Communication Networks

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Communication Networks

• A Second Course

Jean Walrand Department of EECS University of
California at Berkeley
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Graph Theory

• Routing
• Coloring
• QoS in Ad Hoc Neworks
• Percolations

Note Some slides from Abhay Parekh, others from
Rajarshi Gupta
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Routing

• Shortest Path Principle of Optimality
• Algorithms
• Trees

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Shortest Path

• Principle of Optimality

V(E) 6
E
4
10
2
V(B) 4
5
5
4
A
B
D
V(A) 9
F
V(F) 4
V(D) 0
1
10
3
C
V(C) 3
If ABCD is the shortest path from A to D, then
BCD is the shortest path from C to D
V(A) minXL(A, X) V(X) Dynamic
Programming Equation If X achieves minimum,
shortest path goes through X
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Related Idea Optimal Control

• Problem
• Solution

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Related Idea Optimal Control

• Variation
• Solution

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Algorithms
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Dijkstra

• Goal at node 1 Find the shortest paths from 1 to
  all the other nodes.
• Strategy at node 1 Find the shortest paths in
  order of increasing path length

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2
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4
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1
Dijkstra, E.W., A Note on Two Problems in
Connexion with Graphs, Numerische Mathematik,
vol. 1, pp. 269-271, 1959.
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Dijkstra (continued)
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Dijkstra (continued)
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Dijkstra Correctness

• Thm Label v(i) is the shortest path cost to i
• Lemma At one stage, algorithms labels nodes in a
  set P. It then adds nodes to P. Labels in P are
  the shortest path involving nodes in P. Then when
  P V, the theorem holds
• Proof of Lemma
• Suppose we have a shorter path using node in T
• Let p, q in P be two nodes in path, with t in T
  in between
• But if v(p) d(p, t) d(t, q) lt v(p) d(p,
  q),
• then we would have picked t before q
  (Contradiction)

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Bellman-Ford

• Update when receive estimates
• D(i,d) minjeN(i) c(i,j) D(j,d)

3 gets updates from 2 and 5
D(3,1) minc(3,2) D(2,1), c(3,5) D(5,1)
min 3 1 , 1
8 4
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Bellman-Ford

• Focus on destination 1
• Here are the values of D(i,1)

? step
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Counting to Infinity
All links cost 1
A
B
C
3
4
0
A
B
C
5
6
0
Ping-Pong to Eternity
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Bad News Travels Slowly
1
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M
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D(2,1)2, D(3,1)1, D(4,1)2
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Bad News Travels Slowly
1
4
3
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Node 2 takes about M Iterations to figure out
that D(2,1)M
1
1
2
M
1

• Fundamental Cause After a network change, think
  of the network
• protocol running from time 0. The initial
  conditions are arbitrary
• Tricks exist to get around these problems but not
  fool proof

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Asynchronous Bellman Ford

• In general, nodes are using different and
  possibly inconsistent estimates
• If no link changes after some time t, the
  algorithm will eventually converge to the
  shortest path
• No synchronization required at all

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Oscillations

• Link costs must reflect link speed AND congestion
• Under both LSP and DV routing occurs over a tree
• The costs of the links of this tree will increase
• The other links will not be congested
• Their costs will drop
• Routing protocol will shift traffic and create a
  new tree
• This process of shifting and reshifting can be
  severe
• Way out Change congestion costs slowly
  (exponential averaging) Route dampening

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Oscillations - Example
Heavy Load ? High Delay
2
5
5
4
1
Traffic
3
1
1
Light Load ? Low Delay
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Link State vs. Distance VectorNo clear winner

• LS is robust since it each node computes its own
  routes independently
• Suffers from the weaknesses of the topology
  update protocol. Inconsistency etc.
• Excellent choice for a well engineered network
  within one administrative domain
• E. g. OSPF
• DV works well when the network is large since it
  requires no synchronization and has a trivial
  topology update algorithm
• Suffers from convergence delays
• Very simple to implement at each node
• Excellent choice for large networks
• E.g. RIP

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Trees
? Root
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Useful Trees

• Types of Trees
• Binary Tree
• Each non-leaf node has exactly 2 children
• Ternary Tree
• Height Balanced Tree
• Root chosen s.t. height is minimized
• Bipartite Graphs

• Induced Trees
• Formed by choosing a subset of V,E from G
• Rooted Tree
• Rooted at particular node
• Spanning Tree
• Covers every node
• Subset Spanning Tree
• Covers every node in a chosen subset

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Trees in Networks

• Broadcast Tree
• Search Trees
• BFS
• DFS
• Spanning Tree Protocol (STP) in Bridged Networks
  (e.g. Ethernet)
• Multicast Tree
• And many, many other places

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Trees and Ethernet

• Tree Feature
• Connected
• No cycles
• Removing any edge makes it disconnected
• Adding any edge forms a cycle
• Every pair of nodes has unique path

