Theory of Time 2024 (Universal Theory for Everything)
06 routing
1. 1
The Fundamentals of Routing
EE122 Fall 2011
Scott Shenker
http://inst.eecs.berkeley.edu/~ee122/
Materials with thanks to Jennifer Rexford, Ion Stoica, Vern Paxson
and other colleagues at Princeton and UC Berkeley
2. Announcements
• 194 is set up: 4 units: CCN: 25571
– Everyone should be off wait list by now….
– Waiting for more accounts
o Send me email if you need one….
– Will set up bspace for 194 to hand in homework
• Open today with questions about project 1
• Hope you all handed in HW1
– HW2 out on Wednesday
• Survey will be available after Wednesday’s class 2
4. Warning….
• This lecture contains detailed calculations
• Prolonged exposure may induce drowsiness
• Do not operate heavy machinery while listening
4
5. Where Are We?
• Motivated our basic design decisions
– Best-effort, packet-switching, blah, blah, blah
• Last lecture: how to build reliable transport on top
• This lecture: how to build the network itself
• Basic Task: deliver packets
– Need to route them, from anywhere to anywhere
• Today, we focus on routing 5
6. The Traditional Routing Curriculum
• Learning switches
• Link-state routing
– Dijkstra’s Algorithm
• Distance-vector routing
– Bellman-Ford
6
7. I have some bad news…..
• Don’t have anything interesting to say about routing
• Will follow standard curriculum
– Much of it covered in the text
• But will focus more on principles than details
• Will work through two algorithms, but details may
wait until Wednesday…..134 slides!
– Let’s just see how timing goes
– If you don’t want to be bored, ask questions!
7
8. Remember….
• Goal of the first portion of course is conceptual
• Ignore real networks
• Think about the basic concepts
• That’s where we are. Later will deal with reality…
8
10. Addressing (at a conceptual level)
• Assume all hosts have unique IDs
• No particular structure to those IDs
• Later in course will talk about real IP addressing
10
11. Packets (at a conceptual level)
• Assume packet headers contain:
– Source ID, Destination ID, and perhaps other information
11
Destination
Identifier
Source
Identifier
Payload
Why include
this?
12. Switches/Routers
• Multiple ports (attached to other switches or hosts)
• Ports are typically duplex (incoming and outgoing)
12
incoming links outgoing linksSwitch
13. Example of Network Graph
13
Six ports, incoming/outgoing
Four ports, incoming/outgoing
14. A Variety of Networks
• ISPs: carriers
– Backbone
– Edge
– Border (to other ISPs)
• Enterprises: companies, universities
– Core
– Edge
– Border (to outside)
• Datacenters: massive collections of machines
– Top-of-Rack
– Aggregation and Core
– Border (to outside) 14
19. Switches
• Enterprise/Edge: typically 24 to 48 ports
• Aggregation switches: 192 ports or more
• Backbone: typically fewer ports
• Border: typically very few ports
19
20. Forwarding Decisions
• When packet arrives, must choose outgoing port
• Decision is based on routing state (table) in switch
20
incoming links outgoing linksSwitch
Consider
packet header
and routing
table
21. Forwarding Decisions
• When packet arrives..
– Must decide which outgoing port to use
– In single transmission time
– Forwarding decisions must be simple
• Routing state dictates where to forward packets
– Assume decisions are deterministic
• Global routing state means collection of routing
state in each of the routers
– Will focus on where this routing state comes from
– But first, a few preliminaries…. 21
22. Forwarding vs Routing
• Forwarding: “data plane”
– Directing a data packet to an outgoing link
– Individual router using routing state
• Routing: “control plane”
– Computing paths the packets will follow
– Routers talking amongst themselves
– Jointly creating the routing state
• Two very different timescales….
22
24. “Valid” Routing State
• Global routing state is “valid” if it produces
forwarding decisions that always deliver packets to
their destinations
– Valid is my terminology, not standard
• Goal of routing protocols: compute valid state
– But how can you tell if routing state if valid?
