Distributed Construction Of a Planar Spanner

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Distributed Construction Of a Planar Spanner

• Xiaomeng Ban
• Computer Science Dept.

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Content

• Definitions
• Unit Delaunay Triangulation
• Localized Delaunay Triangulation
• Planarized 1-local Delaunay Triangulation
• Routing on PLDel(S)

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Spanner

• G is a t-spanner of H iff.
• (1) V(G)V(H)
• (2) dH(u,v)ltdG(u,v)lttdH(u,v)

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Planar Graph

• A graph that can be embeded in the plane

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K(S)

• Euclidean complete graph of S

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RNG
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Gabriel
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Yao Graph
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Planar Spanner of K(S)

• RNG is planar, not spanner
• Gabriel is planar, not spanner
• Yao graph is spanner, not planar

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Delaunay Triangulation
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Del(S)

• Fact Del(S) is a spanner of K(S)
• Delaunay graphs are almost as good as
  complete graphs, D.P. Dobkin, Discrete
  Computational Geometry
• Pathdirect DT path short cut

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Direct DT path
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Short cut
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UDelS(S)

• Define UDel(S)Del(S) UDG(S)
• To prove UDel(S) is a spanner of K(S)

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UDel(S) is spanner

• Lemma1 For all i, 0ltiltm, bi is contained
  within or on the boundary of disk(u,v)
• Corollary2 all edges of the direct DT path
  connecting u and v have length at most uv.

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UDel(S) is spanner

• Lemma3 All edges of the shortcut path connecting
  bi and bj have length at most uv.
• Theorem4 for any two nodes u and v,

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Local Delaunay Triangulation

• A triangle ?uvw is called a Delaunay triangle if
  the interior of disk(u,v,w) does not contain any
  node of S
• A triangle ?uvw is called a k-localized Delaunay
  triangle if the interior of disk(u,v,w) does not
  contain any node of S that is a k-neighbor of u,v
  or w.

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Delaunay Triangle
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1-localized Delaunay Triangle
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LDel(k)(S)

• LDel(k)(S) Gabriel all k-localized
• Delaunay triangles

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LDel(k)(S) is a spanner
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LDel(k)(S) is a spanner
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LDel(1)(S) is not planar
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Construct PLDel(1)(S)

• Step1 Create LDel(1)(S)
• Step2 Planarize

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Construct PLDel(1)(S)

• Each node broadcasts its identity and location
  and listen to messages from other nodes
• For each node u hears N1(u), compute Delaunay
  triangulation
• Mark Gabriel edges

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Mark Gabriel edges
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Create LDel(1)(S)

• If ?uvw has ?wuvgtpi/3, u broadcasts a message
  proposal(u,v,w) to v and w.
• When u receives a proposal (u,v,w), if ?uvw is
  inside Del(N1(u)), broadcast accept(u,v,w) else,
  broadcast reject(u,v,w)

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Create LDel(1)(S)

• If both v and w has sent proposal(u,v,w) or
  accept(u,v,w), then add uv and uw to incident
  edges

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Planarize LDel(1)(S)

• Each node broadcast its incident Gabriel edges
  and 1-local Delaunay triangles
• Each node removes unqualified triangles

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Planarize LDel(1)(S)
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Planarize LDel(1)(S)

• Each node broadcast its incident Gabriel edges
  and 1-local Delaunay triangles
• Each node removes unqualified triangles
• Broadcast again

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Routing
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Routing
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Routing
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Routing
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Reference

• Distributed Construction of a Palnar Spanner and
  Routing for Ad Hoc Wireless Networks, Xiang-Yang
  Li
• Delaunay graphs are almost as good as complete
  graphs, D.P.Dobkin
• The Delaunay triangulation closely approximates
  the complete euclidean graph, J.M.Keil
• Wiki

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• Thanks!