Quasi-Random Rumor Spreading Benjamin Doerr MPII Saarbrücken joint work with Tobias Friedrich Anna Huber Thomas Sauerwald MPII Saarbrücken MPII Saarbrücken HNI Paderborn
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 Animation: 14 Hungarians Day 5: Let‘s hope the remaining two learn the LLL Day 0 Day 1 Day 2 Day 3 Day 4 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 of the starting vertex � � log � � �� rounds suffice w.h.p.: � Complete graphs: � � �� ([ � ], 2 � � � � � Hypercubes: � � � ({0,1} � , “Hamming distance one”) � Random graphs: � � , � �� �� > � 1+Ɛ �� log � � � / � � For complete graphs, the constant is log 2 ( � ) + 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 � � � � �� �� > � 1 � Ɛ � 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 � � � � �� �� > � 1 � Ɛ � log � � � still hold independent from the structure of the lists �������������� ������������������ ����������� ����������!"�������#�����$���#����� ���� Benjamin Doerr
Semi-Deterministic Rumor Spreading � Results (2): – Random graphs � � � � �� � ��� log � � �� log � log � � ���% � : � 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 – 1 st phase: log( � ) rounds: log( � ) newly informed nodes – 2 nd 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 � � Lists: neighbors sorted in increasing order Average broadcast times: Fully random: 18.09 Quasirandom: 17.63 [Experiments: Marvin Künnemann] Benjamin Doerr
Experimental Results (n=1024) Complete graph � � Hypercube � �� Lists: neighbors sorted in Lists: „inform the neighbor in increasing order dimension 1, 2, 3, ...“ Average broadcast times: Fully random: 18.09 Fully random: 21.11 Quasirandom: 17.63 Quasirandom: 18.71 [Experiments: Marvin Künnemann] Benjamin Doerr
Experimental Results (n=1024) Complete graph � � Hypercube � �� Random graphs � � � � , p such that graph connected w.p.1/2 Lists: neighbors sorted in Lists: „inform the neighbor in Lists: neighbors sorted in increasing order dimension 1, 2, 3, ...“ increasing order Average broadcast times: Fully random: 18.09 Fully random: 21.11 Fully random: 27.31 Quasirandom: 17.63 Quasirandom: 18.71 Quasirandom: 19.48 [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 � Bottom line: Don’t be afraid of dependencies! Köszönöm! Benjamin Doerr
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