1. Network Layer r Introduction r Datagram networks r IP: Internet Protocol m Datagram format m IPv4 addressing m ICMP r What’s inside a router r Routing algorithms m Link state m Distance Vector r Routing in the Internet m RIP m OSPF m BGP r Multicast routing 4-1

2. 1 2 3 0111 value in arriving packet’s header routing algorithm local forwarding table header value output link 0100 0101 0111 1001 32213221 Interplay between routing and forwarding [1] Fact: Forwarding is based on a forwarding/routing table. [2] Question: how do we build up the routing table? Answer: routing alg. 4-2

3. u y x wv z 2 2 1 3 1 1 2 5 3 5 Graph: G = (N,E) N = set of routers = { u, v, w, x, y, z } E = set of links ={ (u,v), (u,x), (v,x), (v,w), (x,w), (x,y), (w,y), (w,z), (y,z) } Graph abstraction Remark: Graph abstraction is useful in other network contexts Example: P2P, where N is set of peers and E is set of TCP connections 4-3

4. Graph abstraction: costs u y x wv z 2 2 1 3 1 1 2 5 3 5 c(x,x’) = cost of link (x,x’) - e.g., c(w,z) = 5 cost could always be 1, or inversely related to bandwidth, or inversely related to congestion Cost of path (x 1, x 2, x 3,…, x p ) = c(x 1,x 2 ) + c(x 2,x 3 ) + … + c(x p-1,x p ) Question: What’s the least-cost path between u and z ? Routing algorithm: algorithm that finds least-cost path 4-4

5. Routing Algorithm classification Global or decentralized information? Global: r all routers have complete topology, link cost info r “link state” algorithms Decentralized: r router knows physically- connected neighbors, link costs to neighbors r iterative process of computation, exchange of info with neighbors r “distance vector” algorithms Static or dynamic? Static: r routes change slowly over time Dynamic: r routes change more quickly m periodic update m in response to topology or link cost changes 4-5

6. Network Layer r Introduction r Datagram networks r IP: Internet Protocol m Datagram format m IPv4 addressing m ICMP r What’s inside a router r Routing algorithms m Link state m Distance Vector r Routing in the Internet m RIP m OSPF m BGP r Multicast routing 4-6

7. A Link-State Routing Algorithm Dijkstra’s algorithm r net topology, link costs known to all nodes m accomplished via “link state broadcast” m all nodes have same info r computes least cost paths from one node (‘source”) to all other nodes m gives forwarding table for that node r iterative: after k iterations, know least cost path to k dests Notation:  c(x,y): link cost from node x to y; = ∞ if not direct neighbors  D(v): current value of cost of path from source to dest. v  p(v): predecessor node along path from source to v  N': set of nodes whose least cost path definitively known 4-7

8. Reliable Flooding of LSP r The Link State Packet includes: m The ID of the router that created the LSP m List of directly connected neighbors, and cost m Sequence number m TTL r Reliable Flooding m Resend LSP over all links other than incident link, if the sequence number is newer. Otherwise drop it. r Link State Detection: m Link layer failure m Loss of “hello” packets 4-8

9. Dijsktra’s Algorithm 1 Initialization: 2 N' = {u} 3 for all nodes v 4 if v adjacent to u 5 then D(v) = c(u,v) 6 else D(v) = ∞ 7 8 Loop 9 find w not in N' such that D(w) is a minimum 10 add w to N' 11 update D(v) for all v adjacent to w and not in N' : 12 D(v) = min( D(v), D(w) + c(w,v) ) 13 /* new cost to v is either old cost to v or known 14 shortest path cost to w plus cost from w to v */ 15 until all nodes in N' 4-9

10. Dijkstra’s algorithm: example Step 0 1 2 3 4 5 N' u ux uxy uxyv uxyvw uxyvwz D(v),p(v) 2,u D(w),p(w) 5,u 4,x 3,y D(x),p(x) 1,u D(y),p(y) ∞ 2,x D(z),p(z) ∞ 4,y u y x wv z 2 2 1 3 1 1 2 5 3 5 4-10

11. Dijkstra’s algorithm: example (2) u y x wv z Resulting shortest-path tree from u: v x y w z (u,v) (u,x) destination link Resulting forwarding table in u: 4-11

12. Dijkstra’s algorithm, discussion Algorithm complexity: n nodes r each iteration: need to check all nodes, w, not in N r n(n+1)/2 comparisons: O(n 2 ) r more efficient implementations possible: O(nlogn) Oscillations possible: r e.g., link cost = amount of carried traffic A D C B 1 1+e e 0 e 1 1 0 0 A D C B 2+e 0 0 0 1+e 1 A D C B 0 2+e 1+e 1 0 0 A D C B 2+e 0 e 0 1+e 1 initially … recompute routing … recompute 4-12

13. Network Layer r Introduction r Datagram networks r IP: Internet Protocol m Datagram format m IPv4 addressing m ICMP r What’s inside a router r Routing algorithms m Link state m Distance Vector r Routing in the Internet m RIP m OSPF m BGP r Multicast routing 4-13

