Donghoon Shin, Saurabh Bagchi and Chih -Chun Wang

1. Donghoon Shin, SaurabhBagchi and Chih-Chun Wang Toward Optimal Sniffer-Channel Assignment for Reliable Monitoring in Multi-Channel Wireless Networks Dependable Computing Systems Lab (DCSL) School of Electrical and Computer Engineering Purdue University

2. Outline • Introduction: Passive Monitoring in Wireless Networks • Existing Works and Motivation • Problem Statement: Optimal Sniffer-Channel Assignment for Reliable Monitoring • Proposed Algorithms • Simulation Results • Conclusion

3. How to assign a set of channels to the sniffers’ radios so as to capture as large an amount of traffic as possible? Introduction • Passive monitoring in wireless networks • A set of sniffers are used to capture and analyze network traffic to estimate network conditions and performance • Sniffers are software or hardware devices that intercept and log packets • Such estimates are utilized for efficient network operation such as: • Resource management • Network configuration • Fault detection/diagnosis • Network intrusion detection • A major issue with passive monitoring in multi-channel wireless networks: “sniffer-channel assignment problem”

4. Existing Studies on Monitoring in Multi-Channel Wireless Networks • [Shin et al, MobiHoc’09]Optimal placement and channel assignment of sniffers in wireless mesh networks • [Chhetriet al, MobiHoc’10] Two models of sniffers that assume different capabilities of sniffers’ capturing traffic • [Aroraet al., INFOCOM’11] Trade-off between assigning sniffers’ radios to the channels known to be busiest based on the current knowledge, versus exploring channels that are under-observed • [Aroraet al., GLOBECOM’11], [Shin et al, INFOCOM’12]Distributed algorithms for optimal sniffer-channel assignment

5. Motivation and Solution Approach • All previous works assumed that sniffers are perfect • In practice, sniffers may probabilistically stop functioning and/or generate erroneous reports on monitoring due to: • Poor reception (due to packet collisions or poor channel conditions) • Compromise by an adversary • Operational failure • Sleep mode for saving energy • In this paper, we allow for imperfect sniffers • For accurate and reliable monitoring, we provide sniffer redundancy to each node • That is, each node has to meet a coverage requirement, i.e., the minimum number of sniffers required to reliably monitor the node

6. Notation & Terminology • S: Set of sniffers • N: Set of nodes • Each node’s radio is tuned to a specific wireless channel • C: Set of available wireless channels • wn: Weight assigned to node n • Captures various application-specific objectives of monitoring • rn: Coverage requirement assigned to node n • Minimum number of sniffers required to reliably monitor node n • Ks,c: Coverage-set of sniffer s on channel c • Contains the nodes that can be overheard by sniffer s operating on channel c • Sniffer-channel assignment: A collection of coverage-sets that include only one coverage-set for each sniffer

7. Channel Assignment for Reliable Monitoring • Full-Coverage Reliable Monitoring (FCRM): Find a sniffer-channel assignment that covers all nodes in the network • A node is covered if it is overhead by at least rn sniffers • Theorem 1: • Complexity grows exponentially with the number of sniffers • Maximum-Coverage Reliable Monitoring (MCRM): Find a sniffer-channel assignment that maximizes the total weight of nodes being covered • Corollary 1: FCRM is NP-hard, even for |C| = 2 and rn = 2 for some node n MCRM is NP-hard, even for |C| = 2 and rn = 2 for some node n

8. For any ε> 0, it is NP-hard to solve MCRM within a factor of 7/8 + ε of the maximum coverage, even for |C| = 2 and rn = 2 for all n Channel Assignment for Reliable Monitoring • Corollary 2: • Theorem 2: • Intuitively, submodularity is a diminishing-return property • Submodularity allows to efficiently find provably (near-)optimal solutions • Similar to convexity in continuous optimization • Known that non-submodular functions are difficult to deal with For MCRM with rn = 1 for all n, the weight function w is submodular. However, MCRM with rn ≥ 2 for some n, the weight function w is not submodular.

