Download Routing Algebras

Metarouting
and Network Optimization
(work in progress)
[email protected]
[email protected]
CISS 2006
The Current Situation
• IP Connectivity is implemented
with dynamic routing protocols
• These protocols are few in
number
• Existing protocols tend to get in
the way of network optimization,
network analysis
Routing Algebras
(Sobrinho)
A  (,,, 0,1 )
1970s 1980s: Path Algebra =
Idempotent semiring.
Carre’, Gondran & Minoux
A  (, ,, 0)
2002: Algebra and Algorithms for
QoS Path Computation and Hop-by-Hop
Routing in the Internet.
A  (, , ,,0)
2003: Network Routing and Path Vector
Protocols: Theory and Applications.
Routing Algebras
Generalized Routing Algebras
(Gurney & Griffin, ongoing work)
A  ( S , L,)
S  (, )
L  ( ,)
This is a pre-order

A semi-group
is a semi-group action on 
(1  2 )   1  (2  )
Big Picture
Routing Algebras
Path Algebras
Generalized Routing Algebras
“Best” Routes

min  (S )  {  S |   S :        }
For finite, non-empty S

Total
order
min( S )  { }
Partial
order
min( S )  {1 ,  2 ,,  n }  i #  j
Total
pre-order
Pre-order
min( S )  {1 ,  2 ,,  n }  i   j
min( S )  S1  S 2    S n
 ,   Si :   
  Si    S j :  # 
Weighted Graph, solution
G  (V , E , w : E  )
 (v, s)  min {w( P) | v 
 s}
P
w( s )   s
w( u, v, )   (u, v)  w( v, )
w( P)   ( P)   s
Generalized Dijkstra
for all v in V: d[v]  0
d[s]   s
QV
while Q not empty:
choose u in Q with d[u] in min{d[v] | v in Q};
Q  Q - {u};
for all v in V with (u, v) in E:
if d[v] > (w(u, v)  d[u]) then:
d[v]  w(u, v)  d[u];
Generalized Bellman-Ford
for all v in V: d[v]  0
d[s]   s
QV
for i in {1, 2, ..., |V| - 1}
for all (u, v) in E
if d[v] > (w(u, v)  d[u]]) then
d[v]  w(u, v)  d[u]
-- Negative weight cycle detection
for (u, v) in E
if d[v] > (w(u, v)  d[u]) then:
return false -- Found “neg-weight” cycle
return true
-- No “neg-weight” cycle
Some Important Properties
Monotonicity (M) :
Strict monotonicity (SM) :
Isotonicity (I) :
Strict isotonicity (SI) :
   
   
(  0)
         
         
What makes these algorithms work (for
bounded path algebras)?
• Dijkstra
– Correctness proof uses monotonicity and
isotonicity,
– Loop-freedom for hop-by-hop forwarding uses
strict monontonicity.
• Bellman-Ford
– Correctness proof uses monotonicity,
– Loop-freedom for hop-by-hop forwarding uses
strict monotonicity
Metarouting
(Griffin & Sobrinho 2005)
• A meta-language for Routing Algebras
– Base algebras
– Constructors
• Property Preservation Rules
– Properties of base algebras known,
– Preservation rules for each constructor
• Can be implemented, standardized
Direct Product
( S A , L A ,  A)  ( S B , L B ,  B )  ( S A  S B , L A  L B ,  A   B )
(1 , 1 )  ( 2 , 2 )  (1  A  2 )  (1 B 2 )
( ,  )

(A , B ) (A A  , B B  )
Direct Product
( S A , L A ,  A)  ( S B , L B ,  B )  ( S A  S B , L A  L B ,  A   B )
(1 , 1 )  ( 2 , 2 )  (1  A  2 )  (1 B 2 )
A
B A B
A
B A B
M
M
M
I
I
I
SM
M
M
SI
I
I
M
SM
M
I
SI
I
SM
SM
SM
SI
SI
SI
Distance x Bandwidth
London
Prague
Moscow
(100, 30)
(311, 70)
(10, 80)
(200, 30)
Paris
Rome
(250, 90)
P = Rome  Prague  Moscow = (300, 30)
Q = Rome  Paris  London  Moscow = (571, 70)
min {w(P), w(Q)} = {(300, 30), (571, 70)}
The corresponding path algebra gives
glb
{(300, 30), (571, 70)} = (300, 70)
Lexicographic Product


( S A , L A ,  A)  ( S B , L B ,  B )  ( S A  S A , L A  L B ,  A   B )
(1 , 1 )  ( 2 , 2 )  (1  A  2 )  ((1  A  2 )  (1 B 2 ))
Property Preservation with Lex Product
A

B A B
M
M
SM
M
SM
A

B A B
M
EQ,SI
I
I
SM
EQ,SI
SI
SI
SM
A design pattern:
EQ
EQ
EQ
( EQ             )
SM
  
  
A1  A2  Ai 1  Ai  An
All at least M
SM
Don’t care!
Local Preference, Origin Preference
A  (S A  ( A ,  A ), L A  ( A , A ), A)
LP( A)  (S A , L  ( A ,),l )
 l   
OP( A)  (S A , L  ({},),r )
 r   
(Always M)
Disjoint Union
( S , L A ,  A)  ( S , L B ,  B )  ( S , L A  L B ,  A   B )


left (A )
A  A 
right (B )
B B 
Disjoint Union : Property Preservation
A
B A B
A
B A B
M
M
M
I
I
I
SM
M
M
SI
I
I
M
SM
M
I
SI
I
SM
SM
SM
SI
SI
SI
Scoped Product


AB  ( A  LP( B))  (OP ( A)  B)
( ,  )

(A ,  )
(A A  , )
( , B )
( , B B  )
Q : What is a good mathematical framework for the analysis of
routing algebra metalanguages?
A : CATEGORY THEORY!
AB
Scoped Product
BGP  EBGPIBGP
Scoped Product : Monotonicity Preservation
A
B
AB
SM
M
M
SM
SM
SM
Dependent Lexicographic Product
Ai  ((i , i ), Li ,i )

A  i {0,1,, n} : Ai


 {( i,  )}
i{0 ,1,, n}  Ai
A  ((, ), (,),)


 {( i,  )}
i{0 ,1,, n}  i
(i,  )  ( j,  )  (i  j )  ((i  j )  ( i  ))
(i,  i  ) i  j
(i,  )  ( j ,  )  
0
i j

An application…
This can be viewed as an instance of

A  i {0,1,, n} : Ai
Operations at the Protocol Level

A0  A1  i {0,1} : Ai
P1  ( A  B, M )
P2  ( A, M A )  ( B, M B )  ( A  B, M A  M B )
B
A B
A
B
A B
A B
A B
A
A
B
A B
A
B
…or
( A, M A )  ( B, M B )  ( A  B, M A  M B )
B
A
B
A
A
A
B
A
Adjacencies of B
supported by connectivity
of provided by A
Think of A = OSPF and
B = IBGP ….
OSPF Revisited

( Areas, bellman )  ( IntraArea, Dijkstra)
Something like

(max{ 0,1})  ( flat ( Seq(2)))
Something like

i {0,1,2} : Ai
where
A0  SP (shortest paths)
A1  OP(...)
A2  OP(...)
Challenge
• If you could “roll your own” routing
protocols, what would you do?
• How does this kind of flexibility
change the way you might think
about network optimization?