Distance Vector Routing

Transcription

1 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Routing in General Distance Vector Routing Jean Yves Le Boudec Fall 22 Contents. Routing in General 2. Distance vector: theory. Distance vector: practice 4. Dynamic Metrics Textbook Section 5.., The control plane 2

2 . Introduction Why were routing protocols invented IP assumes routing tables are maintained at hosts and routers used by Packet Forwarding Routing = control method maintain routing tables automatically in routers Packet Forwarding for every packet done in real time Routing computation of routing tables or data structures for unicast and multicast normally only between routers non real time: latency up to 2 minutes uses dedicated protocols (RIP, OSPF, EIGRP (Cisco) for unicast and DVMRP, M OSPF, PIM) Routing tables in hosts are normally derived from configuration info What a routing protocol does find reachable destinations find best paths towards destinations best = distance to destination is minimal Distance = computed using some metric Static metric does not depend on the network state; for example: number of hops link capacity and static delay Administrative cost Dynamic metric depend on the network state link load current delay see end of section 4 2

3 How routing protocols work There are many different ways, that can be classified as Static / out of band: a separate application writes values of routing tables Distance Vector: Every router knows only: its neighbours + distances to all destination (used inside domains interior routing) this module and lab 4 Link State: Every router has a detailed map of entire network (used inside domains interior routing) module LS Path vector Every router knows only: its neighbours + paths to all destination (used between domains exterior routing) module BGP and lab 5 Source routing Used in some ad hoc mobile networks and in IBM source routing LANs 5 Source Routing SA DA RI data A B A IS A B IS 4 2 IS A B IS 4 B source writes route into packet header (IP header extension) router reads next hop from packet header, moves pointer route discovered by the source, flooding 6

4 What it does: 2. Distance Vector: Theory Computes best paths to all destinations Fully distributed Using as only information the distances from self to all destinations How it works uses Distributed Bellman Ford Note: individual link cost is setup by configuration 7 The Centralized Bellman Ford Algorithm What: Given a directed graph with links costs A(i,j), computes the best path from i to j for any couple (i,j). We assume A(i, j) > and A(i,j) = when i and j are not connected. How: Take for example j= and let p(i) be the cost of the best path from ito. Define p k (i) as the cost of the best path from i to in at most k hops. Let p () =, p (i) = for i. Algorithm BF-C Theorem. If the network is fully connected, the algorithm stops at the latest for k=n and then p k (i)=p(i) for all i 2. The shortest path from i to is defined by pred(i) = Argmin j i [A(i,j) + p(j)]. 8 4

5 Example Apply the theorem: write p k (i) and draw the shortest paths to node for 5. 9 Solution k\i

6 Computation of shortest path tree The predecessor of node is obtained by looking for that minimizes, i p(i) 4 2: 6 7 5: 4 : 5 5 Assume we start from other initial conditions. What will happen? see on graph. Convergence to the correct distances 2. There may or may not be convergence. The algorithm will not converge to the correct distances 4. I don t know 22% 46% % 2% 2 6

7 What is the vector at the end of the first step (? see on graph. (,,,,2) 2. (,,6,,2). (,,,,2) 4. (,,6,6,2) 5. None of the above 6. I don t know 6 8% 8% 8% 5% 8% % Solution k\i i p(i) 4 On this example there is convergence to the correct values 4 7

8 Assume we break some links and start from these initial conditions. What will happen? see on graph. The algorithm will terminate 2. The algorithm may or may not terminate. The algorithm will not terminate 4. I don t know 57% % 6 % 2% What is the vector at the end of the first step (? see on graph. (,,,2,2) 2. (,8,,2,2). (,,,5,) 4. (,,,5,2) 5. None of the above 6. I don t know 5% 6% 9% 7% 7% 7%

9 k\i Solution i p(i) On this example the algorithm does not terminate But if we allow infinite limites, there is convergence to the correct values, i.e. to for nodes and 5 Nodes and 5 increase each other s value (count to infinity) 7 Impact of Initial Condition Theorem The BF C algorithm converges in a finite number of steps to the correct values for all initial conditions and for every node i that is connected to If there is no path from i to, the algorithm lets p k (i) converge to 8 9

10 Proof We do the proof assuming all nodes are connected.. Let p k be the vector p k [i], i=2,. Let B be the mapping that transforms an array x[i] i=2 into the array Bx defined for i by Bx[i]=min j i, j [A(i,j) + x(j)] Let b be the array defined for i by b[i]= A(i,) The algorithm can be rewritten in vector form as () p k = B p k b where is the pointwise minimum 2. Eq () is a min plus linear equation and the operator B satisfies B(x y)= Bx By. Thus, Eq() can be solved using min plus algebra into (2) p k = B k p B k b Bb b. Define the array e for i by e[i]=. Let p =e. Eq (2) becomes () p k = B k b Bb b. Now we have the Bellman Ford algorithm with classical initial conditions, thus, by Theorem : (4) for k n : B k b Bb b = q where q[i] is the distance from i to. 4. We can rewrite Eq(2) for k n as (5) p k = B k p q 5. B k p [i] can be written as A[i,i ]+ A[i,i 2 ]+ + A[i k,i k ]+ p[i k ] thus (6) B k p [i] k a, where a is the minimum of all A[i,j]. Thus B k p [i] tends to when k grows. Thus for k large enough, B k p is larger than q and can be ignored in Eq(5). In other words, for k large enough : (6) p k = q 9 Distributed Bellman Ford BF C can be used to compute p(i) i.e. find the shortest path. However, this is not its main interest, because there is a better algorithm (Dijkstra) that can be used in a centralized way. But: it can be distributed, as follows. Distributed Bellman Ford Algorithm, BF prelim node maintains an estimate of the distance to node ; node also keeps a record of latest values for all neighbors initial conditions are arbitrary but at all steps; from time to time, i sends its value to all neighbours when receives an updated value from neighbor, node recomputes : eq () min A i, j q j pred(i) is set to a value of j that achieves the min in eq() Theorem: if the time to reliably send a message is bounded by, the algo converges to the same result as the centralized version in at most time units (if the network is fully connected). 2

11 Distributed Bellman Ford BF prelim A possible run of algorithm BF prelim. The table shows the successive values of -> 2 2 -> 4 2 -> -> 5 4 -> 5 -> 4 2 -> 5 5 -> 2 5 -> i min,, 2 Super Duper Bellman Ford BF prelim requires a node to remember all previously received estimates q(j) for all neighbours, even if they are not the best ones A possible modification would be to replace eq() by Super Duper Bellman Ford: when receives an updated value from neighbor, node recomputes : eq (s) min,, Node i now needs to store only its own estimate Q. does this work? 22

12 Does Super Duper Bellman Ford work? when receives an updated value from neighbor, node recomputes : eq (s) min,,. Yes, regardless of initial condition 2. Only if initial conditions are, and at time. It may fail, even with initial conditions as in I don t know 44% 7% 6% 2% Solution Super Duper can only decrease. So if we start from initial conditions as in example «Impact of Initial Conditions», the algorithm will not converge to the right value. It gets «stuck» with a low value. It is possible to show that it works if all initial conditions are above the final values, for example and initially. But even then, it will not work if there is a topology change, since this is equivalent to starting from different initial conditions. Indeed, requiring that initial conditions are correct means that we are able to reset the entire network whenever there is a topology change, which is not desirable. 24 2