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

Greedy Routing with Guaranteed Delivery

Using Ricci Flow

Jie GaoStony 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

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Greedy Routing

• Can get stuck

xy z

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Use Ricci flow to make all holes circular

• Greedy routing does not get stuck at holes.

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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

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Part I : Curvature and Ricci Flow

Curvature: κ = 1/RFlatter curves have smaller curvature.

R

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Changing the Curvature

Flatten a curve: κ0

Make a cycle roundκ1/R

R

Discrete Curvature

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Turning angle

Sum of curvature = 2π

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Discrete Curvature in 2D Triangulated Surface

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Discrete Curvature in 2D Triangulated Surface

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

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

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What they look likeNegative Curvature Positive Curvature

Ricci Flow: diffuse curvature

• Riemannian metric g on M: curve length• We modify g by curvature • Curvature evolves:

• Same equation as heat diffusion14

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

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Ricci Flow in our case

• Target: a metric with pre-specified curvature

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.

Discrete Flow : Use Circle Packing Metric

• Circle packing metric: circle of radius γi at each vertex & intersection angle φij.

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 γi at each vertex & intersection angle φij.

• Take • Discrete Ricci flow:

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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

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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)

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Issues with Network Graph

• We need a triangulation s.t. locally we have 2D patches.

Crossing Edgesdegeneracies

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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

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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

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Location Free Triangulation• Landmark-based scheme : planar triangulation. [Funke,

Milosavljevic SODA ’07]

• Set all edge lengths = 1

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 ui = logγi 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.

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Examples

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Routing

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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

Method Delivery Avg Stretch

Max Stretch

Ricci Flow 100% 1.59 3.21

NoGeo 83.66% 1.17 1.54

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Convergence rate

• Curvature error bound ε

• Step size δ• # steps =

O(log(1/ε)/δ)

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

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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?

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Convergence rate

• Curvature error bound ε

• Step size δ• # steps =

O(log(1/ε)/δ)

• Experiments: ε=1e-6. Iterations Vs Number of Nodes