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QuasirandomRumor Spreading
Tobias Friedrich
Max-Planck-Institut für Informatik Saarbrücken
Tobias Friedrich
Rumor Spreading
Rumor Spreading
Rumor Spreading
Outline
Tobias Friedrich
Randomized Rumor Spreading
Deterministic Rumor Spreading
Quasirandom Rumor Spreading
Outline
Tobias Friedrich
Randomized Rumor Spreading
Model (on a graph G):– Start: One node is informed– Each round, each informed node informs a neighbor chosen
uniformly at random– Broadcast time T(G): Number of rounds necessary to inform all
nodes (maximum taken over all starting nodes)
Round 0: Starting node is informedRound 1: Starting node informs random nodeRound 2: Each informed node informs a random nodeRound 3: Each informed node informs a random nodeRound 4: Each informed node informs a random nodeRound 5: Let‘s hope the remaining two get informed...
Tobias Friedrich
Randomized Rumor Spreading
Model (on a graph G):– Start: One node is informed– Each round, each informed node informs a neighbor chosen
uniformly at random– Broadcast time T(G): Number of rounds necessary to inform all
nodes (maximum taken over all starting nodes)
Application:– Broadcasting updates in distributed databases
simple robust self-organized
Tobias Friedrich
Randomized Rumor Spreading
Model (on a graph G):– Start: One node is informed– Each round, each informed node informs a neighbor chosen
uniformly at random– Broadcast time T(G): Number of rounds necessary to inform all
nodes (maximum taken over all starting nodes)
Results [n: Number of nodes]:– T(G) ≥ log(n) for all graphs G
– T(Kn) = O(log(n)) w.h.p. [Frieze, Grimmet’85]
– T({0,1}d) = O(log(n)) w.h.p. [Feige, Peleg, Raghavan, Upfal’90]
– T(Gn,p) = O(log(n)) w.h.p., p > (1+ε) log(n)/n [Feige et al.’90]
Tobias Friedrich
Deterministic Rumor Spreading?
As above, but now with Propp-Machine:– Each node has a list of its neighbors.– Informed nodes inform their neighbors in the order of
this list.
Problem: Might take long...
Here: n-1 rounds .
1 3 4 5 62
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
Tobias Friedrich
Quasirandom Rumor Spreading
As above except:– Each node has a list of its neighbors.– Informed nodes inform their neighbors in the order of
this list, but start at a random position in the list
Tobias Friedrich
Quasirandom Rumor Spreading
As above except:– Each node has a list of its neighbors.– Informed nodes inform their neighbors in the order of
this list, but start at a random position in the list
Results:
Tobias Friedrich
Quasirandom Rumor Spreading
As above except:– Each node has a list of its neighbors.– Informed nodes inform their neighbors in the order of
this list, but start at a random position in the list
Results: The log(n) bounds for – complete graphs,
– random graphs Gn,p, p ≥ (1+ε) log(n)/n,
– hypercubes
still hold...
Tobias Friedrich
Quasirandom Rumor Spreading
As above except:– Each node has a list of its neighbors.– Informed nodes inform their neighbors in the order of
this list, but start at a random position in the list
Results: The log(n) bounds for – complete graphs,
– random graphs Gn,p, p ≥ (1+ε) log(n)/n,
– hypercubes
still hold independent from the structure of the lists[Doerr, F., Sauerwald ‘08]
Tobias Friedrich
Quasirandom Rumor Spreading
Results (cont.):– Random graphs Gn,p, p = (log(n)+log(log(n)))/n:
fully randomized: T(Gn,p) = Θ(log(n)2)
quasirandom: T(Gn,p) = Θ(log(n))
– Complete k-regular trees: fully randomized: T(G) = Θ(k log(n)) quasirandom: T(G) = Θ(k log(n)/log(k))
Algorithm Engineering Perspective:– need fewer random bits– easy to implement: Any implicitly existing permutation of
the neighbors can be used for the lists
Tobias Friedrich
Quasirandom Rumor Spreading Proof ingredients:
– Forward Approximation: O(log n) nodes quickly informed O(log n) phases with a constant number of rounds
– set of newly informed nodes is independent– number of informed nodes doubles per phase
afterwards constant fraction informed
– Backward Approximation: if there is one uninformed vertex at time t, then there are at
least Ω(log n) vertices uninformed O(log n) time steps before
– Coupling w.h.p. one of the Ω(n) informed vertices informes one of the
O(log n) uninformed vertices within a single step
Tobias Friedrich
Summary
Quasirandomness: – Simulate a particular aspect of a random object
Surprising results:– Quasirandom walks (see Talk 61, Sat 13:45)– Quasirandom rumor spreading
For future research:– Good news: Quasirandomness can be analyzed– Many open problems– “What is the right dose of randomness?”
Thank you!!