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

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Tobias Friedrich Rumor Spreading Outline

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Tobias Friedrich Randomized Rumor Spreading Deterministic Rumor Spreading Quasirandom Rumor Spreading Outline

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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...

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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

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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]

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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

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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

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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:

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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...

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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]

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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

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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

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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!!

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