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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Routing Complexity Algorithms

• Overview
• Complexity
• Routing with Multiple Constraints
• Routing and Wavelength Assignments
• QoS routing in Ad Hoc Networks

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Overview

• Easy Problem
• Shortest Path (Dijkstra, Bellman-Ford)
• Difficult Problems
• Finding path with bound on cost and delay
• Assigning wavelengths to paths to maximize the
  number of accepted paths
• Online and Offline
• QoS routing with interference (ad-hoc)
• First two words about complexity

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Complexity

• Examples of Problems
• SSP Subset-Sum Problem Given a set of n
  positive integers, is there a subset of them that
  add up to N? Note easy to verify a solution to
  see if there is one, one must essentially try the
  2n subsets.
• KP Knapsack Problem Given a set of items each
  with a weight, find the number of each item to
  include to achieve the maximum weight less than a
  given value.
• HP Halting Problem Given an algorithm a and its
  input i, will it stop or will it run forever?
• SAT Boolean Satisfiability Problem Given a
  Boolean expression, is there an assignment (true
  or false) of the variables that makes the
  expression true?

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Complexity

• Complexity measures the time (number of steps or
  operations) or the space (memory) required to
  solve a problem by the best possible algorithm
• P (Polynomial) Problems whose solution can be
  found on a sequential deterministic machine in a
  time that is polynomial in the size of the input
• NP (Nondeterministic Polynomial) Problems whose
  solution can be verified in polynomial time on a
  sequential deterministic machine equivalently,
  whose solution can be found on a sequential
  nondeterministic machine in polynomial time.
  Guess and check.
• Open problem P NP?
• NP-Hard A decision problem H such that any
  decision problem in NP can be reduced to H in
  polynomial time
• NP-Complete A problem in NP to which every other
  problem in NP can be reduced in polynomial time
  both NP-Hard and NP
• SAT, KP, SSP are NP-Complete, HP is NP-Hard but
  is not NP

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Halting Theorem

• Theorem (Turing) There is no algorithm that can
  decide, for any algorithm a and any input i
  whether the algorithm a(i) halts.
• Proof
• Assume there is an algorithm with h(a, i) true
  if halts and h(a, i) false otherwise.
• Consider the algorithm t defined so that t(i) is
  as follows if h(i, i) false, then return
  true if h(i, i) true, loop forever.
• Does the algorithm t(t) halt? If it does, then
  h(t, t) false, which says that the algorithm t
  with input t does not halt. On the other hand, if
  t(t) does not halt, then h(t, t) must be true, so
  that t(t) should halt.
• Hence, there cannot be an algorithm h.

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Cooks Theorem

• Theorem (Stephen Cook) SAT is NP-Complete
• Proof
• SAT is NP because a nondeterministic Turing
  machine (NTM) can guess an assignment of
  variables and check.
• Consider a problem that can be solved in p(n) by
  a NTM.
• The NTM corresponds to a tape with n symbols,
  transition rules, and an acceptance rule. Every
  instance of the problem can be encoded into a
  Boolean expression B that is satisfied iff the
  NTM accepts.
• The variables describe the contents of the tape
  and the position of the head at the different
  steps the Boolean expression states that the
  rules of the NTM are followed and that the
  machine finishes in an accepting state. Counting
  shows that B is of size p(n).

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Complexity

• EXPTIME-Complete Problems that require an
  exponential time
• Currently, all know algorithms for solving an
  NP-complete problem require an exponential-time.
• Some problems require e(en) time other
  problems (e.g., halting) cannot be solved in
  finite time
• Note that a problem that requires a time 106n1000
  is P, yet intractable
• The average time of an NP problem might be P

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Routing with Multiple Constraints

• Problem Given a graph where each link has an
  integer cost and an integer delay, find a path
  with cost C and delay D.
• Example

• Find a path from S to D with cost 4 and delay
  3 ? None
• Find a path from S to D with cost 5 and delay
  3 ? SAD
• Find a path from S to D with cost 3 and delay
  4 ? SBD
• Find a path from S to D with cost 5 and delay
  4 ? SAD or SBD

