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July  2019, 15(3): 1085-1100. doi: 10.3934/jimo.2018086

## 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

Received  May 2016 Revised  March 2018 Published  July 2018

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.

Citation: Serap Ergün, Sirma Zeynep Alparslan Gök, Tuncay Aydoǧan, Gerhard Wilhelm Weber. Performance analysis of a cooperative flow game algorithm in ad hoc networks and a comparison to Dijkstra's algorithm. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1085-1100. doi: 10.3934/jimo.2018086
##### References:
 [1] R. Ahuja, T. Magnanti and J. Orlin, Network Flows: Theory, Algorithms, and Applications, PrenticeHall, Upper Saddle River, NJ, 1993. doi: 10.21236/ADA594171. Google Scholar [2] R. K. Ahuja, K. Mehlhorn, J. Orlin and R. E. Tarjan, Faster algorithms for the shortest path problem, Journal of the ACM (JACM), 37 (1990), 213-223. doi: 10.1145/77600.77615. Google Scholar [3] Y. Bachrach and E. Porat, Path disruption games, In Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems, volume 1-Volume 1 (pp. 1123-1130). International Foundation for Autonomous Agents and Multiagent Systems, (2010, May).Google Scholar [4] Z. Barzily, Z. Volkovich, B. Akteke-Öztürk and G. W. Weber, On a minimal spanning tree approach in the cluster validation problem, Informatica, 20 (2009), 187-202. Google Scholar [5] R. Bellman, On a Routing Problem (No. RAND-P-1000), Rand Corp Santa Monica Ca, 1956.Google Scholar [6] P. Borm, H. Hamers and R. Hendrickx, Operations research games: A survey, TOP, 9 (2001), 139-199. doi: 10.1007/BF02579075. Google Scholar [7] R. Branzei, D. Dimitrov and S. Tijs, Models in Cooperative Game Theory: Crisp, Fuzzy And Multi-Choice Games, In: Lecture notes in economics and mathematical systems, Springer, Berlin, vol 556, 2005. Google Scholar [8] T. S. Chandrashekar and Y. Narahari, Economic mechanisms for shortest path cooperative games with incomplete information, In Internet and Network Economics, Springer Berlin Heidelberg, (2005), 70-79.Google Scholar [9] J. H. Chang and L. Tassiulas, Energy conserving routing in wireless ad-hoc networks, In INFOCOM 2000, Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies, Proceedings, IEEE, 1 (2000), 22-31.Google Scholar [10] B. Chen, K. Jamieson, H. Balakrishnan and R. Morris, Span: An energy-efficient coordination algorithm for topology maintenance in ad hoc wireless networks, MobiCom '01 Proceedings of the 7th Annual International Conference on Mobile Computing and Networking, (2001), 85-96. doi: 10.1145/381677.381686. Google Scholar [11] S. M. Choi, X. Huang and W. K. Ching, Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment, Journal of Industrial and Management Optimization, 8 (2012), 299-323. doi: 10.3934/jimo.2012.8.299. Google Scholar [12] R. B. Dial, Algorithm 360: Shortest-path forest with topological ordering [H], Communications of the ACM, 12 (1969), 632-633. doi: 10.1145/363269.363610. Google Scholar [13] E. W. Dijkstra, A note on two problems in connection with graph, Numer. Math., 1 (1959), 269-271. doi: 10.1007/BF01386390. Google Scholar [14] T. S. Driessen, A survey of consistency properties in cooperative game theory, SIAM review, 33 (1991), 43-59. doi: 10.1137/1033003. Google Scholar [15] J. Feigenbaum, C. Papadimitriou, R. Sami and S. Shenker, A BGP-based mechanism for lowest-cost routing, PODC '02 Proceedings of the Twenty-First Annual Symposium on Principles of Distributed Computing, (2002), 173-182. doi: 10.1145/571825.571856. Google Scholar [16] L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, N. J., 1962. Google Scholar [17] V. Fragnelli, I. García-Jurado and L. Méndez-Naya, On shortest path games, Mathematical Methods of Operations Research, 52 (2000), 251-264. doi: 10.1007/s001860000061. Google Scholar [18] M. L. Fredman and R. E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms, Journal