24
25. Necessary and Sufficient Condition
• Global routing state is valid if and only if:
– There are no dead ends (other than destination)
– There are no loops
• A dead end is when there is no outgoing port
– A packet arrives, but the forwarding decision does not
yield any outgoing port
• A loop is when a packet cycles around the same
set of nodes forever
25
26. Necessary: Obvious
• If you run into a deadend before hitting destination,
you’ll never reach the destination
• If you run into a loop, you’ll never reach destination
– With deterministic forwarding, once you loop, you’ll loop
forever (assuming routing state is static)
26
28. Sufficient: Easy
• Assume no deadends, no loops
• Packet must keep wandering, without repeating
– If ever enter same switch from same port, will loop
– Because forwarding decisions are deterministic
• Only a finite number of possible ports for it to visit
– It cannot keep wandering forever without looping
– Must eventually hit destination
28
29. The “Secret” of Routing
• Avoiding deadends is easy
• Avoiding loops is hard
• The key difference between routing protocols is
how they avoid loops!
– Don’t focus on details of mechanisms
– Just ask “how are loops avoided?”
• Will return to this later….
29
30. Making Forwarding Decisions
• Map PacketState+RoutingState into OutgoingPort
– At line rates…..
• Packet State:
– Destination ID
– Source ID
– Incoming Port (from switch, not packet)
– Other packet header information?
• Routing State:
– Stored in router
30
31. Forwarding Decision Dependencies
• Must depend on destination
• Could also depend on :
– Source: requires n2 state
– Input port: not clear what this buys you
– Other header information: let’s ignore for now
• We will focus only on destination-based routing
– But first consider the alternative
31
34. Destination-Based Routing
• Paths to same destination never cross
• Once paths to destination meet, they never split
• Set of paths to destination create a “delivery tree”
– Must cover every node exactly once
– Spanning Tree rooted at destination
34
36. Checking Validity of Routing State
• Focus only on a single destination
– Ignore all other routing state
• Mark outgoing port with arrow
– There can only be one at each node
• Eliminate all links with no arrows
• Look at what’s left….
36
43. Lesson….
• Very easy to check validity of routing state for a
particular destination
• Deadends are obvious
– Node without outgoing arrow
• Loops are obvious
– Disconnected from rest of graph
43
45. Forms of Route Computation
• Learn from observing….
– Not covered in your reading
• Centralized computation
– One node has the entire network map
• Pseudo-centralized computation
– All nodes have the entire network map
• Distributed computation
– No one has the entire network map
45
46. How Can You Avoid Loops?
• Restrict topology to spanning tree
– If the topology has no loops, packets can’t loop!
• Central computation
– Can make sure no loops
• Minimizing metric in distributed computation
– Loops are never the solution to a minimization problem
46
48. Easiest Way to Avoid Loops
• Use a topology where loops are impossible!
• Take arbitrary topology
• Build spanning tree (algorithm covered later)
– Ignore all other links (as before)
• Only one path to destinations on spanning trees
• Use “learning switches” to discover these paths
– No need to compute routes, just observe them 48
53. Flooding on a Spanning Tree
• If you want to send a packet that will reach all
nodes, then switches can use the following rule:
– Ignoring all ports not on spanning tree!
• Originating switch sends “flood” packet out all
ports
• When a “flood” packet arrives on one incoming
port, send it out all other ports
53
56. Flooding on a Spanning Tree
• This works because the lack of loops prevents the
flooding from cycling back on itself
• Eventually all nodes will be covered, exactly once
56
57. This Enables Learning!
• There is only one path from source to destination
• Each switch can learn how to reach a another
node by remembering where its flooding packets
came from!
• If flood packet from Node A entered switch from
port 4, then to reach Node A, switch sends packets
out port 4
57
58. Learning from Flood Packets
58
Node A
Node A can be reached
through this port
Node A can be reached
through this port
Once a node has sent a flood message, all
other switches know how to reach it….
59. General Approach
• Flood first packet
• All switches learn where you are
• When destination responds, all switches learn
where it is…
• Done.