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Routing with Multiple Constraints

• Theorem The problem is NP-Complete
• Proof
• Reduction from SSP (that is, SSP is reduced to a
  routing problem)
• Consider SSP with n integers x1, x2, , xn
• Construct following graph

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Routing with Multiple Constraints
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Routing with Multiple Constraints

• Results (from Tripakis Puri Related results in
  literature)A. Puri and S. Tripakis. Algorithms
  for Routing with Multiple Constraints. In AIPS'02
  Workshop on Planning and Scheduling using
  Multiple Criteria, 2002 Algorithms for routing
  with multiple constraints
• Pseudo-polynomial Algorithm O( VEminC, D)
  steps
• Approximation Result Algorithm that either gives
  a path with cost at most C(1 e) and delay at
  most D(1 e) or states that no such path exists.
  O(V2E(1 1/e))

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Routing and Wavelength Assignments

• Problem Lightpaths in optical network
• Example Consider following network
• Each link can carry N wavelengths
• Requests for connections come from i to j
• When request comes, it can be satisfiedby a
  direct connection if possible, or by
• an indirect one.
• Question Should one be myopic?
• Answer Probably not .

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Routing and Wavelength Assignments

• Example
• Myopic strategy

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Routing and Wavelength Assignments

• Example
• The morale of the story is that one should
  anticipate future arrivals.
• Optimization is difficult. How could we do it?
• Dynamic Programming ? LP
• Size of LP variables are
• a(x) p(accept and state is x)
• ? X O(N3).
• Practical Solution Trunk ReservationAccept long
  route only if there are n free circuits on links

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Routing and Wavelength Assignments

• Some easy results (Ramawami, Sivarajan
  Optical Networks, 98)
• Coloring
• Upper Bound
• Greedy assignment in line network
• Ring Networks

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Routing and Wavelength Assignments

• Coloring
• Assigning walengths is equivalent to coloring the
  path graph
• The minimum number of wavelengths is the minimum
  numberof colors for the path graph NP-Complete

For this path graphs, two colors suffice Let B,
C be yellow and A, D black.This corresponds to
B, C using one wavelength and A, D another one.
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Routing and Wavelength Assignments

• Upper Bound
• Assume a given routing with at most L
  wavelengths/link and at most H hops per path
• The number of necessary wavelengths is at most
• min(L 1)H 1, (2L 1)E0.5 L 2
• Proof
• Each lightpath can intersect at most (L 1)H
  other lightpaths
• The degree of the pathgraph is at most (L 1)H
  and can be colored in a greedy way with (L 1)H
  1 colors
• Assume there are K lightpaths of length ³ E0.5.
  The average number of wavelengths per link is
  then KE0.5/E L, so that K LE0.5.
  Assign LE0.5 different wavelengths to these
  long paths.
• Consider the path with fewer than E0.5 1
  hops. Each intersects with at most (L 1)(E0.5
  1) other such lightpaths and require at most (L
  1)(E0.5 1) wavelengths ? other bound.

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Routing and Wavelength Assignments

• Linear Network Fact Greedy assignment requires
  the minimum number L of wavelengths
• Proof
• Let L be the maximum number of intersecting paths
• Number wavelengths 1 to L
• Proceed from left to right left-most lightpath
  is assigned 1, then next path is assigned
  smallest available number, and so on
• This requires at most L wavelengths

L
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Routing and Wavelength Assignments

• Ring Networks Most optical networks are
  arranged as rings (SONET)
• Cute observation Shortest path routing may
  require more wavelengths, but at most twice as
  many

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Routing and Wavelength Assignments

• Cute observation Shortest path routing may
  require more wavelengths, but at most twice as
  many
• Proof
• Assume shortest path requires k wavelengths
• Consider a link j that uses k wavelengths.
  Reroute n paths that use that link j ? reduce
  wavelengths on j to k n but they now use at
  least N/2 hops and must cross link j N/2 and
  increase its of wavelengths by n
• Thus, the minimum number L of wavelengths must
  be such that
• L ³ minn maxk n, n ³ k/2

