Presentation on theme: "Geometric Algorithms for Coverage in Wireless Sensor Networks"— Presentation transcript:
1 Geometric Algorithms for Coverage in Wireless Sensor Networks Dr. Dinesh DashAsst. Prof.NIT Patna
2 Outline of Talk Introduction Application of geometric algorithms in SensorNetworkGeometric Algorithm for Coverage problem in sensor networkArea CoverageBarrier CoverageLine Coverage
3 IntroductionSensor networks composed of a large number of sensor nodes, which are deployed to sense environmental parameters and send it to a sink.
4 Example of Sensor Network Sensing devices sense environment, do some local processing and send the sensed data to a base station directly or indirectly.Base stationcollator
5 Some ApplicationsSensor networks have been used for habitat monitoring [mainwaring:2002], agriculture monitoring [bilsa:2009], structure monitoring, forest fire detection, object tracking [tsaia:2007], military application, etc.
7 Applications of Geometric Algorithms in Sensor Network RoutingLocalizationCoverage
8 Location Based Routing AssumptionEvery node knows its locationand its neighbors’ locationsThe source knows thelocation of destinationGreedy ForwardingA node always forwardsthe message to a neighborwhose Euclidean distanceto the destination is smallersourcedestination
9 Greedy forwarding may fail The message reaches node x, no next hop can be selected for GreedyForwarding, because both w and y are further away from D than x is.
11 Localized ways to planerize a unit disk graph Gabriel GraphRelative Neighborhood Graph
12 Face Routingfacefacefacefacefacesourcedestination
13 LocalizationDetermines physical / relative positions of sensor nodes in the network based on known informationEssential for:The development of low-cost sensor networks for use in location-aware applicationsGeographic routing
18 Measures of Coverage: Examples Area coverage [thai:2008, bai:2005, huang:2005]Measures coverage of an area/regionCan be full coverage (every point of the area is sensed by at least one sensor) or partial (some fraction of the area is covered)Target coverage [cardei :2005]Given a set of target points, each point in the set is sensed by at least one (or k) sensor
19 Area CoverageIs the entire area covered?1826345
20 Is the entire region k-covered? An area A is k-covered if all intersection pointsamong the sensing circles and area boundariesare k-coveredThere are at most O(n2) intersection pointsEach intersection point can be verify in O(n)An area A is k-covered iff each sensor in A is k perimeter covered
21 1Is the perimeterk-covered?576842Perimeter coverage ismodeled by set of intervalsand can be verified in O(n log n )31092П432156
22 Barrier coverageA rectangular belt region is said to be 1-barrier covered by the deployed sensors if all the crossing paths must intersect at least one sensor’s sensing region [kumar:2005]forts were surroundedby deep trenches
23 Barrier formed by sensors LROpen barrierClosed barrier
24 Graph view of barriersOne vertex for each sensor, two dummy vertices L and R for left and right boundaries in open barrierEdge between two vertices if the sensing regions of the two sensors intersectLR
25 For open barrier a path from L to R in the graph ensures barrier coverage For closed barrier a non-contractible cycle ensures barrier coverage
26 Is a boundary k-barrier covered? Construct a graph G(V, E)V: sensor nodes, plus two dummy nodes L, RE: edge (u,v) if their sensing disks overlapRegion is k-barrier covered iff there are k-disjoint paths between L and RRL
27 Line/path coverage Measures the degree of coverage of lines/paths Some variantsEvery point of a line segment is under the sensing range of at least one sensor [harada:2009]fraction of coverage [harada:2009] : the fraction of the whole path that is within the sensing range of some sensors
28 Path Coverage Maximal breach and support path [megerian:2005 ] Breach pathA path that want to maintaindistance from the sensorsSupport pathA path that want to stayclose to the sensors
29 Breach and SupportBreach value of a path is the minimum distance of any point on the path from the closest sensorSupport value of a path is the maximum distance of any point on the path to its closest sensorBreach value185Pfsupport valueiPif
30 The maximal breach path -> Voronoi diagram The maximal support path -> Delaunay triangulation
31 How to find maximal breach path? Theorem : At least one Maximal Breach Path must lie on the line segments of the bounded Voronoi diagram formed by the locations of the sensors.Apply binary search and breadth first search on the weighted edge of the Voronoi diagram to find a maximal breach path
32 How to find maximal support path? Theorem : At least one maximal Support Path must lie on the edges of the Delaunay triangulation.Apply binary search and breadth first search on the weighted edge of Delaunay triangulation to find a maximal support pathFind Euclidean minimum spanning tree of the set of sensors including initial and final points.
