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Efficient Randomized Broadcasting in Sparse Random Networks

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### Efficient Randomized Broadcasting in Sparse Random Networks

Robert Elsässer

jointworkwith Petra Berenbrinkand Tom Friedetzky

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

- 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 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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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

- 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 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

- 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

- 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)

- 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))

- 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 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

- 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

- 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

- 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

- 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?

Questions?

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