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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
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Exploration of Path Space using Sensor Network Geometry Ruirui Jiang, Xiaomeng Ban, MayankGoswami, Wei Zeng, JieGao, 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? • t • s
k Node Disjoint Paths • Centralized solutions • Flow algorithm, node disjoint paths, O(n3) • 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 • 802.15.4 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 802.15.4 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.
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 • Mӧbius transformation • Explain the idea in a continuous domain • Theoretical guarantee • Implementation issues on a discrete network • Simulation results
Möbius Transform • Möbius transform • Conformal: maps circles to circles • Four basic elements: translation, dilation, rotation, inversions. a, b, c, d are 4 complex numbers, ad ≠ bc
Outline • Mӧbius transformation • Explain the idea in a continuous domain • Theoretical guarantee • Implementation issues on a discrete network • Simulation results
Greedy Paths in Different Circular Domains • Generate disjoint paths using different transformations • t • f1 • s • t • s
Greedy Paths in Different Circular Domains • Generate disjoint paths using different transformations • s • f2 • s • t • t
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 • Mӧbius transformation • Explain the idea in a continuous domain • Theoretical guarantee • Implementation issues on a discrete network • 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 z1, z2, z3, to w1, w2, w3, respectively. • I.e., mapping one circle to another circle
How to compute a Mӧbius Transformation? • New metric can be locally determined. • p • t • f1 • p • s • t • s
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 Möbiustransformation • S.t. the broken link is NOT on the greedy path. • Make big jumps
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. • t • f1 • p • p • s • t • s
Simulation Results • Relatively dense network: • 1K nodes, avgdeg≈ 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] = p1. • Outside a geometric region: prob[failure] = p2. • p1> p2. • 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 • Random walk makes local, random steps and is likely trapped inside the failure region • p1=0.8 p2=0
Simulation Results for Recovery from Failures • Success rate: Circular + Möbius > Circular + RandWalk > Circular Circular + RandWalk Circular + Möbius • p1 p2 Circular 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