1. Lab for Advanced Network Design, Evaluation and Research “Sociological Orbits” Mobility Profiling and Routing for Mobile Wireless Networks Joy Ghosh Ph.D. Dissertation Defense Major Advisor: Dr. Chunming Qiao

2. Lab for Advanced Network Design, Evaluation and Research Outline Mobility - Impact on Routing / Advantages Acquaintance Based Soft Location Management (ABSoLoM) Sociological ORBIT Mobility Framework Mobility Profiling Techniques and Applications Sociological Orbit aware Location Approximation and Routing (SOLAR) – MANET & ICMAN Theoretical Analysis of SOLAR  Routing problem formulation for ICMAN  Approximation algorithm for delivery probability  Mathematical model for computing contact probability  Edge-constrained routing protocol and its performance Concluding Remarks

3. Lab for Advanced Network Design, Evaluation and Research The Overall Picture

4. Lab for Advanced Network Design, Evaluation and Research Mobility Impact on Routing Node Mobility  Dynamic network topology Proactive protocols are inefficient  Need to exchange control packets too often  Leads to congestion  E.g., Distance Vector, Link State Reactive protocols are better suited, but  Locating a node incurs more delay  Route maintenance is tricky as nodes movetricky  E.g., Dynamic Source Routing (DSR), Location Aided Routing (LAR)

5. Lab for Advanced Network Design, Evaluation and Research Framework for analyzing impact of mobility on protocol performance F. Bai, N. Sadagopan, and A. Helmy, “Important: a framework to systematically analyze the impact of mobility on performance of routing protocols for adhoc networks”, Proceedings of IEEE INFOCOM '03, vol. 2, pp. 825-835, March 2003.

6. Lab for Advanced Network Design, Evaluation and Research Greedy Geographic Forwarding Pros  Less affected by mobility than source routes Less affected  Smaller header size (no path cached) Cons  Nodes need to know own location  Needs sufficient node density Workarounds for local maxima  Broadcast  Planar graph perimeter routing (e.g., GPSR)

7. Lab for Advanced Network Design, Evaluation and Research Advantages of Node Mobility – Individual node’s view of network

8. Lab for Advanced Network Design, Evaluation and Research Advantages of Node Mobility – Node’s view of network through “acquaintances”

9. Lab for Advanced Network Design, Evaluation and Research Acquaintance Based Soft Location Management (ABSoLoM) Forming and maintaining acquaintances Limit number of acquaintances Keep updating acquaintances of location Query acquaintances for destination location Query acquaintances Limit query propagation by logical hops On learning of destination, use geographic forwarding to send packets to destination Nosy Neighbors  Can respond to query if destination’s location is known  Caches node locations while forwarding certain packets

10. Lab for Advanced Network Design, Evaluation and Research Performance Analysis Simulated in GloMoSim LAR & DSR borrowed from the GloMoSim distribution Implementation of SLALoM by Dr. Sumesh J. Philip (author) ABSoLoM parameters  Number of friends = 3  Maximum logical hops = 2 100 nodes in 2000m x 1000m for 1000s Random Waypoint mobility  Velocity = 0m/s-10m/s; Pause = 15s Random CBR connections varied in simulation  50 packets per connection; 1024 bytes per packet

11. Lab for Advanced Network Design, Evaluation and Research Results – I.a: Throughput vs. Load

12. Lab for Advanced Network Design, Evaluation and Research Results – I.b: Overhead vs. Load

13. Lab for Advanced Network Design, Evaluation and Research Simulation Results – II (a) Hop Latency vs. Load & (b) Throughput vs. Mobility

14. Lab for Advanced Network Design, Evaluation and Research Parallel growth of models and protocols Practical mobility models  Random Waypoint  simple, but impractical!!  Entity based  individual node movement  Group based  collective group movement  Scenario based  geographical constraints Mobility pattern aware routing protocols  Mobility tracking and prediction  Link break estimation  Choice of next hop

15. Lab for Advanced Network Design, Evaluation and Research Our Motivation Not to suggest only a practical mobility model MANET is comprised of wireless devices carried by people living within societies Society imposes constraints on user movements Study the social influence on user mobility Realization of special regions of some social value Identify a macro level mobility profile per user Use this profile to aid macro level soft location management and routing

