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Benjamin Doerr MPII Saarbrücken joint work with Quasi-Random Rumor Spreading Tobias Friedrich U Berkeley Anna Huber MPII Saarbrücken Thomas Sauerwald U Berkeley Marvin Künnemann U Saarbrücken

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Benjamin Doerr Advertisement: Positions at the MPI 5 Postdocs: – Starting October 2009, deadline: January 31, 5 PhD students: – positions filled continuously All positions have – generous support (travel, computer,...) – no teaching duties, but teaching is possible – are in the “Algorithms&Complexity” group (~40 researchers, mainly theory)

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Benjamin Doerr Quasi-Random Rumor Spreading Outline: – Randomized Rumor Spreading (classical) always contact a random neighbor – Quasirandom Rumor Spreading (new model) less independent randomness – Results Conclusion: dependent random stuff... – can be analyzed – works well

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Benjamin Doerr Randomized Rumor Spreading Model (on a graph G ): – Start: One vertex is knows a rumor (“is informed”) – Each round, each informed vertex contacts a neighbor chosen uniformly at random and informs it (if it wasn’t already) – Problem: How many rounds are necessary to inform all vertices? Stupid animation: G = K n, edges not drawn Round 0: Starting vertex is informedRound 1: Starting vertex informs random vertexRound 2: Each informed vertex informs a random vertexRound 3: Each informed vertex informs a random vertex Round 4: Each informed vertex informs a random vertex Round 5: Let‘s hope the remaining two get informed...

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Benjamin Doerr Randomized Rumor Spreading Model (on a graph G ): – Start: One vertex is knows a rumor – Each round, each informed vertex informs a neighbor chosen uniformly at random – Problem: How many rounds are necessary to inform all vertices? CS-Application: – Broadcasting updates in distributed replicated databases simple robust self-organized Maths-NoApplication: Fun to study

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Benjamin Doerr Randomized Rumor Spreading Model (on a graph G ): – Start: One vertex is knows a rumor – Each round, each informed vertex informs a neighbor chosen uniformly at random – Problem: How many rounds are necessary to inform all vertices? Main results [ n : number of vertices] : – Easy: For all graphs and starting vertices, at least log 2 ( n ) rounds are necessary – Theorem: These graph classes have the property that independent of the starting vertex O ( log ( n )) rounds suffice w.h.p.: Complete graphs: K n = ([ n ], 2 [ n ] ) Hypercubes: H d = ({0,1} d, “Hamming distance one”) Random graphs: G n, p, p ( 1+Ɛ ) log ( n ) / n For complete graphs, the constant is log 2 ( n ) + ln( n ) + o(log( n )) [Frieze&Grimmet (1985), Feige, Peleg, Raghavan, Upfal (1990)]

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Benjamin Doerr Motivation of this Work Observation: – “all decisions independent at random’’ is simple, but efficient Question: Can we do better with more clever (randomized) approaches? – introduce problem-motivated dependencies – concept of quasirandomness [Jim Propp]: Simulate properties of the random object/process deterministically Successful applications: – Quasi Monte Carlo Methods – Propp maschine (quasirandom random walks)

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Benjamin Doerr Deterministic Rumor Spreading? Same model as above, except: – Each vertex has a list of its neighbors. – Informed vertices inform their neighbors in the order of this list Problem: Might take long... [Proof by animation, Graph K n, n = 6 ] Here: n -1 rounds . No hope for cleverness (quasirandomness) here? List:

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Benjamin Doerr Semi-Deterministic Rumor Spreading Same model as above, except: – Each vertex has a list of its neighbors. – Informed vertices inform their neighbors in the order of this list, but start at a random position in the list

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Benjamin Doerr Semi-Deterministic Rumor Spreading Same model as above, except: – Each vertex has a list of its neighbors. – Informed vertices inform their neighbors in the order of this list, but start at a random position in the list Results

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Benjamin Doerr Semi-Deterministic Rumor Spreading Same model as above, except: – Each vertex has a list of its neighbors. – Informed vertices inform their neighbors in the order of this list, but start at a random position in the list Results: The O ( log ( n )) bounds for – complete graphs (including the leading constant), – hypercubes, – random graphs G n, p, p ( 1 + Ɛ ) log ( n ) still hold...

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Benjamin Doerr Semi-Deterministic Rumor Spreading Same model as above, except: – Each vertex has a list of its neighbors. – Informed vertices inform their neighbors in the order of this list, but start at a random position in the list Results: The O ( log ( n )) bounds for – complete graphs (including the leading constant), – hypercubes, – random graphs G n, p, p ( 1 + Ɛ ) log ( n ) still hold regardless of the structure of the lists

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Benjamin Doerr Semi-Deterministic Rumor Spreading Same model as above, except: – Each vertex has a list of its neighbors. – Informed vertices inform their neighbors in the order of this list, but start at a random position in the list Results: The O ( log ( n )) bounds for – complete graphs (including the leading constant), – hypercubes, – random graphs G n, p, p ( 1 + Ɛ ) log ( n ) still hold regardless of the structure of the lists [2 good news: (a) results hold, (b) things can be analyzed in spite of dependencies]

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Benjamin Doerr Semi-Deterministic Rumor Spreading Same model as above, except: – Each vertex has a list of its neighbors. – Informed vertices inform their neighbors in the order of this list, but start at a random position in the list Results: The O ( log ( n )) bounds for – complete graphs (including the leading constant), – hypercubes, – random graphs G n, p, p ( 1 + Ɛ ) log ( n ) still hold regardless of the structure of the lists [2 good news: (a) results hold, (b) things can be analyzed in spite of dependencies] Quasirandom