of the ACM (JACM), 34 (1987), 596-615. doi: 10.1145/28869.28874. Google Scholar [19] H. N. Gabow and R. E. Tarjan, Faster scaling algorithms for network problems, SIAM Journal on Computing, 18 (1989), 1013-1036. doi: 10.1137/0218069. Google Scholar [20] G. Gallo and S. Pallottino, Shortest path algorithms, Annals of Operations Research, 13 (1988), 1-79. doi: 10.1007/BF02288320. Google Scholar [21] J. Gebert, M. Lätsch, E. M. P. Quek and G. W. Weber, Analyzing and optimizing genetic network structure via path-finding Journal of Computational Technologies, 9 (2004).Google Scholar [22] A. V. Goldberg, Scaling algorithms for the shortest paths problem, SIAM Journal on Computing, 24 (1995), 494-504. doi: 10.1137/S0097539792231179. Google Scholar [23] J. Hershberger, J., S. Suri and V. Prices, Shortest Paths: What is an edge worth?, 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, (2001), 252-259. Google Scholar [24] E. Kalai and E. Zemel, Generalized network problems yielding totally balanced games, Operations Research, 30 (1982), 998-1008. doi: 10.1287/opre.30.5.998. Google Scholar [25] E. Kalai and E. Zemel, Totally balanced games and games of flow, Mathematics of Operations Research, 7 (1982), 476-478. doi: 10.1287/moor.7.3.476. Google Scholar [26] J. Leino, Applications of Game Theory in Ad Hoc Networks, Master's Thesis, Department of Engineering Physics and Mathematics, Helsinki University of Technology, (2003).Google Scholar [27] W. Liang, Constructing minimum-energy broadcast trees in wireless ad hoc networks, Proceedings of the 3rd ACM international symposium on Mobile ad hoc networking & computing, (2002), 112-122. doi: 10.1145/513800.513815. Google Scholar [28] A. B. MacKenzie and L. A. DaSilva, Game theory for wireless engineers, Synthesis Lectures on Communications, 1 (2006), 1-86. doi: 10.2200/S00014ED1V01Y200508COM001. Google Scholar [29] N. Megiddo, Computational complexity of the game theory approach to cost allocation for a tree, Mathematics of Operations Research, 3 (1978), 189-196. doi: 10.1287/moor.3.3.189. Google Scholar [30] S. Mehta and K. S. Kwak, Application of Game Theory to Wireless Networks, Convergence and Hybrid Information Technologies: InTech, 2010. doi: 10.5772/9642. Google Scholar [31] S. Moretti, S. Z. A. Gök, R. Branzei and S. Tijs, Connection situations under uncertainty and cost monotonic solutions, Computers & Operations Research, 38 (2011), 1638-1645. doi: 10.1016/j.cor.2011.02.004. Google Scholar [32] F. Nebel, Shortest Path Games: Computational Complexity of Solution Concepts, PhD Thesis, Universiteit van Amsterdam, 2010.Google Scholar [33] N. Nisan and A. Ronen, Algorithmic mechanism design, Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999), ACM, New York, (1999), 129-140. doi: 10.1145/301250.301287. Google Scholar [34] R. C. Prim, Shortest connection networks and some generalizations, Bell System Technical Journal, 36 (1957), 1389-1401. Google Scholar [35] H. Reijnierse, M. Maschler, J. Potters and S. Tijs, Simple flow games, Games and Economic Behavior, 16 (1996), 238-260. doi: 10.1006/game.1996.0085. Google Scholar [36] F. Schulz, D. Wagner and K. Weihe, Dijkstra's algorithm on-line: An empirical case study from public railroad transport, Journal of Experimental Algorithmics (JEA), 5 (2000), Special Issue 2, 23 pp. doi: 10.1145/351827.384254. Google Scholar [37] F. Sha, D. Han and W. Zhong, Bounds on price of anarchy on linear cost functions, Journal of Industrial & Management Optimization, 11 (2015), 1165-1173. doi: 10.3934/jimo.2015.11.1165. Google Scholar [38] R. C. Shah and J. M. Rabaey, Energy aware routing for low energy ad hoc sensor networks, Wireless Communications and Networking Conference, 2002. WCNC2002. IEEE, 1 (2002), 350-355. Google Scholar [39] L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317. Google Scholar [40] J. C. Smith and C. Lim, Algorithms for network interdiction and