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