59
60. 60
Self-Learning Switch
When a packet arrives
• Inspect source ID, associate with incoming port
• Store mapping in the switch table
• Use time-to-live field to eventually forget mapping
A
B
C
D
Packet tells switch
how to reach A.
61. 61
Self Learning: Handling Misses
When packet arrives with unfamiliar destination
• Forward packet out all other ports
• Response will teach switch about that destination
A
B
C
D
When in doubt,
shout!
62. 62
General Rule
When switch receives a packet:
index the switch table using destination ID
if entry found for destination {
if dest on port from which packet arrived
then drop packet
else forward packet on port indicated
}
else flood
forward on all but the interface
on which the frame arrived
Why do this?
63. Summary of Learning Approach
• Avoids loop by restricting to spanning tree
• This makes flooding possible
• Flooding allows packet to reach destination
• And in the process switches learn how to reach
source of flood
• No route “computation”
63
64. Weaknesses of This Approach?
• Requires loop-free topology (Spanning Tree)
• Slow to react to failures (entries time out)
• Very little control over paths
• Spanning Trees suck.
64
67. The Task
• Remove sheet of paper from beanbag, but do not
look at sheet of paper until I say so
• You will have five minutes to complete this task
• Each sheet says:
You are node X You are connected to nodes Y,Z
• Your job: find route from source (node 1) to
destination (node 40) in five minutes
67
68. Ground Rules
• You may not:
– Leave your seat (but you can stand)
– Pass your sheet of paper
– Let anyone copy your sheet of paper
• You may:
– Ask nearby friends for advice
– Shout to other participants (anything you want)
– Curse your instructor (sotto voce)
• You must: Try
68
72. How Can You Avoid Loops?
• Restrict topology to spanning tree
• Central computation (can make sure no loops)
• Minimizing metric in distributed computation
• Any other techniques?
72
73. Organization for Rest of Lecture
• Link-state routing
• Distance-vector routing
• Goal of today: basic idea
– May review details on Wednesday
– Definitely go over details in section
73
75. Another Approach to Loop-Avoidance
• If one has a picture of the entire network, can
compute routing state that does not produce loops
• This route computation can be done in many ways
– Here we will review Dijkstra’s algorithm briefly
– But that’s just because it is expected from such courses
o Snore…..
• The real question is, how do you get picture of
entire network?
75
76. 76
Link-State Routing Is Conceptually Simple
• Each router keeps track of its incident links
• Each router broadcasts the link state
– To give every router a complete view of the graph
• Each router runs Dijkstra’s algorithm
– Compute shortest paths, then construct forwarding table
• Example protocols
– Open Shortest Path First (OSPF)
– Intermediate System – Intermediate System (IS-IS)
• Challenges: scaling, transient disruptions
77. 77
Link State Routing
• Each node floods its local information
• Each node then knows entire network topology
Host A
Host B
Host E
Host D
Host C
N1 N2
N3
N4
N5
N7N6
78. 78
Link State: Each Node Has Global View
Host A
Host B
Host E
Host D
Host C
N1 N2
N3
N4
N5
N7N6
A
B
E
D
C
A
B
E
D
C A
B
E
D
C
A
B
E
D
C
A
B
E
D
C
A
B
E
D
C
A
B
E
D
C
79. How to Compute Routes
• Each node should have same global view
• They each compute their own routing tables
• Using exactly the same algorithm
• Can use any algorithm that avoids loops
• Computing shortest paths is one such algorithm
– Associate a “cost” with each link
– Don’t worry what it stands for….
79
80. Dijkstra’’’’s Shortest Path Algorithm
• INPUT:
– Network topology (graph), with link costs
• OUTPUT:
– Least cost paths from one node to all other nodes
– Produces “tree” of routes
o Different from what we talked about before
o Previous tree was rooted at destination
o This is rooted at source
o But shortest paths are reversible!
80
81. “Least Cost” Routes
• No sensible cost metric will be minimized by
traversing a loop
• “Least cost” routes an easy way to avoid loops
• Least cost routes are also “destination-based”
– i.e., do not depend on the source
– Why is this?