16. Lab for Advanced Network Design, Evaluation and Research 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

17. Lab for Advanced Network Design, Evaluation and Research Illustration of A Random Orbit Model (Random Waypoint + Corridor Path) Conference Track 1 Conference Track 3 Cafeteria Lounge Conference Track 2 Conference Track 4 Posters Registration Exhibits

18. Lab for Advanced Network Design, Evaluation and Research Random Orbit Model

19. Lab for Advanced Network Design, Evaluation and Research Traces Used Profiling techniques applied to ETH Zurich traces  Duration of 1 year from 4/1/04 till 3/31/05  13,620 wireless users, 391 APs, 43 buildings  Grouped users into 6 groups based on degree of activity  Selected one sample (most active) user from each group Mapped APs into buildings based on AP’s coordinates, and each building becomes a “hub”  Converted AP-based traces into hub-based traces Other traces  Expect similar results from Dartmouth’s traces  No sufficient AP location info from other traces  UMass’s traces are for buses, more predictable than users  Need to obtain actual users’ traces with GPS

20. Lab for Advanced Network Design, Evaluation and Research Hub-centric Parameters - I

21. Lab for Advanced Network Design, Evaluation and Research Hub-centric Parameters - II

22. Lab for Advanced Network Design, Evaluation and Research Hub Based Mobility Profiles and 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 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

23. Lab for Advanced Network Design, Evaluation and Research Orbital Mobility Profiling Obtain each user’s daily hub lists as binary vectors Represent each hub list (binary vector) as a point in a n-dimensional space (n = total number of hubs) Cluster these points into multiple clusters, each with a mean  Using the Expectation-Maximization (EM) algorithm based on a Mixture of Bernoulli’s distributionExpectation-Maximization (EM)  Probe other classification methods: Bayesian-Bernoulli’s Each cluster mean represents a mobility profile, described as a probabilistic hub visitation list User’s mobility is aptly modeled using a mixture of mobility profiles with certain “mixing proportions”

24. Lab for Advanced Network Design, Evaluation and Research Profiling illustration Obtain daily hub stay durations Translate to binary hub visitation vectors Apply clustering algorithm to find mixture of profiles

25. Lab for Advanced Network Design, Evaluation and Research Profile parameters for all sample users

26. Lab for Advanced Network Design, Evaluation and Research Hub-based Location Predictions - I Unconditional Hub-visit Prediction  Prediction Error = Incorrect hubs predicted over Total hubs  SPE – Statistical based Prediction Error SPE-ALL: (n+1) th day prediction based on hub-visit frequency from day 1 through day n SPE-W7 : (n+1) th day prediction based on hub-visit frequency within last week, i.e., day (n-7) through day n  PPE – Profile based Prediction Error PPE-W7 : (n+1) th day prediction based on profiles of the last week, i.e., day (n-7) through day n  Prediction Improvement Ration (PIR) PIR-ALL = (SPE-ALL – PPE-W7) / SPE-ALL PIR-W7 = (SPE-W7 – PPE-W7) / SPE-W7

27. Lab for Advanced Network Design, Evaluation and Research Unconditional Prediction Results The profile mixing proportions vary with every window of n days

28. Lab for Advanced Network Design, Evaluation and Research Hub-based Location Predictions - II Conditional Hub-visit Prediction  Improvement given current profile is known/identifiable  It is possible sometimes to infer profile from current hub information alone  Our method effectively leverages information when available Sample user categories Target Hub ID: will the user visit this hub?The current day in questionPredicted probability using visit frequency Indicator (Current) HubCurrent ProfilePredicted probability based on profile Actually visited H t on day D or not

29. Lab for Advanced Network Design, Evaluation and Research Hub-based Location Predictions - III Hub sequence prediction based on hub transitional probability Prediction Accuracy = 1 – (incorrect predictions / total predictions) Scenario 1: only starting hub is known for sequence prediction Scenario 2: hub prediction is corrected at every hub in sequence Better performance with increasing knowledge – intuitive Statistical based Prediction Accuracy (SPA) – no profile informationProfile based Prediction Accuracy (PPA) – no time informationTime based Prediction Accuracy (TPA) – temporal profiles

