computing the relationships between autonomous systems

1
computing the relationships between autonomous
systems

• giuseppe di battista
• maurizio patrignani
• maurizio pizzonia

univ. of rome III http//www.dia.uniroma3.it/comp
unet/
2
overview

• interdomain routing basics
• the problem of determining the relationships
  between autonomous systems
• testing if a solution exists in linear time
• the complexity of finding a good orientation
• an heuristic approach to the general problem
• conclusions

3
the routing process
4
routing protocols

• routing protocols are used to automatically
  update the routing tables
• they fall into two main cathegories
• link-state routing protocols
• approach send information about your neighbors
  to everyone
• each router reconstructs the whole network and
  computes a shortest path tree to all destinations
• examples is-is, ospf
• distance-vector routing protocols
• approach send all your information to your
  neighbors
• each router updates its routing table based on
  the neighbors ones
• examples rip

5
flat routing a single routing algorithm
involving all the organizations
lan send to
router 6
lan send to
router 9
router 1
router 2
router 5
router 3

• it was this way before 1984
• problems slow to converge, difficult to deploy a
  new routing algorithm, difficult to debug and
  configure, does not take into account the
  ownership of the links

6
hierarchical routingmore than one routing
algorithm involved

• approach
• each organization runs a local routing algorithm
  by using an interior gateway protocol (igp)
• external targets are injected into the local
  routing algorithm

7
route injection
lan send to

• the interior routing algorithm spreads into the
  network the local targets as well as the remote
  ones injected by border routers
• border routers are aware of remote targets since
  they talk a higher level routing protocol with
  other border routers

8
exterior gateway protocol

• approach

9
exterior gateway protocol

• approach
• hide the interior part of all organizations

10
represent the internal targets

• each border router represents its internal
  targets as if they were local

11
simplify the graph

• simplify the graph

• consider the (external and internal) reachability
  of the routers

• the graph is actually managed through tcp
  connections called peerings

12
solve the routing problem

• solve the routing problem on the simplified graph
• based on political considerations

13
border gateway protocol

• bgp is an exterior gateway protocol it keeps the
  routing tables updated and propagates routing
  information
• takes into account the willingness of the
  organizations to cooperate in the routing process
  (commercial agreements, local preferences,
  priorities, legal issues, )

14
announcements and traffic flows

• bgp allows a router to offer connectivity to
  another router
• offering connectivity means promising the
  delivery to a specific destination

bgp announcement
195.10.14.0/24
AS100
AS200
ip traffic (to be delivered to 195.10.14.0/24)
15
using AS-path to avoid cycles
AS40
AS31
193.204.161.0/24
AS10
AS60
AS212
legenda
16
using prepending to implement preferences
AS40
AS31
193.204.161.0/24
AS10
AS60
AS212
17
how to ensure cooperation?
AS40
AS31
193.204.161.0/24
AS10
AS60
AS212
why AS212 should propagate AS10 announcement?
18
the scenario
peer-peer
AS2
AS1
provider
customer
peer-peer
AS3
AS5
AS4
sibling-sibling
AS8
AS6
AS7
AS9
not available route
19
overview

• interdomain routing basics
• the problem of determining the relationships
  between autonomous systems
• testing if a solution exists in linear time
• the complexity of finding a good orientation
• an heuristic approach to the general problem
• conclusions

20
problem history

• the problem is introduced by Lixin Gao (On
  Inferring Autonomous System Relationships in the
  Internet, IEEE Trans. Networking, 2001)
• relationships are classified into three
  categories customer-provider, peer-peer, and
  sibling-sibling
• BGP routing tables are used as input
• heuristics are verified with information coming
  from other sources
• further studies on the structural properties of
  the labeled graph (Ge, Figueiredo, Jaiwal, Gao,
  On the Hierarchical Structure of the Logical
  Internet Graph, Proc. SPIE ITCOM, 2001)

21
a simpler problem formulation

• due to Subramanian, Agarwal, Rexford, and Katz
  (Characterizing the Internet Hierarchy from
  Multiple Vantage Points INFOCOM 2002) SARK 2002

from a provider
from a provider
from a provider
from a provider
from a peer
from a peer
from a peer
from a peer
from a customer
from a customer
from a customer
from a customer
peer-peer relationship
customer-provider relationship
22
valid and invalid AS-paths
if all ASes respect the advertisement rules,
AS-path would be all valid
possibly missing
possibly missing
possibly missing
possibly missing
type-2
type-1
type-1 valid AS-path a (possibly missing)
uphill path followed by a (possibly missing)
downhill path type-2 valid AS-path a (possibly
missing) uphill path followed by a peer-peer
edge, followed by a (possibly missing) downhill
path
23
the ToR (Type of Relationships) problem
ToR problem SARK 2002
given an undirected graph G and a set of paths P,
give an orientation to some of the edges of G to
minimize the number of invalid paths in P
invalid paths not valley free or involving
more than one peering