• Therefore, least-cost paths form a spanning tree
81
83. 83
Notation
• c(i,j): link cost from node i
to j; cost infinite if not
direct neighbors; ≥ 0
• D(v): current value of cost
of path from source to
destination v
• p(v): predecessor node
along path from source to
v, that is next to v
• S: set of nodes whose
least cost path definitively
known
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
Source
84. 84
Dijsktra’’’’s Algorithm
1 Initialization:
2 S = {A};
3 for all nodes v
4 if v adjacent to A
5 then D(v) = c(A,v);
6 else D(v) = ;
7
8 Loop
9 find w not in S such that D(w) is a minimum;
10 add w to S;
11 update D(v) for all v adjacent to w and not in S:
12 if D(w) + c(w,v) < D(v) then
// w gives us a shorter path to v than we’ve found so far
13 D(v) = D(w) + c(w,v); p(v) = w;
14 until all nodes in S;
∞
• c(i,j): link cost from node i to j
• D(v): current cost source → v
• p(v): predecessor node along
path from source to v, that is
next to v
• S: set of nodes whose least
cost path definitively known
85. 85
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
D(B),p(B)
2,A
D(C),p(C)
5,A
D(D),p(D)
1,A
D(E),p(E) D(F),p(F)
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
∞ ∞
1 Initialization:
2 S = {A};
3 for all nodes v
4 if v adjacent to A
5 then D(v) = c(A,v);
6 else D(v) = ;
…
∞
86. 86
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
D(B),p(B)
2,A
D(C),p(C)
5,A
…
8 Loop
9 find w not in S s.t. D(w) is a minimum;
10 add w to S;
11 update D(v) for all v adjacent
to w and not in S:
12 If D(w) + c(w,v) < D(v) then
13 D(v) = D(w) + c(w,v); p(v) = w;
14 until all nodes in S;
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
D(D),p(D)
1,A
D(E),p(E) D(F),p(F)
∞ ∞
87. 87
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
AD
D(B),p(B)
2,A
D(C),p(C)
5,A
D(D),p(D)
1,A
D(E),p(E) D(F),p(F)
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
∞ ∞
…
8 Loop
9 find w not in S s.t. D(w) is a minimum;
10 add w to S;
11 update D(v) for all v adjacent
to w and not in S:
12 If D(w) + c(w,v) < D(v) then
13 D(v) = D(w) + c(w,v); p(v) = w;
14 until all nodes in S;
88. 88
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
AD
D(B),p(B)
2,A
D(C),p(C)
5,A
4,D
D(D),p(D)
1,A
D(E),p(E)
2,D
D(F),p(F)
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
∞ ∞
…
8 Loop
9 find w not in S s.t. D(w) is a minimum;
10 add w to S;
11 update D(v) for all v adjacent
to w and not in S:
12 If D(w) + c(w,v) < D(v) then
13 D(v) = D(w) + c(w,v); p(v) = w;
14 until all nodes in S;
89. 89
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
AD
ADE
D(B),p(B)
2,A
D(C),p(C)
5,A
4,D
3,E
D(D),p(D)
1,A
D(E),p(E)
2,D
D(F),p(F)
4,E
∞ ∞
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
…
8 Loop
9 find w not in S s.t. D(w) is a minimum;
10 add w to S;
11 update D(v) for all v adjacent
to w and not in S:
12 If D(w) + c(w,v) < D(v) then
13 D(v) = D(w) + c(w,v); p(v) = w;
14 until all nodes in S;
90. 90
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
AD
ADE
ADEB
D(B),p(B)
2,A
D(C),p(C)
5,A
4,D
3,E
D(D),p(D)
1,A
D(E),p(E)
2,D
D(F),p(F)
4,E
∞ ∞
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
…
8 Loop
9 find w not in S s.t. D(w) is a minimum;
10 add w to S;
11 update D(v) for all v adjacent
to w and not in S:
12 If D(w) + c(w,v) < D(v) then
13 D(v) = D(w) + c(w,v); p(v) = w;
14 until all nodes in S;
91. 91
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
AD
ADE
ADEB
ADEBC
D(B),p(B)
2,A
D(C),p(C)
5,A
4,D