30. Lab for Advanced Network Design, Evaluation and Research Applications of Orbital Mobility Profiles Location Predictions and Routing within MANET and ICMAN Anomaly based intrusion detection  unexpected movement (in time or space) sets off an alarm Customizable traffic alerts  alert only the individuals who might be affected by a specific traffic condition Targeted inspection  examine only the persons who have routinely visited specific regions Environmental/health monitoring  identify travelers who can relay data sensed at remote locations with no APs

31. Lab for Advanced Network Design, Evaluation and Research Profile based Routing within MANET Build a sociological orbit based mobility model (Random Orbit) Assume that mobility profiles are obtained Devise routing protocols to leverage mobility information within MANET setting Key assumption – geographical forwarding is feasible

32. Lab for Advanced Network Design, Evaluation and Research Sociological Orbit aware Location Approximation and Routing (SOLAR) - Basic Every node knows  Own coordinates, Own Hub list, All Hub coordinates Periodically broadcasts Hello  SOLAR-1 : own location & Hub list  SOLAR-2 : own location & Hub list + 1-hop neighbor Hub lists Cache neighbor’s Hello  Build a distributed database of acquaintance’s Hub lists Unlike “acquaintanceship” in ABSoLoM, SOLAR has  No formal acquaintanceship request/response  its not mutual  Hub lists are valid longer than exact locations  lesser updates For unknown destination, query acquaintances for destination’s Hub list (instead of destination’s location), in a process similar to ABSoLoM

33. Lab for Advanced Network Design, Evaluation and Research Sociological Orbit aware Location Approximation and Routing (SOLAR) - Advanced Subset of acquaintances to query  Problem: Lots of acquaintances  lot of query overhead Problem  Solution: Query a subset such that all the Hubs that a node learns of from its acquaintances are covered Solution Packet Transmission to a Hub List  All packets (query, response, data, update) are sent to node’s Hub list  To send a packet to a Hub, geographically forward to Hub’s center  If “current Hub” is known – unicast packet to current Hub  Default – simulcast separate copies to each Hub in list We compared simulcast, unicast, multicast – simulcast had best performance with higher cost of overhead and delay  On reaching Hub, do Hub local flooding if necessary  Improved Data Accessibility – Cache data packets within Hub Data Connection Maintenance  Two ends of active session keep each other informed  Such location updates generate “current Hub” information

34. Lab for Advanced Network Design, Evaluation and Research Sociological Orbit aware Location Approximation and Routing (SOLAR) – Illustration Hub A Hub B Hub C Hub D Hub E Hub I Hub F Hub G Hub H

35. Lab for Advanced Network Design, Evaluation and Research Performance Analysis Metrics Data Throughput (%)  Data packets received / Data packets generated Relative Control Overhead (bytes)  Control bytes send / Data packets received Approximation Factor for E2E Delay  Observed delay / Ideal delay  To address “fairness” issues!fairness

36. Lab for Advanced Network Design, Evaluation and Research Performance Analysis Parameters

37. Lab for Advanced Network Design, Evaluation and Research Results – I.a : Throughput vs. Hubs

38. Lab for Advanced Network Design, Evaluation and Research Results – I.b : Overhead vs. Hubs

39. Lab for Advanced Network Design, Evaluation and Research Results – I.c : Delay vs. Hubs

40. Lab for Advanced Network Design, Evaluation and Research Routing challenges in ICMAN ICMAN  Features of DTN/ICN + MANET Lack of infrastructure and any central control May not have an end-to-end path from source to destination at any given point in time Conventional MANET routing strategies fail User mobility may not be deterministic or controllable Devices are constrained by power, memory, etc. Applications need to be delay/disruption tolerant

41. Lab for Advanced Network Design, Evaluation and Research User level routing strategies Deliver packets to the destination itself Intermediate users store-carry-forward the packets Mobility profiles used to compute pair wise user contact probability P(u,v) via Semi-Markov Process Form weighted graph G with edge weights w(u,v) = log (1/P(u,v)) Apply modified Dijkstra’s on G to obtain k-shortest paths (KSP) with corresponding Delivery probability under following constraints  Paths are chosen in increasing order of total weights (i.e., minimum first)  Each path must have different next hop from source S-SOLAR-KSP (static) protocol  Source only stores set of unique next-hops on its KSP  Forwards only to max k users of the chosen set that come within radio range within time T D-SOLAR-KSP (dynamic) protocol  Source always considers the current set of neighbors  Forwards to max k users with higher delivery probability to destination

