Download presentation

Presentation is loading. Please wait.

Published byPerry Trump Modified over 3 years ago

1
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

2
Benjamin Doerr 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)

3
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

4
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...

5
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

6
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)]

7
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)

8
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? 134562 List: 2 3 4 5 63 4 5 6 14 5 6 1 25 6 1 2 36 1 2 3 41 2 3 4 5

9
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

10
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

11
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...

12
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

13
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]

14
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

15
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

16
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

17
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 ))

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

19
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: 18.09 ± 1.74 Quasirandom: 17.63 ± 1.76 Fully random: 21.11 ± 1.78 Quasirandom: 18.71 ± 0.71 Lists: neighbors sorted in increasing order

20
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: 18.09 ± 1.74 Quasirandom: 17.63 ± 1.76 Fully random: 21.11 ± 1.78 Quasirandom: 18.71 ± 0.71 Fully random: 27.31 ± 50.82 Quasirandom: 19.48 ± 3.07 Lists: neighbors sorted in increasing order

21
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

22
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”...

23
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.

24
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

25
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!

Similar presentations

OK

Recurrences 2008. 1. 11 : 1 Chapter 3. Growth of function Chapter 4. Recurrences.

Recurrences 2008. 1. 11 : 1 Chapter 3. Growth of function Chapter 4. Recurrences.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on earth dam failures Ppt on column chromatography set Ppt on hong kong tourism Ppt on self awareness for students Ppt on difference between mission and vision Ppt on different types of dance forms of the world Ppt on sports day events Ppt on 3 idiots movie last part Ppt on non biodegradable waste example Ppt on child labour in india free download