Communication-constrained p-Center Problem
for Event Coverage in Theme Parks
Gürkan Solmaz*, Kemal Akkaya†, Damla Turgut*
*Department of Electrical Engineering and Computer Science
University of Central Florida - Orlando, FL
†Department of Computer Science
Southern Illinois University, Carbondale, IL
December 11, 2014
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
1 / 19
Outline
1
Introduction
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
2 / 19
Outline
1
Introduction
2
Preliminaries
Background on p-center problem
WSN model
Theme park modeling
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
2 / 19
Outline
1
Introduction
2
Preliminaries
Background on p-center problem
WSN model
Theme park modeling
3
Event coverage with communication-constrained p-center problem
Motivation
Problem formulation
Proposed approach
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
2 / 19
Outline
1
Introduction
2
Preliminaries
Background on p-center problem
WSN model
Theme park modeling
3
Event coverage with communication-constrained p-center problem
Motivation
Problem formulation
Proposed approach
4
Simulation study
Simulation setup and metrics
Performance results
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
2 / 19
Outline
1
Introduction
2
Preliminaries
Background on p-center problem
WSN model
Theme park modeling
3
Event coverage with communication-constrained p-center problem
Motivation
Problem formulation
Proposed approach
4
Simulation study
Simulation setup and metrics
Performance results
5
Conclusion
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
2 / 19
Introduction
Theme parks are crowded regions and have security vulnerabilities
Events: incidents such as violence, robbery or emergency
The aim is to cover events by deploying a WSN with mobile sinks
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Sensors collect information regarding the events and relay to mobile
sinks
Mobile sinks: Limited number of electric safety vehicles controlled by
humans
Crowdsensing via smart phones can also be applied to collect
information
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
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Introduction: Problem
The problem reduces to the effective placement of mobile sinks
Each event should be covered in the minimal time by sending the
closest sink
The problem is modeled as a vertex p-center problem (on a weighted
graph)
Original p-center problem: minimizing the maximum travel time for
each sink
We have additional constraint for communication among the mobile
sinks
The new variant: communication-constrained p-center problem
Solmaz, Akkaya, Turgut (UCF,SIU)
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Preliminaries: Background on p-center problem
p-center problem consists of p facilities and clients (i.e., vertices)
Each client is assigned to a facility
The problem is to minimize maximum distance between a client and
the facility assigned to it
Two variants: absolute and vertex p-center
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Facilities can be located anywhere in absolute p-center
Facilities must be on vertices in vertex p-center
In weighted p-center, weights represent demands of the clients
p-median problem is minimizing sum of the shortest distances
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GLOBECOM 2014
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Preliminaries: WSN model
Static sensor nodes
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are deployed throughout the theme park
detect events in their vicinity
transmit their observations via hop-by-top transmission
stay idle or sense environmental data for regular monitoring if no event
occurs
Mobile sinks
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patrol inside the attractions for data collection and event coverage
have ability to move fast and share data with each other
are responsible for moving to the event region if they are chosen
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Preliminaries: Theme park modeling
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We use a graph model for attractions (vertices) and roads (edges)
Vertex weights are event probabilities of attractions
Edge weights are estimated travel times along the roads
Weights of edges and vertices change throughout the operation due
to crowd flows
Solmaz, Akkaya, Turgut (UCF,SIU)
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Event coverage: Motivation
Mobile sink placement can be solved by one of the existing heuristics
of p-center problem
However, mobile sinks should be directly connected and take
collaborative actions
Mobile sinks always preserve a connected topology as illustrated below
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Solmaz, Akkaya, Turgut (UCF,SIU)
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Event coverage: Motivation
We face with a new variant due to the additional constraint
We call the new variant communication-constrained p-center problem
Communication paths are different than physical paths
Therefore, the solution uses two distinct graphs
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
9 / 19
Event coverage: Problem formulation
Given: Theme park graph G (with set of vertices V ), connectivity
graph G c
Each vertex vi has a weight value w (vi ) and w (vi ) > 0, ∀vi ∈ V
(demand of client)
The objective is to find the subset of vertices F = {f1 , f2 , . . . , fp }
F ⊂ V , |F | = p and we locate the facilities with the following goal:
Min η(F ) s.t. F is connected on G c
,where
η(F ) = max
1≤i≤n
min {(w (vi ) · d(fj , vi )} .
