joy ghosh hung q. ngo , seokhoon yoon, chunming qiao

IEEE Infocom 2007

On a Routing Problem within Probabilistic Graphs

and 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

State University of New York (SUNY) at BUFFALO IEEE Infocom 2007

In many networks such as MANET, DTN, ICMAN, ..., links are probabilistic.

Natural to formulate the following problem.

A Routing Problem on Probabilistic Graphs

s

t

pe

G: directed graph

For each e=(u,v), pe is the probablity that u can deliver a packet to vAll pe are independent

Find a subgraph H maximizing Conn-ProbH(s,t)Subject to application-dependent constraints

Find a subgraph H maximizing Conn-ProbH(s,t)Subject to application-dependent constraints


Natural Questions That Follow (Talk Outline)1. How do we construct G (and compute pe)?

Accuracy of pe routing efficiency

2. What are the constraints for H?

3. Given H, how to compute Conn-ProbH(s,t)?

4. What’s the complexity of finding optimal H?

5. If complexity is too high, how to design good routing algorithms/protocols?

6. How useful is this model, anyhow?


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


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


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


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.


#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: a) 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.

b) If such a decision procedure exists, then we can compute Conn-ProbD(s,t) with a simple binary search


#P-hardness of finding optimal H

Add path with k=|E(D)| edges to D to get G with Πk

i=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


5. Heuristic for finding good H (i.e. routing algo)

Edge-constrained routing EC-SOLAR-KSP


Simulation Parameters


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.


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!


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

IEEE Infocom 2007

Thank You!

Questions?


Mobile Users

• influenced by social routines

• visit a few “hubs” /

places (outdoor/indoor) regularly

• “orbit” around (fine to coarse grained) hubs at several levels

Sociological Orbit Framework


Illustration of A Random Orbit Model(Random Waypoint + Corridor Path)

Conference Track 1

Conference Track 3

Cafeteria

Lounge

Conference Track 2

Conference Track 4

PostersRegistration

Exhibits


Random Orbit Model


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

Real-worldData Traces

Hub-Lists(Binary Vectors)

Users’ Mobility Profiles

Hub TransitionProbabilities

Hub Staying TimeDistributions

Hub Transition Time Distributionss

Model Trained UsingEM-Algorithm

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


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


Profiling illustration

Obtain daily hub stay durations

Translate to binary hub visitation vectors

Apply clustering algorithm to find mixture of profiles


Profile parameters for all sample users


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


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 – Πk

1(1 – pi) Where pi = we(u,vi) * Pd(vi, d) for all edges

(u,vi)


Approximation algorithm for Conn-ProbG(s,d)


Optimal algorithm for delivery probability

Calculate all paths from s to d Apply Algorithm 2 by rules of

inclusion and exclusion


Approximation ratio simulation

joy ghosh hung q. ngo , seokhoon yoon, chunming qiao

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