Chapter 4: Network Layer r 4. 1 Introduction r 4.2 Virtual circuit and datagram networks r 4.3 IP: Internet Protocol  Datagram format  IPv4 addressing r 4.4 Routing: Concepts and Algorithms  Introduction  Math Detour – Graph Theory  Routing via Broadcast – PI, PIF  Connectivity Test – CT  Distributed Routing  Bellman – Ford – Distance Vector  Link State  Optimal Routing Network Model Math Detour – Convexity Flow Deviation Math Detour – Inequality Constraints Optimal Routing – Necessary and Sufficient Conditions r 4.5 Routing in the Internet  Hierarchical Routing  RIP  OSPF  BGP r 4.6 Broadcast and multicast routing Routing4-1

Routing Overview r Recap:  Forwarding: data plane, router uses a forwarding table, “simple” Direct data packets from input interface to output interface  Routing: control plane, router creates the forwarding table, “smart” Compute paths (and forwarding tables) by collaboration with other routers  Name (who)  Address (where)  Route (how) r Concepts  Optimization goal / path selection criteria Least cost / minimal hops / shortest delay / best network utilization  Network topology: nodes, links, characteristics Topology learning, change discovery and update, failure detection and mitigation Routing2

Routing Overview (2) r Optimization Goals  User performance (cost, delay, throughput, loss, jitter, hop count, …)  Network performance (congestion, stability, overheads, load-balanced, …)  Adaptability (mitigate failures, minimize loss, re-optimize) r Challenges  Distributed (cooperation of many nodes, not only neighbors)  Robustness (correct, reliable, stable, convergence)  Performance (it is a core function of the network)  Fairness E.g., maximal utilization ≠ fair Routing3

Routing Algorithm classification Global vs. Decentralized r Global: routers have all info on all links  Complete topology  Complete link cost, delay, etc.  “Link state” algorithms r Decentralized: routers know links to neighbors  Properties known only of physically- connected links  Iterative process of computation, exchange of info with neighbors  “Distance vector” algorithms Static vs. Dynamic r Static:  Routes change slowly over time r Dynamic:  Routes change more quickly  Periodic update  In response to link cost changes Routing4

Example: Routing & Network Performance r Example for throughput: all links have 10 units capacity r Low load: 5 units from 1 to 6 and from 2 to 6  via 3 and 5 respectively: good routing  both via 4 : bad routing (more congestion – higher delay) r High load: 5 units from 1 to 6, 15 units from 2 to 6  if no flow splitting is allowed, ( at least ) 5 units are rejected  if splitting is allowed, we can handle all requirements, e.g. by sending the traffic from 1 via 3, half of the traffic from 2 via 4 and the other half via 5 r Maximum Load:  both flows via 4, max. flow = 10, bad choice  no splitting, max. =20, better routing  with splitting, max = 30 Routing5

u y x wv z Graph abstraction r Graph: G(N,E)  N = set of nodes (routers) = { u, v, w, x, y, z }  E = set of links & costs (u,w), cost(u,w) = 5, (u,v), cost(u,v) = 2, (u,x), cost(u,x) = 1, (v,w), cost(v,w) = 3, (v,x), cost(v,x) = 2, (w,y), cost(w,y) = 1, (w,z), cost(w,z) = 5, (w,x), cost(w,x) = 3, (x,y), cost(x,y) = 1, (y,z), cost(y,z) = 2 r Cost  Could represent different things, e.g., always 1 (# of hops), proportional to delay, money, inversely related to bandwidth / to congestion  Path cost defined as sum of costs on path links  Possible routing goal – least cost path E.g., what is least cost path from u to z ? Remark: Graph abstraction is useful in other network contexts Example: P2P, where N is set of peers and E is set of TCP connections Routing6

Graphs – definitions & concepts r G(N,E) / G(V,E) / G(N,A) / G(N,L)  N / V - collection of nodes or vertices  E / A / L - collection of edges, arches, or links; a link is denoted by (v 1, v 2 ) or by e i r Routes  Walk – a arbitrary collection of successive links  Path - Loop-less walk ( no node appears more than once )  Cycle - walk that starts and ends at the same node r Connected graph  There is a path between every pair of nodes; otherwise, a non-connected graph r Sub-graph  Part of a given graph (some nodes, some links); G’(N’,E’), N’  N, E’  E r Tree  Connected, loop-less graph  Spanning Tree – a tree that is a sub-graph of G and contains all nodes of G Routing7

