Interplay between routing and forwarding routing algorithm Routing - PDF document

Interplay between routing and forwarding routing algorithm Routing Algorithms and Routing local forwarding table header value output link in the Internet 0100 3 0101 2 0111 2 1001 1 value in arriving packet’s header 1 0111 2 3 Network Layer 4-1 Network Layer 4-2 Graph abstraction Graph abstraction: costs 5 5 • c(x,x’) = cost of link (x,x’) 3 v w 3 v w 5 2 5 2 - e.g., c(w,z) = 5 u z 2 1 u z 2 1 3 3 1 1 • cost could always be 1, or 2 x y 2 x y Graph: G = (N,E) 1 inversely related to bandwidth, 1 or inversely related to N = set of routers = { u, v, w, x, y, z } 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 ) E = set of links ={ (u,v), (u,x), (v,x), (v,w), (x,w), (x,y), (w,y), (w,z), (y,z) } Question: What’s the least-cost path between u and z ? Remark: Graph abstraction is useful in other network contexts Routing algorithm: algorithm that finds least-cost path Example: P2P, where N is set of peers and E is set of TCP connections Network Layer 4-3 Network Layer 4-4 Routing Algorithm classification A Link-State Routing Algorithm Global or decentralized Static or dynamic? Dijkstra’s algorithm Notation: information? Static: net topology, link costs known ❒ c(x,y): link cost from node x Global: ❒ to all nodes ❒ routes change slowly over to y; = ∞ if not direct all routers have complete ❒ ❍ accomplished via “link neighbors topology, link cost info time state broadcast” “link state” algorithms ❒ D(v): current value of cost of Dynamic: ❒ ❍ all nodes have same info Decentralized: path from source to dest. v ❒ routes change more quickly computes least cost paths ❒ router knows physically- ❒ p(v): predecessor node along ❒ from one node (‘source”) to all ❍ periodic update connected neighbors, link path from source to v other nodes costs to neighbors ❍ in response to link cost ❒ N': set of nodes whose least ❍ gives forwarding table for iterative process of changes ❒ cost path definitively known that node computation, exchange of info iterative: after k iterations, with neighbors ❒ know least cost path to k “distance vector” algorithms ❒ dest.’s Network Layer 4-5 Network Layer 4-6 1

Dijsktra’s Algorithm Dijkstra’s algorithm: example 1 Initialization: Step N' D(v),p(v) D(w),p(w) D(x),p(x) D(y),p(y) D(z),p(z) 2 N' = {u} 0 u 2,u 5,u 1,u ∞ ∞ 3 for all nodes v 1 ux 2,u 4,x 2,x ∞ 4 if v adjacent to u 2 uxy 2,u 3,y 4,y 5 then D(v) = c(u,v) 3 uxyv 3,y 4,y 6 else D(v) = ∞ 4 uxyvw 4,y 7 5 uxyvwz 8 Loop 9 find w not in N' such that D(w) is a minimum 5 10 add w to N' 11 update D(v) for all v adjacent to w and not in N' : 3 v w 5 12 D(v) = min( D(v), D(w) + c(w,v) ) 2 13 /* new cost to v is either old cost to v or known u z 2 1 3 14 shortest path cost to w plus cost from w to v */ 1 2 x 15 until all nodes in N' y 1 Network Layer 4-7 Network Layer 4-8 Dijkstra’s algorithm, discussion Distance Vector Algorithm (1) Algorithm complexity: n nodes Bellman-Ford Equation (dynamic programming) ❒ each iteration: need to check all nodes, w, not in N Define ❒ n(n+1)/2 comparisons: O(n 2 ) d x (y) := cost of least-cost path from x to y ❒ more efficient implementations possible: O(nlogn) Oscillations possible: ❒ e.g., link cost = amount of carried traffic Then A A A 1 A d x (y) = min {c(x,v) + d v (y) } 1+e 2+e 0 0 2+e 2+e 0 D B D B D 0 B D B 0 1+e 1 0 0 1+e 1 e 0 0 0 1 1+e e 0 C C C C 1 1 where min is taken over all neighbors of x e … recompute … recompute … recompute initially routing Network Layer 4-9 Network Layer 4-10 Bellman-Ford example (2) Distance Vector Algorithm (3) 5 ❒ D x (y) = estimate of least cost from x to y Clearly, d v (z) = 5, d x (z) = 3, d w (z) = 3 3 v w 5 ❒ Distance vector: D x = [D x (y): y є N ] 2 B-F equation says: u 2 z 1 3 ❒ Node x knows cost to each neighbor v: 1 d u (z) = min { c(u,v) + d v (z), 2 x y c(x,v) 1 c(u,x) + d x (z), c(u,w) + d w (z) } ❒ Node x maintains D x = [D x (y): y є N ] = min {2 + 5, ❒ Node x also maintains its neighbors’ 1 + 3, distance vectors 5 + 3} = 4 Node that achieves minimum is next ❍ For each neighbor v, x maintains hop in shortest path ➜ forwarding table D v = [D v (y): y є N ] Network Layer 4-11 Network Layer 4-12 2