3,E
D(D),p(D)
1,A
D(E),p(E)
2,D
D(F),p(F)
4,E
∞ ∞
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
…
8 Loop
9 find w not in S s.t. D(w) is a minimum;
10 add w to S;
11 update D(v) for all v adjacent
to w and not in S:
12 If D(w) + c(w,v) < D(v) then
13 D(v) = D(w) + c(w,v); p(v) = w;
14 until all nodes in S;
92. 92
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
AD
ADE
ADEB
ADEBC
ADEBCF
D(B),p(B)
2,A
D(C),p(C)
5,A
4,D
3,E
D(D),p(D)
1,A
D(E),p(E)
2,D
D(F),p(F)
4,E
∞ ∞
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
…
8 Loop
9 find w not in S s.t. D(w) is a minimum;
10 add w to S;
11 update D(v) for all v adjacent
to w and not in S:
12 If D(w) + c(w,v) < D(v) then
13 D(v) = D(w) + c(w,v); p(v) = w;
14 until all nodes in S;
93. 93
Example: Dijkstra’’’’s Algorithm
Step
0
1
2
3
4
5
start S
A
AD
ADE
ADEB
ADEBC
ADEBCF
D(B),p(B)
2,A
D(C),p(C)
5,A
4,D
3,E
D(D),p(D)
1,A
D(E),p(E)
2,D
D(F),p(F)
4,E
∞ ∞
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
To determine path A → C (say),
work backward from C via p(v)
94. 94
• Running Dijkstra at node A gives the shortest
path from A to all destinations
• We then construct the forwarding table
The Forwarding Table
A
ED
CB
F
2
2
1
3
1
1
2
5
3
5
Destination Link
B (A,B)
C (A,D)
D (A,D)
E (A,D)
F (A,D)
95. Complexity
• How much processing does running the Dijkstra
algorithm take?
• Assume a network consisting of N nodes
– Each iteration: check all nodes w not in S
– N(N+1)/2 comparisons: O(N2)
– More efficient implementations: O(N log(N))
95
96. 96
Flooding the Topology Information
• Each router sends information out its ports
• The next node sends it out through all of its ports
– Except the one where the information arrived
– Need to remember previous msgs, suppress duplicates!
X A
C B D
(a)
X A
C B D
(b)
X A
C B D
(c)
X A
C B D
(d)
97. Making Flooding Reliable
• Reliable flooding
– Ensure all nodes receive link-state information
– Ensure all nodes use the latest version
• Challenges
– Packet loss
– Out-of-order arrival
• Solutions
– Acknowledgments and retransmissions
– Sequence numbers
• How can it still fail?
97
98. When to Initiate Flood?
• Topology change
– Link or node failure
– Link or node recovery
• Configuration change
– Link cost change
– Potential problems with making cost dynamic!
• Periodically
– Refresh the link-state information
– Typically (say) 30 minutes
– Corrects for possible corruption of the data
98
99. 99
Convergence
• Getting consistent routing information to all nodes
– E.g., all nodes having the same link-state database
• Forwarding is consistent after convergence
– All nodes have the same link-state database
– All nodes forward packets on same paths
100. 100
Convergence Delay
• Time elapsed before every router has a consistent
picture of the network
• Sources of convergence delay
– Detection latency
– Flooding of link-state information
– Recomputation of forwarding tables
– Storing forwarding tables
• Performance during convergence period
– Lost packets due to blackholes and TTL expiry
– Looping packets consuming resources
– Out-of-order packets reaching the destination
• Very bad for VoIP, online gaming, and video
101. 101
• Inconsistent link-state database
– Some routers know about failure before others
– The shortest paths are no longer consistent
– Can cause transient forwarding loops
Transient Disruptions
A
ED
CB
F A
ED
CB
F
A and D think that this
is the path to C
E thinks that this
is the path to C
Loop!
