Detecting Cuts in Sensor Networks

1. Detecting Cuts in Sensor Networks Subhash SuriUC Santa Barbara (Joint work with N. Shrivastava and C. Toth)

2. Sensor Networks • Wirelessly networked devices that sense, compute, actuate. • Instrumentation of real world. • Applications in surveillance, tracking, habitat monitoring. • Virtual "eyes and ears" for remote monitoring.

3. Motivation • Operational conditions often severe and hostile. • Devices physically exposed and more vulnerable than wired Internet. • Communication medium fragile. • How can we ensure that any significant damage to our remote eyes-and-ears is quickly and reliably detected?

4. Network Cuts • We focus on a particular kind of failure:network partition by a linear cut. • e-cut: network partition where e-fraction of sensors cut off from the base station. • Linear e-cut: partition defined by a line. • Two equivalent views: physical disabling of sensors, or communication disruption along the cut. Linear e -cut

5. Resource Constraints • Continuously polling each and every sensor is infeasible. • Communication is the largest source of power drain. • Sensors have limited on-board power. • Bandwidth is also limited, as multi-hop routing requires several other nodes to forward a packet. • Need for a low overhead, low latency, high confidence, cut detection scheme.

6. A Model of Monitoring • The base station knows the locations of all sensors. • A small subset of nodes is designated as sentinels. • Each live sentinel node sends a heartbeat message to the base station per epoch. • No message from the dead sentinels. • The live/dead signature vector at the base station is sufficient to decide if an e-cut has occurred. monitoring using Sentinel Set Sentinels

7. Related Work • The sentinel model is inspired by work of Kleinberg. • His setting is a wired network: detect an e-cutthat results from cutting off at most k edges in the graph. • Designate O(poly(k), 1/e) nodes as sentinels, who engage in pairwise communication.

8. Geometric Cuts • Spatially correlatedcuts are more natural in sensor networks. • Geometric cut complexity (linear, circular etc.) more meaningful than edge failures. • A major drawback of Kleinberg scheme is the presence of False Positive. • With high probability, it catches all e-cuts, but many of the reported cuts can be quite small (False Alarms). • In remotely deployed sensor networks, checking a false alarm is expensive. A sensor field Linear e-cut

9. Sampling Methods • By the e-net theory, an O( 1/e log 1/d ) size random sample of nodes can act as a detection set. • But this scheme has a large rate of false positives. • e-approximation sample eliminates the false positives, but it requires too large a sentinel set. • For instance, with e = 0.1 and d = 0.05, the e-approximation has size is at least 10,000

10. e-cuts and approximation guarantees • Cuts must be defined as a fraction of the network size. • Otherwise, catching all cuts of size k requires at least n/k detection nodes. • Sharp threshold impossible as well: catch all e-cuts, but no cuts of size < en. • Such a sharp cutoff requires at least n/2 detection nodes. • Our Result: A detection set of size O(1/e) that detects every e-cut, and every reported cut has size at least en/2. 0 1 n-1 2 n-2 k

11. L Geometry of Network Cuts • Think of sensors as points in the plane. • A linear cut is a line that partitions the point set. • The point-line duality: point (a,b)  line (y = ax - b) L* • It preserves above/below relationship: point p above line L  point L* above line p*.

12. L Geometry of Network Cuts • A line L is an e-cut if the dual point L* lies above en dual lines. • Thus, the set of all linear e-cuts is the region above the en level (symmetrically, below the (n - en) level). • Imagine a polygonal curve (separator) made up of dual lines that lies between en/2 and en levels. • Then, the primal points corresponding to these lines form a detection set. • Two issues: • Is there such a separator using just a few lines? • We don't know the cut line L. How will we decide that L* lies above the separator? L*

13. Complexity of a Separator • A level can have W(nlogn) segments. • Average complexity of a level is Q(n). • We want a separator of size roughly 1/e (independent of n!). • We construct a zig-zag path between levels en/2 and en. • Start at left, follow the edge until top level hit. • Reflect and follow until bottom level hit, reflect and continue. • Claim: At least one zigzag separator path has O(1/e) segments. It lies between en/2 and en levels.

14. Complexity of Separator • Zig zag paths are edge-disjoint. • Total number of bends at most the number of vertices in the top and bottom levels. • We choose two levels that each have O(n) vertices, and are W(en) levels apart. • By the averaging argument, at least one zigzag path will have O(1/e) segments. • The dual points of these lines are our sentinels.

15. Detecting Cuts from Sentinels • Base station computes the dual of the sentinel nodes only. (The arrangement formed by the separator lines.) • Base station learns the dead/alive bit of each sentinel. • For each dead (live) sensor, we know that the dual of the cut L* must be above (below) the line. • In this example, w1, w3, w4 are dead; others alive. • The intersection of these halfspaces gives a convex cell. • If this cell is above the separator, we declare an en. • Otherwise, it's a false alarm.

16. Sample Sentinel Sets US-census data N = 5000,  = 0.01 No. of Sentinels = 12 Uniform N = 5000,  = 0.01 No. of sentinels = 14 Non-uniform N = 5000,  =0.01 No. of sentinels = 14

17. Scalability with Network Size e = 0.01

18. Scalability with e N = 5000

19. k-random directions Sentinels vs. Random Sampling • Two natural random sampling schemes. • Choose as many random nodes as our sentinel schemes. • Random Sampling • k nodes chosen uniformly at random. • Report if more than ek sentinels dead. • Radial Sampling • k directions chosen at random, and for each choose the en extreme vertex. • Report if any of these k dies.

20. False Positives • Generated 250 cuts by picking points randomly between levels 1 and en/2 • These cuts are all below the appr threshold, and should not be reported. • Random and radial sampling schemes misreport some of them as cuts. No False Positives in Sentinel Set

21. False Negatives • 250 cuts by picking points randomly between level en and 2en • These are all above the approximation threshold, and so should be reported. • Random and radial sampling schemes failed to report some of them as cuts. No False Negatives in Sentinel Set

22. Extensions, Future Directions • Robustness to random sensor failures • Make c independent sentinel sets, take the majority vote to detect cuts • Fast randomized algorithm to find sentinel set • Future work • More flexible cut definitions: • More complex shape. • Majority but not all sensors destroyed. • Distributed detection.