Distance vector algorithm (4) Distance Vector Algorithm (5) Iterative, asynchronous: Each node: Basic idea: each local iteration caused ❒ Each node periodically sends its own distance by: wait for (change in local link vector estimate to neighbors ❒ local link cost change cost of msg from neighbor) ❒ When node a node x receives new DV estimate ❒ DV update message from from neighbor, it updates its own DV using B-F neighbor equation: recompute estimates Distributed: D x (y) ← min v {c(x,v) + D v (y)} for each node y ∊ N ❒ each node notifies ❒ Under minor, natural conditions, the estimate if DV to any dest has neighbors only when its changed, notify neighbors D x (y) converge the actual least cost d x (y) DV changes ❍ neighbors then notify their neighbors if necessary Network Layer 4-13 Network Layer 4-14 D x (z) = min{ c(x,y) + D x (y) = min{c(x,y) + D y (y), c(x,z) + D z (y)} D y (z), c(x,z) + D z (z) } Distance Vector: link cost changes = min{2+0 , 7+1} = 2 node x table = min{2+1 , 7+0} = 3 cost to cost to cost to x y z x y z x y z Link cost changes: 1 x 0 2 7 x 0 2 3 x 0 2 3 y from from ❒ node detects local link cost change from y y 2 0 1 4 1 ∞ ∞ ∞ y 2 0 1 z z 7 1 0 ❒ updates routing info, recalculates x ∞ ∞ ∞ z z 3 1 0 50 node y table distance vector cost to cost to cost to y x y z x y z x y z ❒ if DV changes, notify neighbors 2 1 x ∞ ∞ x 0 2 7 ∞ x 0 2 3 z x At time t 0 , y detects the link-cost change, updates its DV, from from from y y 7 2 0 1 y 2 0 1 “good and informs its neighbors. 2 0 1 z z ∞ ∞ ∞ 7 1 0 z 3 1 0 news node z table cost to cost to At time t 1 , z receives the update from y and updates its table. cost to travels x y z x y z It computes a new least cost to x and sends its neighbors its DV. x y z fast” x 0 2 7 x 0 2 3 x ∞ ∞ ∞ from from from At time t 2 , y receives z ’s update and updates its distance table. y y 2 0 1 y 2 0 1 ∞ ∞ ∞ y ’s least costs do not change and hence y does not send any z z z 3 1 0 3 1 0 7 1 0 message to z . time Network Layer 4-15 Network Layer 4-16 Distance Vector: link cost changes Comparison of LS and DV algorithms Message complexity Robustness: what happens if Link cost changes: router malfunctions? 60 LS: with n nodes, E links, O(nE) good news travels fast ❒ y ❒ LS: msgs sent 4 1 bad news travels slow - “count ❒ ❍ node can advertise incorrect DV: exchange between x z ❒ to infinity” problem! link cost 50 neighbors only ❍ each node computes only its 44 iterations before ❒ ❍ convergence time varies own table algorithm stabilizes: see text DV: Speed of Convergence Poissoned reverse: ❍ DV node can advertise LS: O(n 2 ) algorithm requires ❒ If Z routes through Y to get incorrect path cost ❒ O(nE) msgs to X : ❍ each node’s table used by ❍ may have oscillations others ❍ Z tells Y its (Z’s) distance to DV: convergence time varies ❒ • error propagate thru network X is infinite (so Y won’t route ❍ may be routing loops to X via Z) ❍ count-to-infinity problem will this completely solve ❒ count to infinity problem? Network Layer 4-17 Network Layer 4-18 3