102. 102
Reducing Convergence Delay
• Faster detection
– Smaller hello timers
– Link-layer technologies that can detect failures
• Faster flooding
– Flooding immediately
– Sending link-state packets with high-priority
• Faster computation
– Faster processors on the routers
– Incremental Dijkstra algorithm
• Faster forwarding-table update
– Data structures supporting incremental updates
103. 103
Scaling Link-State Routing
• Overhead of link-state routing
– Flooding link-state packets throughout the network
– Running Dijkstra’s shortest-path algorithm
– Becomes unscalable when 100s of routers
• Introducing hierarchy through “areas”
Area 0
Area 1 Area 2
Area 3 Area 4
area
border
router
105. Distributed Computation of Routes
• More scalable than Link-State
– No global flooding
• Each node computing the outgoing port based on:
– Local information (who it is connected to)
– Paths advertised by neighbors
105
106. Distributed Computation
106
I am one hop away
I am one hop away
I am one hop away
I am two hops away
I am two hops away
I am two hops away
I am two hops away
I am three hops away
I am three hops away
107. Loops in a Distributed Computation
• If the nodes choose their paths according to
different criterion, then loops can occur
• Example
– Node A is minimizing hop count
– Node B is minimizing latency
– Node C is maximizing capacity
– Node D is minimizing price
• Any of those goals are fine, if globally adopted
– Only a problem when nodes use different criteria
• Any criteria that results in destination-based
routing is fine…. 107
108. 108
Distance Vector Routing
• Each router knows the links to its neighbors
– Does not flood this information to the whole network
• Each router has provisional “shortest path”
– E.g.: Router A: “I can get to router B with cost 11 via
next hop router D”
• Routers exchange this information with their
neighboring routers
– Again, no flooding the whole network
• Routers update their idea of the best path using
info from neighbors
• This iterative process converges with set of
shortest paths
109. 109
Information Flow in Distance Vector
Host A
Host B
Host E
Host D
Host C
N1 N2
N3
N4
N5
N7N6
110. 110
Information Flow in Distance Vector
Host A
Host B
Host E
Host D
Host C
N1 N2
N3
N4
N5
N7N6
111. 111
Information Flow in Distance Vector
Host A
Host B
Host E
Host D
Host C
N1 N2
N3
N4
N5
N7N6
113. Bellman-Ford Algorithm
• INPUT:
– Link costs to each neighbor
– Not full topology
• OUTPUT:
– Next hop to each destination and the corresponding cost
– Does not give the complete path to the destination
113
114. 114
Bellman-Ford - Overview
• Each router maintains a table
– Best known distance from X to Y,
via Z as next hop = DZ(X,Y)
• Each local iteration caused by:
– Local link cost change
– Message from neighbor
• Notify neighbors only if least cost
path to any destination changes
– Neighbors then notify their neighbors if
necessary
wait for (change in local link
cost or msg from neighbor)
recompute distance table
if least cost path to any dest
has changed, notify
neighbors
Each node:
115. Bellman-Ford - Overview
• Each router maintains a table
– Row for each possible destination
– Column for each directly-attached
neighbor to node
– Entry in row Y and column Z of node X
⇒ best known distance from X to Y, via
