1 / 30

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

evan-hopkins
Download Presentation

On Reliable Broadcast in a Radio Network

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related