Quasi-Random Rumor Spreading Benjamin Doerr MPII Saarbrcken joint - - PowerPoint PPT Presentation

Benjamin Doerr

MPII Saarbrücken

joint work with

Quasi-Random Rumor Spreading

Tobias Friedrich

MPII Saarbrücken

Anna Huber

MPII Saarbrücken

Thomas Sauerwald

HNI Paderborn

Benjamin Doerr

How to spread the LLL

– Day 0: LL discovers the LLL – Day 1: He calls a random Hungarian and explains it to him – Day 2: Both call random Hungarians and teach them the LLL – Day 3, 4, ...: Each Hungarian who knows the LLL calls a random Hungarian and teaches him/her the LLL – Result: After only 40 days, all 10,041,000 Hungarians know the LLL

Day 0 Day 1 Day 2 Day 3 Day 4 Day 5: Let‘s hope the remaining two learn the LLL

Animation: 14 Hungarians

Benjamin Doerr

Randomized Rumor Spreading

Model (on a graph ):

– Start: One vertex is informed – Each round, each informed vertex informs a neighbor chosen uniformly at random – Problem: How many rounds are necessary to inform all other vertices?

CS-Application:

– Broadcasting updates in distributed replicated databases simple robust self-organized

Maths-NoApplication:

– Fun to study

Benjamin Doerr

Randomized Rumor Spreading

Model (on a graph ):

– Start: One vertex is informed – Each round, each informed vertex informs a neighbor chosen uniformly at random – Problem: How many rounds are necessary to inform all other vertices?

Main results [: number of vertices]:

– Easy: For all graphs and starting vertices, at least log() rounds are necessary – Theorem: These graph classes have the property that independent

• f the starting vertex log rounds suffice w.h.p.:

Complete graphs: ([], 2 Hypercubes: ({0,1}, “Hamming distance one”) Random graphs: , > 1+Ɛlog/ For complete graphs, the constant is log2() + ln() + o(log())

[Frieze&Grimmet (1985), Feige, Peleg, Raghavan, Upfal (1990)]

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

Why is this interesting?

– Natural: You don’t inform neighbors twice – Algorithmic aspects: Avoid randomness – Concept: Quasirandomness [Jim Propp]

Simulate a particular property of a random object and often get better results Successful applications:

– Quasi Monte Carlo Methods – Propp machine (quasirandom random walks)

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 , ] Here: -1 rounds . No hope for quasirandomness here?

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

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 (1)

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 (1): The log bounds for

– complete graphs (including the leading constant), – hypercubes, – random graphs >1Ɛ log

still hold...

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 (1): The log bounds for

– complete graphs (including the leading constant), – hypercubes, – random graphs >1Ɛ log

still hold independent from the structure of the lists

• !"#$#

Benjamin Doerr

Semi-Deterministic Rumor Spreading

Results (2):

– Random graphs logloglog%: fully randomized: log necessary to obtain a success probability of 1 – 1% semi-deterministic: log suffice – Complete -regular trees: fully randomized: w.h.p. log rounds necessary/sufficient semi-deterministic: w.p.1, rounds necessary/sufficient, where log%log

Algorithmic Aspects:

– needs fewer random bits – easy to implement: Any implicitly existing permutation of the neighbors can be used for the lists

Benjamin Doerr

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. – 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 on the ?

– Round 0: Startvertex informed – 1st phase: log() rounds: log() newly informed nodes – 2nd phase: log() rounds: Each of the log() newly informed nodes informs a random log() segment of his list. The segments are chosen independently, hence few overlaps. Result: ((log()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

Experimental Results (n=1024)

Complete graph Average broadcast times:

Fully random: 18.09 Quasirandom: 17.63 Lists: neighbors sorted in increasing order

[Experiments: Marvin Künnemann]

Benjamin Doerr

Experimental Results (n=1024)

Complete graph Hypercube

Lists: „inform the neighbor in dimension 1, 2, 3, ...“

Average broadcast times:

Fully random: 18.09 Quasirandom: 17.63 Fully random: 21.11 Quasirandom: 18.71 Lists: neighbors sorted in increasing order

[Experiments: Marvin Künnemann]

Benjamin Doerr

Experimental Results (n=1024)

Complete graph Hypercube Random graphs ,

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 Quasirandom: 17.63 Fully random: 21.11 Quasirandom: 18.71 Fully random: 27.31 Quasirandom: 19.48 Lists: neighbors sorted in increasing order

[Experiments: Marvin Künnemann]

Benjamin Doerr

Summary

Quasirandom rumor spreading:

– Each vertex has a cyclic list of its neighbors. – Once informed, it starts informing from a random position of the list.

Results:

– Independent of the lists, we prove bounds comparable or better than for fully randomized model. – Experiments: Significant improvements for sparse graphs – Theory: A number of tricks to deal with dependencies

Avoid dependencies by only exploiting independent stuff

Köszönöm!

Bottom line: Don’t be afraid of dependencies!