• in SARK 2002 the ToR problem is conjectured to
  be NP-hard

24
a real-life ToR problem instance
six rows extracted from the BGP routing table of
Oregon RouteViews (Apr 18, 2001)
network next hop path
200.1.225.0 167.142.3.6 5056 701 6461 4926 4270 4387 i
200.10.112.0/23 167.142.3.6 5056 701 4926 4926 4926 6461 2914 174 174 174 174 14318 i
204.71.2.0 203.181.248.233 7660 1 5056 701 11334 i
231.172.64.0/19 167.142.3.6 5056 1239 1 1755 1755 1755 1755 3216 13099 i
200.33.121.0 167.142.3.6 5056 1 1239 8151 i
204.71.2.0 144.228.241.81 1239 5056 701 11334 i
5056 701 6461 4926 4270 4387
5056 701 4926 4926 4926 6461 2914 174 174 174 174
14318
7660 1 5056 701 11334
5056 1239 1 1755 1755 1755 1755 3216 13099
5056 1 1239 8151
1239 5056 701 11334
extract AS paths and eliminate prepending
25
a real-life ToR problem instance
six rows extracted from the BGP routing table of
Oregon RouteViews (Apr 18, 2001)
network next hop path
200.1.225.0 167.142.3.6 5056 701 6461 4926 4270 4387 i
200.10.112.0/23 167.142.3.6 5056 701 4926 4926 4926 6461 2914 174 174 174 174 14318 i
204.71.2.0 203.181.248.233 7660 1 5056 701 11334 i
231.172.64.0/19 167.142.3.6 5056 1239 1 1755 1755 1755 1755 3216 13099 i
200.33.121.0 167.142.3.6 5056 1 1239 8151 i
204.71.2.0 144.228.241.81 1239 5056 701 11334 i
extract AS paths and eliminate prepending
5056 701 6461 4926 4270 4387
5056 701 4926 4926 4926 6461 2914 174 174 174 174
14318
6461 2914 174 174 174 174 14318

14318

7660 1 5056 701 11334
5056 1239 1 1755 1755 1755 1755 3216 13099
3216 13099
5056 1 1239 8151
1239 5056 701 11334
26
building the corresponding AS graph
AS8151
AS4387
AS4270
AS1239
AS4926
AS701
AS5056
AS2914
AS3216
AS1755
AS174
AS1
AS6461
AS11334
AS14318
AS7660
AS13099
5056 701 6461 4926 4270 4387
5056 701 4926 6461 2914 174 14318
7660 1 5056 701 11334
5056 1239 1 1755 3216 13099
5056 1 1239 8151
1239 5056 701 11334
27
an orientation for the AS graph
AS8151
AS4387
AS4270
AS1239
AS4926
AS701
AS5056
AS2914
AS3216
AS1755
AS174
AS1
AS6461
AS11334
AS14318
AS7660
AS13099
5056 701 6461 4926 4270 4387
5056 701 4926 6461 2914 174 14318
7660 1 5056 701 11334
5056 1239 1 1755 3216 13099
5056 1 1239 8151
1239 5056 701 11334
28
our contributions

• we show that, although the ToR-problem is
  NP-hard, a solution without invalid paths (if it
  exists) can be found in linear time
• we propose heuristics for the general problem
  based on a novel paradigm and show their
  effectiveness against publicly available data
  sets
• the experiments put in evidence that our
  heuristics performs significantly better than
  state of the art heuristics

29
a hot topic

• independently of our work, Erlebach, Hall, and
  Schank of the Theory of Communication Networks
  Group of Zurich discovered analogous results
  concerning the time complexity of the general
  problem and the linearity in the case of all
  valid paths
• however, while they put more emphasis on the
  approximability of the problem, we focus more on
  the engineering and the experimentation of an
  effective heuristic approach

