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Low Randomness Rumor Spreading via Hashing

Low Randomness Rumor Spreading via Hashing. He Sun Max Planck Institute for Informatics Joint work with George Giakkoupis , Thomas Sauerwald and Philipp Woelfel. Rumor. One guy would like to visit the Statue of Liberty. Q: I am going to find the free woman.

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Low Randomness Rumor Spreading via Hashing

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  1. Low Randomness Rumor Spreading via Hashing He Sun Max Planck Institute for Informatics Joint work with George Giakkoupis, Thomas Sauerwald and Philipp Woelfel

  2. Rumor • One guy would like to visit the Statue of Liberty. • Q: I am going to find the free woman. • A: No woman is free in U.S.

  3. Rumor Spreading (Push Model)Pittel, 1987, Feige, Peleg, Raghavan, Upfal, 1990

  4. Rumor Spreading (Push Model)Pittel, 1987, Feige, Peleg, Raghavan, Upfal, 1990

  5. Rumor Spreading (Push Model)Pittel, 1987, Feige, Peleg, Raghavan, Upfal, 1990

  6. Rumor Spreading (Push Model)Pittel, 1987, Feige, Peleg, Raghavan, Upfal, 1990

  7. Rumor Spreading (Push Model)Pittel, 1987, Feige, Peleg, Raghavan, Upfal, 1990

  8. Rumor Spreading (Push Model) + One of the fundamental protocols in networks Finishes in rounds on a number of network topologies • Complete Graph Pittel 1987 • Hypercube Feige, Peleg, Raghavan, Upfal, 1990 • Graphs with High ExpansionSauerwald and Stauffer 2011 • Graphs with High ConductanceMosk-Aoyama and Shah 2008, Giakkoupis 2011 • Random Graphs Fountoulakis, Huber, Panagiotou 2010 • Random Regular Graphs Fountoulakis, Panagiotou 2010

  9. Rumor Spreading (Push Model) + - The lower bound on the number of random bits is . One of the fundamental protocols in networks Finishes in rounds on a number of network topologies • Complete Graph Pittel 1987 • Hypercube Feige, Peleg, Raghavan, Upfal, 1990 • Graphs with High ExpansionSauerwald and Stauffer 2011 • Graphs with High ConductanceMosk-Aoyama and Shah 2008, Giakkoupis 2011 • Random Graphs Fountoulakis, Huber, Panagiotou 2010 • Random Regular Graphs Fountoulakis, Panagiotou 2010 Needs a lot of randomness

  10. Quasirandom Rumor SpreadingDoerr, Friedrich, Sauerwald, 2008 4 2 7 1 5 6 3

  11. Quasirandom Rumor SpreadingDoerr, Friedrich, Sauerwald, 2008 4 2 7 1 5 6 3

  12. Quasirandom Rumor SpreadingDoerr, Friedrich, Sauerwald, 2008 4 2 7 1 5 6 3

  13. Quasirandom Rumor SpreadingDoerr, Friedrich, Sauerwald, 2008 4 2 7 1 5 6 3

  14. Quasirandom Rumor SpreadingDoerr, Friedrich, Sauerwald, 2008 4 2 7 1 5 6 3

  15. Quasirandom Rumor SpreadingDoerr, Friedrich, Sauerwald, 2008 4 2 7 1 5 6 3 Every node has an arbitrary list of its neighbors. Informed nodes inform their neighbors in the order of this list, but start at a random position in the list.

  16. Quasirandom Rumor Spreading • One of the aims of quasirandom rumor spreading is to “imitate properties of the classical push model with a much smaller degree of randomness.” Doerr, Friedrich, Sauerwald, 2008 • The lower bound for quasirandom protocol is . • Can we further reduce the number of random bits? YES

  17. Results Consider a complete graph with 7, 000, 000, 000 nodes (world population) Every node can be informed within 60 rounds Truly Ran. # of bits: 8, 000, 000, 000, 000 Quasi Ran. # of bits: 230, 000, 000, 000 Expanders General Graphs New protocol. # of bits:36, 000 For all graph families considered so far, pseudorandom protocol runs as fast as quasirandom protocol. Compared with both protocols, pseudorandom protocol obtains exponentialimprovement for the randomness complexity.

  18. Two Techniques Pseudorandom Generators Hashing

  19. Intuition Behind the Algorithms How can I choose them completely randomly? Previous theoretical analyses assume that every neighbor of every vertex is chosen uniformly at random.

  20. Pseudorandom Independent Block Generators Truly random seed G: Polynomial-time deterministic algorithm Sequence that is “close” to uniform distribution

  21. Pseudorandom Independent Block Generators

  22. Construction of PIBGs (contd.)

  23. Construction of PIBGs (contd.)

  24. Construction of PIBGs (contd.)

  25. PIBG-Based Protocol PIBG

  26. ID Distribution

  27. PIBG-Based Protocol PIBG

  28. PIBG-Based Protocol (contd.)

  29. Analysis of a Single Round Truly random seed PIBG

  30. Analysis of a Single Round (contd.) Informed nodes Non-informed nodes

  31. Summary & Open problems Thank you A general framework for reducing the randomness complexity in rumor spreading. For a large family of graphs, we obtain an exponentialimprovement in terms of the number of random bits. Conjecture: For any graph, pseudorandom protocol is asymptotically as fast as truly random protocol. Design better space-bounded pseudorandom generators for distributed algorithms (e.g. load balancing).

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