34 A line segment is said to be k-covered if it is sensed by k sensors 3-uncoveredline segment
35 Definition Smallest k-covered line segment Given a sensor deployment, the minimum length line segment that intersects at least k sensors’ sensing regionsLongest k-uncovered line segmentGiven a sensor deployment, the maximum length line segment that intersects at most k-1 sensors’ sensing regionsMaximal 3-uncoveredline segmentMinimal 3-coveredline segment
36 Problems AddressedDesigned algorithms for finding smallest k-covered and longest k-uncovered line segment given a sensor deployment in a bounded rectangular region R forAxis-parallel line segments in O((n+ χ) log n) time and linear space, where n is the number of sensors and χ is the number of intersections between the circles corresponding to the sensor’s sensing regionsArbitrary line segment starting from a given point in O((n+χ) log n) time and linear spaceICDCN 2012(Poster)
37 Arbitrary line segments Smallest k-covered segment in O(χ2logn+n4) time and linear spaceLongest k-uncovered segment in O((χ2+n3)logn) time and linear spaceJPDC 2014
38 Algorithm for Smallest k-Covered Axis-Parallel Line Segment Overall approachFind the smallest k-covered horizontal line segmentFind the smallest k-covered vertical line segmentChoose the one with the minimum length among the twoOnly the algorithm for horizontal line segments discussed, the algorithm for vertical line segments is the same
39 Some Definitions Top end point Right-half circle Left-half circle Bottom-end point
40 Left-left intersection Left-right intersectionRight-right intersectionNote: Two equal radius circles can make only one left-left intersection and one right-right intersection
41 Minimal length k-covered horizontal segment: A k-covered horizontal line segment/interval such that no subinterval of it is k-covered.Minimal length 3-coveredHorizontal segmentNon Minimal length 3-coveredHorizontal segment
42 Midpoint between Two Circles Midpoint between two circles Cp and Cq is the y-coordinate where the horizontal distance between the two circles is minimum. For unit circle, it is at (yp+yq)/2 where yp and yq are the y–coordinates of the centers of Cp and Cq respectively(xp,yp)ymid = (yp+yq) /2(xq,yq)
43 Approach Use plane sweep paradigm Horizontal sweep line moves from top to bottom of regionProved that minimal length k-covered horizontal line segments are only created or deleted when the sweep line touches a top-endpoint, a bottom-endpoint, or a left-left or right-right intersectionKeep track of all such minimal length k-covered horizontal segments createdDetermine the minimum length attained by each segment (using the mid-point events)Update the global minimum as necessary
44 Sweep-Line Touches the Top-Endpoint of a Circle [c0r,c2l][c1r,c4l][c4r,c6l][c2r,c5l]c2c5c3[c1r,c3l][c2r,c4l][c3r,c5l]At the top end point of C3 [c1r,c4l], [c2r,c5l] segments are deleted. At the same time [c1r,c3l], [c2r,c4l], [c3r,c5l] new segments are generated.Before touching top-endpoint of c3 , the set of minimal length3-covered segments, are [c0r,c2l], [c1r,c4l], [c2r,c5l], [c4r,c6l]As it touches top-endpoint of c3Existing segments [c1r,c4l], [c2r,c5l] are deletedNew segments [c1r,c3l], [c2r,c4l], [c3r,c5l] are created
45 Lemma : When the sweep-line touches the top-endpoint of a circle, at most k new minimal length k-covered segments can be created on the sweep line. Moreover, if x (x>0) k-covered segments are created, then exactly x−1 existing k-covered segments will be deleted.
46 Sweep-Line Touches the Bottom-Endpoint of a Circle [c1r,c3l][c2r,c4l][c3r,c5l][c4r,c6l]c2c5c3[c2r,c5l][c3r,c6l]Before touching bottom-endpoint of c4 , the set of minimal length3-covered segments, are [c1r,c3l] , [c2r,c4l], [c3r,c5l] , [c4r,c6l]As it touches the bottom-endpoint of c4Existing segments [c2r,c4l], [c3r,c5l] , [c4r,c6l] are deletedNew segments [c2r,c5l], [c3r,c6l] are created
47 Lemma : When the sweep-line touches the bottom-endpoint of a circle, at most k existing k-covered segments are deleted from the sweep line. Moreover, if x (x>0) existing k-covered segments are deleted, then exactly x−1 new k-covered segments will be created.
48 Sweep-Line Crosses Right-Right Intersection of Two Circles [c2r,c3l][c0r,c2l]c1c0[c3r,c5l][c1r,c4l]c5c3c4c2[c1r,c3l][c2r,c4l]Right-right intersectionBefore touching right-right intersection between c1 & c2 , the set of minimal length 3-covered segments are [c0r,c2l] , [c2r,c3l], [c1r,c4l] , [c3r,c5l]As it crosses the right-right intersection between c1 & c2Existing segments [c2r,c3l] & [c1r,c4l] are deletedNew segments [c1r,c3l] & [c2r,c4l] are created
49 Sweep-Line Touches Left-Left or Right-Right Intersection Lemma : When the sweep-line touches the left- left or right-right intersection point between two circles then it creates at most two new k-covered segments. Moreover, if x (0 ≤ x ≤ 2) new k- covered segments are created, then exactly x existing k-covered segments are deleted.
50 Sweep-line touches left-right intersection of two circles [c0r,c2l]c3c0c1[c1r,c3l][c0r,c2l][c1r,c3l]left-right intersectionBefore touching left-right intersection between c1 & c2 , the set of minimal length 3-covered segments are [c0r,c2l] , [c1r,c3l]As it touches the intersection no existing segments are deleted as well as no new segments are createdWhen sweep-line crosses any left-right intersection point no new k-covered intervals are created or deleted
51 Algorithm Outline Move the sweep line from top to bottom Identify when it touches a top-endpoint, bottom-endpoint, left-left intersection, right-right intersection or the mid-point where the length of a k-covered line segment is minimized (events)Keep track of the minimal length k-covered segments formed, deleted, and their minimum lengths on each such event
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