CS547: Wireless Networking

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CS547 Wireless Networking

• Lecture 2 Broadcast (Radio/TV) Networks

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Broadcast (Radio/TV) Networks

• One-way communication
• Coverage by a broadcast station

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Simplified Interference Model

• Assumption
• Stations in a plane
• Coverage by each station u is a circular disk
  with radius ru
• Stations u and v interfere with each other ?
  Their coverages overlap? uv ? ru rv

u
v
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Conflict-free channel assignment

• Given the locations and coverage radii of a set
  of stations, assign the fewest channels to
  stations s.t. interfering stations receive
  distinct channels

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Computation complexity and approximation
algorithms

• Complexity class of optimization problems
• Polynomial-time solvable
• NP-hard no polynomial-time algorithm unless PNP
• Approximation algorithms tradeoff between time
  and performance
• Polynomial time
• Approximation ratio the largest ratio of the
  approximate (resp., optimal) solution to optimal
  (resp., approximate) solution for a minimization
  (resp., maximization)

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Interference Topology

• G (V, E) V stations, E pairs of
  interfering stations
• Disk graph
• Unit-disk graph all stations have the same radii

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Subgraph

• Subgraph ? subset of stations U, and subset of
  some pairs of interfering stations in U
• Induced subgraph ? subset of stations U and
  subset of all pairs of interfering stations in U

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Neighbor and node degree

• Neighbor u is a neighbor of v ? u interferes v
• degG(v) degree of v ? number of stations
  interfering v
• Handshaking lemma total degree 2E
• ?(G) min degG(v) v?V
• ?(G) max degG(v) v?V
• deg (v1)6, deg (v2)deg(v4)deg(v5)deg(v6)3,
  deg(v3)deg(v7)4
• ?3, ?6

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Independence number

• Independent set (IS) ? subset of pairwise
  non-interfering stations
• ? subset of stations that can share a channel
• e.g. v3, v5, v2, v4,v6
• Maximal IS
• e.g. v1, v2, v5
• Maximum IS
• e.g. v2, v4,v6
• Independence number ?(G) size of a maximum IS
• Number of channels ? V/?(G)

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Clique number

• Clique ? subset of pairwise interfering stations
• ? subset of stations that must use distinct
  channels
• e.g. v1, v3, v4, v1, v3, v4
• Maximal clique
• e.g. v1, v3, v4
• Maximum IS
• e.g. v1, v2, v3, v7
• Clique number ?(G) size of a maximum clique
• Number of channels ? ?(G)

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Chromatic number

• Proper node coloring assign colors to nodes s.t.
  neighboring nodes receive different colors ?
  conflict-free channel assignment
• ? partition of nodes into independent sets
• Chromatic number ?(G)smallest colors needed by
  any (proper) node coloring
• ?(G) ? max V/?(G), ?(G)

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FIRST-FIT Coloring

• FIRST-FIT in a node ordering ltv1, v2, ?, vngt
• For i 1 to n, assign vi with the smallest color
  number without interference
• If ltv1, v2, ?, vngt has inductivity q, then the
  number of colors is at most q1

ltv1, v7, v6, v5, v4, v3, v2gt
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Inductivity of a vertex ordering

• The inductivity of ltv1, v2, ?, vngt is
• A vertex ordering is q-inductive if ? its
  inductivity is at most q? each vertex is
  adjacent to at most q pre-ordered vertices.
• every vertex ordering is ?(G)-inductive.

ltv1, v2, v3, v4, v5, v6, v7gt 3 ltv2, v3, v4, v5,
v6,v7 , v1gt 7
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A lower bound on inductivity of a vertex ordering

• The inductivity of any vertex ordering is at
  least ?(G) max?(GU) U?V.
• Let U?V be such that ?(GU) is the largest among
  all subsets U. Then in any ordering, the vertex
  in U appearing last has at least ?(GU)
  prior-neighbors.

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Smallest-last ordering

• ltvn, ?, v2, v1gt vi is a vertex of smallest
  degree in Gv1,v2, ?,vi
• The inductivity of the smallest-last ordering is
  at most ?(G)
• for each node vi, the number of prior neighbors
  of vi is ?(Gv1,v2,?,vi).
• smallest-last ordering has the smallest
  inductivity

ltv1, v7, v6, v5, v4, v3, v2gt
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Inductivity of a graph

• ?(G) is called the inductivity of G
• A graph G is said to be q-inductive if there is a
  q-inductive vertex ordering.
• trees are 1-inductive
• planar graphs are 5-inductive
• triangle-free planar graphs are 3-inductive
• outerplanar graphs are 2-inductive

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Neighborhood of the node u with smallest radius

• Let v and w be two neighbors of u with ?vuw ?
  60o. Then v and w are neighbors to each other.
• All neighbors of u lying in any 60o-sector form a
  clique
• The degree of u is at most 6?(G)-7
• u has at most five independent neighbors
• The degree of u is at most 5?(G)-5

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Upper bound on inductivity

• Arbitrary radii
• Radius-decreasing min6?(G)-7, 5?(G)-5
• Smallest-degree-last same bound holds
• Uniform radii
• Arbitrary ordering min6?(G)-7, 5?(G)-5
• Lexicographic/Distance-increasing/Smallest-degree-
  last 3?(G)-3

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Performance Bound on FF Coloring

• Arbitrary radii
• FF in radius-decreasing min6?(G)-6, 5?(G)-4
• Smallest-degree-last same bound holds
• Uniform radii
• Arbitrary ordering min6?(G)-6, 5?(G)-4
• Lexicographic/Distance-increasing/Smallest-degree-
  last 3?(G)-2

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Strip Coloring

• Assume radius 1/2
• Partition into -strips
• IG over a strip co-comparability graph
• Perfect
• Optimal coloring by maximum matching
• Total number of colors ?