Shortest Path Routing r Input: graph, G(V,E), link costs, cost: E  ℝ +  Costs are static (do not change with load)  Costs are positive  Path cost is sum of link costs, cost(P) =  e  P cost(e) No loops (always add cost anyway) Other routing goals, such as bandwidth constraints, may be based on bottleneck (min) r Goal: least cost path to each destination  Destination-based forwarding  We focus on single destination from all sources Single source to all destinations is similar (how?) u w v x t y s z Routing8

Optimal sub-structure r If a shortest path from u to v P uv goes through node w, then the paths P uw and P wv are also shortest paths  Proof – otherwise replace them with shortest paths to obtain better path from u to v.  Key property for efficient SP algorithm (remark: for greedy algorithms) – combine shortest paths to create longer ones r Special case: if a shortest path from u to v P uv starts with the link (u,w), then P wv is also a shortest paths  If we know all the shortest paths from u ’s neighbors {P wv } (u,w)  E then cost(P uv ) = min (u,w)  E [ cost(u,w) + cost(P wv ) ], and the shortest path goes through w which achieves the minimum Routing9

Bellman-Ford Algorithm r Define distance function to v for a source node u  V  D u (v) = cost of shortest path from u to v, i.e., cost(P uv ) r Initialization, for each source node u  V  D 0 u = { D 0 u (v) = 0 if v=u; D 0 u (v)=∞ if v≠u } r Update DV operation for node u  V  D i+1 u (v) = min (u,w)  E [ cost(u,w) + D i w (v) ] and set min. w as next hop towards v  Only if better (less) than D i u (v) r Stop if D i+1 (v) u = D i (v) u for all u  V r Notes  i is number of hops used  dynamic program with entries per (i,u) Routing10 D u (z) = min{ cost(u,v) + D v (z), cost(u,w) + D w (z) } u w v x t y s z

Example of Bellman-Ford Routing11

Bellman-Ford (analysis) r Distance vector can never increase with iterations  D i+1 u (v) ≤ D i u (v) r D i u (v) is the cost of the shortest path from u to v with at most i hops  Can prove with induction  At most |V|-1 iterations At most H iterations, where H is max hop count r Computing D i u (v) in each iteration  O( number of outgoing links from u)  O(|E| ), aggregate per iteration over all source nodes in V r Overall  O(|V|·|E|), O(H·|E|) Routing12

Bellman-Ford Algorithm (distributed) r Each node u maintains distance vectors (DVs)  its own distance vector D u = { D u (v) } v  V (distances to all other nodes)  distance vector D w for each neighbor w (from w to all other nodes) r Initialization:  D u = { D u (u) = 0; D u (v)=∞ v≠u }  Send D u to all neighbors (assume updates are reliable and in-order) r Update D w (v) ( v≠u ) or update cost(u,w)  If [ cost(u,w) + D w (v) ] < D u (v) then update D u (v), set w as next hop Send D u (v) to all neighbors  Otherwise, if current next hop is not w then do nothing (no improvement)  Otherwise, ( D u (v) got worse….) If saved all D w (v) then can find new optimum, and update all Otherwise, update all and hope they send you something better r Periodically send D u to all neighbors Routing13

Bellman-Ford Algorithm (synchronized) r Distributed, but simulates centralized r Use some synchronization mechanism  Maintain iteration # as and makes sure all nodes agree on it  Iteration does not change until all nodes finish what they need to do  Iteration does not change until updates reach all nodes r Still maintains distance vectors (DVs) and same initialization  Send D u to all neighbors once per iteration r Update D w (v)  Wait for updates from all neighbors (sent in previous iteration)  Find best neighbor (same as centralized)  Send D u to all neighbors r Update cost(u,w)  Reset iterations and start over…. r Note: Bellman-Ford is correct without synchronization but can take time to converge Routing14

Example of update step Table at node J Routing15

stability r unbounded number of steps  ∞ means “a bounded large number” r to overcome the problem :  Run in parallel a protocol with weights=1 and stop when variables reach |N| Routing16

Loops r Problem: Detecting Situations when old information is irrelevant and should not be used S a b Routing17

Loops Routing S a b c 1 1

Poisoned reverse r If Z routes through Y to get to X then:  Z sends to all neighbors, except Y, the true estimated distance to X  Z sends to Y an infinite distance (so Y won’t route to X via Z) r Obviously, this solves only 2-node loops r Problem: entries for unreachable destinations Routing19

Distance Vector Protocol r Distributed Asynchronous Bellman-Ford is also called Distance Vector Protocol r The Distributed Asynchronous Bellman-Ford forms the basis for the Routing Information Protocol (RIP) used in Internet and other Networks Routing20

