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

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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.

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

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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 Expansion Sauerwald and Stauffer 2011 –Graphs with High Conductance Mosk-Aoyama and Shah 2008, Giakkoupis 2011 –Random Graphs Fountoulakis, Huber, Panagiotou 2010 –Random Regular Graphs Fountoulakis, Panagiotou 2010 +

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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 Expansion Sauerwald and Stauffer 2011 –Graphs with High Conductance Mosk-Aoyama and Shah 2008, Giakkoupis 2011 –Random Graphs Fountoulakis, Huber, Panagiotou 2010 –Random Regular Graphs Fountoulakis, Panagiotou 2010 Needs a lot of randomness + - The lower bound on the number of random bits is.

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Quasirandom Rumor Spreading Doerr, Friedrich, Sauerwald, 2008 1 2 3 4 5 6 7 235 16 73 45 61423 71

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1 3 5 6 7 235 16 73 45 61423 71 2 4

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1 3 5 6 7 235 16 73 45 61423 71 2 4

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1 3 5 6 7 235 16 73 45 61423 71 2 4

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1 3 5 6 7 235 16 73 45 61423 71 2 4

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1 3 5 6 7 235 16 73 45 61423 71 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. 2 4

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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

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For all graph families considered so far, pseudorandom protocol runs as fast as quasirandom protocol. Compared with both protocols, pseudorandom protocol obtains exponential improvement for the randomness complexity. Graph FamilyRumor Spreading TimeRandom Bits Complete Graphs General Graphs Expanders 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 New protocol. # of bits: 36, 000

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Two Techniques Pseudorandom Generators Hashing

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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.

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

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Pseudorandom Independent Block Generators

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Construction of PIBGs (contd.)

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PIBG-Based Protocol PIBG

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ID Distribution

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PIBG-Based Protocol PIBG

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PIBG-Based Protocol (contd.)

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Analysis of a Single Round Truly random seed PIBG

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

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Summary & Open problems A general framework for reducing the randomness complexity in rumor spreading. For a large family of graphs, we obtain an exponential improvement 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). Thank you

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Benjamin Doerr Max-Planck-Institut für Informatik Saarbrücken Quasirandomness.

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