42. Lab for Advanced Network Design, Evaluation and Research Hub level routing strategy Deliver packets to the hubs visited by destination Intermediate users store-carry-forward the packets Packet stored in a hub by other users staying in that hub (or using a fixed hub storage device if any) Mobility profiles used to obtain delivery probabilities (DP), not the visit probability, of a user to a given hub  i.e. user may either directly deliver to hub by traversing to the hub, or may pass onto other users who can deliver to the hub Fractional data delivered to each hub proportional to the probability of finding the destination in it Routing Strategy  SOLAR-HUB protocol

43. Lab for Advanced Network Design, Evaluation and Research SOLAR-HUB Protocol P d n i h j : delivery probability (DP) of user n i to hub h j P t n i h j : probability of user n i to travel to hub h j h(n i ): hub that user n i is going to visit next P c n i n k (h j ): probability of contact between users n i & n j in hub h j N(n i ): neighbors of user n i P d n i h j = max(P t n i h j, max k (P c n i n k (h(n i ))*P t n k h j )) Source n s will pick n i as next hop to hub h j as:  {n i | max(P d n i h j ), n i Є N(n s )} iff P d n i h j > P d n s h j Packet Delivery Scheme  Source transmits up to k copies of message k/2 to neighbors with higher DP to “most visited” hub k/2 to neighbors with higher DP to “2nd most visited” hub  Downstream users forward up to k users with higher DP to the hub chosen by upstream node

44. Lab for Advanced Network Design, Evaluation and Research Simulation Parameters for GloMoSim

45. Lab for Advanced Network Design, Evaluation and Research Performance – Number of Hubs Overhead of EPIDEMIC is much more than others and had to be omitted from plot Overall D-SOLAR-KSP performs best

46. Lab for Advanced Network Design, Evaluation and Research Performance – Number of Users Overhead of EPIDEMIC is much more than others and had to be omitted from plot Overall D-SOLAR-KSP performs best like before because it is the most opportunistic in forwarding to any of its current neighbors

47. Lab for Advanced Network Design, Evaluation and Research Performance – Cache Size (Only SOLAR) All versions fair better with more cache Overall D-SOLAR-KSP performs best

48. Lab for Advanced Network Design, Evaluation and Research Performance – Cache Timeout (Only SOLAR) All versions fair better with larger timeout Overall D-SOLAR-KSP performs best

49. Lab for Advanced Network Design, Evaluation and Research Routing problem in probabilistic graphs Objective: maximize delivery probability from nodes s to t under various constraints G = (V,E) be a complete directed graph  V = ICMAN users; E = probabilistic contact between users Let A be a routing algorithm and G(A) be the delivery sub-graph induced by A Delivery probability is then s,t-connectedness probability (two- terminal reliability) denoted by Conn 2 (G(A)) Goal is to find a delivery sub-graph G(A) to maximize Conn 2 (G(A))  we have shown it to be #P-hard#P-hard 2 Possible approaches  Approximate Conn 2 (G(A)) by another polynomial time function  Develop heuristics for A for which Conn 2 (G(A)) can be approximated in polynomial time

50. Lab for Advanced Network Design, Evaluation and Research Approximation algorithm G = (V, E) where edge probability between nodes u and v is p e (u,v)  (a) In G, starting from s, all nodes choose at most k downstream edges to get G k = (V, E k )  (b) Weight of each edge in G k is set to  w e (u,v) = -1 * log (p e (u,v)) to get G’ k say Compute shortest path from s to all nodes in G’ k to get G sp = (V, E sp ) & assign BFS level #s  (c) Reset w e (u,v) = p e (u,v) & add all edges (v,d) that were in G to get G’ = (V, E’)  (d) Let P d (u,v) be delivery probability of node u to v Apply Algorithm 1 to G’ to get P d (s,d)  Start with any u ≠ d with maximum level #  P d (u,d) = 1 – Π k 1 (1 – p i )  Where p i = w e (u,v i ) * P d (v i, d) for all edges (u,v i )

51. Lab for Advanced Network Design, Evaluation and Research Algorithms for delivery probability Calculate all paths from s to d Apply Algorithm 2 by rules of inclusion and exclusion

52. Lab for Advanced Network Design, Evaluation and Research Performance comparison of approximation algorithm with optimal

