paths are preferred over peer paths at nodes, 2 and. Ranking backup paths based on path length would have avoided these problems, at the expense of imposing significant restrictions on backup policies. The shortest-path policy would force a node to prefer a path with a large number of steps with a small number of steps, whenever the path with more steps has a shorter overall length. Instead, we consider another approach that ranks paths based on the number of steps. Among paths with the same number of steps, paths are ranked higher than peer and paths, consistent with the approach in Section II-B. This policy is consistent with the commercial relationships between nodes and also ensures that the system is inherently safe. Informally, ranking paths with fewer steps higher than paths with more steps reduces the problem to a collection of subproblems, each corresponding to paths with a fixed number of steps. The safety of each of these subproblems is assured by the prefer- requirement, as shown in the next section. IV. GENERALIZED BACKU ROUTING This section proposes a general technique for dividing the permissible paths into avoidance levels. aths with a lower avoidance level are preferred over paths with a higher avoidance level; within an avoidance level, paths are preferred over peer and paths. In addition to capturing the peer-peer backup relationship (i.e., steps), this model incorporates multihomed backup relationships and load balancing while ensuring that the system remains inherently safe. The idea of counting the number of steps in a path suggests a general routing model based on avoidance levels, where each node assigns a higher rank to paths with a lower avoidance level. The assignment of an avoidance level to each path should preserve the maximum flexibility for each node to apply local policies for backup routing and load balancing. The notion of an avoidance level is more general than counting steps in a path in two ways. First, the avoidance level does not have to increase by unit increments. For example, a path with two steps could have a higher avoidance level than another path with three steps. Second, the avoidance level can increase for any edge in the path. For example, a - edge can increase the avoidance level of a path. Conceptually, a backup edge with respect to a path is any edge that increases the avoidance level. The model does not prescribe how much to increase the avoidance level for each backup edge; in practice, this may depend on local policies. However, some basic restrictions on avoidance levels are necessary to ensure that the global system is inherently safe. An edge =0f that contributes to a step is a must be a backup edge; that is, the path =0J O must have a higher avoidance level than the subpath O. Any other edge may also be a backup edge with respect to a path. We formalize these notions using a non negative function \$OU, called an avoidance classifier, that obeys the following conditions: (a) If O and R are paths permitted at node and \$OU ]RU, then ]RU \$OU. (b) If O, R, and R5O are permitted paths, then \$OU R5OU. An avoidance level is any value in the range of. Condition (a) states that a path O with a lower avoidance level is preferred over a path R with a higher avoidance level. Condition (b) states that the avoidance level cannot decrease as a path traverses additional edges. The weight of an edge with respect to a path O is equal to =0a.]OU \$OU. An edge "0S is a backup edge with respect to path O if the weight of edge 0S. with respect to O is non-zero. Note that this generalizes the notion of edge weights in two ways. First, a given edge can have different weights with respect to different paths. Second, an edge may have a weight of zero with respect to certain paths. An edge 0S. is a manda- if it matches any tory backup edge with respect to path =< ]O of the three conditions outlined in Figure 5. In particular, this means that (a) "0S is a - edge and <` is a peer-peer edge, or (b) "0S is a peer-peer edge and " < is a - edge, or (c) 0S. and " < are peer-peer edges. An avoidance classifier is step aware if for any O permitted at < and =0a <` O permitted at 0 we have =< ]OU% =0a <` OU whenever "0S is a mandatory backup edge with respect to path \$ <` O. If is an avoidance classifier for S, then ranking function is said to be prefer- with respect to if within a set of paths of the same avoidance level s paths are preferred. More formally, if for every O and R permitted at node with \$OU h, if O is a path to one of s s and R is not, then RU OU. Based on these definitions, we have the following theorem:

7 Theorem IV.: Consider an S with the acyclic and novalley properties with a step-aware avoidance classifier. If the ranking function prefers s with respect to, then is inherently safe. The proof appears in Appendix VII. V. THINK GLOBALLY, ACT LOCALLY We now consider how safe backup routing might be implemented in BG with locally-implementable routing policies that do not require global knowledge. The goal is to define a set of recommendations that, if followed by all ASes, would allow for a very flexible framework for implementing backup routes and load balancing, while at the same time provide a guarantee of the inherent safety of interdomain routing. Our recommendations will ensure that all conforming interdomain BG routing policies could be translated into inherently safe instances of the Stable aths roblem for every network destination. In particular, we require that () valleys are not permitted, (2) we can define an avoidance classifier on the associated S that is step aware, () the ranking of paths prefers s with respect to the avoidance levels of. We do not explicitly enforce the condition that the digraph is acyclic, depending instead on natural market considerations. Note that we are not advocating the the translation of BG policies into instances of the S. The S formalism is serving only as a simple semantic framework in which we can more easily prove inherent safety. Section V-A first describes how routing policies are implemented with BG and how BG policies can be translated to an instance of the Stable aths roblem. Section V-B then outlines how the rules of [0] can be implemented in BG. Section V- C proposes a new attribute for BG, called the avoidance level, and describes a very natural extension to the implementation of [0] that covers the inherently safe backup routing defined the current paper. Since the avoidance level attribute does not yet exist in BG, Section V-D describes how it can be simulated using the BG communities attribute, at the cost of introducing some complexity into the definition of routing policies. A. BG Routing olicies and Translation to S BG exchanges route announcements between BG speaking routers. These announcements are records containing a destination network (also called an address block, CIDR block, or I prefix) and attributes associated with this destination. Route announcements include the following attributes. : I prefix : next hop (address of next hop router) : ordered list of ASes traversed : local preference : set of community values The local preference attribute is not passed between autonomous systems, but is used internally within an autonomous system to assign a local degree of preference. Community values [5] are typically used in routing policies for deciding on the value of or on filtering. The BG attributes are used by import policies and export policies at each router to implement its routing policies. These policies can delete, insert, and modify some attribute values in route announcements. They can also filter out announcements. As a BG announcement moves from AS 0 to AS it undergoes three transformations. First, 0 applies its export policies. If the route announcement is not filtered out, then a BG-specific transformation occurs, which adds 0 to the of the announcement, and filters out the announcement if the contains. Finally, the import policies of are applied to the announcement. This is the point that a local preference value is assigned. The BG route selection procedure selects best route by first preferring highest values of local preference, then shortest AS paths, and so on. For a complete description of this process, see [2], [], []. Informally, the process of transforming a set of BG policies into an instance of S could proceed as follows. Designate any AS as the origin (. A distinct instance of the S would be constructed for each prefix originated by an (. For each AS we then enumerate every possible simple path in the AS graph from AS ( to AS. For each such path O, we start at ( and compose the export policies, BG transformation, and import policies, marching along the path in a hop-by-hop fashion toward. If we do not make it to the end of the path (that is, the route was filtered out somewhere along the path), then the path is not in the set of permitted paths at. Otherwise, the path O is in the set of permitted paths at. For each permitted path O this procedure will produce a BG route announcement *X\$OU. We then define to be any ranking function that the obeys the rules () ` O A `\$O, if and only if the BG route selection process will prefer *X\$O - over *X O A, and (2) '\$O A '\$O, if and only if the BG route selection process does not prefer one of *X\$O over *g OCAE the other. In case (2), the details of the BG selection process dictate that both *X O and *X\$O A must come from the same next-hop router. Since a BG speaking router only announces at most one route to any prefix (its best route), there is no possibility of a router simultaneously having two routes of the same rank. (Note that this accounts for the strictness condition in the definition of a ranking function for Stable aths roblems, given in Section II-A.) Observe that many distinct sets of BG routing policies could translate to the same instance of S. ut another way, the set of S permitted paths at node and the ranking of those paths via does not simply represent the routing policy of AS, but rather the composition of all routing policies along paths from the origin to. B. Implementing Simple Routing We now describe how the simple routing of [0] and Section II-B can be implemented in BG. The following table indicates whether or not an AS announces a route to its neighbor depending on its relationship to the AS that sent the route. A Y indicates that the route may be announced, while a N indicates that it may not be. For example, an announcement from a peer may be passed along to a but not to another peer.