9. Greedy Approach • Naïve greedy algorithms: at each iteration, pick one coverage-set that maximizes: • Coverage improvement • Sum of the weights of the hitherto uncovered nodes • Look-ahead greedy algorithms: consider combinations of multiple coverage-sets at each step • Look-t-steps-ahead greedy algorithm • At each step, picks one coverage-set through the following procedure: • Find a collection of t + 1 coverage-sets that achieve the maximum coverage improvement for the current step and the next t steps • Among the coverage-sets in the selected collection, picks one coverage-set that maximizes coverage improvement at the current step • t-sniffers-at-one-step greedy algorithm • At each step, picks a collection of at most t coverage-sets that maximize the per-sniffer coverage improvement

10. Relaxation-and-Rounding Approach • Steps for relaxation-and-rounding algorithms to solve MCRM • Formulate MCRM into an integer program (IP) • Transform the IP into a relaxed program by removing the integer constraints • Find as tight a relaxed program as possible, while keeping the relaxed program solvable in polynomial time • Solve the relaxed program to find the optimal fractional solution • Round the non-integer values from Step 3 to obtain an integer solution feasible for the original IP • In rounding, the goal is to minimize the degradation of the quality of the resulting integer solution • Two relaxations devised • Linear Program (LP) relaxation • SemiDefinite Program (SDP) relaxation tighter relaxation

11. Relaxation-and-Rounding Approach • Two rounding algorithms designed • Randomized Rounding Algorithm (RRA) • Probabilistically round the optimal LP/SDP solution {ys,c*} such that: whereYs,c is the integer value resulted from rounding • Greedy Rounding Algorithm (GRA) • At each iteration, rounds (at least) one fractional value as the followings: • For each sniffer-channel pair (s, c) whose value is not rounded to an integer, adjust the fractional values of the sniffer s according to: • Find the sniffer-channel pair (s#, c#) whose associated adjusted values achieve the maximum coverage improvement • Update the fractional values of sniffer s# to the adjusted values ys,c* = 1indicates that sniffer stunes its radio to channel c P(Ys,c = 1) = ys,c*

12. Simulation Settings • Two metrics • Coverage • Running time • Two kinds of networks • Random network: Nodes are randomly deployed in the network with a uniform distribution • Scale-free network: Nodes are deployed such that the distribution of the nodes with degree d follows a power law in a form of d-r • Parameter settings • Number of nodes: 40 • Number of channels: 3 • All nodes have the same weight of one (i.e., wn = 1) and the same coverage requirement of two (i.e., rn = 2)

13. Coverage in Random Network ILP optimum (maximum coverage) Rounding by GRA Look-ahead greedy algorithms Naïve greedy algorithm-2(which picks the coverage-set that achieves the maximum total weight of the uncovered nodes) Rounding by RRA Naïve greedy algorithm-1(which picks the coverage-set of the maximum coverage improvement) • Naïve greedy algorithm-2 shows reasonable coverage, while naïve greedy algorithm-1 shows poor coverage • SDP + GRA and LP + GRA show coverage comparable to the maximum achievable coverage (i.e., at least 95% and 94% of maximum coverage) • Look-ahead greedy algorithms show reasonably good performance (at least 92% of maximum coverage) • After rounding, GRA maintains the solution quality closer to the maximum coverage, while RRA results in the degradation of the solution quality

14. Coverage in Scale-free Network Gap from the upper bound by LP relaxation Gap from the upper bound by SDP relaxation SDP-relaxation based algorithms LP-relaxation based algorithms • SDP-based algorithms achieve a higher coverage improvement (by 2~5%) compared to LP-based algorithms, than in random network • SDP relaxation shows a noticeable improvement on the upper bound of the maximum achievable coverage (by 4~7%)

15. LP-relaxation based algorithms Running Time in Random Network SDP-relaxation based algorithms • LP-relaxationandrounding algorithms show the fastest running time Look-ahead greedy algorithms y-axis for the other algorithms y-axis for look-ahead greedy algorithms(5x left y-axis) • SDP-relaxation and rounding algorithms show reasonably fast running time CPU: 2.4 GHz Memory: 4 GB Bus: 1.07 GHz • Look-ahead greedy algorithms show the slowest running time • Grow rapidly as the number of sniffers increases • Running time of the t-sniffers-at-one-step greedy algorithm is almost half of the running time of the look-t-steps-ahead greedy algorithm

16. Summary of Simulation Results • SDP + GRA achieves the highest coverage close to the maximum coverage, but shows a (relatively) slow running time • Favored, especially, for monitoring applications where a higher coverage is more emphasized (e.g., critical security monitoring) • LP + GRA attains the coverage comparable to the coverage of the SDP + GRA, and also shows a fast running time • A good compromise between coverage and running-time • Favored for monitoring applications requiring fast running-time (e.g., monitoring dynamic network environments)

17. Conclusion • Studied the optimal sniffer-channel assignment problem for reliable monitoring in multi-channel wireless networks • Showed that the problem is fundamentally differs from the previously studied problems that assume perfect sniffers and thus do not need to consider sniffer redundancy • Proposed various approximation algorithms based on two basic approaches: • Greedy • Relaxation and rounding • Present a comparative analysis of the proposed algorithms through simulations

18. Thank You Contact Info: Donghoon Shin(donghoon.shin.2@asu.edu) Questions?