Joy Ghosh Hung Q. Ngo , Seokhoon Yoon, Chunming Qiao

1. On a Routing Problem within Probabilistic Graphsand its application to Intermittently Connected Networks Joy Ghosh Hung Q. Ngo, Seokhoon Yoon, Chunming Qiao Messenger Server Department of Computer Science and Engineering, Yahoo! Inc. State University of New York at Buffalo, Sunnyvale, CA Buffalo, NY

2. A Routing Problem on Probabilistic Graphs • In many networks such as MANET, DTN, ICMAN, ..., links are probabilistic. • Natural to formulate the following problem. G: directed graph pe s For each e=(u,v), pe is the probablity that u can deliver a packet to v All pe are independent t Find a subgraph H maximizing Conn-ProbH(s,t) Subject to application-dependent constraints

3. Natural Questions That Follow (Talk Outline) • How do we construct G (and compute pe)? • Accuracy of pe routing efficiency • What are the constraints for H? • Given H, how to compute Conn-ProbH(s,t)? • What’s the complexity of finding optimal H? • If complexity is too high, how to design good routing algorithms/protocols? • How useful is this model, anyhow?

4. 1. How do we construct G (and compute pe)? • Short answer: application dependent • Long answer: • Don’t care (e.g. Epidemic, randomized flooding, Spray-and-Wait, …) • Locally estimate delivery predictability/frequencies (e.g., ProPHET, ZebraNet, …) • Assume an Oracle (e.g., Spyropoulos et al. in Secon’04, Jain et al. Sigcomm’05, …) • Mobility modeling/profiling (e.g., random waypoint, group mobility, freeway mobility, …) • We use random orbit model from our SOLAR framework • Main reasons: model built on real-world data traces, and we already have the simulation code for it

5. 2. Constraints for the delivery subgraph H • Short answer: application dependent • Long answer: • No constraint (blind flooding) • Acyclic (Epidemic) • Threshold on expected delivery time (e.g. some works on DTN) • … • Our proposal: |E(H)| ≤ given threshold k, because this will reduce • Contention, thus message drops and retransmissions • Data and bandwidth overheads

6. 3. Given H, how to compute Conn-ProbH(s,t)? • Bad news: #P-complete • Classic s,t-reliability problem • Shown #P-complete by Valiant (1979) • Good news: • Can be approximated with our simple and efficient heuristics to within about 85%-90% accuracy on average • Note: • May have an FPRAS using Markov-chain Monte Carlo (along the line of Jerrum-Sinclair-Vigoda’s work, Journal of the ACM, 2004) • But: Long Standing Open Problem

7. 4. What’s the complexity of finding optimal H? • Somewhat subtle: hard to compute objective function does not imply hard to optimize • Bad News: #P-hard, as we showed in the paper • Good News: • Our heuristic can find reasonably good H • Note: • A wide-open research direction: approximation algorithms for #P-optimization problems.

8. #P-hardness of finding optimal H • Given directed graph D = (V,E) with edge probabilities pe, source s, destination t; computing Conn-ProbD(s,t) is #P-complete in 1979 • Let c be the least common multiple of all denominators of the pe • We show that: • If the optimal H can be found for any given G, then an efficient decision procedure for deciding if Conn-ProbD(s,t) ≤ c’/c can be designed for any c’ ≤ c. • If such a decision procedure exists, then we can compute Conn-ProbD(s,t) with a simple binary search

9. #P-hardness of finding optimal H • Add path with k=|E(D)| edges to D to get G with Πki=1pi = c’/c + ε, for any ε < 1/c • Suppose finding optimal H can be done efficiently • If H is the • Upper part  Conn-ProbD(s,t) > c’/c • Lower part  Conn-ProbD(s,t) ≤ c’/c

10. 5. Heuristic for finding good H (i.e. routing algo) Edge-constrained routing EC-SOLAR-KSP

11. Simulation Parameters

12. Edge-constrained routing – EC-SOLAR-KSP • EC-SOLAR-KSP1  L = |E| • EC-SOLAR-KSP2  L = 0.8 * |E| • EC-SOLAR-KSP3  L = 0.6 * |E| • PROB-ROUTE  Pinit = β = γ = 0.5; • A. Lindgren, A. Doria, and O. Schelen, “Poster: Probabilistic routing in intermittently connected networks,” Proceedings of The Fourth ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc 2003), June 2003. • EPIDEMIC  Not shown in 2nd figure as its Network Byte Overhead was much higher • A. Vahdat and D. Becker, “Epidemic routing for partially connected ad hoc networks,” Technical Report CS- 00006, Duke University, April 2000.

