Performance analysis of a cooperative flow game algorithm in ad hoc networks and a comparison to Dijkstra's algorithm

Serap Ergün; Sirma Zeynep Alparslan Gök; Tuncay Aydoǧan; Gerhard Wilhelm Weber

doi:10.3934/jimo.2018086

Article Contents

2019, Volume 15, Issue 3: 1085-1100. Doi: 10.3934/jimo.2018086

This issue Previous Article Optimization of a condition-based duration-varying preventive maintenance policy for the stockless production system based on queueing model Next Article Unified optimality conditions for set-valued optimizations

Performance analysis of a cooperative flow game algorithm in ad hoc networks and a comparison to Dijkstra's algorithm

1.
Süleyman Demirel University, Faculty of Technology, Department of Software Engineering, Isparta, Turkey
2.
Süleyman Demirel University, Arts and Sciences Faculty, Department of Mathematics, Isparta, Turkey
3.
Poznan University of Technology, Faculty of Engineering Management, Poznan, Poland

* Corresponding author: Serap Ergün
* Corresponding author: Serap Ergün

Received: May 2016

Revised: March 2018

Early access: July 2018

Published: April 2019

Abstract / Introduction Full Text(HTML) Figure(4) / Table(5) Related Papers Cited by

Abstract

The aim of this study is to provide a mathematical framework for studying node cooperation, and to define strategies leading to optimal node behaviour in ad hoc networks. In this study we show time performances of three different methods, namely, Dijkstra's algorithm, Dijkstra's algorithm with battery times and cooperative flow game algorithm constructed from a flow network model. There are two main outcomes of this study regarding the shortest path problem which is that of finding a path of minimum length between two distinct vertices in a network. The first one finds out which method gives better results in terms of time while finding the shortest path, the second one considers the battery life of wireless devices on the network to determine the remaining nodes on the network. Further, optimization performances of the methods are examined in finding the shortest path problem. The study shows that the battery times play an important role in network routing and more devices provided to keep the network. To view the time performance analysis of the methods MATLAB is used. Also, considering the cooperation between the nodes, it is envisaged that using cooperative game theory brings a new approach to network traffic engineering and routing methods.

Keywords:

Ad hoc networks,
Dijkstra's algorithm,
Dijkstra's algorithm with battery times,
cooperative flow game algorithm,
simulation,
optimization.

Mathematics Subject Classification: Primary: 91A80, 90B18; Secondary: 68W40.

Citation:

Full Text(HTML)

Figure 1. An ad hoc network

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Figure 2. The network example

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Figure 3. Time Performances of three algorithms

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Figure 4. The lifetime of the nodes

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Table 1. The pseudo code of Dijkstra's Algorithm

1 function Dijkstra(Graph, source):

2 dist[source] := 0 // Distance from source to source

3 for each vertex v in Graph: // Initializations

4 if v

$\mathit{\boldsymbol{\neq}}$

source

5 dist[v] := infinity // Unknown distance function from source to v

6 previous[v] := undefined // Previous node in optimal path from source

7 end if

8 add v to Q // All nodes initially in Q (unvisited nodes)

9 end for

11 while Q is not empty: // The main loop

12 u := vertex in Q with min dist[u] // Source node in first case

13 remove u from Q

15 for each neighbor v of u: // where v has not yet been removed from Q.

16 alt := dist[u] + length(u, v)

17 if alt $<$ dist[v]: // A shorter path to v has been found

18 dist[v] := alt

19 previous[v] := u

20 end if

21 end for

22 end while

23 return dist[], previous[]

24 end function

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Table 2. The pseudo code of Dijkstra's Algorithm with Battery Times

1 function DijkstraBatteryTime(Graph, source):

2 dist[source] := 0 // Distance from source to source

3 for each vertex v in Graph: // Initializations

4 if v

$\mathit{\boldsymbol{\neq }}$

source

5 dist[v] := infinity // Unknown distance function from source to v

6 previous[v] := undefined // Previous node in optimal path from source

7 end if

8 add v to Q // All nodes initially in Q (unvisited nodes)

9 end for

10 while Q is not empty: // The main loop

11 u := vertex in Q with min dist[u] // Source node in first case

12 remove u from Q

13 for each neighbor v of u: // where v has not yet been removed from Q.

