1. Metarouting and Network Optimization timothy.griffin@cl.cam.ac.uk CISS 2006 alex.gurney@cl.cam.ac.uk (work in progress)

2. 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

3. Routing Algebras (Sobrinho) 2002: Algebra and Algorithms for QoS Path Computation and Hop-by-Hop Routing in the Internet. 2003: Network Routing and Path Vector Protocols: Theory and Applications. Routing Algebras 1970s 1980s: Path Algebra = Idempotent semiring. Carre’, Gondran & Minoux

4. Generalized Routing Algebras (Gurney & Griffin, ongoing work) is a semi-group action on  A semi-group This is a pre-order

5. Big Picture Path Algebras Generalized Routing Algebras Routing Algebras

6. “Best” Routes For finite, non-empty Total order Partial order Total pre-order Pre-order

7. Weighted Graph, solution

8. Generalized Dijkstra for all v in V: d[v]  d[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];

9. Generalized Bellman-Ford for all v in V: d[v]  d[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

10. Some Important Properties Monotonicity (M) : Strict monotonicity (SM) : Isotonicity (I) : Strict isotonicity (SI) :

11. 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

12. 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

13. Direct Product

14. M SM M M M M M M I SI I I I I I I

15. Distance x Bandwidth LondonMoscow Prague Rome Paris (250, 90) (311, 70) (100, 30) (200, 30)(10, 80) P = Rome  Prague  Moscow = (300, 30) Q = Rome  Paris  London  Moscow = (571, 70) min {w(P), w(Q)} = {(300, 30), (571, 70)} {(300, 30), (571, 70)} = (300, 70)The corresponding path algebra gives

16. Lexicographic Product

17. Property Preservation with Lex Product M SM M M M EQ,SI II SI A design pattern: SI EQ All at least MSM Don’t care! SM

18. Local Preference, Origin Preference (Always M)

19. Disjoint Union

20. Disjoint Union : Property Preservation M SM M M M M M M I SI I I I I I I

21. Scoped Product Q : What is a good mathematical framework for the analysis of routing algebra metalanguages? A : CATEGORY THEORY!

22. Scoped Product

23. Scoped Product : Monotonicity Preservation SM M M

24. Dependent Lexicographic Product

25. An application… This can be viewed as an instance of

26. Operations at the Protocol Level A A B B A B A B

27. …or A A A A A B B B Adjacencies of B supported by connectivity of provided by A Think of A = OSPF and B = IBGP ….

28. OSPF Revisited Something like where (shortest paths) Something like

29. 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?