L11: Algebraic Path Problems with applications to Internet Routing Lectures 7 and 8

Transcription

1 L: Algebraic Path Problems with applications to Internet Routing Lectures 7 and 8 Timothy G. Grifn timothy.grifn@cl.cam.ac.uk Computer Laboratory University of Cambridge, UK Michaelmas Term, 27 tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 7 and / 33 8 Recall our basic iterative algorithm A closer look... A xy I A xk`y AA xky I A xk`y pi, jq Ipi, jq à Api, uqa xky pu, jq u Ipi, jq à Api, uqa xky pu, jq pi,uqpe This is the basis of distributed Bellman-Ford algorithms (as in RIP and BGP) a node i computes routes to a destination j by applying its link weights to the routes learned from its immediate neighbors. It then makes these routes available to its neighbors and the process continues... tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 72 and / 33 8

2 What if we start iteration in an arbitrary state M? In a distributed environment the topology (captured here by A) can change and the state of the computation can start in an arbitrary state (with respect to a new A). Theorem For ď k, A xy M M A xk`y M AA xky M I A xky M If A is q-stable and q ă k, then Ak M A pk q A xky M Ak M A tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 73 and / 33 8 RIP-like example (see RFC 58) Adjacency matrix A Adjacency matrix A f f f f Using AddZerop8, pn, min, `qq but ignoring inl and inr tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 74 and / 33 8

3 RIP-like example counting to convergence (2) The solution A The solution A f f f f tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 75 and / 33 8 RIP-like example counting to convergence (3) The scenario: we arrived at A, but then links tp, 3q, p3, qu fail. So we start iterating using the new matrix A 2. xky Let B K represent A 2 M, where M A. tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 76 and / 33 8

4 RIP-like example counting to convergence (4) B f f B f f B f f B f f B f f B f f tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 7 and / 33 8 RIP-like example counting to convergence (5) B f f B f f B f f B f f B f f tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 78 and / 33 8

5 RIP-like example counting to innity () The solution A The solution A f f f f xky Now let B K represent A 3 M, where M A. tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routingc 27 Lectures 79 and / 33 8 RIP-like example counting to innity (2) B B B f f f f f f.. B B f f f f tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures 7 and / 33 8

6 RIP-like example What s going on? Recall A xky M pi, jq Ak Mpi, jq A pi, jq A (i, j) may be arrived at very quickly but A k Mpi, jq may be better until a very large value of k is reached (counting to convergence) or it may always be better (counting to innity). Solutions? RIP: 8 6 tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures 7 and / 33 8 Other solutions? The Border Gateway Protocol (BGP) BGP exchanges metrics and paths. It avoids counting to innity by throwing away routes that have a loop in the path. The plan... Starting from pn, min, `q we will attempt to construct a semiring (using our lexicographic operators) that has elements of the form pd, Xq, where d is a shortest-path metric and X is a set of paths. Then, by succsessive renements, we will arrive at a BGP-like solution. tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures2 7 and / 33 8

7 A useful construction: The Lifted Product Lifted product semigroup Assume ps, q is a semigroup. Let liftps, q pnp2 S q, ˆ q where Xˆ Y tx y x P X, y P Y u. t, 3, 7u ˆ` t, 3, 7u t2, 4, 6, 8, 2, 34u tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures3 7 and / 33 8 Some rules (remember 2 ď S ) ASpliftpS, qq ô ASpS, q IDpliftpS, qq ô IDpS, q pˆα tαuq ANpliftpS, qq ô TRUE pω tuq CMpliftpS, qq ô CMpS, q SLpliftpS, qq ô ILpS, q _IRpS, q _ pipps, q ^ S 2q IPpliftpS, qq ô SLppS, qq ILpliftpS, qq ô FALSE IRpliftpS, qq ô FALSE tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures4 7 and / 33 8