Dijkstra’s shortest path algorithm r Uses same update process as Bellman-Ford  D i+1 u (v) = min (u,w)  E [ cost(u,w) + D i w (v) ] r But, each iterates over nodes in specific order  Maintain estimated D u (v) for all nodes  Always selects node with minimal D u (v)  Searches for paths with increasing cost (rather than hop count) r Each iteration finalizes D u (v) for one node (the selected min)  That is, D u (v) will not change in later iterations  Implies at most |V| iterations Routing21

Dijsktra’s Algorithm r Initialization:  P = {v}  D u (v) = cost(u,v) iff (u,v)  E, otherwise ∞ r Loop 1. find w not in P with the smallest D w (v) 2. add w to P 3. update D u (v) = min [D u (v), cost(u,w) + D w (v)] for all u s.t. (u,w)  E and u not in P 4. until all nodes in P 22

Dijsktra’s Algorithm -- analysis r Correctness  D w (v) ≤ D u (v) if w  P and u not in P Prove by induction  D w (v) = cost of shortest path from w to u only through nodes in P Prove by induction, show that holds when node added to P r Complexity  Key: efficiently find w not in P with the smallest D w (v)  update is O(|V|)  O(|V|) iterations 23

Example of Dijkstra Routing24

Link State Routing Protocol r Basis for the OSPF ( Open Shortest Path First) algorithm  Each node periodically update all other nodes on network information Sequence number Identity Identity of neighbors Cost to each neighbor Time-to-live  Each node maintains an updated picture of the network topology and costs  Each node performs a Dijkstra algorithm to find forwarding table r Questions:  How to update? – by flooding  What if nodes don’t agree on information? Routing25

Comparison of LS and DV algorithms LS r O(nE) msgs to learn  Each nodes knows all network  Can be reduced to O(E) msgs r O(n log n + E) algorithm  Simplified needs O(n 2 )  may have oscillations r Node malfunctions?  Might advertise incorrect link cost  Computes only its own table DV r O(nE) msgs to run  Each nodes only talks to neighbors  O(h degree ) msgs per node r O(h) iterations  Might have routing loops  Count-to-infinity problem r Node malfunctions?  Might advertise incorrect path cost  Node’s table used by others => error propagate through network Routing26

Spanning Trees r Tree  Connected, loop-less graph r Spanning Tree  a tree  a sub-graph of G  contains all nodes of G r Minimal sub-graph that contains all nodes  Minimal in terms of links What about paths hop count?  Minimal in terms of cost? Total cost? Path cost? Routing27

Spanning Tree Algorithm r Given a connected graph G(V,E)  1. Select an arbitrary node v, T( {v},  )  2. If V  T stop. T is the Spanning Tree.  3. Select a link (i,j), such that i  T, j  T; add (i,j) to T  4. Go to 2 r Proof:  There always exists a link for selection in step 3. ( G is connected )  T is always a tree (connected, loopless)  The final T is spanning tree r Properties for any connected graph G(V,E)  G has a spanning tre  |E|  |V| -1; |E| = |V| -1 iff G is a tree Routing28

Routing in Networks of Bridged LAN’s r Network made of LAN segments and bridges  LANs are the “nodes”  Bridges are the “links” r The routing (forwarding by bridges) is done on a Spanning Tree, because:  Need a connected graph Reach each LAN from each LAN  No loops LAN segment is a broadcast medium – all traffic forwarded to entire segment – including any attached bridges Routing29

Minimum Spanning Tree (MST) r Many spanning trees r Goal: find a spanning tree with minimal overall cost  Assume each link (u,v) has a known cost c uv  Definition: “tree cost” = sum of tree link costs. Cost(T) =  e  T c e r Formal problem defintion: Given a graph G(V,E), with link costs {c ij }, select a tree T that  Is a spanning tree of G  cost(T)  cost(T’) for any other spanning tree T’ of G r Definitions:  segment = sub-graph of a MST  outgoing link = link with exactly one end in the segment Routing30

Minimum Spanning Tree - continued Lemma: Given a segment F of an MST, let e=(i,j) be the outgoing link from F, with minimum cost. Then F  {e} is also a segment of an MST Proof: r By definition F belongs to some MST M; if e  M then done; otherwise r Consider M  {e}  This is connected but not a tree, so contains a loop  The loop includes the link e – an outgoing link from F, so is through a node of e not in F  Traversing the loop from that node in the other direction  We must reach another outgoing link from F – denote that link by a r Consider M’ = M  {e} - {a}  M’ has no loops, so M’ is a spanning tree (just from counting links)  F is a segment of M’, F  {e} is a segment of M’ r Examine cost(M’)  By definition cost(e)  cost(a) thus, cost(M’)  cost(M) r M’ is an MST Routing31 F M e a

