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A Group-theoretic Framework for Rendezvous in Heterogeneous Cognitive Radio Networks

Lin Chen∗, Kaigui Bian∗, Lin Chen†

Cong Liu#, Jung-Min Jerry Park♠, and Xiaoming Li∗

∗ Peking University, Beijing, China † University Paris-Sud, Orsay, France

# Sun Yat-Sen University, Guangzhou, China ♠ Virginia Tech, Blacksburg, VA, USA

ACM MobiHoc 2014

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What is the Rendezvous problem

Rendezvous dilemma, rendezvous search game

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Rendezvous is a problem about “dating”… Two young people want to date (meet or rendezvous) in a

large park, where N places are suitable for dating. [Steve Alpern, 1976]

They need a strategy to visit these N places for early rendezvous.

A

B C

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It is NOT a challenging problem today… They can call each other directly by cell phone

A

B C

Let’s meet at “C”At 10AM!

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No hidden assumptions here E.g., no cell phones!

That means, no pre-shared knowledge Places can be unavailable (due to congestion) Clocks can be asynchronous No pre-assigned roles (i.e., the strategy should be

the same for two people)

It is challenging as a math problem

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Rendezvous problem in multi-channel wireless networks

Rendezvous channel = control channel Link establishment and control message exchange, etc. Subject to congestion, attack, primary user traffic, etc

So, it is needed to rendezvous on multiple channels

SERIAL ETHERNET

Ch 2

Ch 1

Rdv ch Rdv Data

SERIAL ETHERNET

Rdv Data

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Q1: How fast can they achieve rendezvous? Is there a minimum, bounded latency?

Q2: What is the max # of rendezvous channels? What if a given rendezvous channel is unavailable?

Two interesting questions

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Existing research

Channel hopping (CH) can create rendezvous

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C1 C2C0 C1 C2C0 C1 C2C0

C1 C2C0 C1 C2C0 C1 C2C0

Random, common channel hopping

Random hopping： unbounded TTR

Common hopping: clock sync. required

C1 C2C0

C1 C2 C0

C1 C2 C0

C1C2 C0

…...

…...

A

B C

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Sequence based channel hopping

Interleaving, [Dyspan08]

Modular clock, [MobiCom04, Infocom11, MobiHoc13]

Single rendezvous channel

C1 C2C0 C1 C2C0 C1 C2C0C0 C1 C2

C1 C2C0 C1 C2C0 C1 C2C0C0 C1 C2

C1 C2C0 C1 C2C0 C0 C0 C0

C1 C2C0 C1 C2C0 C0 C0 C0

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Different sensing channel sets [MobiHoc13] No common channel index, no integer channel indices

Node i Node j

x y a b c

Channel hopping over heterogeneous channel sets

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A lower bound for rendezvous latency

Q1: how fast to rendezvous?

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Nodes i has a number of Ni channels, in chan set Ci Nodes j has a number of Nj channels, in chan set Cj

Theorem 1: to rdv on every channel in Ci ∩ Cj

Two nodes need at least Ni Nj time slots

Intuition: Elements in group ZNi⊕ZNj enumerate all

possible pairs of rendezvous channels in Ci ∩ Cj

A lower bound of rdv latency (TTR)

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Max # of rendezvous channels = |Ci ∩ Cj|

Q2: what is the max # of rdv channels?

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3 steps of creating channel hopping sequences

Three channels: Everyone has two short sequences: fast and slow Choice bit sequence: 0/1 sequence Interleave fast and slow sequence

If 0, pick fast; if 1, pick slow.

10 2 10 2

10 10 10

10 1 00 0

1 0

10 1 10

12

0

0 2Fast seq

Slow seq

Choice bit seq

Final seqused for rdv

0 1 22

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Fast hopping: hop across Ni channels by Ni slots Slow hopping: stay on channel h for Ni slots

However, two nodes use different strategies!

C1 C1 C1 C0C0C0C2 C2C2

C1 C1 C1 C0C0C0C2 C2C2

Step 1: Rdv between fast and slow sequences

C1 C2 C1 C1C0 C0 C0C2 C2 C1 C2 C1 C1C0 C0 C0C2 C2Fast seq.

Slow seq.

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Step 2: Creating choice bit sequences

Node has its ID as , then create its choice seq. Any and are at least one bit different after any cyclic

rotation Symmetrization map: a unique ID a unique bit-string Example:

Assign 01010 to node and 10101 to node

𝜔 (𝑖 )

𝜔 ( 𝑗 )

10 1 00

1 1 00 1

10 1 00

1 1 00 1

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Step 3: Interleaving fast and slow seqs for rdv

10 2 10 2 10 2

10 210 210 2

1 1 00

10 1 00 10 1 00 …

1 0 0 2

10 21 10

12

0

0 2 1

2 2Node i3 chans

Node j2 chans

Fast

SlowChoice

Final seq

1

2 0

0 20 2

0

0 2

2

0 2

Fast

Slow

Choice

0 2

1 1 00 1 …

Final seq

2 002 2 002 2 002

0 20 20 20 2

0 0 0 0 22 22 …

0

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Simulation results

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Legend of our protocol is “Adv rdv” by light blue curve

Small TTR (left) + Max robustness (right)

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Conclusion

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Conclusion

We formulate the rendezvous problem in heterogeneous cognitive radio networks.

We derive the lower bound of rdv latency in the heterogeneous environment.

By symmetrization and interleaving fast/slow seqs, we devise a near-optimal rdv protocol. Max # of rdv channels is |Ci ∩ Cj| Achieve max rdv with a bounded latency ~ O(Ni Nj )

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any questions?

Thanks & 感谢观看

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Assignment of Choice Sequence

Symmetrization

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Finished!Assignment of choice seq.

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Two distributed assignment algorithms

symmetrization map

symmetrization map

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Suppose the length of ID is . Just append to it.

Length of choice seq.:

symmetrization

11000101Node ’s 8-bit ID 1000000000001