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Greedy Routing with Guaranteed Delivery Using Ricci Flow Jie Gao Stony Brook University Rik Sarkar, Xiaotian Yin, Feng Luo, Xianfeng David Gu

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Greedy Routing Assign coordinates to nodes Message moves to neighbor closest to destination Simple, easy, compact 2

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Greedy Routing Can get stuck 3 x y z

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Use Ricci flow to make all holes circular Greedy routing does not get stuck at holes. 4

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Talk Overview Theory on Ricci flow --smoothing out surface curvature Distributed algorithm on sensor network – Extract a triangulation from unit disk / quasi unit disk graphs – Ricci flow: distributed iterative algorithm 5

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Part I : Curvature and Ricci Flow 6 Curvature: κ = 1/R Flatter curves have smaller curvature. R

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Changing the Curvature 7 Flatten a curve: κ 0 Make a cycle round κ 1/R R

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Discrete Curvature 8 Turning angle Sum of curvature = 2π

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Discrete Curvature in 2D Triangulated Surface 9

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10 Deviation from straight line For an interior vertex For a vertex on the boundary Deviation from the plane corner angle

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Total Discrete Curvature Gauss Bonnet Theorem: total curvature of a surface M is a topological invariant: Ricci flow: diffuse uneven curvatures to be uniform 11 Euler Characteristic: 2 – 2(# handles) – (# holes)

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Sanity check: total curvature h holes, Euler characteristics = 2-(h+1) Outer boundary: curvature 2π Each inner hole: -2π Interior vertex: 0 Total curvature: (1-h)2π 12

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What they look like 13 Negative CurvaturePositive Curvature

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Ricci Flow: diffuse curvature Riemannian metric g on M: curve length We modify g by curvature Curvature evolves: Same equation as heat diffusion 14 Δ: Laplace operator

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Ricci Flow: diffuse curvature “Ricci energy” is strictly convex unique surface with the same surface area s.t. there is constant curvature everywhere. Conformal map: angle preserving 15

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Ricci Flow in our case Target: a metric with pre-specified curvature 16 Boundary cycle

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Discrete Ricci Flow Metric: edge length of the triangulation – Satisfies triangle inequality Metric determines the curvature Discrete Ricci flow: conformal map that modifies edge length to smooth out curvature. 17

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Discrete Flow : Use Circle Packing Metric Circle packing metric: circle of radius γ i at each vertex & intersection angle φ ij. Remark: no need of an embedding (vertex location) 18 With γ, φ one can calculate the edge length l and corner angle θ, by cosine law Discrete Ricci flow modifies γ, preserves φ

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Discrete Flow : Use Circle Packing Metric Circle packing metric: circle of radius γ i at each vertex & intersection angle φ ij. Take Discrete Ricci flow: 19 Target curvature Current curvature

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Background [Hamilton 82, Chow 91 ] : Smooth curvature flow flattens the metric [Thurston 85, Sullivan and Rodin 87 ] : Circle packing – discrete conformal maps [He and Schramm 93, 96 ] : Discrete flow with non-uniform triangulations [Chow and Luo 03 ] : Discrete flow, existence of solutions, criteria, fast convergence 20

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Part II : Ricci Flow in Sensor Networks Build a triangulation from network graph – Requirement: triangulation of a 2D manifold Compute virtual coordinates – Apply distributed Ricci flow algorithm to compute edge lengths d(u, v) with the target curvature (0 at interior vertices and 2π/|B| at boundary vertices B) – Calculate the virtual coordinates with the edge lengths d(u, v) 21

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Issues with Network Graph We need a triangulation s.t. locally we have 2D patches. 22 Crossing Edges degeneracies

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Location-based Triangulation Local delaunay triangulations with locations [Gao, Guibas, Hershberger, Zhang, Zhu - Mobihoc 01] Compute Delaunay triangulations in each node’s neighborhood Glue neighboring local triangulations Remove crossing edges We now extend it to quasi-UDG 23

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Location-based triangulation Handling Degeneracies : Insert virtual nodes Locations not required for Ricci flow algorithm Set all initial edge lengths = 1 24

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Location Free Triangulation Landmark-based scheme : planar triangulation. [Funke, Milosavljevic SODA ’07] Set all edge lengths = 1 25

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Ricci Flow in Sensor Networks All edge length = 1 initially – Introduces curvature in the surface Use Ricci flow to reach target curvature κ’ – Take tangent circle packing metric: γ=1/2, φ=0 initially – Interior nodes: target curvature = 0 – Nodes on boundary C: target curvature = - 2π/|C| – Modify u i = logγ i by δ (κ’-κ) – Until the curvature difference < ε |C|: Length of the boundary cycle

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Ricci Flow in Sensor Networks Ricci flow is a distributed algorithm – Each node modifies its own circle radius. Compute virtual coordinates – Start from an arbitrary triangle – Iteratively flatten the graph by triangulation. 27

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Examples 28

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Routing 29

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Experiments and Comparison NoGeo : Fix locations of boundary, replace edges by tight rubber bands : Produces convex holes [Rao, Papadimitriou, Shenker, Stoica – Mobicom 03] Does not guarantee delivery: cannot handle concave holes well MethodDeliveryAvg StretchMax Stretch Ricci Flow100% NoGeo83.66%

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Convergence rate Curvature error bound ε Step size δ # steps = O(log(1/ε)/δ) 31 Iterations Vs error

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Theoretical guarantee of delivery Theoretically, Ricci flow is a numerical algorithm. Triangles can be skinny. In our simulations, we have not encountered any delivery problems In theory, one can refine the triangulation to deal with skinny triangles 32

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Summary Deformation of network metric by smoothing out curvatures. All holes are made circular. Greedy routing always works. Future work: – Guarantee of stretch? – Load balancing? 33

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Mobius Ambiguity Many different deformations are conformal – Equivalent under mobius transform Local operation : need to fix target deformation 34

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Convergence rate Curvature error bound ε Step size δ # steps = O(log(1/ε)/δ) Experiments: ε=1e Iterations Vs Number of Nodes

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