Geometric Algorithms for Coverage in Wireless Sensor Networks Dr. Dinesh Dash Asst. Prof.NIT Patna

Outline of Talk Introduction Application of geometric algorithms in Sensor Network Geometric Algorithm for Coverage problem in sensor network Area Coverage Barrier Coverage Line Coverage

Introduction Sensor networks composed of a large number of sensor nodes, which are deployed to sense environmental parameters and send it to a sink.

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 station collator

Some Applications Sensor 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.

Applications of Geometric Algorithms in Sensor Network Routing Localization Coverage

Location Based Routing Assumption Every node knows its location and its neighbors’ locations The source knows the location of destination Greedy Forwarding A node always forwards the message to a neighbor whose Euclidean distance to the destination is smaller source destination

Greedy forwarding may fail The message reaches node x, no next hop can be selected for Greedy Forwarding, because both w and y are further away from D than x is.

Planer Graph and Face Routing helps

Localized ways to planerize a unit disk graph Gabriel Graph Relative Neighborhood Graph

Face Routing face face face face face source destination

Localization Determines physical / relative positions of sensor nodes in the network based on known information Essential for: The development of low-cost sensor networks for use in location-aware applications Geographic routing

Beacon Based Algorithm

Coverage Coverage problem measures how well a set of deployed sensors cover(sense) an area or a set of objects

Different measures of coverage is possible depending on its applications Area coverage, target coverage, barrier coverage, trap coverage, perimeter coverage, line coverage, breach and support etc.

Coverage Algorithms

Measures of Coverage: Examples Area coverage [thai:2008, bai:2005, huang:2005] Measures coverage of an area/region Can 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

Area Coverage Is the entire area covered? 1 8 2 6 3 4 5

Is the entire region k-covered? An area A is k-covered if all intersection points among the sensing circles and area boundaries are k-covered There are at most O(n2) intersection points Each intersection point can be verify in O(n) An area A is k-covered iff each sensor in A is k perimeter covered

1 Is the perimeter k-covered? 5 7 6 8 4 2 Perimeter coverage is modeled by set of intervals and can be verified in O(n log n ) 3 10 9 2П 4 3 2 1 5 6

Barrier coverage A 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 surrounded by deep trenches

Barrier formed by sensors L R Open barrier Closed barrier

Graph view of barriers One vertex for each sensor, two dummy vertices L and R for left and right boundaries in open barrier Edge between two vertices if the sensing regions of the two sensors intersect L R

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

Is a boundary k-barrier covered? Construct a graph G(V, E) V: sensor nodes, plus two dummy nodes L, R E: edge (u,v) if their sensing disks overlap Region is k-barrier covered iff there are k-disjoint paths between L and R R L

Line/path coverage Measures the degree of coverage of lines/paths Some variants Every 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

Path Coverage Maximal breach and support path [megerian:2005 ] Breach path A path that want to maintain distance from the sensors Support path A path that want to stay close to the sensors

Breach and Support Breach value of a path is the minimum distance of any point on the path from the closest sensor Support value of a path is the maximum distance of any point on the path to its closest sensor Breach value 18 5 Pf support value i Pi f

The maximal breach path -> Voronoi diagram The maximal support path -> Delaunay triangulation

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

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 path Find Euclidean minimum spanning tree of the set of sensors including initial and final points.

Line Coverage Problem

A line segment is said to be k-covered if it is sensed by k sensors 3-uncovered line segment

Definition Smallest k-covered line segment Given a sensor deployment, the minimum length line segment that intersects at least k sensors’ sensing regions Longest k-uncovered line segment Given a sensor deployment, the maximum length line segment that intersects at most k-1 sensors’ sensing regions Maximal 3-uncovered line segment Minimal 3-covered line segment

Problems Addressed Designed algorithms for finding smallest k-covered and longest k-uncovered line segment given a sensor deployment in a bounded rectangular region R for Axis-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 regions Arbitrary line segment starting from a given point in O((n+χ) log n) time and linear space ICDCN 2012(Poster)

Arbitrary line segments Smallest k-covered segment in O(χ2logn+n4) time and linear space Longest k-uncovered segment in O((χ2+n3)logn) time and linear space JPDC 2014

Algorithm for Smallest k-Covered Axis-Parallel Line Segment Overall approach Find the smallest k-covered horizontal line segment Find the smallest k-covered vertical line segment Choose the one with the minimum length among the two Only the algorithm for horizontal line segments discussed, the algorithm for vertical line segments is the same

Some Definitions Top end point Right-half circle Left-half circle Bottom-end point

Left-left intersection Left-right intersection Right-right intersection Note: Two equal radius circles can make only one left-left intersection and one right-right intersection

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-covered Horizontal segment Non Minimal length 3-covered Horizontal segment

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)

Approach Use plane sweep paradigm Horizontal sweep line moves from top to bottom of region Proved 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 intersection Keep track of all such minimal length k-covered horizontal segments created Determine the minimum length attained by each segment (using the mid-point events) Update the global minimum as necessary

Sweep-Line Touches the Top-Endpoint of a Circle [c0r,c2l] [c1r,c4l] [c4r,c6l] [c2r,c5l] c2 c5 c3 [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 length 3-covered segments, are [c0r,c2l], [c1r,c4l], [c2r,c5l], [c4r,c6l] As it touches top-endpoint of c3 Existing segments [c1r,c4l], [c2r,c5l] are deleted New segments [c1r,c3l], [c2r,c4l], [c3r,c5l] are created

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.

Sweep-Line Touches the Bottom-Endpoint of a Circle [c1r,c3l] [c2r,c4l] [c3r,c5l] [c4r,c6l] c2 c5 c3 [c2r,c5l] [c3r,c6l] Before touching bottom-endpoint of c4 , the set of minimal length 3-covered segments, are [c1r,c3l] , [c2r,c4l], [c3r,c5l] , [c4r,c6l] As it touches the bottom-endpoint of c4 Existing segments [c2r,c4l], [c3r,c5l] , [c4r,c6l] are deleted New segments [c2r,c5l], [c3r,c6l] are created

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.

Sweep-Line Crosses Right-Right Intersection of Two Circles [c2r,c3l] [c0r,c2l] c1 c0 [c3r,c5l] [c1r,c4l] c5 c3 c4 c2 [c1r,c3l] [c2r,c4l] Right-right intersection Before 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 & c2 Existing segments [c2r,c3l] & [c1r,c4l] are deleted New segments [c1r,c3l] & [c2r,c4l] are created

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.

Sweep-line touches left-right intersection of two circles [c0r,c2l] c3 c0 c1 [c1r,c3l] [c0r,c2l] [c1r,c3l] left-right intersection Before 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 created When sweep-line crosses any left-right intersection point no new k-covered intervals are created or deleted

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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