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QoS in Ad-Hoc Networks

• Ad-Hoc Networks
• QoS
• Model and Related Work
• Row Constraints
• Clique Constraints
• Implementation of Algorithms
• Interference-based QoS Routing

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QoS in Ad-Hoc Networks

• Ad-Hoc Networks
• QoS
• Model and Related Work
• Interference
• Row Constraints
• Clique Constraints
• Implementation of Algorithms
• Interference-based QoS Routing

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Ad-Hoc Networks

• No base station
• Multi-hop transmissions
• Distributed and dynamic operations

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QoS in Ad-Hoc Networks

• Ad-Hoc Networks
• QoS
• Model and Related Work
• Interference
• Row Constraints
• Clique Constraints
• Implementation of Algorithms
• Interference-based QoS Routing

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QoS Model, related work

• Want to support flows with quality (bandwidth)
  requirements
• Aspects of the problem
• Maximum capacity in a network
• Interference
• Feasibility of a given set of flows
• Available capacity once flows are assigned
• Routing a given set of flows

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QoS Model, related work

• Capacity of ad-hoc networks
• Random/homogenous topology, traffic matrix
• Asymptotic bounds on capacity (see next lecture)
• Our Approach
• Arbitrary topology, traffic matrix
• Graph theoretic model
• Feasibility of given set of flows
• Distributed, localized and dynamic algorithm

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QoS in Ad-Hoc Networks

• Ad-Hoc Networks
• QoS
• Model and Related Work
• Interference, conflict graph, independent sets
• Row Constraints
• Clique Constraints
• Implementation of Algorithms
• Interference-based QoS Routing

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Interference

• In wired networks, all links may be used
  simultaneously
• In Ad-Hoc networks, neighboring links interfere
• Interference Range gt Transmission Range

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Interference Conflict Graph
Three Links F1 F2 lt C and F2 F3 lt C
Single Link F1 lt C
Two Links F1 F2 lt C
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Independent Set Solution

• Construct Conflict Graph

• Identify All Maximal Independent Sets
• L1, L3

, L1, L4
L2, L4 , L2, L5 , L3, L5

• Write Constraints such that
• Only one Independent Set on at a time
• QoS requirements met for flow at each link

A New Model for Packet Scheduling in Multihop
Wireless Networks, H. Luo, S. Lu, and V.
Bhargavan, ACM Mobicom 2000.
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Issues with Independent Sets

• Shown to be necessary and sufficient for
  existence of global feasible schedule
• But scales poorly
• Need centralized information
• Finding all maximal independent sets is
  exponential
• Takes 10s of minutes for simple graph (lt100
  links)
• Want distributed and sufficient constraints that
  can be computed quickly in a large network

"Impact of Interference on Multi-hop Wireless
Network Performance, K. Jain, J. Padhye, V. N.
Padmanabhan, and L. Qiu, ACM Mobicom 2003.
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Overview

• Introduction and Motivation
• QoS in Ad-Hoc Networks
• Model and Related Work
• Interference
• Row Constraints
• Clique Constraints
• Implementation of Algorithms
• Interference-based QoS Routing

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Conflict Graph
Necessary and Sufficient F1 F2 lt C and F2
F3 lt C
Single Link F1 lt C
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Row Constraints

• Each row in the Conflict Graph incidence matrix
  yields a constraint

• At Node 2 F2 F1 lt C
• At Node 1
• F1 F2 F3 F4 F5 lt C

• Sufficient for existence of feasible schedule
• Often too pessimistic
• F2 F3 F4 F5 C possible
• Row constraints allow only F2 F3 F4 F5
  C/4

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Sufficiency of Row Constraints Proof

• Assume each weight Fi is integral
• Transform CG ? CGF
• Replace each node i with Ki fully connected nodes
• Color this graph
• Each node will be scheduled for requisite number
  of slots
• Neighboring nodes will be scheduled for disjoint
  slots
• Need to achieve coloring in T colors/slots
• Greedy algorithm
• Color each node with smallest available color
• Can always find such a color since degree (row
  constraints) lt T

