low randomness rumor spreading via hashing he sun max planck institute for informatics joint work...
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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
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.
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
+
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 .
Quasirandom Rumor Spreading Doerr, Friedrich, Sauerwald, 2008
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Quasirandom Rumor Spreading Doerr, Friedrich, Sauerwald, 2008
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72 3 5
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Quasirandom Rumor Spreading Doerr, Friedrich, Sauerwald, 2008
2 4
1
3
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72 3 5
1 6 7 3
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6 1 4 2 3
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Quasirandom Rumor Spreading Doerr, Friedrich, Sauerwald, 2008
2 4
1
3
5
6
72 3 5
1 6 7 3
4 5
6 1 4 2 3
7 1
Quasirandom Rumor Spreading Doerr, Friedrich, Sauerwald, 2008
2 4
1
3
5
6
72 3 5
1 6 7 3
4 5
6 1 4 2 3
7 1
Quasirandom Rumor Spreading Doerr, Friedrich, Sauerwald, 2008
• 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.
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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
• 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 Family Rumor Spreading Time Random 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
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.
Pseudorandom Independent Block Generators
G: Polynomial-time deterministic algorithm
Truly random seed
Sequence that is “close” to uniform distribution
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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