MST algorithms r Prim-Dijkstra  Select any node as a first segment.  Keep enlarging the segment, by selecting the minimum outgoing edge. r Kruskal  start with |V| segments, each composed of one node.  Select a minimum cost link and combine two segments.  Convenient for distributed computation. Routing32

Example MST algorithms Kruskal Topology Prim Dijkstra A B Routing33

Routing via Broadcast r Problem: Source node wants to transmit same information to all nodes in the network (broadcast)  Best performance: on Spanning Tree, if one is available, if not, easiest to “flood” r Flooding: each node sends information to all its neighbors r Advantages:  No need to know network current topology or costs  Reliable and fast: Every connected node will get the information at the earliest possible time. r Assumptions and notations:  every node knows its adjacent topology (neighbors)  every node assigns local id’s to its adjacent links  MSG(info) - message carrying the information info Routing34

Protocol PI ( Propagation of Information ) r Every node i performs the following (source receives START):  initialization: m  0  upon START or receipt of MSG(info) if m=0 then m  1 Accept info Send MSG(info) to all neighbors r Notes:  Completely distributed; every node works independently with own schedule  The variable m marks that message was accepted Makes sure that nodes will transfer and accept information no more than once.  Last line can be changed to: send MSG(info) to all neighbors except the one MSG was received from Routing35

Properties of PI r During the operation of the protocol, exactly one message travels on each link in each direction r All nodes connected to the source will accept the info, exactly once r For each node i, let p i be the neighbor from which node i receives the first MSG:  The collection {(i,p i )} forms a Spanning Tree  It is the shortest (delay) path tree from source to all node r If several nodes have the same info and start PI asynchronously, their PI’s converge and all properties above hold,  Except that the collection {(i,pi)} will form a Spanning Forest (collection of non-overlapping trees whose union spans all nodes) r The source does not know the termination time, i.e. some time when it can be sure that all nodes have received the information.  It knows the information will be received and accepted by all nodes, but it doesn’t know when. Routing36

Broadcast with termination feedback (PIF) r Idea:  Use PI-type to broadcast information and to form Spanning Tree  Use Spanning Tree to collect termination information r Method:  Node that accepts info, forwards it to all neighbors, except p i  Upon receipt of MSG from all neighbors, send to p i and node is done Info backwards on the Spanning Tree ( towards source ).  Termination indication = source is done r Notations:  m i = flag that indicates participation in protocol  N i (j) = flag that indicates that the node has received MSG from neighbor j  p i = neighbor from which MSG was received first  receive MSG from nil = START Routing37

PIF algorithm r Every node i performs the following ( source receives START):  initialization: m i  0, p i  nil, N i (j)  0  upon receipt of MSG from j N i (j)  1 If m i =0 then(propagation) –m i  1; p i  j –Accept info and send MSG(info) to all neighbors except j If N i (k) for all neighbors k of i then (feedback) –Send MSG(info) back to p i –m i  0; N i (k)  0 for all neighbors k of i(reset) r Properties:  All connected nodes will accept message exactly once  One MSG on each link in each direction  Collection {(i, p i )} is a spanning tree all nodes  All nodes complete protocol and reset (m i  0)  i completes protocol before p i, source completes last (termination indication) Routing38

source PIF Example Routing39

Example source Routing40

Connectivity Test r Goal: learn network nodes r Protocol CT1:  every node sends its id in PI  nodes wake up when they get the first message, at which time they start their own PI r Properties:  can be started asynchronously at several nodes  every node will receive the identity of every other connected node  disconnected nodes will not know of each other  a node cannot determine a termination time, i.e. a time when it knows for sure the identities of all connected nodes Routing41

CT2 r Idea: use PIF’s instead of PI’s r Protocol CT2:  A node starts its own PIF when: gets a START upon receiving the first message ( of some PIF started by another node )  Note: PIF per node  must track m, p, N per source PIF r Properties:  can be started asynchronously at several nodes  every node will receive the identity of every other connected node  disconnected nodes will not know of each other  a node i can determine a termination time, i.e. a time when it knows for sure the identities of all connected nodes. When i completes its own PIF Follows from the fact that the propagation phase of PIF j starts before j enters feedback for PIF i. Routing42

CT3 r Idea: same as CT1, except that nodes broadcast their neighbors’ identities along with their own identity  Assume node knows identities of neighbors r Properties:  all “good” properties of CT1  Termination signal: When a PI j from j arrives, i now knows all j’s neighbors  i now expects PI from each of these neighbors When all expected identities have arrived can terminate r Additional variations on CT1, CT2, CT3:  encoded information to reduce MSG contents  reduction of number of messages  pruning of MSG contents Routing43