8 To peer Y Y Y From peer Y N N Y N N These export rules ensure that no permitted path will have a step (N in peer-peer, peer-, and -peer entries) or a valley (N in - entry). In addition to these rules, each AS must ensure that all values of local preference assigned to announcements are greater than all local preference values assigned to peers and s. C. Safe Backup Routing with an Avoidance Level Attribute The easiest way to describe inherently safe backup routing is to imagine that BG is extended with a new attribute, the avoidance level of a route. The idea is that the higher this value, the less preferable a route. The avoidance level attribute must be considered before any other attribute in the BG decision process. Other approaches that affect later stages of the decision process are not sufficient. One example is the commonly-used technique of AS prepending (inflating AS path length by repeating an AS number multiple times) that affects the decision process after the local preference attribute. We can now extend the export rules to allow the three cases covered by Figure 5: To peer Y Y Y From peer Y Y Y Y Y N In order to make sure that are avoidance classifier is step aware, we must impose rules concerning the use of the avoidance level attribute. For each Y of the export table, the following table indicates when the avoidance level attribute must be increased. The entry O indicates that its increase is optional, while a R indicates that its increase is required To peer O O O From peer O R R O R Note that the optional entries in this table allow for very flexible backup routing and load balancing. The more liberal export rules allow the possibility of forming a valley involving one or more steps. To avoid this, an AS must never send to a an upstream route that was received from a peer. An AS must mark routes that it receives from a before it sends them to a peer. An AS should never export such a marked route to a. A special community value could be employed between peers to signal that a route was received from an upstream. Local preference must be assigned in a way that ensures that no route can get a value less than or equal to a peer or route. This ensures that s are preferred within an avoidance level since the selection process will consider local preference only after the avoidance level. Together these simple rules ensure that the conditions of Theorem IV. are met. D. Simulating Avoidance Levels with Communities Values RFC 998 [7] suggests using BG communities as a way to influence a neighbor s routing policies. It applies this technique to the implementation of peer-peer backup. We extend this approach to simulate inherently safe backup routing described in Section V-C. Since we will use communities to carry several bits of information, it is perhaps best to think of them as extended community values [6]. We assume that each AS 0 has defined the following community values, and that the semantics of these labels is shared with 0 s neighbors. =0g& tag for backup route of avoidance level, =0-. tag for upstream routes (used between peers). Routes that have not been tagged with an avoidance level community value are treated as being in avoidance level. We also assume that for any neighbor, and any avoidance level, that '=0S represents the local preference assigned to routes from received at 0. We assume that for each 0 this function conforms to the following rules: If 6 ) +L(B"*E+\$05 <B6 '*,+=0S `*(<g!!"*,+=05, then for each level we have '\$0S <' '=0S. For each A, <, if A, then `\$0S E '=0S <` A. That is, within each avoidance level routes are preferred over peer and routes, and routes of level lower avoidance levels are always preferred over routes of higher avoidance levels. In practice, this scheme requires imposing some limit the number of avoidance levels. If an AS uses values of local preference to differentiate between routes of the same avoidance level, then the function must take on distinct values. This complicates configuration of routing policies, particularly given the limitations of today s router configuration languages. VI. CONCLUSIONS Selecting efficient, stable routes between each pair of hosts is a major challenge for the distributed Internet routing infrastructure. In and of itself, BG does not ensure that hosts can communicate, even if the network is connected. In addition, conflicting local policies amongst a collection of ASes can cause the protocol to oscillate. This paper has has presented a model for backup routing that increase global network reliability without compromising the stability of the routing protocol. Several issues remain to be addressed. First, the interaction of backup routing with prefix aggregation needs to be studied. Second, the models presented in this paper are simplified in that they ignore the internal structure of ASes. It should be noted that we are ignoring the configuration of BG within each AS, and so our recommendations will not exclude all types of BG divergence. This follows from the fact that some types of internal BG configurations can lead to BG divergence within a single AS. For example, the interaction of BG route reflectors, the processing of BG routes using multi-exit discriminator values, and internal routing metrics, is not guaranteed to converge (see for example Scudder [7]). Routing policies may not actually be AS-wide, but may vary between border routers, mostly on

