April 21, 2023 1
Broadcasting with Bounded Number of Redundant
Transmissions
Majid Khabbazian
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Outline
Assumptions
Objectives
Classifications
The proposed algorithm
Algorithm’s characteristics
Conclusion
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Assumptions
Single message broadcast
Nodes are distributed in 2-D space
The transmission range of each node is RWe 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
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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
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Flooding: A Simple Solution
FloodingEvery 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
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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 networkLocal Broadcast Algorithms
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Local Broadcast Algorithms
ClassificationsStatic (Proactive)Dynamic (Reactive)
Static ApproachA backbone is constructed firstThe 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 dynamicThe backbone is fixed in the static network
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Local Broadcast Algorithms (Con’d)
Dynamic ApproachThere 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
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Further Assumptions
Each node has the list of its 1-hop neighborsExchanging “hello” messages
Geographical information is availableE.g., Using GPS
Relative distance may suffice
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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
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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
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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
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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)
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Example
H
G
F
A
EB
C
DA 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
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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 setS={(u,v)| u and v are neighbors, u has received the message, v has not received the message}
S is not empty
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Full Delivery (Con’d)
S is not emptyThere 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 conditionu’ must have transmitted the message
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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
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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
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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
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Relaxing Some of the Assumptions
Similar results can also be achieved whenNodes are distributed in 3-dimensional space
Nodes can have different transmission ranges
Nodes don’t have IDs
Geographical information is not accurateError 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
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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
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Simulation (cont’d)
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Example
#nodes: 400
Trans. range: 300meter
#broadcasting nodes: 10
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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..
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Thank you