Efficient randomized broadcasting in sparse random networks
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Efficient Randomized Broadcasting in Sparse Random Networks. Robert Elsässer joint work with Petra Berenbrink and Tom Friedetzky. Overview. Models und definitions Random phone call model (address-oblivious case) Power of “four different choices”

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Efficient Randomized Broadcasting in Sparse Random Networks

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Efficient randomized broadcasting in sparse random networks

Efficient Randomized Broadcasting in Sparse Random Networks

Robert Elsässer

jointworkwith Petra Berenbrinkand Tom Friedetzky


Overview

Overview

  • Models und definitions

  • Random phone call model (address-oblivious case)

  • Power of “four different choices”

  • Oblivious vs. “almost” oblivious broadcasting

  • Conclusion and further research


Models and definitions

Models and Definitions

  • Let G=(V,E) be a connected undirected graph of size n

  • V – set of nodes in G

  • E – set of edges in G

  • Broadcasting Problem

    • A node of G has a message M

    • How many steps and message transmissions are needed to spread the message to all nodes of G using local communications only?


Deterministic vs randomized algorithms

Deterministic vs. Randomized Algorithms

  • Deterministic Algorithms

    • Very fast (if the structure of the graph is known)

    • Cannot efficiently handle node and/or edge failures, or topological changes in the network

  • Randomized Algorithms

    • Robust against node and/or edge failures

    • Cope with topological changes

    • Runtime/number of message transmissions?


Replicated dns server

Replicated DNS Server


Dns updates

DNS Updates

Update

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Update

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Update

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Update

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Update

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Update

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Update

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Updates occur

Updates Occur

Update

www.inc1.de

Update

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Update

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Update

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Update

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Step 1

Step 1


Step 2

Step 2


Step 3

Step 3


Step 4

Step 4


Final state

Final state


Random phone call model

Random Phone Call Model


Initial

Initial


Step 11

Step 1

push

pull


Step 21

Step 2


Step 31

Step 3


Final state1

Final state


Related work

Related Work

  • W.h.p., O(log n) push iterations are enough in complete graphs and random graphs (of certain density) of size n

    [Frieze&Grimmett-DAM’85, Pittel-SIAM J. Appl. Math’87, Feige et al.-RS&A‘90]

  • W.h.p., O(log n) push iterations are enough in Hypercubes, Star graphs, and certain Cayley graphs [Feige et al.-RS&A’90, E.&Sauerwald-WG’05, E.&Sauerwald-STACS‘07]

  • W.h.p., O(D) push iterations are enough in bounded degree graphs of diameter D[Feige et al.-RS&A‘90]

  • W.h.p., O(n log log n) transmissions are enough in complete graphs of size n (push&pull) [Karp et al.-FOCS’00]


Random regular graph model

Random Regular Graph - Model

  • Gn,d=(V,E), a graph is chosen uniformly at random from the space of all d-regular graphs.

  • Random regular graphs with (sub-)logarithmic degree are of interest in the context of P2P networks:

    • Overlay networks

    • Good expansion and connectivity properties, small diameter…


Random phone call model history

Random Phone Call Model – History

  • Random phone call model in complete graphs:

    • Θ(n log log n) message transmissions, w.h.p. [Karp et al., FOCS`00]

  • Random phone call model in random graphs:

    • In Gn,p graphs with p > log2n / n:

      • Θ (log n) time steps

  • Θ (n (log log n + log n / log (pn))) message transmissions [E., SPAA`06]

  • Modified random phone call model in random graphs:

    • If p > log2n / n, then a modified random phone call algorithm exists:

      • Θ (log n) time steps

  • Θ (n log log n) message transmissions [E.&Sauerwald, SODA`08]


Communication complexity

Communication Complexity

  • If |I(t)| < n/2, then |I(t+1)| - |I(t)| = Ω(|I(t)|)

  • If |I(t)| < n - n/d,then |H(t+1)| = Ω(|H(t)|2/n)

  • If |H(t)| < n/d, then |H(t+1)| = Ω(|H(t)|/d)

    Θ(n log n/log d) transmissions

Gn,d


Four choices model

Four Choices Model

  • A piece of information is placed on one of the nodes at time 0.

  • In each succeeding step, any node:

    • chooses four different neighbors, uniformly at random,

      • informed nodes may push the message, and

      • informed nodes may perform pull transmissions to the neighbors which have chosen this node in the current step


Four choices algorithm d log log n 2

Four Choices Algorithm (d > (log log n)2)

  • For O(log n) steps, all informed nodes push M exactly once, directly after they receive the message

  • In the following O(log log n) steps, all informed nodes push M in all these steps

  • In the following O(log log n) steps all informed nodes perform push&pull transmissions of M


Four choices algorithm d o log log n 2

Four Choices Algorithm (d = O((log log n)2))

  • For O(log n) steps, all informed nodes push M exactly once, directly after they receive the message

  • In the following O(log log n) steps, all informed nodes push M in all these steps

  • In the next step all informed nodes perform a single pull transmission

  • In the following O(log n) steps all nodes informed in phases 3 or 4 push M


Phases of the algorithm

Phases of the Algorithm


Phases of the algorithm1

Phases of the Algorithm

w

u


Four choices model on g n d

Four Choices Model on Gn,d

  • Termination mechanism:

    • In any Gn,d graph a four choices algorithm exists with

      • runtime: O(log n),

      • # message transmissions: O(n log logn).

    • Communication overhead decreases significantly.

  • These results are asymptotically optimal


Message transmissions different choices

Message Transmissions – Different Choices

  • Onechoice (withprobability1-n-1):

    • Running time: O(log n)

    • # messagetransmissions: Θ(n log n / log d)

  • Twochoices (withprobability1-o(1)):

    • Running time: O(log n)

    • # messagetransmissions: Θ(n (log n / log d)1/2)

  • Threechoices (withprobability1-n-Ω(1)):

    • Running time: O(log n)

    • # messagetransmissions: O(n log logn)


Almost oblivious algorithm

„Almost“ Oblivious Algorithm

  • A pieceofinformationisplaced on oneofthenodesat time 0.

  • In eachsucceedingstept, anynode:

    • choosesoneneighbor, uniformlyatrandom, amongthenodes not chosen in stepst-1, t-2, andt-3. Then,

      • informednodesmay push theinformation

      • informednodesmayperform pull transmissionstotheneighborswhichhavechosenthisnode in thecurrentstep


Power of memory

Power of Memory

  • Termination mechanism:

    • In any Gn,dgraph there is an almost oblivious algorithm with

      • running time: O(log n),

      • # message transmissions: O(n log logn).

    • Communication complexity decreases significantly compared to oblivious algorithms.

  • These results are asymptotically optimal (even if the

    nodes are allowed to remember all the previous steps).


Conclusion

Conclusion

  • Random phone call model in random regular graphs

  • Power of “four choices” in randomized broadcasting

  • Oblivious vs. almost oblivious algorithms

  • Further graph classes

    • Random power law graphs with dmin = Ω(log logn)

    • Power ofmemory in certainclassesofexpanders

  • Runtime vs. # message transmissions?


Efficient randomized broadcasting in sparse random networks

Thank you!

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