53. Lab for Advanced Network Design, Evaluation and Research Contact Probability using a Semi-Markov Chain Hub transitional probability of user X from hubs h to h’ = β X hh’ >0, Σ h≠h’ β X hh’ =1 Inter-hub transition time  exponential with mean λ X hh’ X t be the hub X is in at time t E t be the hub stay time at X t-1 before coming to X t  distributed as power law with exponent λ X h This movement can be modeled with a Semi-Markov Chain (SMC) State space of X: I x = S U { (h, h’) | h,h’ Є S, h ≠ h’}  Where, S = set of X’s hubs, (h, h’) = movement of X from h to h’  Holding times at states in S are power law distributed  Holding times at states in (h, h’) are exponentially distributed State transitional probability p X ij  = β X hh’ when i = h and j = (h,h’)  = 1 when i = (h,h’) and j = h’  = 0 otherwise

54. Lab for Advanced Network Design, Evaluation and Research Contact Probability using a Semi-Markov Chain We consider similar formulation for user Y with hub set T  Let R = S ∩ T ≠ 0 Objective  Find probability of X meeting Y at time t (~equilibrium)  Find probability of X meeting Y at a particular hub h Є R, at time t Combined SMC: {Z t | t ≥ 0}  Cartesian product of SMCs of X and Y  State space I = I X x I Y ; states (x, y)  x Є I X, y Є I Y Sojourn times at x and y are either exponential, or power law with known parameters  Sojourn time at (x, y) may be calculated with simple exercises Jumping probabilities  If sojourn time T i at state i of X < sojourn time T i’ at state i’ of Y p XY (i,i’)(j,i’) = p X ij  If sojourn time T i at state i of X > sojourn time T i’ at state i’ of Y p XY (i,i’)(i,j’) = p Y ij EMC of Z is ergodic as long as EMC of X and Y are ergodic Find only occupancy probabilities π XY (h, h) at equilibrium for state (h, h), hЄR Probability that X meets Y at equilibrium  Σ hЄR π XY (h, h)

55. Lab for Advanced Network Design, Evaluation and Research 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|

56. Lab for Advanced Network Design, Evaluation and Research Concluding Remarks - Contributions Use of acquaintances for soft location management Sociological ORBIT framework and mobility models Profiling user mobility and predicting locations Using mobility profiles for routing within MANET and ICMAN Formulation and analysis of a novel routing problem within probabilistic graphs

57. Lab for Advanced Network Design, Evaluation and Research Concluding Remarks – Future work More efficient profiling techniques  Overcome shortcomings – bias towards hub visits  Use other tools like time series analysis Profile exchange and management  Profile lifetime in cache  Distribution of profile to minimize query radii Solutions to our routing optimization problem  Develop an optimal routing algorithm that gives a delivery sub-graph which maximizes the delivery probability

58. Lab for Advanced Network Design, Evaluation and Research Related Publications Journal  Joy Ghosh, Sumesh J. Philip, Chunming Qiao, "Sociological Orbit aware Location Approximation and Routing (SOLAR) in MANET" - Accepted for publication in ELSEVIER Ad Hoc Networks Journal, Nov 2005 Workshops  Joy Ghosh, Hung Q. Ngo, Chunming Qiao, "Mobility Profile based Routing within Intermittently Connected Mobile Ad hoc Networks (ICMAN)" - Accepted for publication in IWCMC 2006 Delay Tolerant Mobile Networks workshop, Vancouver, Canada, July 2006  Joy Ghosh, Matthew J. Beal, Hung Q. Ngo, Chunming Qiao, "On Profiling Mobility and Predicting Locations of Wireless Users" - Accepted for publication in ACM/SIGMOBILE REALMAN 2006 workshop at ACM Mobihoc '06, Florence, Italy, May 2006 Conferences  Joy Ghosh, Cedric Westphal, Hung Ngo, Chunming Qiao, "Bridging Intermittently Connected Mobile Ad hoc Networks (ICMAN) with Sociological Orbits" - Poster at INFOCOM '06, Barcelona, Spain 2006 (April)  Joy Ghosh, Sumesh J. Philip, Chunming Qiao, "Sociological Orbit aware Location Approximation and Routing in MANET" - Proceedings of IEEE Broadnets, Boston, MA, 2005 (October)  Joy Ghosh, Sumesh J. Philip, Chunming Qiao, "Poster Abstract: Sociological Orbit aware Location Approximation and Routing (SOLAR) in MANET" - Poster at ACM International Symposium on Mobile Ad Hoc Networking and Computing, MobiHoc 2005 (May)  Joy Ghosh, Sumesh J. Philip, Chunming Qiao, "Acquaintance Based Soft Location Management (ABSLM) in MANET" - Proceedings of IEEE Wireless Communications and Networking Conference 2004 (March) Technical Reports  http://www.cse.buffalo.edu/~joyghosh/solar.html