14. Distance Vector Algorithm Bellman-Ford Equation (dynamic programming) Define d x (y) := cost of least-cost path from x to y Then d x (y) = min {c(x,v) + d v (y) } where min is taken over all neighbors v of x v 4-14

15. Bellman-Ford example u y x wv z 2 2 1 3 1 1 2 5 3 5 Clearly, d v (z) = 5, d x (z) = 3, d w (z) = 3 d u (z) = min { c(u,v) + d v (z), c(u,x) + d x (z), c(u,w) + d w (z) } = min {2 + 5, 1 + 3, 5 + 3} = 4 Node that achieves minimum is next hop in shortest path ➜ forwarding table B-F equation says: 4-15

16. Distance Vector Algorithm r D x (y) = estimate of least cost from x to y r Distance vector: D x = [D x (y): y є N ] r Node x knows cost to each neighbor v: c(x,v) r Node x maintains D x = [D x (y): y є N ] r Node x also maintains its neighbors’ distance vectors m For each neighbor v, x maintains D v = [D v (y): y є N ] 4-16

17. Distance vector algorithm (4) Basic idea: r Each node periodically sends its own distance vector estimate to neighbors r When a node x receives new DV estimate from neighbor, it updates its own DV using B-F equation: D x (y) ← min v {c(x,v) + D v (y)} for each node y ∊ N  Under minor, natural conditions, the estimate D x (y) converge to the actual least cost d x (y) 4-17

18. Distance Vector Algorithm (5) Iterative, asynchronous: each local iteration caused by: r local link cost change r DV update message from neighbor Distributed: r each node notifies neighbors only when its DV changes m neighbors then notify their neighbors if necessary wait for (change in local link cost of msg from neighbor) recompute estimates if DV to any dest has changed, notify neighbors Each node: 4-18

19. x y z x y z 0 2 7 ∞∞∞ ∞∞∞ from cost to from x y z x y z 0 2 3 from cost to x y z x y z 0 2 3 from cost to x y z x y z ∞∞ ∞∞∞ cost to x y z x y z 0 2 7 from cost to x y z x y z 0 2 3 from cost to x y z x y z 0 2 3 from cost to x y z x y z 0 2 7 from cost to x y z x y z ∞∞∞ 710 cost to ∞ 2 0 1 ∞ ∞ ∞ 2 0 1 7 1 0 2 0 1 7 1 0 2 0 1 3 1 0 2 0 1 3 1 0 2 0 1 3 1 0 2 0 1 3 1 0 time x z 1 2 7 y node x table node y table node z table D x (y) = min{c(x,y) + D y (y), c(x,z) + D z (y)} = min{2+0, 7+1} = 2 D x (z) = min{c(x,y) + D y (z), c(x,z) + D z (z)} = min{2+1, 7+0} = 3 4-19

20. Distance Vector: link cost changes Link cost changes: r node detects local link cost change r updates routing info, recalculates distance vector r if DV changes, notify neighbors “good news travels fast” x z 1 4 50 y 1 At time t 0, y detects the link-cost change, updates its DV, and informs its neighbors. At time t 1, z receives the update from y and updates its table. It computes a new least cost to x and sends its neighbors its DV. At time t 2, y receives z’s update and updates its distance table. y’s least costs do not change and hence y does not send any message to z. 4-20

21. Bellman-Ford Algorithm Questions: 1. How long can the algorithm take to run? 2. How do we know that the algorithm always converges? 3. What happens when link costs change, or when routers/links fail? Topology changes make life hard for the Bellman-Ford algorithm… 4-21

22. A Problem with Bellman-Ford R4R4 R3R3 R2R2 R1R1 111 “Bad news travels slowly” Consider the calculation of distances to R 4 : ………… 5,R 2 4,R 3 5,R 2 3 3,R 2 4,R 3 3,R 2 2 2,R 3 3,R 2 1 1, R 4 2,R 3 3,R 2 0 R3R3 R2R2 R1R1 Time “Counting to infinity” R 3 R 4 fails 4-22

23. Counting to Infinity Problem Solutions 1. Set infinity = “some small integer” (e.g. 16). Stop when count = 16. 2. Split Horizon: Because R 2 received lowest cost path from R 3, it does not advertise cost to R 3 3. Split-horizon with poison reverse: R 2 advertises infinity to R 3 m R2 gets to R4 thru R3 4. There are many problems with (and fixes for) the Bellman-Ford algorithm. 4-23

24. Comparison of LS and DV algorithms Message complexity r LS: with n nodes, E links, O(nE) msgs sent r DV: exchange between neighbors only m convergence time varies Speed of Convergence r LS: O(n 2 ) algorithm requires O(nE) msgs m may have oscillations r DV: convergence time varies m may be routing loops m count-to-infinity problem Robustness: what happens if router malfunctions? LS: m node can advertise incorrect link cost m each node computes only its own table DV: m DV node can advertise incorrect path cost m each node’s table used by others error propagate thru network 4-24

25. Comparison of LS and DV algorithms Space requirement r LS: Maintain entire topology r DV: Maintain only neighbor state 4-25