14 alt := dist[u] + length(u, v)

15 battime:=dist[u]+length(u, v)

16 if alt $<$ dist[v]: // A shorter path to v has been found

17 if battime$<$dist[v:] // A shorter path to v has been found

18 dist[v]:=battime

19 previous [v]:=u

20 dist[v] := alt

21 previous[v] := u

22 end if

23 end if

24 end for

25 end while

26 return dist[], previous[]

27 end function

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Table 3. The pseudo code of cooperative flow game algorithm

1 function CooperativeFlowGame(Graph, source):

2 dist[source] := 0 // Distance from source to source

3 for each vertex v in Graph: // Initializations

4 if v

$\mathit{\boldsymbol{\neq }}$

source

5 dist[v] := infinity // Unknown distance function from source to v

6 previous[v] := undefined // Previous node in optimal path from source

7 end if

8 add v to Q // All nodes initially in Q (unvisited nodes)

9 end for

10 subset[vs]; // Calculate all the subset's (coalitions) values

11 while Q is not empty: // The main loop

12 u := vertex in Q with min dist[u] // Source node in first case

13 remove u from Q

14 for each coalition vs of u: // where vs has not yet been removed from Q.

15 alt := dist[u] + length(u, vs)

16 if alt $<$ dist[vs]: // A shorter path to vs has been found

17 dist[vs] := alt // Choose this coalition

18 previous[vs] := u

19 end if

20 end for

21 end while

22 return dist[], previous[]

23 end function

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Table 4. The marginal vectors of the cooperative flow game

$\sigma $	$m_{1}^{\sigma }$	$m_{2}^{\sigma }$	$m_{3}^{\sigma }$	$ m_{4}^{\sigma }$
$\sigma _{1}=(1, 2, 3, 4)$	$0$	$0$	$0$	$12$
$\sigma _{2}=(1, 2, 4, 3)$	$0$	$0$	$12$	$0$
$\sigma _{3}=(1, 3, 2, 4)$	$0$	$0$	$0$	$12$
$\sigma _{4}=(1, 3, 4, 2)$	$0$	$7$	$0$	$5$
$\sigma _{5}=(1, 4, 2, 3)$	$0$	$0$	$12$	$0$
$\sigma _{6}=(1, 4, 3, 2)$	$0$	$7$	$5$	$0$
$\sigma _{7}=(2, 1, 3, 4)$	$0$	$0$	$0$	$12$
$\sigma _{8}=(2, 1, 4, 3)$	$0$	$0$	$12$	$0$
$\sigma _{9}=(2, 3, 1, 4)$	$0$	$0$	$0$	$12$
$\sigma _{10}=(2, 3, 4, 1)$	$5$	$0$	$0$	$7$
$\sigma _{11}=(2, 4, 1, 3)$	$0$	$0$	$12$	$0$
$\sigma _{12}=(2, 4, 3, 1)$	$5$	$0$	$7$	$0$
$\sigma _{13}=(3, 1, 2, 4)$	$0$	$0$	$0$	$12$
$\sigma _{14}=(3, 1, 4, 2)$	$0$	$7$	$0$	$5$
$\sigma _{15}=(3, 2, 1, 4)$	$0$	$0$	$0$	$12$
$\sigma _{16}=(3, 2, 4, 1)$	$5$	$0$	$0$	$7$
$\sigma _{17}=(3, 4, 1, 2)$	$-1$	$7$	$0$	$6$
$\sigma _{18}=(3, 4, 2, 1)$	$5$	$1$	$0$	$6$
$\sigma _{19}=(4, 1, 2, 3)$	$0$	$0$	$12$	$0$
$\sigma _{20}=(4, 1, 3, 2)$	$0$	$7$	$5$	$0$
$\sigma _{21}=(4, 2, 1, 3)$	$0$	$0$	$12$	$0$
$\sigma _{22}=(4, 2, 3, 1)$	$5$	$0$	$7$	$0$
$\sigma _{23}=(4, 3, 1, 2)$	$-1$	$7$	$6$	$0$
$\sigma _{24}=(4, 3, 2, 1)$	$5$	$1$	$6$	$0$

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Table 5. The comparision of three methods

Time performances ^*	$CFGA\ <DA\ <DWBT$
Lifetime of nodes ^**	$CFGA\ >DWBT>DA$

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