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

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

Greedy Routing Can get stuck 3 x y z

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

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

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

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

Discrete Curvature 8 Turning angle Sum of curvature = 2π

Discrete Curvature in 2D Triangulated Surface 9

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

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)

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

What they look like 13 Negative CurvaturePositive Curvature

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Examples 28

Routing 29

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%

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

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

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

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

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