Minimalism in Sensor Networks Subhash Suri UC Santa Barbara and ETH Zurich
Moving from Big to Small Today: Few, big, powerful, global sensors Tomorrow: many small, weak, local sensors Pervasive computing, sensing, monitoring, actuation. Challenge: compose global picture from local data
Some Applications • Structural monitoring (civilian, industrial infrastructure faults) • Agriculture (soil moisture, pesticide levels) • Environmental monitoring (wildfires, hazmat) • Habitat monitoring (Great Duck Island, UCB Redwoods) • Surveillance (target tracking, border policing) How to achieve large scale with mote-caliber devices
Scale Requires Minimalist Design • Scaling in Space (size) – Sensors have small coverage area (e. g. , bio or chemical) – Large areas must be covered – Large deployments must be automated • Minimalist network protocols – Location-based naming and addressing – Geographical routing (e. g. GPSR, GLIDER) A B (x, y)
Scaling in Time • Sensor nodes have limited battery life – On/off scheduling and power management – Need minimalist models of energy consumption – Processing/communication tradeoffs • transmit/compute cost ratio > 1000 – Tiny. DB, Cougar style in-network processing – Lightweight algorithms for lifetime maximization • Today’s Talk [IPSN 2005] – Minimalist self-monitoring – Detect large network failures – Joint work with Shrivastava and Toth
Scaling in Functionality • Need minimalist sensing models – Small, inexpensive, noisy, failure-prone micro-sensors – Simple broadly applicable architectures – Fundamental limits of performance • Today’s Talk [ACM Sen. Sys ‘ 06] – Tracking with binary proximity sensors – Sensors detect presence or absence of target – How well can a target be tracked? – Joint work with Shrivastava, Mudumbai, Madhow
Monitoring the Network Health • Deployment conditions for sensor can be harsh and adversarial: need to remotely monitor the overall health of the network. • Continuously monitoring state of each sensor too expensive: need lightweight mechanisms. • Focus on significant damage to the network: large network disconnection. • -cut: partition where fraction of sensors are cut off from the rest. • Sensors embedded in physical space, so significant failures often spatially correlated.
Linear Cuts and a Minimalist Model • Linear -cut: partition by a line. Base Station • Base station: a distinguished node. • Assume linear cut disconnects -fraction of sensors from the base. • Designate a small number of nodes as sentinels, and track only their health. • Minimalist Model: – Each sentinel sends to the base station a bit to indicate it is alive; absence of bit indicates death. – Separation of routing from the monitoring. • How many sentinels needed to detect every linear -cut? A linear cut
Sampling and VC-dimension • O(1/ log 1/ ) size random sample is an -net with prob 1 -. • A related problem studied by Kleinberg: detect -cuts in wired network caused by failure of k edges or nodes. • Kleinberg shows this set system has VC dimension poly(k), and so number of sentinels is poly(k, 1/ ). • Improved bounds and variations in A. Gupta and Fakcharoenphol. • However, these methods give 1 -sided guarantees: every -cut is detected, but not every cut found is necessarily an -cut. • False positives in (remote) sensor networks can be a major nuisance. • The -approximation is not scalable: requires (d/ log 1/ ) nodes. • Is it possible to do scalable, minimalistic monitoring with the 2 -sided guarantee?
Desiderata • Cuts must be defined as a fraction of the network size. • Otherwise, catching all cuts of size k requires at least n/k sentinel nodes. • Sharp threshold impossible as well: catch all -cuts, but no cuts of size < n. • Such a sharp cutoff requires at least n/2 sentinel nodes. n-2 n-1 0 1 2 k
Main Result • Theorem: A sentinel set of size O(1/ ) that deterministically detects every cut, and every reported cut has size at least n/2.
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 q (1, 2) L p* q* p (1, 0 L* ) • It inverts the above/below relationship: point p above line L <-> point L* above line p*
Geometry of Network Cuts • L is an -cut if the dual point L* lies above n dual lines: Cut – set of all linear -cuts is the region above the n level (symmetrically, below the (n - n) level). • Imagine a polygonal curve (separator) made up of dual lines that lies between n/2 and n levels: – primal points corresponding to these lines form a sentinel 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* Level 0 L
Complexity of a Separator • A level can have size (nlogn). • Average complexity of a level is (n). • We want a separator of size roughly 1/ (independent of n!). • Fact: Total size of first k levels is O(nk). • We can choose two levels a and b s. t. – Each has size O(n), and – |b - a| ( n).