59. Lab for Advanced Network Design, Evaluation and Research Thank You! Questions?

60. Lab for Advanced Network Design, Evaluation and Research Source Routing (DSR, LAR) Return

61. Lab for Advanced Network Design, Evaluation and Research Geographic Forwarding may help (nodes must know own location) Return

62. Lab for Advanced Network Design, Evaluation and Research Send request to all All pending acquaintances Few accepted request Time0: some nodes move out Time1: timeout terminates acquaintance Time2: some move back in Time3: some move out again Time4: timeout terminates acquaintance Forming & maintaining acquaintances Non AcqntncePending AcqntnceAccepted Acqntnce Return

63. Lab for Advanced Network Design, Evaluation and Research Querying Acquaintances Return

64. Lab for Advanced Network Design, Evaluation and Research Fairness in Delay Comparison return

65. Lab for Advanced Network Design, Evaluation and Research Expectation-Maximization (EM) Hub lists y (1) = “110011” y (2) = “110000” y (3) = “000011” y (4) = “101010” y (5) = “010101” Daily hub listCluster mean 2-D example view Initializations Weighted means ρ (1) = “0.7, 0.8, 0.2, 0.3, 0.7, 0.7” ρ (2) = “0.1, 0.3, 0.9, 0.8, 0.3, 0.2” Mixing proportions π = {π 1, π 2 } = {0.5, 0.5} r (i) j C 1 C 2 y (1) 0.90.1 y (2) 0.60.4 y (3) 0.60.4 y (4) 0.50.5 ……… ρ (1) = “0.9, 0.9, 0.1, 0.1, 0.9, 0.9” ρ (2) = “0.1, 0.1, 0.9, 0.9, 0.1, 0.1” π = {π1, π2} = {0.7, 0.3} return

66. Lab for Advanced Network Design, Evaluation and Research Problem complexity: #P-hard ! Valiant proved Conn 2 (G) to be #P-complete in 1979; we reduce it to our problem In directed graph D = (V,E) let p ij be all edge probabilities with source s and destination t; let c be the LCM of all denominators of p ij (c is polynomial in input size) If we have a procedure to compute Conn 2 (G) ≤ c’/c for any c’ ≤ c, we can compute Conn 2 (G) by simple binary search Our objective: find routing algorithm A, which finds delivery sub-graph D = G(A) to maximize Conn 2 (D) – a solution to this can be used to decide if Conn 2 (G) ≤ c’/c !! Add path with k edges to D to get G with Π k i=1 p i = c’/c + ε, where ε < 1/c Our aim:  find sub-graph H of G with |E(H)| ≤ k (edge constraint) Routing algorithm A returns  Upper part  Conn 2 (D) ≤ c’/c  Lower part  Conn 2 (D) > c’/c “A” can be used to decide if Conn 2 (G) ≤ c’/c “A” is at least as hard as Conn 2 (G) return

67. Lab for Advanced Network Design, Evaluation and Research Acquaintance A i has a Hub list H i = {h 1, h 2, …, h m } where h i is a Hub H = {H 1, H 2, …, H n } is the set of Hub lists covered by A 1, A 2, …, A n C = H1 U H2 U … U Hn is the set of all Hubs covered by A 1, A 2, …, A n Objective: find a minimum subset This is a minimum set cover problem – NP Complete We use the Quine-McCluskey optimization techniqueQuine-McCluskey Subset of acquaintances to query Return

68. Lab for Advanced Network Design, Evaluation and Research Quine-McCluskey optimization Acquaintance _ a Example: A = {1,2}, B = {2,3,4}, C = {1,3}  A, B, C are Prime acquaintances  B is an Essential Prime acquaintance Choose all the Essential Prime acquaintances first If any Hub is still uncovered, iteratively choose non-essential Prime acquaintances that cover the max number of remaining Hubs, till all Hubs are covered Return