Re: [Lsr] Questions about draft-ietf-lsr-dynamic-flooding-04

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Hi Xuesong,


> Thank you for your work about Flooding Reduction mechanism. I think it is very useful for IGP scalability.


Thank you, we very much appreciate that.


> When Sarah giving presentation for draft-chen-lsr-dynamic-flooding-algorithm-00 in LSR interim meeting, I raised a question about how to handle multipoint failure in the flooding topology. I remembered that draft-ietf-lsr-dynamic-flooding-04 was mentioned. I have read this draft, especially section 6.8.11 and temporary flooding relevant content. I think it is a valid method logically. (We have tried to raise some examples to challenge this method, but in all these cases, this method works) Considering that reliability of IGP is crucial for deployment, we are still wondering whether some mathematical proof is necessary (or possible) to guarantee that, in all the possible scenarios, proper convergence could be achieved when using flooding reduction algorithm, just as traditional IGP mechanism. The proof may be case by case depending on the flooding reduction algorithm, but we think even some example or clue about how to do this will be very helpful.


A proof? Ok, I can try to be rigorous, but it’s been a very long time. I apologize in advance to Kireeti, Mr. Myers, Mr. Douglas, and Prof. Greever for what follows.


Definition: Let G=(V, E) be a finite graph. F=(V, E’) is a ‘flooding topology’ on G if E’ is a subset of E and F is connected.


We normally have other properties for flooding topologies, such as being bi-connected, relatively sparse, minimal diameter, and minimal degree, but those properties are not relevant for this proof.


Definition: Let C=(V*, E*), V* a subset of V, E* a subset of E, be the ‘cut’ of G. This represents the failures in the topology. Let G’ = G - C, the topology after failure, and let F’ = F - C, be the flooding topology after failure. 


Definition: Let R = E - E* - E’. This is the set of 'recovery edges': edges that have not failed and are not part of the flooding topology.


Corollary: R is finite. 

Proof: G is finite, by definition. Therefore E is finite, E* must be finite, and E’ is finite.


Lemma 1: If G’ is connected, then F’ + R is connected.

Proof: 

G’ = G - C = (V, E) - (V*, E*) = (V - V*, E - E*).  

F’ + R = (F - C) + (E - E* - E’) = (V, E’) - (V*, E*) + (E - E* - E’) = ( V - V*, E’ - E* ) + ( E - E* - E’) = ( V - V*, E’ - E* + E - E* - E’ ) = ( V - V*, E - E* - E* ) = ( V - V*, E - E* ) = G’.

Q.E.D.


Lemma 2: If G’ is connected, then incrementally adding edges from R to F’ will yield a connected flooding topology.

Proof: By induction. Let R’ be a subset of R. Induction property: Either F' + R’ is connected, or R’ is a proper subset of R.

Base case: R’ is the empty set. If F’ is connected, then F’ + R’ is connected and the property is satisfied. If F’ + R’ is not connected, R’ is empty, which is a proper subset of R.

Induction case: Let r be a recovery edge in R - R’ (i.e., a new edge to add).  If F’ + R’ is connected, then the property is satisfied already. If F’ + R’ + r is connected, the property is satisfied. If R’ + r = R, then by Lemma 1, the property is satisfied. Otherwise, R’ + r != R, and the property is satisfied. By our corollary, the induction must terminate.

Q.E.D.



Bugs, questions, and comments are welcome.

Regards,
Tony