1. computing the relationships between autonomous systems
giuseppe di battista
maurizio patrignani
maurizio pizzonia
univ. of rome III

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

3. the routing process router 2 router 3 router 1
lan send to
directly connected
router 1
everything else
router 2
router 1
router 3
lan send to
directly connected
router 4
router 2
lan send to
router 5
router 3
router 1
lan send to
router 6
router 2
directly connected
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

6. hierarchical routing more 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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

7. route injection
lan send to
router 1
router 2
router 1
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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

8. exterior gateway protocol
approach:
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

9. exterior gateway protocol
approach:
hide the interior part of all organizations
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

10. represent the internal targets
each border router represents its internal targets as if they were local
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

12. solve the routing problem
solve the routing problem on the simplified graph
based on political considerations
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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, …)
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
/24
AS100
router 1
router 2
AS200
ip traffic
(to be delivered to /24)
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

15. using AS-path to avoid cycles
/24
/24
40 10
/24 10
/24
AS10
/24
AS60
/24
/24 10
AS212
legenda:
prefix
as-path
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

16. using prepending to implement preferences
AS40
AS31
/24
/24
40 10
/24 10
/24
AS10
/24
AS60
/24
AS212
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

17. how to ensure cooperation?
AS40
AS31
/24
/24
40 10
/24 10
/24
AS10
AS60
/24 10
/24
212 10
AS212
why AS212 should propagate AS10 announcement?
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

18. the scenario not available route peer-peer provider customer peer-peer
AS1
AS2
provider
customer
peer-peer
AS3
AS4
AS5
sibling-sibling
AS6
AS7
AS8
AS9
not available route
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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)
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
customer-provider relationship
peer-peer relationship
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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-1
type-2
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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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 i
/23 i i
/19 i i i
extract AS paths and eliminate prepending
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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 i
/23 i i
/19 i i i
extract AS paths and eliminate prepending
14318
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

26. building the corresponding AS graph
This is the graph corresponding to the previous BGP table lines. If you give an arbitrary orientation to some edges of this graph, some paths are valid, some other paths are not.
AS11334
AS14318
AS7660
AS13099
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

27. an orientation for the AS graph
This is the graph corresponding to the previous BGP table lines. If you give an arbitrary orientation to some edges of this graph, some paths are valid, some other paths are not.
AS11334
AS14318
AS7660
AS13099
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
The classification of AS relationships in the Internet is a hot research topic. Its relevance has been confirmed by the interest of other research groups on the same subject
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
There may be AS-paths that touch one of the two ASes without using the peer-peer edge, and these are not affected by the label we give to the edge itself. All AS-paths traversing the edge, though, must arrive the two ASes from left and right through uphill paths. It follows that the edge could be arbitrarily directed without producing invalid paths.
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

32. example of orientable AS graph
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

33. an orientation leaving all valid paths
AS8709
AS8938
AS3561
AS6893
AS4197
AS3967
AS8843
AS7018
AS15493
AS3582
AS1740
AS10311
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

34. a different representation
AS3561
AS7018
AS3967
AS1740
AS8709
AS10311
AS8938
AS3582
AS4197
AS8843
AS6893
AS15493
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

35. example of not orientable AS graph
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

36. “chestnuts” and contradictions
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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)
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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 v1 v2 v3 v4 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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

40. construction of the 2SAT instance
x1,2
x2,3
x3,4
x4,5 v1 v2 v3 v4 v5
(x1,2  x2,3)  (x2,3  x3,4)  (x3,4  x4,5)
x1,2
x2,3
x3,4
x4,5 v1 v2 v3 v4 v5
x1,2
x2,3
x3,4
x4,5 v1 v2 v3 v4 v5
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
x3,4
x4,5
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)
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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(n+m+q) 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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

45. AS graphs from the various sources http://www. cs. berkeley
AS Name
Apr 18, 2001
Apr 6, 2002
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
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
13,004
59,745
12,518
15,628
5388
Energis Squared
58,832
10,411
13,259
117,003
12,659
16,822
7018
AT&T
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
2516
KDDI
126,414
10,790
17,735
2548
MaeWest
80,549
10,583
15,249
6893 CW
70,265
10,523
14,359
telnet sources
web sources
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

46. solvable without invalid paths?
AS #
AS name
Apr 18, 2001
Apr 6, 2002
orientable w/o invalid paths 1
Genuity
yes
1740
CERFnet
not available
3549
Globalcrossing
3582
Univ.of Oregon no
3967
Exodus Comm.
4197
Global Online Japan
5388
Energis Squared
7018
AT&T
8220
COLT Internet
8709
Exodus, Europe
1755
Ebone
2516
KDDI
2548
MaeWest
6893 CW
telnet sources
web sources
the TOR problem has a solution without invalid paths for several BGP routing tables
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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) > f(x), false otherwise 1 2 3 4
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

49. the general problem is NP-hard
MAX2SAT
instance
MAX2SAT
assignment
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

50. example (x1  x2)  (x2  x3)  (x2  x4)  (x1  x4)  (x3  x4)
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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) x x y y
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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 = x*2
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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
0.13
3582
Univ.of Oregon
0.57
3967
Exodus Comm.
0.42
4197
Global Online Japan
0.46
5388
Energis Squared
7018
AT&T
0.63
0.21
8220
COLT Internet
0.22
8709
Exodus, Europe
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
noi abbiamo lavorato sui loro dati, preprocessati (insiemi di path)
loro hanno già tolto I duplicati consecutivi
web sources
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

56. a deeper comparison SARK algorithm SAT 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
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
SAT algorithm
number of nodes
10897
number of edges
23690
peering edges
percentage of total 0%
path covering distribution
minimum: 1
average: 80
maximum: 19918
strongly connected components
10312
scc's to edges ratio
size distribution:
10311 scc's of size 1
1 scc of size 586
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
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
size distribution:
18 cc's of size 2
1 cc of size 453
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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 4 v6 v3 6 5 v5 v4
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
AT&T
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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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?
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

63. links
Lixin Gao, “On Inferring Autonomous System Relationships in the Internet”
Subramanian, Agarwal, Rexford, and Katz “Characterizing the Internet Hierarchy from Multiple Vantage Points”
Erlebach, Hall, and Schank, “Classifying Customer-Provider Relationships in the Internet”
website dedicated to the algorithms described in this presentation
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

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
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

65. questions?
Seminario Interdipartimentale di Algoritmica – 17 Marzo 2003

66. computing the relationships between autonomous systems
giuseppe di battista
maurizio patrignani
maurizio pizzonia
univ. of rome III