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Benjamin Doerr

Quasi-Random Rumor Spreading. Benjamin Doerr. MPII Saarbrücken joint work with. Tobias Friedrich U Berkeley. Anna Huber MPII Saarbrücken. Marvin Kü nnemann U Saarbrücken. Thomas Sauerwald U Berkeley. Advertisement: Positions at the MPI. 5 Postdocs:

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Benjamin Doerr

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

  2. Advertisement: Positions at the MPI • 5 Postdocs: • Starting October 2009, deadline: January 31, 2009. • 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) Benjamin Doerr

  3. 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 Benjamin Doerr

  4. 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 = Kn, edges not drawn Round 4: Each informed vertex informs a random vertex Round 5: Let‘s hope the remaining two get informed... Round 2: Each informed vertex informs a random vertex Round 3: Each informed vertex informs a random vertex Round 1: Starting vertex informs random vertex Round 0: Starting vertex is informed Benjamin Doerr

  5. 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  Benjamin Doerr

  6. 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 log2(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: Kn= ([n],2[n]) • Hypercubes: Hd= ({0,1}d, “Hamming distance one”) • Random graphs: Gn,p, p  (1+Ɛ) log(n)/n • For complete graphs, the constant is log2(n) + ln(n) + o(log(n)) [Frieze&Grimmet (1985), Feige, Peleg, Raghavan, Upfal (1990)] Benjamin Doerr

  7. 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) Benjamin Doerr

  8. 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 Kn, n = 6] • Here: n -1rounds . • No hope for cleverness (quasirandomness) here? 1 2 3 4 5 6                List: 2 3 4 5 6 3 4 5 6 1 4 5 6 1 2 5 6 1 2 3 6 1 2 3 4 1 2 3 4 5 Benjamin Doerr

  9. 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 Benjamin Doerr

  10. 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 Benjamin Doerr

  11. 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 Gn,p, p (1+Ɛ) log(n) still hold... Benjamin Doerr

  12. 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 Gn,p, p (1+Ɛ) log(n) still hold regardless of the structure of the lists Benjamin Doerr

  13. 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 Gn,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] Benjamin Doerr

  14. Quasirandom 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 Gn,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] Benjamin Doerr

  15. 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 Benjamin Doerr

  16. 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 Benjamin Doerr

  17. Runtime: Proven bounds • “As fast as independent”: The O(log(n)) bounds hold for • complete graphs (including the leading constant), • hypercubes, • random graphs Gn,p, p (1+Ɛ) log(n) • “Slightly faster than independent”: • Random graphs Gn,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(klog(n)) rounds necessary/sufficient • quasirandom: w.p.1, r rounds necessary/sufficient,where r=Θ(klog(n)/log(k)) Benjamin Doerr

  18. Runtime: Experimental Results (n=1024) Complete graph Kn Lists: neighbors sorted in increasing order Average broadcast times: Fully random: 18.09 ± 1.74Quasirandom: 17.63 ± 1.76 Benjamin Doerr

  19. Runtime: Experimental Results (n=1024) Complete graph Kn Hypercube H10 Lists: neighbors sorted in increasing order Lists: “inform the neighbor in dimension 1, 2, 3, ...” Average broadcast times: Fully random: 18.09 ± 1.74Quasirandom: 17.63 ± 1.76 Fully random: 21.11 ± 1.78Quasirandom: 18.71 ± 0.71 Benjamin Doerr

  20. Runtime: Experimental Results (n=1024) Complete graph Kn Hypercube H10 Random graphs Gn,p, p such that graph connected w.p.1/2 Lists: neighbors sorted in increasing order Lists: “inform the neighbor in dimension 1, 2, 3, ...” Lists: neighbors sorted in increasing order Average broadcast times: Fully random: 18.09 ± 1.74Quasirandom: 17.63 ± 1.76 Fully random: 21.11 ± 1.78Quasirandom: 18.71 ± 0.71 Fully random: 27.31 ± 50.82Quasirandom: 19.48 ± 3.07 Benjamin Doerr

  21. 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 timelog2(1+p)-1 log2(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 Benjamin Doerr

  22. 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 Kn? • Round 0: Startvertex informed • 1st phase: log(n) rounds: log(n) newly informed nodes • 2nd 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”... Benjamin Doerr

  23. 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. Benjamin Doerr

  24. 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! Benjamin Doerr

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