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Overview

• Introduction and Motivation
• QoS in Ad-Hoc Networks
• Model and Related Work
• Row Constraints
• Clique Constraints
• Implementation of Algorithms
• Interference-based QoS Routing

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Cliques

• Definitions
• Clique Complete Subgraph
• Maximal Clique Clique not a subset of any other

• Observe
• Cliques in CG are local structures (IS are
  global)
• Only one node in a clique may be active at once

Maximal Cliques ABC, BCEF, CDF
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Clique Constraints
Clique

• Identify All Maximal Cliques
• L1, L2, L1, L5 , L2, L3, L3, L4, L4,
  L5
• Write Constraints
• Only one member of a Clique can be on at once
• F1 F2 lt C, F1 F5 lt C, ...
• Necessary conditions for a feasible schedule

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Insufficiency of Clique Constraints

• But, clique constraints are not sufficient
• F1F2F3F4F5 C/2 satisfy clique constraints
• But, we see that only 2 of 5 nodes may be on at
  once
• F1F2F3F4F5 2C/5 is the max possible
  allocation
• Sufficient only for Perfect Graphs

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Imperfection Ratio

• Imperfection Ratio is the ratio between the
  weighted Chromatic and Clique numbers
• Supremum over all weight (flow) vectors
• Bounded when the underlying graph is UDG
• Feasible schedule exists if scaled clique
  constraints are satisfied on a conflict graph
• Scale capacity of each link by
• So,

Graph Imperfection I, S. Gerke and C.
McDiarmid, Journal of Combinatorial Theory,
Series B, vol. 83 (2001), pp. 58-78.
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Extensions to Realistic Networks

• Earlier results valid for CG that are unit disk
  graph
• Variance in interference range
• Model interference range varying between x,1
• Then, need to scale the clique constraints by
• Obstructions in network
• Consider virtual CGV without obstructions
• Feasible schedule in CGV implies schedule in CG
• Satisfy scaled clique constraints in CGV

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Overview

• Introduction and Motivation
• QoS in Ad-Hoc Networks
• Model and Related Work
• Row Constraints
• Clique Constraints
• Implementation of Algorithms
• Interference-based QoS Routing

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Choose Destination
Click on bar to choose flow rate
Routing
Choose Source
Y position in km
X position in km
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Choose Next Source
Choose Destination
Click on bar to choose flow rate
Routing
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(No Transcript)
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Choose Next Source
Choose Destination
Click on bar to choose flow rate
Flow Rejected. Insufficient Resources
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Overview

• Introduction and Motivation
• QoS in Ad-Hoc Networks
• Model and Related Work
• Row Constraints
• Clique Constraints
• Computing Cliques
• Implementation of Algorithms
• Simulations of 802.11b
• Interference-based QoS Routing

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Shortest Path Methods ??

• 1-3 is widest path from node 1 to 3
• Consider path from 1 to 5
• Path 1-3-4-5 FAFDFEltC, so fltC/3
• Path 1-2-3-4-5 FBFCltC, FCFDltC, FDFEltC,
  so fltC/2

• Violates Bellmans principle of optimality
• Does not conform to distributed algorithm
  extending path hop by hop
• Distributed algorithm unlikely to be optimal
• Work with distributed heuristic algorithms

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Source Routing

• Link state exchange allows src to know
• Topology
• Available capacity on all links i
• New flow (src, dest, bw) arrives
• Choose several candidate paths by source routing
• Shortest Path (SP)
• SP complement
• Approximation of Shortest Widest Path
  (ASWP)(evaluate local constraints row or
  clique keep n-best paths)
• Send probe packets along each path
• Final path chosen and confirmed by destination

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SWP Tradeoffs (contd)

• Short Paths
• Take least resources
• Tend to crowd middle of network
• Wide Paths
• Use up too much resources
• Computation intensive
• ASWP seems pretty good in simulations