13. 6. How useful is this model, anyhow? • Short answer: we don’t know yet • Long answer: • The good • This model is potentially useful • There are many interesting open questions • Good solution can serve as benchmark • The bad and the ugly • Edge probabilities may not be independent • Practical applications can’t afford to use a centralized algorithm • Intermediate vertices may have more information than the source at the point they are about to deliver packets • This may be a good thing, may be a bad thing!

14. Concluding Remarks • Contributions • Formulatied of a new routing problem within probabilistic graphs • Addressed several aspects of the problem: graph construction, complexity, routing heuristics • Open problems • Approximation algorithm for this #P-Hard problem • Better distributive algorithm • Better Heuristics for this and other mobility models

15. Thank You! Questions?

16. Sociological Orbit Framework • Mobile Users • influenced by social routines • visit a few “hubs” / places(outdoor/indoor) regularly • “orbit” around (fine to coarse grained) hubs at several levels

17. Illustration of A Random Orbit Model(Random Waypoint + Corridor Path) Conference Track 2 Conference Track 1 Exhibits Lounge Conference Track 3 Registration Posters Conference Track 4 Cafeteria

18. Random Orbit Model

19. How to compute model’s parameters? • ETH Zurich traces • 1 year from 4/1/04 till 3/31/05 • 13,620 wireless users, 391 APs, 43 buildings • Mapped APs into buildings based on AP’s coordinates, and each building becomes a “hub” • Converted AP-based traces into hub-based traces • Each Profile is a weighted Hublist, e.g.Profile = (0.4, 0.5, 0.9) • Each profile is a cluster mean obtained via the Expectation Maximization (EM) Algorithm Real-world Data Traces Hub-Lists (Binary Vectors) Model Trained UsingEM-Algorithm Users’ Mobility Profiles Hub Transition Probabilities Hub Staying Time Distributions Hub Transition Time Distributionss

20. Examples of Hub-Lists and Mobility Profiles • On any given day, a user may regularly visit a small number of “hubs” (e.g., locations A and B) • Each mobility profile is a weighted list of hubs, where weight = hub visit probability (e.g., 70% A and 50% B) • In any given period (e.g., week), a user may follow a few such “mobility profiles” (e.g., P1 and P2) • Each profile is in turn associated with a (daily) probability (e.g., 60% P1 and 40% P2) • Example: P1={A=0.7, B=0.5} and P2={B=0.9, C=0.6} • On an ordinary day, a user may go to locations A, B and C with the following probabilities, resp.: 0.42 (=0.6x0.7), 0.66 (= 0.6x0.5 + 0.4+0.9) and 0.24 (=0.4x0.6) • 20% more accurate than simple visit-frequency based prediction • Knowing exactly which profile a user will follow on a given day can result in even more accurate prediction On any given day, a user may regularly visit a small number of “hubs” (e.g., locations A and B) Each mobility profile is a weighted list of hubs, where weight = hub visit probability (e.g., 70% A and 50% B) In any given period (e.g., week), a user may follow a few such “mobility profiles” (e.g., P1 and P2) Each profile is in turn associated with a (daily) probability (e.g., 60% P1 and 40% P2) • Example: P1={A=0.7, B=0.5} and P2={B=0.9, C=0.6} • On an ordinary day, a user may go to locations • A, B & C with the following probabilities: • 0.42 (=0.6x0.7), 0.66 (= 0.6x0.5 + 0.4+0.9), 0.24 (=0.4x0.6) • 20% more accurate than simple visit-frequency based prediction • Knowing exactly which profile a user will follow on a given day • can result in even more accurate prediction

21. Profiling illustration Translate to binary hub visitation vectors Apply clustering algorithm to find mixture of profiles Obtain daily hub stay durations

22. Profile parameters for all sample users

23. Approximation algorithm for Conn-ProbG(s,d) • G = (V, E) where edge probability between nodes u and v is pe(u,v) • In G, starting from s, all nodes choose at most k downstream edges to get Gk = (V, Ek)  (b) • Weight of each edge in Gk is set to • we(u,v) = -1 * log (pe(u,v)) to get G’k say

24. Approximation algorithm for Conn-ProbG(s,d) • Compute shortest path from s to all nodes in G’k to get Gsp = (V, Esp) & assign BFS level • Reset we(u,v) = pe(u,v) & add all edges (v,d)that were in G to get G’ = (V, E’)  (d) • Let Pd(u,v) be delivery probability of node u to v • Apply Approximation Algorithm to G’ to get Pd(s,d) • Start with any u≠ d with maximum level # • Pd(u,d) = 1 – Πk1(1 – pi) • Where pi = we(u,vi) * Pd(vi, d) for all edges (u,vi)

25. Approximation algorithm for Conn-ProbG(s,d)

26. Optimal algorithm for delivery probability • Calculate all paths from s to d • Apply Algorithm 2 by rules of inclusion and exclusion

27. Approximation ratio simulation