• Ethernet Behavior
• Need all LANs to talk
• Loops cause broadcast storms
• Any bridge/port failure needs recomputation
• Need to disable all redundant ports
• Guarantees path with STP

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Concrete Example

• Weight balanced Spanning Tree (WBST)
• Each edge has cost
• e.g. 10/100/1000 Mbps
• Choose spanning tree that has min difference
  between min and max edge cost
• Want to minimize buffering requirements as
  packets move through the network. Also control
  jitter.
• Complexity

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Recall Spanning Tree Algos

• Kruskals Greedy Algorithm to compute Spanning
  Tree
• 1. Order the edges in term of weight
• 2. Add lowest cost edge (provided no loops)
• 3. Check if all vertices are connected
• 4. If not, return to step 2
• Any search (BFS/DFS) can check for connectedness
• BFS/DFS may be performed in O(m)

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WBST Algorithm

• Sort edges in increasing order of weight
  c1ltc2ltltcm
• Initialize diff cm-c1, low 1, high 1
• Take Glow,high G (V, ci low?i?high)
• if Glow,high is connected
• if diff gt (chigh clow)
• diff chigh clow
• Remember low, high
• low low 1
• repeat
• else
• high high 1
• repeat

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WBST Correctness

• Spanning Tree exists
• By checking Glow,high is connected gt spanning
  tree must exist
• Do this using DFS. Also finds the WBST
• Why balanced
• Keep adding low cost edges till graph connected
• Then try to discard as many low cost edges
• Move both low and high pointers to consider
  other ranges
• Remember best diff

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WBST Complexity

• In each iteration, we either increment low, or
  increment high
• So max number of iterations 2m
• Each iteration needs to check connectedness DFS
  O(m)
• So entire algo is O(m2)
• Think How would we make this algorithm
  distributed

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Multicast Trees

• Application Want to send packets from a source
  (e.g. internet radio station) to many hosts
• Sending n copies of the stream is wasteful
• Solution Form tree rooted at source that spans
  all member nodes

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Distributed Algorithm

• Need to know address of group
• First Join
• First member sends route request
• Request will be forwarded all the way to source
• Confirmation flows back to member
• All routers in path will add entry (mc_addr
  port)
• Subsequent Join
• New member send route request
• Request forwarded to router that has mc_addr
  entry
• Send confirmation from this point on
• Augment route entry (mc_addr port1, port2)
• Think Optimal multicast tree

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Multicast Algorithm (contd)

• Leave/Pruning
• Node leaving sends leave message to router
• Router removes port from route entry
• If last port entry deleted, send prune message to
  parent
• Multi-source multicast tree
• Less optimal packets between two branches travel
  long distance
• But need to minimize state in routers

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Coloring History

• Map coloring Color all countries on a map using
  fewest colors
• Model Each country is a node, edge if share
  boundary
• This forms a planar graph (edges dont intersect)

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4 5 Color Theorems

• Five color theorem Every planar graph is
  5-colorable
• Heawood, 1890
• Four Color Theorem Every planar graph is
  4-colorable
• Conjectured for many years (since 1890)
• Proved by Appel and Haken (1977)

• Can do no better than 4 colors
• K4 (complete graph in 4 vertices) is planar

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Independent Sets (IS)

• Set of nodes that do not have an edge between
  them
• Set of nodes that can have the same color
• Maximal IS not contained in any other IS
• Result Number of maximal IS in a graph is
  exponential

• Graph coloring algorithms find IS
• Finding the chromatic number of a graph is
  NP-hard
• Many approximations exist

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Approximate Coloring

• Greedy Algorithm
• Order nodes by degree
• Add first node to IS
• Discard all neighbors into a Future bin
• Repeat till set empty
• Start with future bin
• Correctness
• Are the sets generated independent ?
• Are they maximal ?

• Complexity
• Each iteration adds one node, so O(m)
• Discarding neighbors O(n)
• Total O(mn)
• Follow up How far is this from optimal coloring
• Arbitrary or Bounded
• Many approx algos give coloring within 4X, 6X of
  optimal

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Cliques

• Definitions
• Clique Complete Subgraph
• Maximal Clique Clique not a subset of any other
• Clique Number ? size of largest clique

? 4

• Observe
• Each node in a clique needs a different color
• So ? ? ?
• Clique is complement of IS

Maximal Cliques ABC, BCEF, CDF
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Perfect Graphs

• A graph is said to be perfect when its
• Chromatic Number ?
• Clique Number ?
• Are equal
• For all induced subgraphs
• Strong Perfect Graph Theorem
• Chudnovsky, Robertson, Seymour, Thomas (2002)
• A graph is perfect if and only if it has no odd
  holes or odd anti-holes
• Odd-hole is a odd cycle with no chord
• Odd anti-hole is complement of odd-hole