Complexity of Separator • Construct a zig-zag path between levels a and b: – Start at left, follow the edge until top level hit. – Reflect and follow until bottom level hit, reflect and continue. • These paths are edge-disjoint. • Total number of bends at most the number of vertices in the top and bottom levels. • By the pigeon-hole principle, at least one of the O( n) paths has O(1/ ) segments. • The dual points of these lines are our sentinels.
Detecting Cuts from Sentinels • Base station stores the arrangement formed by the separator lines • For each dead (live) sensor, we know that L* must be above (below) the line. • Intersection of these halfspaces a convex cell. • If this cell is above the separator, we declare an cut. • Otherwise, it's a false alarm. • In this example, w 1, w 3, w 4 are dead; others alive.
Cut Detection Guarantee • Separator lies below level n, so if intersection cell below it, must be smaller than a cut. • Separator lies entirely above level n so if intersection cell above it, must be at least a ( cut.
Simulation Results Non-uniform Uniform N = 5000, = 0. 01 No. of sentinels = 14 US-census data N = 5000, = 0. 01 No. of sentinels = 14 No. of Sentinels = 12
Scalability with Network Size = 0. 0
Scalability with N = 5000
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 k sentinels dead. • Radial Sampling – k directions chosen at random, and for each choose the en extreme vertex. – Report if any of these k dies. k-random directions
False Positives • Generated 250 cuts by picking points randomly between levels 1 and n/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
False Negatives • 250 cuts by picking points randomly between level n and 2 n • 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
Target Tracking with Binary Sensors • Minimalist model – Single bit output: presence/absence of target. – No information about position, distance, angle etc. – Idealization: perfect detection, circular range. – Simple, broadly applicable, robust model – Appropriate for large-scale deployments Ex. Scale project at OSU) • Fundamental limits of network sensing – Spatial resolution – Minimal path descriptions – Efficient geometric algorithms (e. g. ,
The Geometry of Binary Sensing Sensor Outputs Target Path Localization patches
Target Localization • Sensing output is a binary vector. Ex. F 2 = (1, 1, 0). • Each vector localizes the target to a localization patch. • The accuracy (max error) of localization is function of the size of localization patches. • What are the minimalist parameters to study this?
Tracking Resolution Theorem • sensor density (#sensors per unit area) • R: sensing radius • Theorem: If sensors have sensing radius R and the field has sensor density then the target can be tracked with spatial resolution (1/ R and this is the best possible. • Cor: In d-space, the resolution is O(1/ Rd-1)
Upper Bound on Attainable Resolution • Consider 2 concentric circles C 1 (radius R) and C 2 (radius 2 R), with center x. • Only sensors in C 2 can detect a target in C 1. • Assume C 2 has avg sensor density, so at most N = (4 R 2) sensors in it. • N sensing ranges form N 2 -N+2 patches. • At least one of these patches must have area >= ( R 2)/N 2 = (1/ 2 R 2). • Worst-case localization accuracy is the diameter of this patch.
Achieving the Resolution • Uniform placement of sensors in a grid achieves the resolution (1/ R). • Geometric and probabilistic analysis shows the same bound for uniform random distribution.
Localizing a Trajectory • Target localized to a time-ordered sequence of patches. • Any path inside this “tube” is within the resolution guarantee. • What is a good representative path?
Occam. Track: Minimal Representation • Use Occam’s principle of minimal representation. • Geometric algorithm computes polygonal path of minimum number of line segments through the resolution tube.
Spatial Low Pass Filtering • Sensors act like a low pass filter. • Local rapid variations invisible. • Estimating velocity: only average across patches. • Target localized to a 1 -dim arc at sensor boundary crossings. • Interpolate the velocity across multiple patches. • Velocity estimation reliable only over long path segments • Minimal representation helps again!
Velocity Estimation Theorem: To achieve relative velocity error , length of approximating segments must be L is the spatial resolution (patch size). For 10% error, L 5 ; for 5% error, L 6. 32
Simulation Results: Path Representation Weighted Centroid Output (Kim et al, IPSN 2005) 1000 vertices Occam. Track Output 50 vertices
Velocity Estimation
Fundamental Resolution Limits Theoretical resolution attained by both regular and random deployment
Lab-Scale Mote Experiment Non-ideal sensor response Occam. Track with ideal sensing Particle Filter + Geometric
Research Problems • Network monitoring – More general failure models – Large but not total failure • Location based routing – 3 -dimensional networks • Tracking – Beyond idealized sensing – Multiple targets
Sensor Nodes: Motes ØLightweight tiny devices, run on batteries Mica motes with light, acoustic, acceleration sensors
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