9 ! to meet traffic engineering goals. The interaction and tradeoffs between external routing (both normal and backup) and internal routing should be explored. REFERENCES [] Y. Rekhter and T. Li, A border gateway protocol, RFC 77 (BG version ), March 995. [2] Bassam Halabi, Internet Routing Architectures, Cisco ress, 997. [] John W. Stewart, BG: Inter-Domain Routing in the Internet, Addison- Wesley, 998. [] Geoff Huston, Interconnection, peering, and settlements, in roc. INET, June 999. [5] C. Alaettinoglu, Scalable router configuration for the Internet, in roc. IEEE ICN, October 996. [6] C. Labovitz, R. Wattenhofer, S. Venkatachary, and A. Ahuja, The impact of Internet policy and topology on delayed routing convergence, in roc. IEEE INFOCOM, April 200. [7] E. Chen and T. Bates, An application of the BG community attribute in multi-home routing, Request for Comments 998, August 996. [8] Kannan Varadhan, Ramesh Govindan, and Deborah Estrin, ersistent route oscillations in inter-domain routing, Tech. Rep. 96-6, USC/ISI, February 996. [9] T. G. Griffin and G. Wilfong, An analysis of BG convergence properties, in roc. ACM SIGCOMM, September 999. [0] Lixin Gao and Jennifer Rexford, Stable Internet routing without global coordination, in roc. ACM SIGMETRICS, June [] T. G. Griffin, F. B. Shepherd, and G. Wilfong, olicy disputes in pathvector protocols, in roc. Inter. Conf. on Network rotocols, November 999. [2] Timothy Griffin and Gordon Wilfong, A safe path vector protocol, in roc. IEEE INFOCOM, March [] T. G. Griffin, F. B. Shepherd, and G. Wilfong, The stable paths problem and interdomain routing, Submitted for publication. [] Lixin Gao, On inferring autonomous system relationships in the Internet, in roc. IEEE Global Internet Symposium, November [5] R. Chandra,. Traina, and T. Li, BG communities attribute, RFC 997, August 996. [6] Srihari Ramachandra and Daniel Tappan, BG extended communities attribute, Internet Draft draft-ramachandra-bgp-ext-communities-07.txt, Work in progress, expires June 200. [7] John Scudder, BG update, Talk at NANOG, October 2, See VII. ROOFS This appendix presents the proofs of all of results stated in the body of the paper. Some definitions are repeated here for ease of reading. A. Dispute Wheels We first introduce a sufficient condition that guarantees that an instance of the Stable aths roblem is inherently safe. This definition is taken from [], [2]. A dispute wheel of size F is a triple U, where is a sequence of F nodes D- A A, is a sequence of F non-empty paths R D R A R A, and is a sequence of F non-empty paths D A A. This triple is such that for each F' we have () ZK is a path from K to K A, (2) R K and K R K A are permitted at K, () ]R K K R K A. All subscripts are to be interpreted modulo F. See Figure 9 for an illustration of a dispute wheel. An S is wheel-free if it does not contain a dispute wheel. It follows from the results of [], [2] that if is wheel-free, then it has a unique solution and is inherently safe. B. Avoidance Classifiers Fig. 9. A dispute wheel of size. A function \$OU on permitted paths O is called an avoidance classifier, if the following conditions hold. (a) If O and R are paths permitted at node and \$OU ]RU, then ]RU \$OU. (b) If O, R, and R5O are permitted paths, then \$OU R5OU. An avoidance level is any value in the range of. If is an avoidance level and is an S, A dispute wheel is said to be of level if for each! and, ]R K R A. Lemma VII.: Suppose is a dispute wheel for S and is an avoidance classifier for. Then there must exist an avoidance level such that is of level. roof: Let be a dispute wheel. For any! we have R K K R K A. By property ` of an avoidance classifier, this implies that ZK]R5K A" ]R5K. From property &" we derive R5K A" ZK R5K AE. Together, these give us R5K AE ZK]R5K A" ]R5K for each!. By composing these equations we obtain (again, all subscripts are taken to be mod F ), ]R K A R K ]R A A" K A R K, ]R K A K R K A ]R K This proves the equalities claimed. Corollary VII.2: Suppose is an S and is an avoidance classifier for. If for every avoidance level there is no dispute wheel of level, then is wheel-free. C. Backup Edges Suppose " < is an edge in the S graph. The - digraph can impose an orientation on this edge. If the edge is oriented, we use the notation \$ <. We say that = <` is a - edge if 6) +L(B"*E+\$< or a - edge if :6&`*(<g!!"*,+=<`. If is an edge