30
formulation as a decision problem

• the ToR minimization problem corresponds to the
  following decision problem

ToR-D problem
given an undirected graph G, a set of paths P,
and an integer k, test if it is possible to give
an orientation to some of the edges of G so that
number of invalid paths in P is at most k
31
simplifying the decision problem
suppose to have a solution to the TOR problem
with all valid paths and consider an edge labeled
as a peer-peer edge
peer-peer
32
example of orientable AS graph
AS8709
AS8938
AS3561
AS6893
AS3967
AS8843
AS4197
AS7018
AS15493
AS3582
AS10311
AS1740
33
an orientation leaving all valid paths
AS8709
AS8938
AS3561
AS6893
AS3967
AS8843
AS4197
AS7018
AS15493
AS3582
AS10311
AS1740
34
a different representation
AS3561
AS7018
AS3967
AS1740
AS8709
AS10311
AS8938
AS4197
AS3582
AS8843
AS6893
AS15493
35
example of not orientable AS graph
AS8151
AS4387
AS4270
AS1239
AS4926
AS701
AS5056
AS2914
AS3216
AS1755
AS174
AS1
AS6461
AS11334
AS14318
AS7660
AS13099
5056 701 6461 4926 4270 4387
5056 701 4926 6461 2914 174 14318
7660 1 5056 701 11334
5056 1239 1 1755 3216 13099
5056 1 1239 8151
1239 5056 701 11334
36
chestnuts and contradictions
chestnut
contradiction!
37
overview

• interdomain routing basics
• the problem of determining the relationships
  between autonomous systems
• testing if a solution exists in linear time
• the complexity of finding a good orientation
• an heuristic approach to the general problem
• conclusions

38
testing if a solution exists with all valid paths
observation
a path p v1, , vn is valid if and only if it
does not have a vertex vi such that the two edges
of p incident on vi are directed away form vi
vi

• based on this observation the problem can be
  mapped to 2SAT

given a set X of Boolean variables and a formula
in conjunctive normal form composed by clauses of
two literals, where a literal is a variable or a
negated variable, find a truth assignment for the
Boolean variables in X such that the formula is
satisfied
example (x1 ? x2) ? (?x2 ? ?x3) ? (x3
? ?x1)
39
mapping the problem to 2SAT

• associate each edge with a Boolean variable
• provide each edge with an arbitrary (say random)
  orientation

x1,2
x2,3
x3,4
x4,5
v3
v2
v4
v1
v5

• a truth assignment for the Boolean variables
  corresponds to an orientation for all the edges
  of the AS-graph (and vice versa)
• if the variable is true, the associated edge
  preserves its original direction
• if the variable is false, the associated edge is
  reversed

40
construction of the 2SAT instance
x1,2
x2,3
x3,4
x4,5
v3
v2
v4
v1
v5
(x1,2 ? x2,3) ? (?x2,3 ? ?x3,4) ? (x3,4 ?
?x4,5)
x1,2
?x2,3
?x3,4
?x4,5
v3
v2
v4
v1
v5
x1,2
?x2,3
?x3,4
x4,5
v3
v2
v4
v1
v5
41
solving a huge 2SAT
given a 2SAT formula
(x1,2 ? x2,3) ? (?x2,3 ? ?x3,4) ? (x3,4 ?
?x4,5) ? (x4,5 ? ?x2,3)
compute the graph of the literals
?x3,4
?x4,5
?x1,2
x2,3
x1,2
?x2,3
x4,5
x3,4

• there is a direct path between a literal and its
  opposite and vice versa iff a solution without
  invalid paths does not exist (Aspvall, Plass,
  Tarjan, IPL 79)

42
strongly connected components

• compute the strongly connected components of the
  graph of the literals
• strongly connected component maximal set of
  vertices such that for each pair u, v of vertices
  of the set there exists a directed path from u to
  v and vice versa

43
testing if a solution exists

• a solution for the 2SAT problem does not exists
  if and only if the two literals of the same
  Boolean variable (edge) fall into the same
  strongly connected component
• the test takes O(nmq) time
• where n is the number of ASes, m is the number of
  edges, and q is the sum of the lengths of the
  AS-paths

?x
x
44
the experimental setting

• same setting used by SARK 2002
• ten telnet looking glasses are selected as
  sources of data
• all the AS-paths put together are used to test
  the algorithms
• four web sources are used to validate the output

