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

2. 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

3. Introduction
Sensor 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 station
collator

5. 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.

7. Applications of Geometric Algorithms in Sensor Network
Routing
Localization
Coverage

8. 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

9. 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.

10. Planer Graph and Face Routing helps

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

12. Face Routing
face
face
face
face
face
source
destination

13. 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

14. Beacon Based Algorithm

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

16. 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.

17. Coverage Algorithms

18. 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

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

20. 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

21. 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

22. 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

23. Barrier formed by sensorsL R
Open barrier
Closed barrier

24. 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

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, 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

27. 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

28. 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

29. 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

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

33. Line Coverage Problem

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

35. 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

36. 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)

37. 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

38. 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

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

40. 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

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-covered
Horizontal segment
Non Minimal length 3-covered
Horizontal 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 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

44. 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

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] 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

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] 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

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] 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

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

52. References
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56. Thanks!