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On Reliable Broadcast in a Radio Network

On Reliable Broadcast in a Radio Network. Vartika Bhandari & Nitin H. Vaidya. The Reliable Broadcast Problem. Given a communication network (maybe multi-hop). If source s sends a message: All non-faulty nodes must agree on a single value for that message

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On Reliable Broadcast in a Radio Network

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  1. On Reliable Broadcast in a Radio Network Vartika Bhandari & Nitin H. Vaidya

  2. The Reliable Broadcast Problem • Given a communication network (maybe multi-hop) If source s sends a message: All non-faulty nodes must agree on a single value for that message If s is non-faulty, the agreed value must be the one sent by s s

  3. Communication Model: Radio Network • Abstraction of Wireless Channel • If a node S transmits a message: • All neighbors of S are certain to receive the message ( “reliable local broadcast” assumption) • Address-spoofing or collisions not allowed • Pre-determined transmission schedule • MAC/PHY free of Byzantine Faults • Authenticity of source of local broadcast is never in doubt

  4. The Radio Network Model Time t’ > t Time t (S, y) (S, x) B B S S (S, y) (S, x) A A Duplicity not possible!

  5. Network Topology: Infinite Integer Grid • Nodes located at integer lattice sites • Nodes u and v are neighbors if: • d(u, v) ≤ r (r=transmission range) • Distance metrics considered: L∞, L2 • In n-dim space, Lp distance = (∑xip)1/p, where (x1, .., xn) is distance vector r r L∞ L2

  6. Fault Model: Locally Bounded • Number of faults per neighborhood ≤ t • Type of fault: Byzantine and Crash-Stop • Bounds fault density • Allows a large number • of faults to be dispersed • throughout the network r

  7. Locally Bounded Fault Model • First introduced in [Koo:04] • Lower and Upper Bounds proved for grid network model • Further explored in [Pelc:05] • Graph-theoretic conditions for possibility/impossibility obtained for general graphs • Shown that topology-unaware algorithms are less powerful than topology-aware ones Exact Threshold not obtained by either work! Our Contribution: Exact Thresholds for Grid Network Model

  8. Earlier Results [Koo:04] (Byzantine Failures Considered) • Reliable broadcast impossible in L∞ metric if: • Reliable broadcast possible in L∞ if: Bounds do not match: Region Of Uncertainty Approx. 1/4 fraction of neighborhood Approx. 1/8 fraction

  9. Our Results (Byzantine Faults) • An “exact” threshold for L∞: • Upper bound on t proved in [Koo:04] is tight in L∞ • Reliable broadcast achievable if t < ½ r(2r+1) • A protocol that achieves the bound: • uses localized indirect reports • exploits local connectivity • An approximate argument for L2 (Euclidean metric) • Same protocol

  10. The Proposed Protocol (Tolerance threshold: t faults per neighborhood) • Direct neighbors of s commit to the first message value they hear from s • All other nodes commit to a value v if they reliably determine that at least t+1 nodes lying in a single neighborhood committed to v s

  11. The Proposed Protocol • Reliable determination implies: • hearing the committed value directly, or • hearing reports of the same value through t+1 node-disjoint paths that all lie entirely within one single neighborhood Example: t=1 v can reliably determine value committed to by u m v u

  12. The Proposed Protocol • When a node u commits to a value, it does a local broadcast of a COMMIT message u v Locally broadcast COMMIT

  13. The Proposed Protocol • All nodes hearing COMMIT, broadcast a HEARD after appending their identity HEARD messages sent out u v

  14. The Proposed Protocol • HEARD messages propagate upto four hops of u, accumulating path information en-route HEARD messages get sent upto four hops of u u v

  15. Proof Idea • Induction: • If all neighbors of node at (a, b) can commit to the correct value, so can all neighbors of nodes at (a-1, b), (a+1, b), (a, b-1), (a, b+1) • Base case: • All neighbors of source hear message directly and can commit to it! (a, b)

  16. Proof Idea If a node can reliably determine the value committed to by (2t+1) neighbors of (a, b), it is sure to determine thecorrect value P N (a, b) If node P has 2t+1 node-disjoint paths to node N, lying in a single neighborhood, it is sure to reliably determine N’s committed value r(2r+1) nodes Illustration of Worst Case Scenario Show existence of sufficient local connectivity

  17. Local 2t+1 Connectivity Exists! P 4 sets of node-disjoint paths between P and N: N-->A-->P N-->B1-->B2-->P N-->C1-->C2-->P N-->D1-->D2-->D3-->P All lying within the neighborhood of (a, b+r+1) D3 C2 D2 D1 B2 A B1 C1 (a, b) N

  18. L∞ Case: Summary • Square neighborhoods allow for precise counting of nodes/paths • Exact threshold derived But what of the Euclidean (L2) case?

  19. Euclidean(L2) Metric • Difficult to formulate exact threshold • Difficult to count exact number of lattice points falling into a region bounded by arcs • [Koo:04] provides a loose bound based on inclusion of a smaller L∞/L1 neighborhood in aL2 neighborhood • We derive an approximate argument to arrive at a much tighter estimated bound • Intuition derived from L∞ • Same Protocol • Estimate of t is 0.23 πr2 • (slightly less than 1/4 fraction of neighborhood)

  20. Euclidean Metric: An Approximate Argument Illustration for Worst Case P Q Similar Inductive Approach: If all neighbors of node at (a, b) can commit to the correct value, so can all neighbors of nodes at (a-1, b), (a+1, b), (a, b-1), (a, b+1)

  21. Further Results • Byzantine Faults: • A simpler connectivity condition and protocol for Byzantine agreement [Technical Report] • Indirect reports need to propagate only upto two hops

  22. Further Results • Crash-Stop Faults: • Tolerable t for crash-stop faults is twice that for Byzantine faults, under the given model • Reliable Broadcast is possible in L∞ metric if t <r(2r+1) • (Approximate) Reliable Broadcast is possible in L2 metric if t < 0.46πr2

  23. Summary • Exact Thresholds obtained for Grid Network Model with L∞ metric • Approximate thresholds for Euclidean metric • Proofs provide insight into spatial progress of broadcast propagation

  24. Discussion • Outstanding Issue: The gap between theory and practice • Wireless communication is lossy and interference-prone! • How to realize the “reliable local broadcast” assumption in real wireless networks? • Probabilistic local broadcast primitive? • A proof-of-concept approach proposed [Technical Report] for networks of moderate density

  25. Thank You! Related Documents available at: http://www.crhc.uiuc.edu/wireless

  26. Proof Details P P R Region R comprises neighbors of P Worst Case: A Corner Node P

  27. Proof Details P U N S1 S2 Illustration for Node N in U already provided Nodes in regions U, S1 and S2 possess r(2r+1) node-disjoint paths to P lying in 1 nbd

  28. Proof Details P P K2 R U J K1 S1 S2 N Non-worst Case Location of P: Consider Translated Regions Illustration for a Node N in S1

  29. Crash-Stop Results • Reliable Broadcast is possible in L∞ metric if t <r(2r+1) • (Approximate) Reliable Broadcast is possible in L2 metric if t < 0.46πr2

  30. Simple Protocol [Koo:04] • Protocol Used: • Simple, “re-broadcast only if certain” approach • Initially all neighbors of source hear direct message: • Commit to it • Re-broadcast • Thereafter, if a node hears same value from at least t+1 neighbors • Commits to it • Re-broadcasts

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