Spherical Representation _ Polyhedron Routing for Load Balancing by hcj


									 Spherical Representation &
   Polyhedron Routing for
       Load Balancing
in Wireless Sensor Networks
          Xiaokang Yu
         Xiaomeng Ban
            Wei Zeng
           Rik Sarkar
       Xianfeng David Gu
            Jie Gao
   Load Balanced Routing in Sensor
• Goal: Min Max # messages any node delivers.
  – Prolong network lifetime
• A difficult problem
  – NP-hard, unsplittable flow problem.
  – Existing approximation algorithms are centralized.
  – Practical solutions use heuristic methods.
     • Curveball Routing [Popa et. al. 2007]
     • Routing in Outer Space [Mei et. al. 2008]
               A Simple Case
• A disk shape network.
• greedy routing (send to neighbor closer to
  ≈ Shortest path routing
• Uniform traffic: All pairs of node have 1
• Center load is high!
           Curveball Routing
• Use stereographic projection and perform
  greedy routing on the sphere
• The center load is alleviated.

• But greedy routing may fail on sparse
          Routing in Outer Spaces
            i.e., Torus Routing
•   A rectangular network
•   Wrapped up as a torus.              Flip

•   Route on the torus.
•   With equal prob to each of
    the 4 images.                Flip

• Again, delivery is not
               Our Approach
• Embed the network as a convex polytope
  (Thurston’s theorem)
  – Greedy routing guarantees delivery
• Embedding is subject to a Möbius
  transformation f
  – Optimize f for load balancing.
• Explore different network density, battery
  level, traffic pattern, etc.
             Thurston’s Theorem
• Koebe-Andreev-Thurston
  Theorem: Any 3-connected
  graph can be embedded as
  a convex polyhedron
  – Circle packing with circles on
  – all edges are tangent to a unit
• Compared to stereographic
  mapping, vertices are lifted
  up from the sphere.
            Polyhedron Routing
• [Papadimitriou & Ratajczak]
  Greedy routing with
    d(u, v)= – c(u) · c(v)
  guarantees delivery.

             3D coordinates of v

• Route along the surface of a
  convex polytope.
 Compute Thurston’s Embedding
1. Extract a planar graph G of a sensor network
  – Many prior algorithms exist.
2. Compute a pair of circle packings, for G and
   its dual graph Ĝ using curvature flow.
  – Variation definition of the Thurston’s embedding
  – Vertex circle is orthogonal to the adjacent face
  – Use Curvature flow on the reduced graph = G +
    Prepare the Reduced Graph
• Input graph
     Prepare the Reduced Graph
                           Vertex node
• Overlay G and the                 Edge node
  dual graph Ĝ, add
                                     Face node
  intersection vertices
                                Edge node
  as edge nodes.
• Each “face” becomes
  a quadrilateral
• Triangulate each
  quadrilateral by
  adding a virtual edge.
       Compute Circle Packing Using
            Curvature Flow
 • Goal: find radius of vertex circle and the radius
   of the face circle that are orthogonal &
   embedding is flat on the plane.

Idea: start from some
initial values that
guarantee orthogonality
& run Ricci flow to flatten
         Circle Packing Results
• Use stereographic projection to map circles to the
• Compute the supporting planes of the face circles
• Their intersection is the convex polytope
 Different Möbius transformation
• Möbius transformation preserves the circle
• Optimize for “uniform vertex distribution” ≈
  uniform vertex circle size.
• Compare with Curveball Routing and Torus
 Delivery Rate and Load Balancing
• Delivery Rate:
  – Dense network: all methods can deliver.

• Load balancing, tested on dense network
  – Torus routing: most uniform load; but avg load is
    80% higher than simple greedy methods.
  – Ours v.s Curveball: slightly higher avg load, but
    solves the center-dense problem better.
   Adjust Node Density wrt Battery Level
   • Find the Möbius transformation st circle size ~
     battery level.

                                               With optimization
Battery level: High to Low   No optimization   Routes prefer high battery
 Network with Non-Uniform Density
• Dense region spans wider area.

                  Birdeye view     Uniform density
     Conclusion & Future Work
• Bend a network for better load balancing.
• Open Question: How to deform a surface such
  that the geodesic paths have uniform density?
  – Saddles attract geodesic paths, peaks/valleys
  – Uniformizing curvature always leads to better load
Questions and Comments?

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