CS547: Wireless Networking

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CS547 Wireless Networking

• Lecture 9 Cellular Networks

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Cellular Networks
Telephony Networks
Switching/Routing
Switching/Routing
Data Networks
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Cellular Hexagonal Geometry
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Channel assignment s.t. co-channel separation

• All bases stations sharing the same channel have
  to be apart by a specified geographic distance.
• The total number of channels is minimized.
• The adj-channel separation is maximal.

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Lattice

• A lattice with minimal base e1 and e2 L(e1,e2)

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Oblique and Cartesian coordinates

• The oblique coordinates of the point xe1ye2 is
  .
• oblique ? Cartesian
• Cartesian ? oblique

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Fundamental domains

• A set V?R2 is a fundamental domain of ? if the
  translates of V by the lattice points of ?, i.e.
  the set uV u??, form a tiling (or partition)
  of the plane
• fundamental parallelogram a half-closed
  half-open parallelogram
• R(e1,e2)?e1?e2 0 ? ?, ? lt 1
• fundamental hexagon half-closed half-open
  hexagon
• area e1?e2

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Number of lattice points in a half-closed and
half open parallelogram

• The number of lattice points in R(p1,p2) is
  detp1,p2

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Regular lattice

• Regular
• If p , then p² x² xy y².
• A number of the format x² xy y² with x, y?Z
  is called rhombic number.
• The first few rhomic numbers are 1, 3, 4, 7, and
  9.
• The squared distance between any pair of
  transmitters is rhombic.

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Characterization of rhombic numbers

• A positive integer is rhombic if and only if,
  after removing all square factors, its prime
  decomposition contains no prime other than 3 and
  primes of the form 6k1 (k?Z).
• The product of two rhombic numbers is also
  rhombic.

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Representations of a rhombic number

• Positive representation with 0 ? x ? y.
• Rotations by 60o
• Reflections

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Number of representations

• Suppose that in the prime decomposition of a
  rhombic number n, each prime of the form 6k1
  appears ?k times. Then the number of
  representations of n is 6?k(?k 1).

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Min. channel assigment s.t. cochannel separation
d

• n ? the rhombic number up-rounded from d².
• p1 ? a positive representation of the n
• p2 ? the counterclockwise rotation of p1 by 60o
• The lattice points in R(p1,p2) and C(p1,p2)
  receive distinct channels
• Repeat the same assignment in other translates of
  R(p1,p2)

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An example
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Optimality

• The number of channels n
• n is also a lower bound due to Thue theorem
• If a convex domian D in the plane contains k
  2 non-overlapping unit-diameter circular disks,
  then the area of D is greater than k .

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Adj-channel separation
Jumps
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Adj-channel separation
Jumps
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Generating jumps and Hamiltonian sequences

• Given a regular sublattice L(p1,p2), a set of
  jumps is generating if each lattice point in
  R(p1,p2) can be reached from the origin by
  following a sequence of some of these jumps with
  possible wrapping-arounds.
• Given a regular sublattice L(p1,p2), a sequence
  of jumps is Hamiltonian if all points in
  R(p1,p2) are visited exactly once by following
  this sequence with possible wrapping-arounds.

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Selection of generating jumps

• Let
• ? (j - k) mod 3 ? 0,1,-1
• g gcd (j, k)
• Selection depends on ?

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Generating jumps ? 0
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Generating jumps ? 1
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Generating jumps ? -1
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Iterative construction of Hamiltonian sequence
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Heterogeneous channel assignment

• Different base stations may have disparate
  channel demands
• Adjacent base stations receive different channels
• Minimize the total number of assigned channel

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Minimum weighted coloring

• Input
• An induced subgraph G of regular lattice
• Channel/color demand c(v) of each node v
• Output coloring of the nodes such that
• Each node v receives c(v) different colors
• Adjacent nodes receive different channels
• Cost total number of colors

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Weighted chromatic/clique number

• Weighted chromatic number ?c minimum number of
  colors
• NP-hard
• Weighted clique number ?c maximum weight of a
  clique
• computable in polynomial time
• ?c ? ?c

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Weighted coloring of bipartite graphs

• Let G(U,V E) be a bipartite graph
• Compute

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Fixed quota of colors for each node

• Compute ?c and set
• Partition the nodes into three classes the
  class number of a node is
  f(v)(x2y)mod3
• Each node receives a quota of k colors
• (f(v), 0), (f(v), 1), , (f(v), k-1)
• Totally 3k colors (i,j) 0?i ?2, 0?j ?k-1

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Channel borrowing by a node v

• v has demands more than quota c(v)gtk
• The straight right, up-left, and down-left
  neighbors of v all have demands less than quota
• m(v) the largest demands of these three
  neighbors. Then m(v)ltk
• v borrows b(v)minc(v)-k,k- m(v) colors from
  these three neighbors (f(v)1, k- b(v)), ,
  (f(v)1, k-1)

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Two-phased algorithm

• Phase 1 for each node v,
• receives the first minc(v), k colors from its
  quota
• borrows the last b(v) colors, if b(v)gt0, from the
  straight right, up-left and down-left neighbors
• moves to next phase if still not fully colored
• Those nodes U induce a forest, which is a
  bipartite graph
• Phase 2. Apply bipartite coloring to U using
  colors disjoint from those used in phase 1.

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Residue demand of a node v moving to Phase 2

• v receives maxk,2k-m(v) colors totally in Phase
  1
• c(v)gtk
• If m(v) ? k, v cannot borrow and receives k
  maxk,2k-m(v) colors
• If m(v) gt k, v borrows b(v) k-m(v) channels and
  receives 2k-m(v) colors
• Residue demand c(v) c(v) - maxk,2k-m(v)

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Weighted clique number of GU

• Triangle-free (i.e., no 3-clique)
• For otherwise, the 3-clique would have weight ?
  3(k1)gt ?c
• 1-clique v c(v) ? c(v) - (2k-m(v)) c(v)
  m(v) - 2k ? ?c - 2k
• 2-clique u,v c(u) c(v) ? c(u) - k c(v) -
  k ? ?c - 2k

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GU is a forest

• Each node of U has at most one right neighbor in
  GU
• Prove by contradiction
• v has 2 right neighbors. Label the neighbors of
  v as below
• By ?-free property of GU, v has 2 right
  neighbors x and y.
• Let u be the one among x, y and z with the
  largest demand m(v).
• u, v and one of x and y form a 3-clique, whose
  weight in G is ?m(v)c(v)(k1) gt
  m(v)2k-m(v)(k1)3k1 ? ?c

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Total number of used colors

• Phase 1 ? 3k
• Phase 2 ? ?c - 2k
• Total ? ?c k
• 4/3-pproximation