Hop-limited f looding over d ynamic networks

1. Hop-limited flooding over dynamic networks M. Vojnović and A. Proutiere Microsoft Research IEEE Infocom 2011, Shanghai, April 2011

2. Introduction • Disseminate a message to all nodes using transmissions between pairs of nodes • Dynamic network • Communication link between a pair of nodes alternates between active and inactive state • Desired for the dissemination to be time efficient and of low cost • Application scenarios: • Mobile networks • Peer-to-peer networks

3. Related work • Parsimonious flooding [Baumann et al, PODC 2009] • Message offered by a node only within some fixed time since the message was received by this node • k-copy forwarding [Chainterau et al, ToN 2007] • Each node relays the message to at most k other nodes • The diameter of opportunistic mobile networks [Chainterau et al, CONEXT 2007] • Characterized expected number of paths between two end nodes within given time for a dynamic network similar to ours (assumed no hop limit constraints) • Coupon collector problem • Special case of 1-hop limited flooding

4. k-hop limited flooding • Lazy k-hop limited flooding • k-hop limited flooding • Message can be relayed by a node only if this node observed a copy of the message transferred through less than k hops 0 0 1 1 2 2 1 2 • Message can be relayed by a node only if it was first received by this node through at most k hops

5. Main questions • Q1: What is the completion time of k-hop limited flooding? • Completion time defined as the time for the message to reach given fraction of all nodes • Q2: What is the communication cost of k-hop limited flooding? • Communication cost defined as the maximum number of message transmissions per node • Q3: How much worse is the lazy version?

6. Assumptions • n nodes • Two nodes in contact at instances of a Poisson process with rate • Message initially held by nodes • Completion at the smallest time at which all but nodes have not yet received the message

7. The limit of many nodes • k-hop limited flooding: • Lazy k-hop limited flooding: fraction of nodes that observed a copy of the message that was transferred through less than i hops by time t fraction of nodes that first received the message through at most i hops by time t * The paper also contains some additional characterizations of the completion time by studying the underlying Markov processes (not in this slide deck)

8. Performance measures • Completion time • Communication cost 1 t

9. Special cases(an = bn = n) • No hop limits • 1-hop limit coupon collector

10. Completion time lower bound • For every such that

11. Completion time • k-hop limited flooding:Suppose = and =, then If, in addition,, then

12. Completion time (cont’d) • Lazy k-hop limited flooding: Suppose =, then

13. Completion time comparison

14. Communication cost • k-hop limited flooding • Lazy k-hop limited flooding

15. Communication cost comparison

16. Summary of results k-hop limited flooding Lazy k-hop limited flooding No hop limits Completion time Communication cost

17. Dissemination delay vs. hops • Diminishing improvement with the number of hops

18. Convergence Lazy k = 3 Lazy k = 2 • Accurate asymptotes already for small number of nodes

19. Conclusion • Under the assumed dynamic network and the hop limit constraint, for both variants of k-hop limited flooding: • Completion time optimal up to poly-log factors • Communication cost optimal up to constant factors • Lazy version slower for at most factor • Lazy version more expensive for at most factor • Open problem: performance under more general dynamic networks? • Ex 1 Poisson but with node-pair specific rates • Ex 2 Correlated link activation process, e. g. a Markov process