10 and O is a permitted path, then is called a backup edge if \$OU \$OU. Let \$0J <` O be a permitted path at 0. If any of the following conditions hold, then edge "0S is called a mandatory backup edge: (a) \$0S is a - edge and <` is a peer-peer edge, (b) "0S is a peer-peer edge and = <` is a - edge, (c) 0S. and " < are peer-peer edges. An avoidance classifier is step aware if it treats any mandatory backup edge as a backup edge. That is, = <` OU =0a <` OU for each of the cases (a)-(c). Lemma VII.: Let be an S with a step aware avoidance classifier. Suppose has a dispute wheel of avoidance level. Then no K contains a mandatory backup edge. roof: Suppose that K O A O, where is a mandatory backup edge. From the definition of an avoidance classifier we have O R5K A \$OCA"O RSK AE AR5K A. Since is step aware, we have R K A ' \$O R K A. Together these facts imply that ]R K A K R K A. This contradicts the assumption that is of one avoidance level. Lemma VII.: Let be a valley-free S, a step aware avoidance classifier for, and is a dispute wheel for. () If there is some ZK in where ZK \$'K-< ]O and \$ K < is a - edge, then every edge of 5K is a edge. (2) If there is some K in where K O=<\ K A and =<' K A is a - edge, then every edge of K is a - edge. roof: (). Let be any edge in K O A O. Assume, without loss of generality, that all edges of O A are - edges. Suppose that is a - edge. This is a contradiction, since this would mean that K R K A contains a valley. On the other hand, suppose that is a peer-peer edge. Let be the last edge in O A (a - edge). This is then a mandatory backup edge (condition (a)), which is prohibited by Lemma VII.. Therefore, can be only a - edge. (2). Let K O=< K A and be any edge in OQ8O O. Assume without loss of generality that all edges of O are - edges. Suppose that is a - edge. This is a contradiction, since this would mean that the permitted path K R K A contains a valley. On the other hand, suppose that is a peer-peer edge. Then is a mandatory backup edge (condition (b)), and is prohibited by Lemma VII.. Therefore, can be only a - edge. D. reference for Customers If is an avoidance classifier for S, then ranking function is said to be preferring with respect to if within a set of paths of the same avoidance level s paths are preferred, or put more formally, if for every O and R permitted at node with \$OU h, if O is a path to one of s s and R is not, then RU \$OU. Theorem VII.5: Let be a valley-free S with an acyclic - digraph. Suppose that is a step aware avoidance classifier for, and that has a preferring ranking with respect to. Then is inherently safe. roof: From Corollary VII.2 and the results of [], [2] it is enough to show that for every avoidance level there is no dispute wheel of level. Suppose is a dispute wheel of level. ick any K 6. Let R K A R and A. We now perform a case analysis on all of the combinations of edges A and. First, suppose that A is a - edge. Since prefers s, and R[KL K]R5K A, it must be that ZK starts with a - edge. Lemma VII. () tells us that all the edges in K are - edges. We now consider the first edge of R K A. This cannot be a edge, since this would imply that K]R5K A contains a valley. And this edge cannot be a peer-peer edge, since this would imply that the last edge of K is a mandatory backup edge (condition (a)), which is prohibited by Lemma VII.. Therefore the first edge of R5K A must be a a - edge. We can now repeat this argument step-by-step clockwise around the dispute wheel and conclude that the entire rim is made up of - edges. This contradicts the assumption that the - digraph is acyclic. Therefore, we need not consider further the case where -A is a - edge. This leaves the following cases to consider. Case : is a peer-peer edge. There are two subcases to consider. Case.:,A is a peer-peer edge. This means that is a mandatory backup edge (condition (c)), which is prohibited by Lemma VII.. Case.2: A is a - edge. This means that is a mandatory backup edge (condition (b)), which is prohibited by Lemma VII.. Case 2: is a - edge. There are two subcases to consider. Case 2.:,A is a peer-peer edge. This means that is a mandatory backup edge (condition (a)), which is prohibited by Lemma VII.. Case 2.2: A is a - edge. This contradicts the assumption that no permitted path contains a valley. Case : is a - edge. Since this is the only remaining case, this must be true for the last edge in every path in the dispute wheel. Then we know, by Lemma VII. (2), that all edges on the rim of the wheel are - edges. This contradicts the assumption that the - digraph is acyclic.

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