Broadcasting with Bounded Number of Redundant Transmissions

1. Broadcasting with Bounded Number of Redundant Transmissions Majid Khabbazian

2. Outline • Assumptions • Objectives • Classifications • The proposed algorithm • Algorithm’s characteristics • Conclusion

3. Assumptions • Single message broadcast • Nodes are distributed in 2-D space • The transmission range of each node is R • We can use Unit Disk Graph (UDG) to model the network • No Synchronization • Perfect Medium Access Control (MAC) • No errors or collisions • Neighbors don’t transmit at the same time • Nodes are static during the broadcast

4. Objectives • End-to-end delay is NOT a concern • What do we care about? • Full delivery • Reducing the number of transmissions • Each node has a local view of the network

5. Flooding: A Simple Solution • Flooding • Every node transmits the first copy of received message • Pros. • A simple solution • No need to have neighbor information • Requires almost no computation • Cons. • All the nodes transmit the message • It can cause a large number of redundant transmissions • It can lead to significant performance degradation and network congestion

6. A Question • Can we minimize the total number of transmissions? • This is related to fining a Minimum Connected Dominating Set (MCDS) • Finding MCDS is NP-hard even for UDGs • Good approximation algorithms? • Case 1: The whole topology is known • Case 2: Each node has a local view of the network • Local Broadcast Algorithms

7. Local Broadcast Algorithms • Classifications • Static (Proactive) • Dynamic (Reactive) • Static Approach • A backbone is constructed first • The backbone is a Connected Dominating Set • Pros. • Can be used for both broadcasting and unicasting • Cons. • May not be good where the network topology is dynamic • The backbone is fixed in the static network

8. Local Broadcast Algorithms (Con’d) • Dynamic Approach • There is no backbone • Nodes decide “on-the-fly” based on their local view • Pros. • The backbone changes from one network-wide broadcast to another (even for the single source) • More robust against failures than static approach • Cons. • Constructed backbone may not be stable

9. Further Assumptions • Each node has the list of its 1-hop neighbors • Exchanging “hello” messages • Geographical information is available • E.g., Using GPS • Relative distance may suffice

10. Static Approach • A small size backbone can be easily constructed • Regionalizing the network • Selecting a constant number of nodes in each region • Example: • Divide the network into square cells with diameter 1 • At most 20 nodes have to be selected in each cell

11. Dynamic Approach • Can we reduce the total number of transmissions in the worst case? • Is constant approximation factor achievable? • Our proposed algorithm is proven to achieve: • Full delivery • Constant approximation factor

12. Proposed Algorithm • Each node decides on its own whether or not to transmit • Before transmitting, the node removes the information attached to the message and adds the list of its 1-hop neighbors to the message • The decision is made based on a self-pruning condition called the responsibility condition • The closer, the more responsible

13. Responsibility Condition • A node u has to transmit the message if it has a neighbor v s.t. • v has not received the message AND • There is no node w such that w has received the message and dist(wv )< dist(uv)

14. Example • A receives the message from H • A knows that E, F and G have received the message and B, C and D have not • Based on the responsibility condition A does not need to transmit the message G D F C H A B E

15. Full Delivery • It achieves full delivery • Proof by contradiction: • The broadcast will eventually terminate • Suppose there is a node that has not received the message • Consider the set • S={(u,v)| u and v are neighbors, u has received the message, v has not received the message} • S is not empty

16. Full Delivery (Con’d) • S is not empty There exists a pair (u’,v’) in S such that Dist(u’,v’)<= dist(u,v) for any pair (u,v) in S. • u’ has the highest responsibility toward v’ • v’ has not receive the message • Based on the responsibility condition • u’ must have transmitted the message

17. Approximation Factor • The proposed algorithm achieves a constant approximation factor Sketch of proof • There are at most a constant number of transmissions in each disk with radius ¼ • Transmission coverage of each node is a disk with radius 1 • Each node has a constant number of neighbors that transmit the message • The number of transmission has to be within a constant factor of the optimum

18. Approximation Factor (Con’d) • Transmitters: Blue nodes • Blue nodes are neighbors • All the nodes in the white disk will get the message after the first transmission • Blue nodes are aware of this fact

19. Approximation Factor (Con’d) • Every blue node is responsible for a unique red node • The distance between a blue and a red node is at least ½ • The number of red nods must be constant

20. Relaxing Some of the Assumptions • Similar results can also be achieved when • Nodes are distributed in 3-dimensional space • Nodes can have different transmission ranges • Nodes don’t have IDs • Geographical information is not accurate • Error must be less than ~0.1 • Geographical information can be represented using a constant number of bits • Key Idea: Each node required to report its position to its neighbors

21. Simulation • We compared the performance of the proposed algorithm with • Liu’s algorithm [Infocom 2006 ] • A ratio-8 approximation algorithm [Infocom 2002 ] • Used as a benchmark

22. Simulation (cont’d)

23. Example • #nodes: 400 • Trans. range: 300meter • #broadcasting nodes: 10

24. Conclusion • Reactive broadcast algorithms are in fact powerful • Question: Can we do this without using geographical info. (or relative distances)? • The answer is YES. This can be the subject of a future talk..

25. Thank you