8 Turn the Crank... ô ô ô ô IPpliftpliftptt, f u, ^qq SLpliftptt, f u, ^q ILptt, f u, ^q _IRptt, f u, ^q _ pipptt, f u, ^q ^ tt, f u 2q FALSE _FALSE _ ptrue ^TRUEq TRUE Note This kind of calculation become more interesting as we introduce more complex constructors and consider more complex properties... tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures5 7 and / 33 8 Let s use lift to construct bi-semigroups... Union with lifted product Assume ps, q is a semigroup. Let union_liftps, q pp n psq, Y, ˆ q where Xˆ Y tx y x P X, y P Y u, and X, Y P P n psq, the set of nite subsets of S. tu is the if has idenity α, then tαu is the a non-empty sequence will be written as rs, s,, s k s when does it have an annihilator for Y? we really need some inference rules here... tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures6 7 and / 33 8

9 pathspeq over graph G pv, Eq pathspeq union_liftpe, q where is sequence concatenation. spwp For our graph G pv, Eq, we will build a shortest paths with paths semiring spwp AddZerop, pn, min, `q ˆ pathspeqq Given an arc pi, jq we will give it a weight of the form Api, jq inlpn, trpi, jqsuq for some n P N. However, for ease of reading we will write and so on. Api, jq pn, trpi, jqsuq tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures7 7 and / 33 8 Recall : RIP-like counting to convergence The solution A The solution A f f Imagine that we have augmented this example with paths. f f tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures8 7 and / 33 8

10 B Let B A be the solution for the (path augmented) matrix A : 2 3 p, tǫuq p, trp, qsuq p, trp, 2qsuq p2, trp, q, p, 3qsuq p, trp, qsuq p, tǫuq p, trp, 2qsuq p, trp, 3qsuq 2 p, trp2, qsuq p, trp2, qsuq p, tǫuq p2, trp2, q, p, 3qsuq 3 p2, trp3, q, p, qsuq p, trp3, qsuq p2, trp3, q, p, 2qsuq p, tǫuq tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures9 7 and / 33 8 B Now, arcs p, 3q and p3, q vanish! With the new (path-augmented) matrix A 2, we start the iteration in the old state B. After one iteration we have B p, 3q p2, trp, q, p, 3qsuq B p, 3q p3, trp, q, p, q, p, 3qs, rp, 2q, p2, q, p, 3qsuq B p2, 3q p2, trp2, q, p, 3qsuq B p3, q p, trp3, 2q, p2, qsuq (stable!) B p3, q p, trp3, 2q, p2, qsuq (stable!) B p3, 2q p, trp3, 2qsuq (stable!) tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures2 7 and / 33 8

11 B 2 After another iteration we have B 2 p, 3q p3, trp, 2q, p2, q, p, 3qsuq B 2 p, 3q p3, trp, q, p, q, p, 3qs, rp, 2q, p2, q, p, 3qsuq B 2 p2, 3q p3, trp2, q, p, q, p, 3qsuq tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures2 7 and / 33 8 B 3 After another iteration we have B 3 p, 3q p4, trp, q, p, q, p, q, p, 3qs, rp, q, p, 2q, p2, q, p, 3qs, rp, 2q, p2, q, p, q, p, 3qsuq B 3 p, 3q p4, trp, q, p, 2q, p2, q, p, 3qs, rp, 2q, p2, q, p, q, p, 3qsuq B 3 p2, 3q p4, trp2, q, p, 2q, p2, q, p, 3qs, rp2, q, p, q, p, q, p, 3qs, rp2, q, p, 2q, p2, q, p, 3qsuq tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures22 7 and / 33 8

12 B Finally, the problem is resolved at the -th iteration: 2 3 p, tǫuq p, trp, qsuq p, trp, 2qsuq p, trp, 2q, p p, trp, qsuq p, tǫuq p, trp, 2qsuq p, trp, 2q, p 2 p, trp2, qsuq p, trp2, qsuq p, tǫuq p, trp2, 3 3 p, trp3, 2q, p2, qsuq p, trp3, 2q, p2, qsuq p, trp3, 2qsuq p, tǫuq tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures23 7 and / 33 8 Recall RIP-like counting to innity It should be clear that we have not yet reached our goal. This example will lead to sets of ever longer and longer paths and never terminate. How do we x this? tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures24 7 and / 33 8

