Communication complexity of randomized rumor spreading

1. Communication complexity ofrandomized rumor spreading George Giakkoupis joint work with Pierre Fraigniaud LIAFA, Université Paris 7

2. The rumor spreading problem [FG85] • A town contains n people, one of whom knows a rumor. • At the first stage he tells someone chosen randomly from the town. • At each stage, each person who knows the rumor tells someone else, chosen randomly and independently of all other choices. • What is the number of stages until the whole town knows the rumor? • … lgn + lnn + O(1), wp 1-o(1)[FG85, Pit87] PROSE meeting, January 2010

3. A more general problem statement • n players communicate in parallel rounds. • In each round every player calls a randomly selected player. • Player u is allowed to exchange messages during a round only with the player that u called, and with all the (0 or more) players that u received calls from, in that round. • … the random phone-call communication model • In every round, a (possibly empty) set of rumors is generated. • Each of the rumors is initially placed in a subset of the players, the sources of that rumor. • Goal: each rumor be distributed among all players, with high probability, within a small number of rounds, using a small amount of communication between players. PROSE meeting, January 2010

4. Applications • Maintenance of replicated databases, e.g., on name servers in a large corporate network [DGH+87,FPRU90]. • Why randomized communication? • simplicity • scalability • robustness PROSE meeting, January 2010

5. Rumor-spreading algorithms: Push • The algorithm proposed in original problem statement: • In each round, every player u who knows rumor r, forwards the rumor to the player v that u called in that round --- upushesr to v • The age of the r is also transmitted together with r • The distribution of r terminates after Θ(lgn) rounds • Time-optimal  • Communication-heavy: Θ(nlgn) transmissions of each rumor  PROSE meeting, January 2010

6. Push-pull algorithm [KSSV00] • In each round, every player u who knows rumor r, pushes the rumor to the player v that u called in that round, and also forwards r to all the (0 or more) players w who called u---ris pulled from u to w • The age of the r is transmitted together with r • The distribution of r terminates after lg3n + Θ(lglgn) rounds • Time-optimal  • Communication: Θ(nlglgn) transmissions of each rumor  • one source per rumor is assumed • A variation of this protocol: • uses a less sensitive termination criterion, and • achieves Θ(nlglgn) transmissions even for many sources per rumor. PROSE meeting, January 2010

7. Lower bounds [KSSV00] • No (distributed) algorithm is both time-optimal, requiring O(lgn) round, and message-optimal, using O(n) messages. • For the special class of address-oblivious algorithms, O(nlglgn) messages are required (regardless of the number of rounds). • -> the push-pull algorithm is asymptotically optimal. PROSE meeting, January 2010

8. Our work: bit communication complexity • Instead of counting messages, we count bits. • #of bits more relevant that #of messages for some applications, especially when rumors are large or too many. • Assumptions: • A rumor is a binary string, of arbitrary length. • Any binary string is a valid rumor. • Any number of rumors can be generated in each round. PROSE meeting, January 2010

9. Our work • Recall: No (distributed) algorithm is both time-optimal, requiring O(lgn) round, and message-optimal, using O(n) messages [KSSV00]. • There is an algorithm that is both time-optimal, and bit-communication-optimal, for all but very small rumor sizes: b « lglgn lglglgn. • We describe an address-oblivious algorithm that requires O(lgn) rounds and uses O(nb+nlglgnlgb) bits, per b-bit rumor. • We show a lower bound of O(nb+nlglgn) bits, for any address-oblivious algorithm using O(lgn) rounds. • Our algorithm is a push-pull algorithm with “concise” feedback. • The original Push-pull algorithms send O(nblglgn) bits per rumor. PROSE meeting, January 2010

10. Graph-based variation of the problem • n players communicate in parallel rounds. • The players correspond to the vertices of an n-vertex graph G. • In each round every player calls a randomly selected player, among its neighbors in G. • Player u is allowed to exchange messages during a round only with the player that u called, and with all the (0 or more) players that u received calls from, in that round. • … the random phone-call communication model (for graphs) • In every round, a (possibly empty) set of rumors is generated. • Each of the rumors is initially placed in a subset of the players, the sources of that rumor. • For G=Kn the problem is the same as before. PROSE meeting, January 2010

11. Graph-based variation: existing results • Runtime of Push on general graphs, d-regular graphs, the hypercube, and random graphs [FPRU90]. • Runtime and message complexity of rumor spreading on random graphs [Els06, ES08], d-regular graphs [BEF08], and scale-free graphs [Els06]. • Runtime of Push for a quasi-random analogue to random phone-call model [DFS08,DFS09]. PROSE meeting, January 2010

12. Thank you! PROSE meeting, January 2010