Scalable Routing in 3D High Genus
Sensor Networks Using Graph
Embedding
Xiaokang Yu1, Xiaotian Yin2, Wei Han2, Jie Gao3,
Xianfeng David Gu3
1Shandong University, PRC
2Harvard University
3Stony Brook University
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Routing in a high genus 3D network
• Sensors monitoring underground tunnels
– water, sewer, gas system; coal mine [Li, Liu 2009]
• Non-planar, 3D network
• Sparse with complex shape
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Geographical routing in 2D
• Greedy routing: next hop closer to dest
• Face routing: tour around a planar face
towards dest
S
t
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Geographical routing in 3D
• 2D geographical routing
– Greedy routing works in any dimension
– Face routing is unique to 2D
• Geo-routing in 3D is strictly harder.
– Analogy of “faces” is “voids” in 3D; how to get
around a void is not obvious.
– In 3D unit ball graph, no deterministic algorithm
can guarantee delivery with O(1) state. [DKN08]
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Geographical routing in 3D
• Ideas to do geo-routing in 3D:
– Handle special cases of dense networks [XYW+11].
– Project to 2D. [KFO05, LCW+08]
– Use hull tree [ZCL+10]
– Random walk [FW08]
– Face routing on a general graph [ZLJ+07].
• Our idea:
– Embed a graph on a general surface
– Apply greedy routing with guaranteed delivery
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Embedding a graph on a surface
• Every graph can be embedded on a (high
genus) surface without crossing edges
• Genus: # handles.
– # times one can cut the surface without
disconnecting it
genus=1
genus=2
genus=3
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Special case: planar graphs
• Combinatorial: no K5, or K3, 3 minor.
• K3, 3 cannot be embedded in the plane without
crossing edges.
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Special case: planar graphs
• Combinatorial: no K5, or K3, 3 minor.
• K3, 3 can be embedded on a torus
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Special case: planar graphs
• Planar graphs can always be embedded on a
sphere (genus 0), e.g., in Thuston’s embedding
Point of infinity
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Red: planar graph; Blue: dual graph
Example: embedding on a torus
Planar illustration:
Left & right
boundaries are
identified; Top &
bottom boundaries
are identified.
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Example: embedding on a torus
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Red: planar graph; Blue: dual graph
Algorithm
1. Define planar faces for a general graph
– In a combinatorial sense.
– Same idea used in FaceTracing [ZLJ+07]
2. Embed these faces on a general surface
3. Use the embedding for greedy routing
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Define faces on a general graph
• Rotation system: around each vertex, define a
(arbitrary) cyclic order of the edges.
• Trace edges of a face: come in from edge i of a
vertex v, leave at edge i+1 at v.
• Each edge only belongs to two faces, once in
each direction.
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2
1
4
5
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Genus of the embedding surface
• Determining the embedding surface with
minimum genus is NP-hard. [T89]
• We tested a heuristic algorithm:
– Swap the orders of two adjacent edges.
– Keep it if the genus goes down.
– In practice we can bring down the genus to about
1/3 the original.
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Compute the embedding
• Determine the edge lengths for such a surface
embedding
• Introduce the dual graph
• Introduce a vertex for
intersection of a primal &
its dual edge
• This defines a partitioning
into 4-gons, each with
one primal vertex, one
dual vertex, two edge
vertices.
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Compute the embedding
• Use circle packing metric
• Red circles: primal
graph, tangent to
neighbors.
• Green circles: dual
graph, tangent to
neighbors
• Red circle orthogonal
to adjacent green
circle.
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Compute the embedding
• Existence of such a metric is guaranteed in
theory.
– Genus=0, spherical metric, curvature > 0
– Genus=1, Euclidean metric, curvature = 0
– Genus=2, hyperbolic metric, curvature < 0
• Use Ricci flow (curvature flow) to compute the
final metric
– Distributed algorithm introduced in [SYG+08]
– Convergence is guaranteed by theory.
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Algorithm outline
1. Define planar faces for a general graph
2. Embed these faces on a general surface
3. Use the embedding for greedy routing
1. Define virtual coordinates.
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Route on the embedded surface
• Decompose into |2-2 genus| “pants” (genus 0
with 3 boundaries)
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Route on the embedded surface
• Each pant can be partitioned into two
hexagonal pieces (half pants).
• With the computed embedding, each piece is
convex  inside which greedy routing
guarantees delivery!
Embedding in hyperbolic space
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Route on the embedded surface
• Routing in two stages
• Global planning:
– Find the sequence of hexagonal pieces to visit
– Each node stores the adjacency of the half pants.
– Storage = O(genus of the embedding)
• Local routing:
– Perform greedy routing inside each piece
– Each node stores virtual coordinate on its half
pants
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Example: pants decomposition
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Example: global planning phase
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Example: local routing phase
• Greedy routing to the closest point on the
boundary to the next piece to visit.
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Example: entire path
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Performance comparison
• Compare with
1. Face tracing [ZLJ+08]: Use rotation system to
define faces, do DFS on faces
2. Shortest path routing
3. Greedy routing using geographical coordinates
• Assume the ideal setting of no transmission
failures.
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Performance comparison
• Sensors densely deployed on two surfaces.
genus=4
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Simulation results
• Delivery rate
• Average path lengths
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Path length histogram
Global guidance
is helpful in
finding paths in
the correct
direction.
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Conclusion and open questions
• A general solution for routing on an arbitrary
graph.
• Storage size of each node:
– adjacency of pants decomposition
– virtual coordinates in each half pants
– O(genus of the embedding).
• Open: approximation algorithm for
determining the embedding surface with
minimum genus?
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Questions and Comments?
• http://www.cs.sunysb.edu/~jgao
• [email protected]
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