13 Reductions If ps,, bq is a semiring and r is a function from S to S, then r is a reduction if for all a and b in S rpaq rprpaqq 2 rpa bq rprpaq bq rpa rpbqq 3 rpa b bq rprpaq b bq rpa b rpbqq Note that if either operation has an identity, then the rst axioms is not needed. For example, rpaq rpa q rprpaq q rprpaqq tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures25 7 and / 33 8 Reduce operation If ps,, bq is semiring and r is a reduction, then let red r psq ps r, r, b r q where S r ts P S rpsq su 2 x r y rpx yq 3 x b r y rpx b yq Is the result always semiring? tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures26 7 and / 33 8

14 Elementary paths reduction Recall pathspeq pathspeq union_liftpe, q where is sequence concatenation. A path p is elementary if no node is repeated. Dene the reduction rpxq tp P X p is an elementary pathu Semiring of Elementary Paths epathspeq red r ppathspeqq tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures27 7 and / 33 8 Starting in an arbitrary state? tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures28 7 and / 33 8

15 Starting in an arbitrary state? No! using AddZerop8, pn, min, `q ˆ epathspeqq.. B p, tǫuq p, trp, qsuq p, trp, 2qsuq p999, tuq p, trp, qsuq p, tǫuq p, trp, 2qsuq p999, tuq 2 p, trp2, qsuq p, trp2, qsuq p, tǫuq p999, tuq p, tǫuq. f f tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures29 7 and / 33 8 Solution: use another reduction! r pinrp8qq inrp8q " inrp8q if W tu rpinlqps, W q inlps, W q otherwise Now use this instead red r paddzerop8, pn, min, `q ˆ epathspeqqq tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures3 7 and / 33 8

16 Starting in an arbitrary state? B and B 2 3 p, tǫuq p, trp, qsuq p, trp, 2qsuq p2, trp, q, p, 3qsuq p, trp, qsuq p, tǫuq p, trp, 2qsuq p, trp, 3qsuq 2 p, trp2, qsuq p, trp2, qsuq p, tǫuq p2, trp2, q, p, 3qsuq 3 p2, trp3, q, p, qsuq p, trp3, qsuq p2, trp3, q, p, 2qsuq p, tǫuq 2 3 p, tǫuq p, trp, qsuq p, trp, 2qsuq p2, trp, q, p, 3qsuq p, trp, qsuq p, tǫuq p, trp, 2qsuq 8 2 p, trp2, qsuq p, trp2, qsuq p, tǫuq p2, trp2, q, p, 3qsuq p, tǫuq f f tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures3 7 and / 33 8 Starting in an arbitrary state? B 2 and B p, tǫuq p, trp, qsuq p, trp, 2qsuq p3, trp, 2q, p2, q, p, 3qsuq p, trp, qsuq p, tǫuq p, trp, 2qsuq 8 2 p, trp2, qsuq p, trp2, qsuq p, tǫuq p3, trp2, q, p, q, p, 3qsuq p, tǫuq f f 2 3 p, tǫuq p, trp, qsuq p, trp, 2qsuq 8 p, trp, qsuq p, tǫuq p, trp, 2qsuq 8 2 p, trp2, qsuq p, trp2, qsuq p, tǫuq p, tǫuq f f tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures32 7 and / 33 8

17 Homework 3 (due 6 November) Prove SLpliftpS, qq ô ILpS, q _IRpS, q _ pipps, q ^ S 2q Characterise exactly when union_liftps, q is a semiring. tgg22 (cl.cam.ac.uk) L: Algebraic Path Problems with applications to Internet T.G.Grifn Routing c 27 Lectures33 7 and / 33 8