Exploration of Path Space using Sensor Network Geometry Ruirui Jiang, Xiaomeng Ban, Mayank Goswami, Wei Zeng, Jie Gao, Xianfeng David Gu Stony Brook University

Prob1: Scalable Multipath Routing Deliver data using multiple (disjoint) paths – Improving throughput – Lower delay – Improving data security Encode data and send different segments along different paths Q: How to generate multiple paths? s t

k Node Disjoint Paths Centralized solutions – Flow algorithm, node disjoint paths, O(n 3 ) Distributed solutions – Only exist for 2 node disjoint paths. – Relaxation: braided multi-paths. – High discovery costs.

Prob2: Fast Recovery from Failures Sudden node or link failure – Link quality fluctuates. – Unpredictable inference, e.g., hidden terminal problem – networks interfering with WiFi [MT 08] – Jamming attacks Q: how to quickly generate an alternative path? R. Musaloiu-E, A. Terzis, Minimising the effect of WiFi interference in wireless sensor networks, International Journal of Sensor Networks, 3(1):43-54, 2008.

Prior Work IP fast re-routing schemes – Avoid loop? Path splicing [SIGCOMM 08] – Perturb edge weights – Compute multiple shortest path trees for each root – Switch to another SPT under in-transit failures – Storage requirement is too high for sensornet. Motiwala, Elmore, Feamster, Vempala, Path splicing, SIGCOMM Comput. Commun,. Rev., 2008

Understanding of the Path Space Where are the paths connecting source and destination? How to quickly find them? How to minimize storage/computation costs? Multipath routing using greedy routing? – Find an embedding, route to the neighbor closer to destination.

Our Solution: Using Circular Domains Our previous work: deform a network into circular domain [IPSN’09]. Greedy routing guarantees delivery. 

Circular Domains are not Unique Embedding into a circular domain is not unique, they differ by a Möbius transformation. 8 

Main Idea: Use Different Metrics Find multiple paths: – Embed to a circular domain D. – Apply Mӧbius transformation f on D: f(D) – Find greedy routing in f(D). – Goal: find disjoint paths. Recover from link failure: – Apply a Mӧbius transformation f on D: f(D) – Goal: greedy routing on f(D) does not use the broken link.

Outline 1.Mӧbius transformation 2.Explain the idea in a continuous domain – Theoretical guarantee 3.Implementation issues on a discrete network 4.Simulation results

Möbius Transform Möbius transform – Conformal: maps circles to circles – Four basic elements: translation, dilation, rotation, inversions. 11 a, b, c, d are 4 complex numbers, ad ≠ bc

Outline 1.Mӧbius transformation 2.Explain the idea in a continuous domain – Theoretical guarantee 3.Implementation issues on a discrete network 4.Simulation results

Greedy Paths in Different Circular Domains Generate disjoint paths using different transformations s t s t f1f1

Greedy Paths in Different Circular Domains Generate disjoint paths using different transformations s t s t f2f2

Isn’t This Just Routing along Arcs? One can greedily route along an arc. [NN, 03] – But there is no guarantee of delivery! Our method follows an arc, which is actually the greedy route in another circular domain. – Delivery is always guaranteed. Nath and Niculescu, Routing on a curve, SIGCOMM Comput. Commun. Rev., 2003.

Networks with Holes When a circular arc route hits a hole, it is diverted along the boundary – Two paths can converge on the boundary s t

Finding Disjoint Paths Each hole defines two angle ranges Inside each range there can only be one path Goal: find a max number of paths.

Finding Disjoint Paths – Project all intervals on a unit circle – Order the endpoints angularly. – Adjacent endpoints define canonical segments. – Remove canonical segments NOT covered by any range

Finding Disjoint Paths Greedy algorithm: – For each segment as the starting segment, – Choose it, move clockwise. – Choose the first one not in conflict – Ex: choose 5.

Finding Disjoint Paths Greedy algorithm: – Different starting segments give different solns. – Take the solution with maximum # segments Theorem: GREEDY is optimal.

Outline 1.Mӧbius transformation 2.Explain the idea in a continuous domain – Theoretical guarantee 3.Implementation issues on a discrete network 4.Simulation results

Preprocessing Into a Circular Domain Embed the network into a circular domain – Follows [IPSN’09] – Compute a triangulation locally – Non-triangular faces are “holes” – Nodes locally compute curvature – Modify edge lengths iteratively – When converge, obtain the requested embedding Distributed, gossip-style algorithm.

How to compute a Mӧbius Transformation? Uniquely determined by mapping 3 points z 1, z 2, z 3, to w 1, w 2, w 3, respectively. I.e., mapping one circle to another circle

How to compute a Mӧbius Transformation? New metric can be locally determined. s t s t f1f1 p p

Generate Multiple Paths No holes: – Evenly spread the tangent vectors of the paths at source and dest. – Heuristic – Bounded degree. With holes: – Know: locations & sizes of holes – Very limited global info s t

How to Apply the Mӧbius Transformation in Routing? A packet carries the four parameters a, b, c, d as a matrix. Composition of 2 transformations is simply matrix multiplication

Recovery from Temporary Link Failure We compute a Möbius transformation – S.t. the broken link is NOT on the greedy path. – Make big jumps 27

Recovery from Temporary Link Failure Map the “live” neighbor p to be on the straight path – On-demand recovery of in-transit failures. – Möbius transform attached to packet. 28 s ts t f1f1 p p

Simulation Results Relatively dense network: – 1K nodes, avg deg ≈ 20. Multi-path routing – Compare with centralized flow algorithm. – Vary parameter m: # paths we seek. – Check # disjoint paths we got – Compare with the OPT.

Multipath Routing Results Max # disjoint paths

Simulation Results for Recovery from Failures Regional failure – Inside a geometric region: prob[failure] = p 1. – Outside a geometric region: prob[failure] = p 2. – p 1 > p 2. Compare with – Greedy routing with geographical coordinates – Greedy routing in a circular domain – Recovery with Möbius transformation – Recovery with random walk.

Simulation Results for Recovery from Failures Circular + Möbius > Circular + RandWalk > Circular >> Geographical + RandWalk ≈ Geographical p 1 =0.8 p 2 =0 Random walk makes local, random steps and is likely trapped inside the failure region

Simulation Results for Recovery from Failures Success rate: Circular + Möbius > Circular + RandWalk > Circular Circular + Möbius Circular + RandWalk Circular p 1 p 2 Consistently better by making big jumps

Conclusion and Future Work Regulating a sensor network shape – Helps to explore path space with limited global info Open problem: bridging the gap – Provable results in the continuous space – Heuristic methods in the discrete network

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