Z as next hop = DZ(X,Y)
A C
12
7
B D3
1
B C
B 2 8
C 3 7
D 4 8
Node A
Neighbor
(next-hop)
Destinations DC(A, D)
116. Bellman-Ford - Overview
• Each router maintains a table
– Row for each possible destination
– Column for each directly-attached
neighbor to node
– Entry in row Y and column Z of node X
⇒ best known distance from X to Y, via
Z as next hop = DZ(X,Y)
A C
12
7
B D3
1
B C
B 2 8
C 3 7
D 4 8
Node A
Smallest distance in row Y = shortest
Distance of A to Y, D(A, Y)
117. 117
Distance Vector Algorithm (cont’’’’d)
1 Initialization:
2 for all neighbors V do
3 if V adjacent to A
4 D(A, V) = c(A,V);
5 else
6 D(A, V) = ∞;
7 send D(A, Y) to all neighbors
loop:
8 wait (until A sees a link cost change to neighbor V /* case 1 */
9 or until A receives update from neighbor V) /* case 2 */
10 if (c(A,V) changes by ±d) /* ⇐ case 1 */
11 for all destinations Y that go through V do
12 DV(A,Y) = DV(A,Y) ± d
13 else if (update D(V, Y) received from V) /* ⇐ case 2 */
/* shortest path from V to some Y has changed */
14 DV(A,Y) = DV(A,V) + D(V, Y); /* may also change D(A,Y) */
15 if (there is a new minimum for destination Y)
16 send D(A, Y) to all neighbors
17 forever
• c(i,j): link cost from node i to j
• DZ(A,V): cost from A to V via Z
• D(A,V): cost of A’’’’s best path to V
118. Example:1st Iteration (C A)
A C
12
7
B D3
1
B C
B 2 8
C ∞ 7
D ∞ 8
Node A
A C D
A 2 ∞ ∞
C ∞ 1 ∞
D ∞ ∞ 3
Node B
Node C
A B D
A 7 ∞ ∞
B ∞ 1 ∞
D ∞ ∞ 1
B C
A ∞ ∞
B 3 ∞
C ∞ 1
Node D
7 loop:
…
13 else if (update D(A, Y) from C)
14 DC(A,Y) = DC(A,C) + D(C, Y);
15 if (new min. for destination Y)
16 send D(A, Y) to all neighbors
17 forever
DC(A, B) = DC(A,C) + D(C, B) = 7 + 1 = 8
DC(A, D) = DC(A,C) + D(C, D) = 7 + 1 = 8
119. Example: 1st Iteration (B A)
A C
12
7
B D3
1
B C
B 2 8
C 3 7
D 5 8
Node A
A C D
A 2 ∞ ∞
C ∞ 1 ∞
D ∞ ∞ 3
Node B
Node C
A B D
A 7 ∞ ∞
B ∞ 1
D ∞ ∞ 1
Node D
7 loop:
…
13 else if (update D(A, Y) from B)
14 DB(A,Y) = DB(A,B) + D(B, Y);
15 if (new min. for destination Y)
16 send D(A, Y) to all neighbors
17 forever
DB(A, C) = DB(A,B) + D(B, C) = 2 + 1 = 3
DB(A, D) = DB(A,B) + D(B, D) = 2 + 3 = 5
B C
A ∞ ∞
B 3 ∞
C ∞ 1
120. Example: End of 1st Iteration
A C
12
7
B D3
1
B C
B 2 8
C 3 7
D 5 8
Node A Node B
Node C
A B D
A 7 3 ∞
B 9 1 4
D ∞ 4 1
Node D
B C
A 5 8
B 3 2
C 4 1
End of 1st Iteration
All nodes knows the
best two-hop paths
A C D
A 2 3 ∞
C 9 1 4
D ∞ 2 3
121. Example: 2nd Iteration (A B)
A C
12
7
B D3
1
B C
B 2 8
C 3 7
D 5 8
Node A Node B
Node C
A B D
A 7 3 ∞
B 9 1 4
D ∞ 4 1
Node D
B C
A 5 8
B 3 2
C 4 1
A C D
A 2 3 ∞
C 5 1 4
D 7 2 3
7 loop:
…
13 else if (update D(B, Y) from A)
14 DA(B,Y) = DA(B,A) + D(A, Y);
15 if (new min. for destination Y)
16 send D(B, Y) to all neighbors
17 forever
DA(B, C) = DA(B,A) + D(A, C) = 2 + 3 = 5
DA(B, D) = DA(B,A) + D(A, D) = 2 + 5 = 7
122. Example: End of 2nd Iteration
A C
12
7
B D3
1
B C
B 2 8
C 3 7
D 4 8
Node A
A C D
A 2 3 11
C 5 1 4
D 7 2 3
Node B
Node C
A B D
A 7 3 6
B 9 1 4
D 12 4 1
Node D
B C
A 5 4
B 3 2
C 4 1
End of 2nd Iteration
All nodes knows the
best three-hop paths
123. Example: End of 3rd Iteration
A C
12
7
B D3
1
B C
B 2 8
C 3 7
D 4 8
Node A
A C D
A 2 3 6
C 5 1 4
D 7 2 3
Node B
Node C
A B D
A 7 3 5
B 9 1 4
D 11 4 1
Node D
B C
A 5 4
B 3 2
C 4 1
End of 2nd Iteration:
Algorithm
Converges!