45
AS graphs from the various sourceshttp//www.cs.b
erkeley.edu/sagarwal/research/BGP-hierarchy/
AS AS Name Apr 18, 2001 Apr 18, 2001 Apr 18, 2001 Apr 6, 2002 Apr 6, 2002 Apr 6, 2002
AS AS Name Paths Vertices Edges Paths Vertices Edges
1 Genuity 58,156 10,203 13,001 63,744 12,700 15,946
1740 CERFnet 70,830 10,007 13,416 not available not available not available
3549 Globalcrossing 60,409 10,288 13,039 76,572 12,533 16,025
3582 Univ.of Oregon 2,584,230 10,826 22,440 4,600,981 13,055 27,277
3967 Exodus Comm. 254,123 10,387 18,401 399,023 12,616 21,527
4197 Global Online Japan 55,060 10,288 13,004 59,745 12,518 15,628
5388 Energis Squared 58,832 10,411 13,259 117,003 12,659 16,822
7018 ATT 120,283 9,252 12,117 170,325 11,706 15,429
8220 COLT Internet 46,606 8,376 10,932 154,855 12,660 18,421
8709 Exodus, Europe 114,931 10,333 15,006 126,370 12,555 18,175
1755 Ebone 23,469 10,540 13,898 not available not available not available
2516 KDDI 126,414 10,790 17,735 not available not available not available
2548 MaeWest 80,549 10,583 15,249 not available not available not available
6893 CW 70,265 10,523 14,359 not available not available not available
46
solvable without invalid paths?
AS AS name Apr 18, 2001 Apr 6, 2002
AS AS name orientable w/o invalid paths orientable w/o invalid paths
1 Genuity yes yes
1740 CERFnet yes not available
3549 Globalcrossing yes yes
3582 Univ.of Oregon no no
3967 Exodus Comm. yes yes
4197 Global Online Japan yes yes
5388 Energis Squared yes yes
7018 ATT yes yes
8220 COLT Internet yes yes
8709 Exodus, Europe yes yes
1755 Ebone yes not available
2516 KDDI no not available
2548 MaeWest yes not available
6893 CW yes not available

47
computing a solution without invalid paths

• consider the directed acyclic graph of the
  connected components of the graph of the literals
• compute a topological sorting of the connected
  components and assign an integer to each
  component
• call f(x) the index of the component to which the
  literal x belongs
• a true value is assigned to variable x if f(x) gt
  f(?x), false otherwise

1
2
3
4
48
overview

• interdomain routing basics
• the problem of determining the relationships
  between autonomous systems
• testing if a solution exists in linear time
• the complexity of finding a good orientation
• an heuristic approach to the general problem
• conclusions

49
the general problem is NP-hard
MAX2SAT instance
MAX2SAT assignment
50
example
(x1 ? ?x2) ? (x2 ? ?x3) ? (x2 ? x4) ? (?x1 ? ?x4)
? (?x3 ? x4)
x1
x2
x3
x4
51
overview

• interdomain routing basics
• the problem of determining the relationships
  between autonomous systems
• testing if a solution exists in linear time
• the complexity of finding a good orientation
• an heuristic approach to the general problem
• conclusions

52
our heuristic approach

• size of the problem
• 3,423,460 AS-paths
• 10,916 vertices
• 23,761 edges
• very simple approach
• find a very large set of AS-paths admitting an
  orientation
• try to reinsert the kept out AS-paths