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Benjamin Doerr Quasirandom Rumor Spreading Same model as above, except: – Each vertex has a list of its neighbors. – Informed vertices inform their neighbors in the order of this list, but start at a random position in the list Natural Property: – A vertex never informs a neighbor twice (unless it informed all neighbors) Algorithmic aspects: – If results hold for all lists, then lists already present for technical reasons can be used – Less random bits needed

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Benjamin Doerr Intra-Talk Summary Randomized rumor spreading: – Informed vertices inform neighbors chosen uniformly at random Quasirandom rumor spreading – Each vertex has an arbitrary list of its neighbors – Informed vertices inform their neighbors in the order of this list, starting at a random position in the list – Some nice properties Remainder of the talk: Results! – Runtime – Robustness – Some proof ideas

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Benjamin Doerr Runtime: Proven bounds “As fast as independent”: The O ( log ( n )) bounds hold for – complete graphs (including the leading constant), – hypercubes, – random graphs G n, p, p ( 1 + Ɛ ) log ( n ) “Slightly faster than independent”: – Random graphs G n, p, p = ( log ( n )+ log ( log ( n )))/ n : independent: Θ ( log ( n ) 2 ) necessary to obtain a success probability of 1 – 1 / n quasirandom: Θ ( log ( n )) suffice – Complete k -regular trees: independent: w.h.p. Θ ( k log ( n )) rounds necessary/sufficient quasirandom: w.p.1, r rounds necessary/sufficient, where r = Θ ( k log ( n )/ log ( k ))

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Benjamin Doerr Runtime: Experimental Results (n=1024) Complete graph K n Average broadcast times: Fully random: ± 1.74 Quasirandom: ± 1.76 Lists: neighbors sorted in increasing order

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Benjamin Doerr Runtime: Experimental Results (n=1024) Complete graph K n Hypercube H 10 Lists: “inform the neighbor in dimension 1, 2, 3,...” Average broadcast times: Fully random: ± 1.74 Quasirandom: ± 1.76 Fully random: ± 1.78 Quasirandom: ± 0.71 Lists: neighbors sorted in increasing order

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Benjamin Doerr Runtime: Experimental Results (n=1024) Complete graph K n Hypercube H 10 Random graphs G n, p, p such that graph connected w.p.1/2 Lists: “inform the neighbor in dimension 1, 2, 3,...” Lists: neighbors sorted in increasing order Average broadcast times: Fully random: ± 1.74 Quasirandom: ± 1.76 Fully random: ± 1.78 Quasirandom: ± 0.71 Fully random: ± Quasirandom: ± 3.07 Lists: neighbors sorted in increasing order

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Benjamin Doerr Robustness Robustness: How well does the protocol work if some transmissions fail? – Failure model: Each transmission fails with a (1- p ) chance (independently). The sender does not get to know this. – Referee question: Quasirandom could be less robust? – ‘Theorem’ [not yet written up]: W.h.p., both models need time log 2 (1+ p ) -1 log 2 ( n ) + p -1 ln( n ) + o(log( n )) on the complete graph. – Experiments: Average broadcast times ± standard deviations for hypercube and complete graph, n =4096, p =1/2

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Benjamin Doerr Delaying&Ignoring: Some proof ideas... Proceed in phases of several rounds: – Assume pessimistically that nodes informed in this phase start rumor spreading only in the next phase (delaying). – Next phase: Only the nodes newly informed in the last phase spread the rumor (ignore the rest). – Cool: They still have their independent random choice! How does is work for the Θ(log( n )) bound for the K n ? – Round 0: Startvertex informed – 1 st phase: log( n ) rounds: log( n ) newly informed nodes – 2 nd phase: log( n ) rounds: Each of the log( n ) newly informed nodes informs a random log( n ) segment of his list. The segments are chosen independently, hence few overlaps. Result: Θ(log( n ) 2 ) newly informed nodes. – Phases until 1% informed: 8 rounds per phase. Half of the newly informed inform at least 4 new ones. Result: Twice as many newly informed nodes. – “Endgame”...

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Benjamin Doerr Delaying&Ignoring... Delaying: Delay independent random decisions until you have enough of them – admits Chernoff bounds Ignoring: Ignore nasty stuff to make the rest independent. Problem: To get the leading constant, in average only – a o(1) fraction of the decisions may be delayed; – a o(1) fraction of the informed vertices may be ignored. Solution: Busy phases – vertices informed in the phase do inform others in this phase – reduce dependencies by ignoring “overtaking”: If A calls B in the phase (determined by A’s random decision), then we ignore that A might call C and C might call B earlier than A. – yields an only (1-o(1)) slowdown of the process.

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Benjamin Doerr Analyzing the Experiments More numbers: and their story: – The quasirandom models finds it much easier to inform the remaining nodes of small degree Expected number of uninformed nodes in a random graph G(n,p), n = 4096, p = ln(n)/n, after t time steps Average broadcast times ± standard deviations for some 4096-vertex graphs

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Benjamin Doerr Summary Results: – Theory: Guarantee that things work fine for all list structures good broadcast times & robustness for many graphs better broadcast times for some graphs – Experiments: The lists we tried yield better results reduced broadcast times broadcast times stronger concentrated – General: No need to be afraid of dependencies ! Outlook: – Try to “mathematically” see the differences seen in the experiments. – Open problem: Are some lists structures better or worse than others? Grazie mille!

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