124. Intuition
• Initial state: best one-hop paths
• One round: best two-hop paths
• Two rounds: best three-hop paths
• …
• Kth round: best (k+1) hop paths
• This must eventually converge….but how does it
respond to changes in cost?
124
125. 125
Distance Vector: Link Cost Changes
A C
14
50
B
1
“good
news
travels
fast”
A C
A 4 6
C 9 1
Node B
A B
A 50 5
B 54 1
Node C
Link cost changes here
time
Algorithm terminates
loop:
8 wait (until A sees a link cost change to neighbor V
9 or until A receives update from neighbor V) /
10 if (c(A,V) changes by ±d) /* ⇐ case 1 */
11 for all destinations Y that go through V do
12 DV(A,Y) = DV(A,Y) ± d
13 else if (update D(V, Y) received from V) /* ⇐ case 2 */
14 DV(A,Y) = DV(A,V) + D(V, Y);
15 if (there is a new minimum for destination Y)
16 send D(A, Y) to all neighbors
17 forever
A C
A 1 6
C 9 1
A B
A 50 5
B 54 1
A C
A 1 6
C 9 1
A B
A 50 2
B 51 1
A C
A 1 3
C 3 1
A B
A 50 2
B 51 1
126. 126
DV: Count to Infinity Problem
A C
14
50
B
60
“bad
news
travels
slowly”
Node B
Node C
Link cost changes here
time
…
loop:
8 wait (until A sees a link cost change to neighbor V
9 or until A receives update from neighbor V) /
10 if (c(A,V) changes by ±d) /* ⇐ case 1 */
11 for all destinations Y that go through V do
12 DV(A,Y) = DV(A,Y) ± d
13 else if (update D(V, Y) received from V) /* ⇐ case 2 */
14 DV(A,Y) = DV(A,V) + D(V, Y);
15 if (there is a new minimum for destination Y)
16 send D(A, Y) to all neighbors
17 forever
A C
A 4 6
C 9 1
A B
A 50 5
B 54 1
A C
A 60 6
C 9 1
A B
A 50 5
B 54 1
A C
A 60 6
C 9 1
A B
A 50 7
B 101 1
A C
A 60 8
C 9 1
A B
A 50 7
B 101 1
127. 127
Distance Vector: Poisoned Reverse
A C
14
50
B
60• If B routes through C to get to A:
- B tells C its (B’s) distance to A is infinite
(so C won’t route to A via B)
- Will this completely solve count to infinity
problem?
Node B
Node C
Link cost changes here; C updates D(C, A) = 60 as
B has advertised D(B, A) = ∞
time
Algorithm terminates
A C
A 4 6
C 9 1
A B
A 50 5
B ∞ 1
A C
A 60 6
C 9 1
A B
A 50 5
B ∞ 1
A C
A 60 6
C 9 1
A B
A 50 ∞
B ∞ 1
A C
A 60 51
C 9 1
A B
A 50 ∞
B ∞ 1
A C
A 60 51
C 9 1
A B
A 50 ∞
B ∞ 1
128. Aside: Another Way to Avoid Loops?
• In Distributed Computation?
• Exchange entire paths, not just cost of paths
• Nodes can use arbitrary metrics to choose paths
– No need to agree on single metric
• No loops, but algorithm might not converge
• Will discuss later in semester…
128
129. Routing: Just the Beginning
• Link state and distance-vector (and path vector)
are the deployed routing paradigms
• But we know how to do much, much better…
• Stay tuned for a later lecture where we:
– Reduce convergence time to zero
– Respond to failures instantly!
129
130. Next Lecture
• Review computations (if needed)
• What pieces are we missing?
– Naming
– Security
– Content retrieval
– ???
130