1≤j≤p
(1)
The minimum value of η(F ) gives the optimal subset F for facility
locations
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
10 / 19
Event coverage: Proposed approach
We propose an exact algorithm for solving the new variant
The algorithm has two steps: (see Algorithm 1)
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Step 1: Compute the connected subgraphs with p vertices using G c
Step 2: Place the facilities to the subgraph which minimizes the
maximum distance
We implemented two strategies
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P-center positioning (PcP): Minimize the maximum distance from an
attraction to the closest sink
P-median positioning (PmP): Minimize the sum of distances from
attractions to the closest sinks
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
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Event coverage: Proposed approach
G c is a static graph since attraction locations are static while G is
dynamic
G c is initially provided to mobile sinks
Mobile sinks have discrete location update times because of weight
changes in G
At each update time, weight values are shared with a chosen mobile
sink (master)
Master runs Algorithm 1 (p-center alg.) and assigns new positions to
other sinks (slaves)
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December 11, 2014
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Event coverage: Proposed approach
PcP
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PmP
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Difference between PcP and PmP is illustrated for one mobile sink
Assume the distances are relative and attractions’ event probabilities
are equal
PcP minimizes the max distance, PmP minimizes sum of all distances
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
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Simulation study: Simulation setup
simulation time (T )
terrain size
number of attractions (n)
min distance among attractions
max distance among attractions
node degree of G
sink update time t
sink transmission range
event probability change rate
expected number of events
min active time of events
max active time of events
edge weight change rate
max edge weight difference
max mobile sink speed
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
10 hours
500 x 500 m
15
50 m
250 m
4
30 min
100 m
0.01
100
200 sec
600 sec
0.20
400%
1.00 m/sec
December 11, 2014
14 / 19
Simulation study: Baselines and performance metrics
Four strategies are compared PcP, PmP, WP, RP
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WP: Weighted positioning according to vertex weights
RP: Random mobile sink positioning on vertices
We simulated scenarios in which communication constraint does (w/
CC) or does not (w/o CC) exist
w/o CC assumes no need for communication (global knowledge),
relaxes the problem but it is not practical
Two metrics:
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Average event handling time: travel time to reach the events on
average
Success ratio: ratio of successfully reaching events before they expire
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
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Simulation study: Performance results
a) Biased event distribution with 5 sinks
100
w/ CC
w/o CC
w/ CC
w/o CC
80
400
Success ratio (%)
Average event handling time (sec)
500
300
200
40
20
100
0
60
RP
WP
PcP
PmP
Sink positioning strategies
0
RP
WP
PcP
PmP
Sink positioning strategies
Events are generated based on attraction weights
PcP and PmP are the winners while WP is slightly better than RP
PmP performs slightly better than PcP, since we observe average
handling time (not the maximum time)
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
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Simulation study: Performance results
b) Random event distribution with 5 sinks
100
w/ CC
w/o CC
w/ CC
w/o CC
80
400
Success ratio (%)
Average event handling time (sec)
500
300
200
40
20
100
0
60
RP
WP
PcP
PmP
Sink positioning strategies
0
RP
WP
PcP
PmP
Sink positioning strategies
Events are randomly generated on attractions
Compared to the previous case, the gap between PcP and PmP is
almost negligible
Placing based on attraction weights (WP, PmP) is not good for
random scenario
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
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Simulation study: Performance results
c) 1 to 14 sinks, random event distribution
100
PcP − w/ CC
PmP − w/ CC
PcP − w/o CC
PmP − w/o CC
500
400
300
200
60
40
PcP − w/ CC
PmP − w/ CC
PcP − w/o CC
PmP − w/o CC
20
100
0
1
80
Success ratio (%)
Average event handling time (sec)
600
2
4
6
8
10
Number of sinks
12
14
0
1
2
4
6
8
10
Number of sinks
12
14
Better results with the increasing number of mobile sinks
Performance gap between PmP and PcP decreases as the number of
sinks increases
As the network becomes saturated with more sinks, the gap between
w/ CC and w/o CC diminishes
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
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Conclusion
We focused on the event coverage in theme parks
We proposed a new variant of p-center problem
We proposed PcP and PmP approaches for sink positioning
The evaluation indicated significant success compared to WP and RP
Future work:
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Developing a heuristic algorithm with reduced complexity
Heuristic is needed for theme parks with large number of attractions
Solmaz, Akkaya, Turgut (UCF,SIU)
GLOBECOM 2014
December 11, 2014
19 / 19