Theory for Sensor Networks

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Minimum Dominating Set Approximation in Graphs of
Bounded Arboricity
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Minimum Dominating Sets (MDS)

• important in theory and practice

minimum dominating set
dominating set in a social network

• graph G(V,E)
• N(A) denotes inclusive neighborhood of AµV
• DµV is dominating set (DS) iff VN(D)
• minimum dominating set is DS of minimum size

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MDS on General Graphs

• finding an MDS is NP-hard
• ) we're looking for approximations
• O(log ?) approx. in O(log n) rounds
• ...but for reasonable message size O(log2 ?)
  rounds
• o(log ?) approx. is NP-hard
• polylog. approx. needs ?(log ?) and ?(log1/2 n)
  rounds
• ) maybe "simpler" graphs are easier?

Kuhn al., SODA '06
Garey Johnson, '79
Raz Safra, STOC '97
Feige, JACM '98
Kuhn al., PODC '04
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MDS on Restricted Families of Graphs
planar
O(1) approx. O(1) rounds
(1²) approx. polylog n rounds
general
bounded degree
T(log n) approx. O(log2 ?) rounds ?(log ?) rounds
O(1) approx. O(1) rounds
unit disc
O(1) approx. O(log n) rounds
L. et al SPAA '08
O(1) approx. T(log n) rounds
e.g. Luby SIAM J. Comp. '86
Czygrinow Hanckowiak, ESA '06
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What's a Good Compromise?

• ...or what have many "easy" graphs in common?
• ) They are sparse!
• This is not good enough

O(n) edges


same lower bounds as in general case
arbitrary graph n1/2 nodes difficult to handle
star graph n-n1/2 nodes center covers all
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Arboricity

• A "good" property is preserved under taking
  subgraphs.
• ) Demand sparsity in every subgraph!
• This property is called bounded arboricity.

3-forest decomp. of the Peterson graph...
...whose arboricity is however only 2.

• graph G(V,E)
• partition E E1 E2 ... Ef into f forests
• minimum number of forests is arboricity A of G

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Where are Graphs of Bounded Arboricity?
no o(A) approx. in o(log n) rounds

• arboricity 2 permits Kvn minor
• no strong lower bounds
• o(log A) approx. is NP-hard
• no (5-²) approximation in o(log n) time

bounded arboricity
excluded minor
bounded arboricity
planar
general
bounded degree
unit disc
Czygrinow al., DISC '08
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Be Greedy!

• sequentially add nodes covering most others
• ) yields O(log ?) approx.
• ...but in parallel?
• ) Just take all high-degree nodes!
• repeat until finished

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Why does Greedy-By-Degree work?
V

• D nodes of (current) max. deg. ?
• C nodes (freshly) covered by D
• M optimum solution
• D?/2 E(CD) lt A(CD) A(CD)
• ) (?/2-A)D lt AC A(?1)M
• if ? 4A and A 2 O(1)
• ) D 2 O(M)

D
C
M
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Greedy-By-Degree Details
Q What about ? lt 4A ? A Each c2C elects one
deg. ? neighbor into D! Q How avoid time
complexity ?(?)? A Take all nodes of degree ?/2
at once! Q How deal with unknown ?? A It's
enough to check up to distance 2! ) uniform
O(log ?) approx. in O(log ?) rounds
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Neat, but...

• ...we would like to have an O(1) approx. for A 2
  O(1)
• What about using a (rooted) forest decomposition?
• decomposition into f 2 O(A) forests takes T(log
  n) time
• note we cannot handle each forest individually

Barenboim Elkin, PODC '08
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How to use a Forest-Decomposition

• For an MDS M, (A1)M nodes are not covered by
  parents.
• ) These have A(A1)M parents.
• ) Let's try to cover all nodes (that have one) by
  parents!
• ) set cover instance with each element in A sets

)
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Acting Greedily again

• sequentially, an A approx. is trivial
• pick any uncovered node
• choose all of its parents
• repeat until finished
• for every node, one of its parents is in an
  optimum solution

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1
1,3,7
9
1,10
6
2
5
9,10
7
10
3,6,10
8
9
3,5,9
3
4
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And now more quickly...

• any sequence of nodes that share no parents is
  feasible
• the order is irrelevant for the outcome
• define H(V,E') by v,w 2 E' , v and w share a
  parent
• ) we need a maximal independent in H

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Algorithm Parent Dominating Set

• compute O(A) forest decomp. (O(log n) rounds)
• simulate MIS algorithm on H (O(log n) rounds
  w.h.p.
• output parents of MIS nodes and nodes w/o parents
• ) O(A2) approx. in O(log n) rounds w.h.p.

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Greedy-By-Degree Pros'n'Cons
general graphs O(log2 ?)
very simple running time O(log ?) message
size O(log log ?) uniform deterministic - O(A
log ?) approx.
general graphs O(log ?)
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Parent Dominating Set Pros'n'Cons
general graphs O(log ?)

• simple
• O(A2) approx. (deterministic)
• /- running time O(log n) (randomized)
• open question
• Are there faster O(1) approx. for A2O(1)?

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Thank You!Questions Comments?