53
1) find a very large set of paths admitting an
orientation
1.a) rank the edges with respect to the number of
paths using them 1.b) construct the graph of the
literals 1.c) compute the strongly connected
components of it 1.d) find out all the variables
whose two literals fall into the same strongly
connected component 1.e) consider the
corresponding edges and eliminate the one with
the highest rank (and all the paths using it,
too)
54
2) try to reinsert the left out AS-paths
2.a) set x 1 2.b) add x paths 2.c) if it is
still solvable commit the path addition x
x2 else if x is 1 discard the path
else x x/2 2.d) repeat 2.b and 2.c
until no path is left
some figures out of 3,423,460 paths, we removed
246,835 paths with the first step, ending with
had 3,176,625 paths after the first step we
reinserted 222,764 with the second step, ending
with 3,399,389 valid paths
55
comparison with the SARK paper
two peerings allowed
recomputed
AS AS Name SARK Err. SAT Err.
1 Genuity 0.65 0.45
1740 CERFnet n.a. 0.36
3549 Globalcrossing n.a. 0.13
3582 Univ.of Oregon n.a. 0.57
3967 Exodus Comm. n.a. 0.42
4197 Global Online Japan n.a. 0.46
5388 Energis Squared n.a. 0.46
7018 ATT 0.63 0.21
8220 COLT Internet n.a. 0.22
8709 Exodus, Europe n.a. 0.21
1755 Ebone 2.89 1.52
2516 KDDI 8.97 4.95
2548 MaeWest 1.49 0.19
6893 CW 2.92 0.64
SARK Err.
16.85
6.29
17.28
30.52
29.65
13.67
55.37
23.30
12.62
11.68
6.32
76.36
2.71
23.63
SARK Err.
0.86
0.96
3.44
19.20
5.95
2.83
53.63
0.73
6.32
3.98
3.33
20.80
0.99
11.84
telnet sources
web sources
56
a deeper comparison
SAT algorithm SAT algorithm
number of nodes 10897
number of edges 23690
peering edges 0
percentage of total 0
path covering distribution minimum 1 average 80 maximum 19918
strongly connected components 10312
scc's to edges ratio 0.4352892
size distribution 10311 scc's of size 1 1 scc of size 586
SARK algorithm SARK algorithm
number of nodes 10923
number of edges 23757
peering edges 1136
percentage of total 5
path covering distribution minimum 0 average 50 maximum 22327
strongly connected components 10302
scc's to edges ratio 0.4336406
size distribution 10096 scc's of size 1 176 scc's of size 2 21 scc's of size 3 4 scc's of size 4 1 scc of size 6 1 scc of size 7 1 scc of size 19 1 scc of size 86 1 scc of size 278
57
graph of the differences
we inserted one undirected edge (and the two end
vertices if needed) into the graph of the
differences for every oppositely directed edge
graph of the differences graph of the differences
number of nodes 489
number of edges 1148
path covering distribution minimum 1 average 37 maximum 3420
connected components 19
cc's to edges ratio 0.0165505
size distribution 18 cc's of size 2 1 cc of size 453
58
discovering the peer-peer relationships
given an oriented graph, discovering peer-peer
relationships is a hard problem if you want to
maximize them (MAX-INDEPENDENT-SET can be mapped
to it)
v1
v2
1
3
2
v3
4
v6
6
5
v4
v5
59
degrees of freedom in determining the peering
relationships
AS AS Name edges candidates peer-peer peer-peer
1 Genuity 13,001 11,395 8,573 65.94
1740 CERFnet 13,416 10,741 7,931 59.12
3549 Globalcrossing 13,039 10,851 8,250 63.28
3582 Univ.of Oregon
3967 Exodus Comm. 18,401 14,562 11,626 63.18
4197 Global Online Japan 13,004 9,794 7,573 58.24
5388 Energis Squared 13,259 8,969 7,383 55.68
7018 ATT 12,117 9,664 7,672 63.32
8220 COLT Internet 10,932 7,970 6,616 60.52
8709 Exodus, Europe 15,006 10,079 8,477 56.49
all the ten together 23,761 4,148 3,936 16.56

• need for more semantic information for
  determining peer-peer relationships

60
overview

• interdomain routing basics
• the problem of determining the relationships
  between autonomous systems
• testing if a solution exists in linear time
• the complexity of finding a good orientation
• an heuristic approach to the general problem
• conclusions

61
conclusions

• we show that the ToR-problem is NP-hard but
• if a solution without invalid paths exists, it
  can be found in linear time by mapping the
  problem to 2SAT
• we propose heuristics for the general problem
  based on the 2SAT mapping
• we show the effectiveness of the heuristics
  against publicly available data sets
• the experiments put in evidence that our
  heuristics performs significantly better than
  state of the art heuristics
• we show that discovering peer-peer relationships
  is a hard problem if one wants to maximize them

62
issues

• how to determine peer-peer relationships once the
  graph is oriented?
• each AS-path is weighted one, irrespectively of
  the size of its prefix. To what extent is this
  correct?
• could snapshots of the same BGP table taken at
  different dates help in better understanding the
  relationships between ASes?
• what if we knew in advance the orientation of
  some of the edges?

63
links

• Lixin Gao, On Inferring Autonomous System
  Relationships in the Internethttp//www-unix.ecs
  .umass.edu/lgao/
• Subramanian, Agarwal, Rexford, and Katz
  Characterizing the Internet Hierarchy from
  Multiple Vantage Pointshttp//www.cs.berkeley.ed
  u/sagarwal/research/BGP-hierarchy/
• Erlebach, Hall, and Schank, Classifying
  Customer-Provider Relationships in the Internet
  http//www.tik.ee.ethz.ch/the/
• website dedicated to the algorithms described in
  this presentation http//www.dia.uniroma3.it/comp
  unet/relationships/

64
acknowledgements

• Subramanian, Agarwal, Rexford, and Katz for
  sharing their data and explanations
• Thomas Erlebach, Alexander Hall, and Thomas
  Schank for their insight
• Debora Donato, Andrea Vitaletti for useful
  discussions
• Massimo Rimondini for computing statistics on the
  graphs

65
questions?
66
computing the relationships between autonomous
systems

• giuseppe di battista
• maurizio patrignani
• maurizio pizzonia

univ. of rome III http//www.dia.uniroma3.it/comp
unet/