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

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?
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updates occur
Updates Occur

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step 11
Step 1

push

pull

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
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?
slide33

Thank you!

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