Dear All,
since our previous works in remote LFA analysis (papers already sent to
these mailing lists) assumed unit cost networks, therefore as I
promised, I calculated the corresponding LFA and rLFA coverages in those
networks with their original link costs.
These networks were inferred from Rocketfuel dataset (/Mahajan, R.,
Spring, N., Wetherall, D., Anderson, T.://
//Inferring link weights using end-to-end measurements. In: ACM IMC, pp.
231–236 (2002)/), SNDLib (http://sndlib.zib.de), and
TopologyZoo(http://www.topology-zoo.org). The found coverages and some
details are found in the table below, where /n/ and /m/ denote the
number of nodes and the number of links, respectively. The other columns
mark the different coverages obrtained by simple LFA and remote LFA,
where LP indicates the link-protecting case, while NP notes the case of
node protection.
Topologies marked with an asterisk(*) did not have inferred real link
costs from the datasets, so their costs were initially set to 1 (unit
costs).
I believe that these results computed on real-world networks could
significantly improve the rlfa draft, especially Sec. 9.3., and the
advantages of remote LFA over simple LFA would be more emphasized.
+==============+====+====+==========+===========+============+============+
| Topology | n | m | LFA_LP | LFA_NP | rLFA_LP | rLFA_NP |
+==============+====+====+==========+===========+============+============+
| AS1221 | 7 | 9 | 0.809 | 0.25 | 0.809 | 0.25 |
+--------------+----+----+----------+-----------+------------+------------+
| AS1239 | 30 | 69 | 0.8735 | 0.7554 | 1 | 0.9795 |
+--------------+----+----+----------+-----------+------------+------------+
| AS1755 | 18 | 33 | 0.8725 | 0.7741 | 0.9967 | 0.9959 |
+--------------+----+----+----------+-----------+------------+------------+
| AS3257 | 27 | 64 | 0.923 | 0.7186 | 0.99 | 0.8472 |
+--------------+----+----+----------+-----------+------------+------------+
| AS3967 | 21 | 36 | 0.7857 | 0.6460 | 1 | 0.9325 |
+--------------+----+----+----------+-----------+------------+------------+
| AS6461 | 17 | 37 | 0.9338 | 0.6933 | 0.9963 | 0.7075 |
+--------------+----+----+----------+-----------+------------+------------+
| Abilene* | 12 | 15 | 0.5606 | 0.6078 | 0.9090 | 0.8725 |
+--------------+----+----+----------+-----------+------------+------------+
| Arnes* | 41 | 57 | 0.6225 | 0.3518 | 0.7487 | 0.4562 |
+--------------+----+----+----------+-----------+------------+------------+
| AT&T | 22 | 38 | 0.8225 | 0.5647 | 1 | 0.8497 |
+--------------+----+----+----------+-----------+------------+------------+
| Deltacom | 113| 161| 0.5771 | 0.4910 | 0.8539 | 0.8148 |
+--------------+----+----+----------+-----------+------------+------------+
| Gambia | 28 | 28 | 0.037 | 0.04 | 0.1851 | 0.12 |
+--------------+----+----+----------+-----------+------------+------------+
| Geant | 37 | 55 | 0.6906 | 0.3977 | 0.8303 | 0.6582 |
+--------------+----+----+----------+-----------+------------+------------+
| Germ_50 | 50 | 88 | 0.9004 | 0.8381 | 1 | 0.9995 |
+--------------+----+----+----------+-----------+------------+------------+
| Germany* | 27 | 32 | 0.6948 | 0.599 | 1 | 0.9549 |
+--------------+----+----+----------+-----------+------------+------------+
| InternetMCI | 19 | 33 | 0.9035 | 0.6798 | 0.9415 | 0.9136 |
+--------------+----+----+----------+-----------+------------+------------+
| Italy* | 33 | 56 | 0.784 | 0.5741 | 1 | 0.9269 |
+--------------+----+----+----------+-----------+------------+------------+
| NSF* | 26 | 43 | 0.86 | 0.6347 | 1 | 1 |
+--------------+----+----+----------+-----------+------------+------------+
Furthermore, after reading the draft again-and-again, I have found that
it should be more emphasized in the draft that calculating or seeking
for a remote LFA staging point should be done IF AND ONLY IF no simple
LFA were found. This is important, since normally people obviously think
and observe that the failure coverage of remote LFA should be greater
than or equal to the coverage obtained by simple LFA. Moreover, people
also think that LFAs produce a subset of remote LFAs. However, if after
a failure only P-spaces and Q-spaces are taken into account in order to
seek a (remote) loop-free alternate, then it is possible that a simple
LFA would not be found resulting unprotected node-pairs. But, this case
could only happen when link costs are not unit costs. For an easier
understanding, consider the network depicted below:
1 1
F--------S-----------X--------E
\ / \
10 \ / 4 \ 1
\ / 1 \
N ---------------------------C
| |
1 | | 1
| |
| 1 1 |
A--------------B-------------D
Assume that S wishes to send a packet to D, and the shortest path goes
through E, therefore it is S-E-C-D. Suppose that the link (s,e)
fails, or even the node e itself fails. In this case, since LFA (and its
calculation) only consider all other neighbors of s, then node N would
be an easy LFA for this failure, since dist(N,D) < dist(N,S)+dist(S,D).
