Efficient Randomized Broadcasting in Sparse Random Networks

1. Efficient Randomized Broadcasting in Sparse Random Networks Robert Elsässer jointworkwith Petra Berenbrinkand Tom Friedetzky

2. 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

3. 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?

4. 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?

5. Replicated DNS Server

6. DNS Updates Update www.inc1.de Update www.inc3.de Update www.inc2.de Update www.inc4.de Update www.inc7.de Update www.inc6.de Update www.inc5.de

7. Updates Occur Update www.inc1.de Update www.inc3.de Update www.inc2.de Update www.inc4.de Update www.inc7.de Update www.inc6.de Update www.inc5.de

8. Step 1

9. Step 2

10. Step 3

11. Step 4

12. Final state

13. Random Phone Call Model

14. Initial

15. Step 1 push pull

16. Step 2

17. Step 3

18. Final state

19. 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]

20. 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…

21. 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]

22. 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

23. 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

24. 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

25. 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

26. Phases of the Algorithm

27. Phases of the Algorithm w u

28. 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

29. 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)

30. „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

31. 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).

32. 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?

33. Thank you! Questions?