1. Quasirandom Rumor Spreading Tobias Friedrich Max-Planck-Institut für Informatik Saarbrücken

2. Tobias Friedrich  Rumor Spreading Outline

3. Tobias Friedrich  Randomized Rumor Spreading  Deterministic Rumor Spreading  Quasirandom Rumor Spreading Outline

4. 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 node Round 4: Each informed node informs a random node Round 5: Let‘s hope the remaining two get informed...

5. 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

6. 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(K n ) = 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(G n,p ) = O(log(n)) w.h.p., p > (1+ ε ) log(n)/n [Feige et al.’90]

7. 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 . 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

8. 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

9. 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:

10. 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 G n,p, p ≥ (1+ε) log(n)/n, –hypercubes still hold...

11. 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 G n,p, p ≥ (1+ε) log(n)/n, –hypercubes still hold independent from the structure of the lists [Doerr, F., Sauerwald ‘ 08]

12. Tobias Friedrich Quasirandom Rumor Spreading  Results (cont.): –Random graphs G n,p, p = (log(n)+log(log(n)))/n:  fully randomized:T(G n,p ) = Θ(log(n) 2 )  quasirandom:T(G n,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

13. 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

14. 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!!