However, if we only seek possible remote LFAs, than according to the
(r)SPF calculations, or taking into account our distance-function
conditions will result that D's Q-space will contain node N (besides
some other nodes), but S's P-space won't (it will only contain node F
and what is more, if node F does not exist, then S's P-space would be
empty), therefore no intersection of the two spaces will exist, leaving
this network vulnerable to the failure of link (s,e).
This results that a neighbor should be always reached by the neighboring
link, even if there exists a shorter, but definitely bigger in hop-count
path to it.
According to this case (when seeking simple LFA is missed), the
condition of extended P-space described by our distance functions,
should be modified a bit:
For source /s/, destination /d/, and next-hop /e/, some node /n != s,d/
is an extended link-protecting remote LFA for the /s-d/ pair if and only if
/∃v ∈ neigh(s) : dist(v, n) < dist(v, s) + dist(s, e) + dist(e, n)//*&&
dist(s,v) < dist(s,e) + dist(e**,v)*/**
/dist(n, d) < dist(n, s) + dist(s, d) . /
One can easily observe that emphasizing more that remote LFA seeking
process is only "executed" after no simple LFA is found could much more
ease the understanding and won't result a headache to the reader who
accidentally wants to calculate rLFA coverage in such network.
Please let me know, if my interpretation is not correct.
Thanks.
Best,
Levente
On 07/02/2013 07:07 PM, Stewart Bryant wrote:
On 26/06/2013 14:13, Levente Csikor wrote:
Levente
In each case below the conditions are surely fulfilled if n=e
thus I think that in each case the condition needs to be
changed to :
some node n!={s or e}
In order to not to read our full papers and searching the answers, I
copied here the relevant parts:
*Link-protecting rLFA condition:*
For source /s/, destination /d/, and next-hop /e/, some node /n/ /!=
s,d/ is a link-protecting remote LFA for the /s-d/ pair if and only if
/dist(s, n) < dist(s, e) + dist(e, n)/ (1)
/dist(n, d) < dist(n, s) + dist(s, d)/ (2)
In these equations, one can easily see, that (1) defines the P-space,
while (2) is the condition of Q-space. Furthermore, with these
formalized conditions, one can easily observe, that (2) is actually
the basic loop-free criterion of pure LFA.
*Link-protecting rLFA condition with extended P-space:*
For source /s/, destination /d/, and next-hop /e/, some node /n !=
s,d/ is an extended link-protecting remote LFA for the /s-d/ pair if
and only if
/∃v ∈ neigh(s) : dist(v, n) < dist(v, s) + dist(s, e) + dist(e, n)///
/dist(n, d) < dist(n, s) + dist(s, d) . /
*Node-protecting rLFA condition:*
For source /s/, destination /d/, and next-hop /e/, some /n != s,d/ is
a node-protecting remote LFA for the /s-d/ pair if and only if
/dist(s,n) < dist(s,e) + dist(e,n)/// (3)
/dist(n,d) < dist(n,e) + dist(e,d) / (4)
As it was in the case of link protection, here, (3) defines the
P-space, while (4) characterize the Q-space.
Here, two important observations can be made, which are the followings:
- P-space does not depend on the protection scheme (i.e., link or
node protection)
SB> That falls directly out of the definition of P-space
SB> since in link you cannot traverse the link to e and in
SB> node only get to e if you traverse the link to e
SB> thus the exclusion of the link to e applies in both cases
- (4) again is the basic node-protecting loop-free criterion of pure
LFA.
*Node-protecting rLFA condition with extended P-space:*
For source /s/, destination /d/, and next-hop /e/, some node /n !=
s,d/ is an extended node-protecting remote LFA for the /s-d/ pair if
and only if
/∃v ∈ neigh(s) : dist(v, n) < dist(v, e) + dist(e, n)/
/dist(n, d) < dist(n, e) + dist(e, d) ./
Despite the fact that we only considered unit cost networks, the
formal definitions above are *true for any arbitrary weighted network.*
SB> I do not see where the unit costs come into the text above. It looks
SB> like it is already expressed in terms of arbitrary cost.
SB> Additionally in order to limit the number of SPFs to a practical
SB> level, we normally suggest that the repair target in not d, but
SB> instead is e (in the link case) or next hop of e (node case).
SB> Anyway I will work on some text.
- Stewart
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