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Exploration of Path Space using Sensor Network Geometry

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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
                                                       t
  – Lower delay
  – Improving data security         s
     • Encode data and send different segments along
       different paths


• Q: How to generate multiple paths?
         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.




                      



                                               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.



a, b, c, d are 4 complex
numbers, ad ≠ bc


                                                          11
                    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

                       f1                   t
     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
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 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
                      f1                  t
                                 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:                          s                t
  – Know: locations & sizes of holes
  – Very limited global info
        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.


                         f1                  t
         p                          p
     s             t            s
                                                 28
           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] = 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 + Circular +
                      p1 p2   Möbius RandWalk 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
Questions & Comments?

				
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posted:4/1/2013
language:English
pages:35