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.
collator
collator
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
source
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
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
3
4
6
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
Perimeter coverage is
modeled by set of intervals
and can be verified in O(n log n )
2
3
9
4
0
5
3
1
2
6
5
4
2П
10
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
L


R
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


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
L
R
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


i
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
Pi
Pf
support
value
f
a) The maximal breach path -> Voronoi diagram
b) 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-covered
line segment
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 kuncovered 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

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
Left-half circle
Right-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 ycoordinate 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
(xq,yq)
(xp,yp)
ymid = (yp+yq) /2
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 bottomendpoint, 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 TopEndpoint of a Circle
Sweep line
c0
r
l
[c0r,c2l] [c1 ,c4 ]
c1
[c1r,c3l] c2
c4
[c2r,c5l]
[c2r,c4l]
c3
[c3r,c5l]
[c4r,c6l]
c6
c5
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 BottomEndpoint of a Circle
Sweep line
c0
c1
[c3r,c5l]
[c2r,c4l]
[c1r,c3l]
c2
[c2r,c5l]
c4
[c4r,c6l]
c6
c5
c3
[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 kcovered segments are deleted, then exactly
x−1 new k-covered segments will be created.
Sweep-Line Crosses Right-Right
Intersection of Two Circles
Sweep line
c0
[c0r,c2l]
c1
[c2r,c3l]
[c1r,c4l]
[c1r,c3l]
c2
[c3r,c5l]
c3
c4
c5
[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 leftleft 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 kcovered segments are created, then exactly x
existing k-covered segments are deleted.
Sweep-line touches left-right
intersection of two circles
Sweep line
c0
[c0r,c2l]
[c0r,c2l]
c2
c1
[c1r,c3l]
[c1r,c3l]
c3
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 kcovered intervals are created or deleted
Algorithm Outline



Move the sweep line from top to bottom
Identify when it touches a top-endpoint, bottomendpoint,
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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Thanks!