fortification games, Pareto Optimality, Game Theory and Equilibria, 17 (2008), 609-644. doi: 10.1007/978-0-387-77247-9_24. Google Scholar [41] V. Srivastava, J. Neel, A. B. MacKenzie, R. Menon, L. A. DaSilva, E. H. Hick, J. H. Reed and R. P. Gilles, Using game theory to analyze wireless ad hoc networks, IEEE Communications Surveys and Tutorials, 7 (2005), 46-56. Google Scholar [42] S. Tijs, Introduction to Game Theory, SIAM Hindustan Book Agency, India, 2003. Google Scholar [43] C. K. Toh, Maximum battery life routing to support ubiquitous mobile computing in wireless ad hoc networks, Communications Magazine, IEEE, 39 (2001), 138-147. Google Scholar [44] University of Waterloo, Combinatorics & Optimization. Discrete Optimization research group, https://math.uwaterloo.ca/combinatorics-and-optimization/research/areas/discrete-optimization.Google Scholar [45] M. Voorneveld and S. Grahn, Cost allocation in shortest path games, Mathematical methods of operations research, 56 (2002), 323-340. doi: 10.1007/s001860200222. Google Scholar [46] M. H. Xu, Y. Q. Liu, Q. L. Huang, Y. X. Zhang and G. F. Luan, An improved Dijkstra's shortest path algorithm for sparse network, Applied Mathematics and Computation, 185 (2007), 247-254. doi: 10.1016/j.amc.2006.06.094. Google Scholar [47] Y. Zhao, S. Jin and W. Yue, Adjustable admission control with threshold in centralized CR networks: Analysis and optimization, Journal of Industrial & Management Optimization, 11 (2015), 1393-1408. doi: 10.3934/jimo.2015.11.1393. Google Scholar [48] L. Zhou, A new bargaining set of an n-person game and endogenous coalition formation, Games and Economic Behavior, 6 (1994), 512-526. doi: 10.1006/game.1994.1030. Google Scholar

show all references

##### References:
 [1] R. Ahuja, T. Magnanti and J. Orlin, Network Flows: Theory, Algorithms, and Applications, PrenticeHall, Upper Saddle River, NJ, 1993. doi: 10.21236/ADA594171. Google Scholar [2] R. K. Ahuja, K. Mehlhorn, J. Orlin and R. E. Tarjan, Faster algorithms for the shortest path problem, Journal of the ACM (JACM), 37 (1990), 213-223. doi: 10.1145/77600.77615. Google Scholar [3] Y. Bachrach and E. Porat, Path disruption games, In Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems, volume 1-Volume 1 (pp. 1123-1130). International Foundation for Autonomous Agents and Multiagent Systems, (2010, May).Google Scholar [4] Z. Barzily, Z. Volkovich, B. Akteke-Öztürk and G. W. Weber, On a minimal spanning tree approach in the cluster validation problem, Informatica, 20 (2009), 187-202. Google Scholar [5] R. Bellman, On a Routing Problem (No. RAND-P-1000), Rand Corp Santa Monica Ca, 1956.Google Scholar [6] P. Borm, H. Hamers and R. Hendrickx, Operations research games: A survey, TOP, 9 (2001), 139-199. doi: 10.1007/BF02579075. Google Scholar [7] R. Branzei, D. Dimitrov and S. Tijs, Models in Cooperative Game Theory: Crisp, Fuzzy And Multi-Choice Games, In: Lecture notes in economics and mathematical systems, Springer, Berlin, vol 556, 2005. Google Scholar [8] T. S. Chandrashekar and Y. Narahari, Economic mechanisms for shortest path cooperative games with incomplete information, In Internet and Network Economics, Springer Berlin Heidelberg, (2005), 70-79.Google Scholar [9] J. H. Chang and L. Tassiulas, Energy conserving routing in wireless ad-hoc networks, In INFOCOM 2000, Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies, Proceedings, IEEE, 1 (2000), 22-31.Google Scholar [10] B. Chen, K. Jamieson, H. Balakrishnan and R. Morris, Span: An energy-efficient coordination algorithm for topology maintenance in ad hoc wireless networks, MobiCom '01 Proceedings of the 7th Annual International Conference on Mobile Computing and Networking, (2001), 85-96. doi: 10.1145/381677.381686. Google Scholar [11] S. M. Choi, X. Huang and W. K. Ching, Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment, Journal of Industrial and Management Optimization, 8 (2012), 299-323. doi: 10.3934/jimo.2012.8.299. Google Scholar [12] R. B. Dial, Algorithm 360: Shortest-path forest with topological ordering [H], Communications of the ACM, 12 (1969), 632-633. doi: 10.1145/363269.363610. Google Scholar [13] E. W. Dijkstra, A note on two problems in connection with graph, Numer. Math., 1 (1959), 269-271. doi: 10.1007/BF01386390. Google Scholar [14] T. S. Driessen, A survey of consistency properties in cooperative game theory, SIAM review, 33 (1991), 43-59. doi: 10.1137/1033003. Google Scholar [15] J. Feigenbaum, C. Papadimitriou, R. Sami and S. Shenker, A BGP-based mechanism for lowest-cost routing, PODC '02 Proceedings of the Twenty-First Annual Symposium on Principles of Distributed Computing, (2002), 173-182. doi: 10.1145/571825.571856. Google Scholar [16] L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, N. J., 1962. Google Scholar [17] V. Fragnelli, I. García-Jurado and L. Méndez-Naya, On shortest path games, Mathematical Methods of Operations Research, 52 (2000), 251-264. doi: 10.1007/s001860000061. Google Scholar [18] M. L. Fredman and R. E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms, Journal of the ACM (JACM), 34 (1987), 596-615. doi: 10.1145/28869.28874. Google Scholar [19] H. N. Gabow and R. E. Tarjan, Faster scaling algorithms for network problems, SIAM Journal on Computing, 18 (1989), 1013-1036. doi: 10.1137/0218069. Google Scholar [20] G. Gallo and S. Pallottino, Shortest path algorithms, Annals of Operations Research, 13 (1988), 1-79. doi: 10.1007/BF02288320. Google Scholar [21] J. Gebert, M. Lätsch, E. M. P. Quek and G. W. Weber, Analyzing and optimizing genetic network structure via path-finding Journal of Computational Technologies, 9 (2004).Google Scholar [22] A. V. Goldberg, Scaling algorithms for the shortest paths problem, SIAM Journal on Computing, 24 (1995), 494-504. doi: 10.1137/S0097539792231179. Google Scholar [23] J. Hershberger, J., S. Suri and V. Prices, Shortest Paths: What is an edge worth?, 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, (2001), 252-259. Google Scholar [24] E. Kalai and E. Zemel, Generalized network problems yielding totally balanced games, Operations Research, 30 (1982), 998-1008. doi: 10.1287/opre.30.5.998. Google Scholar [25] E. Kalai and E. Zemel, Totally balanced games and games of flow, Mathematics of Operations Research, 7 (1982), 476-478. doi: 10.1287/moor.7.3.476. Google Scholar [26] J. Leino, Applications of Game Theory in Ad Hoc Networks, Master's Thesis, Department of Engineering Physics and Mathematics, Helsinki University of Technology, (2003).Google Scholar [27] W. Liang, Constructing minimum-energy broadcast trees in wireless ad hoc networks, Proceedings of the 3rd ACM international symposium on Mobile ad hoc networking & computing, (2002), 112-122. doi: 10.1145/513800.513815. Google Scholar [28] A. B. MacKenzie and L. A. DaSilva, Game theory for wireless engineers, Synthesis Lectures on Communications, 1 (2006), 1-86. doi: 10.2200/S00014ED1V01Y200508COM001. Google Scholar [29] N. Megiddo, Computational complexity of the game theory approach to cost allocation for a tree, Mathematics of Operations Research, 3 (1978), 189-196. doi: 10.1287/moor.3.3.189. Google Scholar [30] S. Mehta and K. S. Kwak, Application of Game Theory to Wireless Networks, Convergence and Hybrid Information Technologies: InTech, 2010. doi: 10.5772/9642. Google Scholar [31] S. Moretti, S. Z. A. Gök, R. Branzei and S. Tijs, Connection situations under uncertainty and cost monotonic solutions, Computers & Operations Research, 38 (2011), 1638-1645. doi: 10.1016/j.cor.2011.02.004. Google Scholar [32] F. Nebel, Shortest Path Games: Computational Complexity of Solution Concepts, PhD Thesis, Universiteit van Amsterdam, 2010.Google Scholar [33] N. Nisan and A. Ronen, Algorithmic mechanism design, Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999), ACM, New York, (1999), 129-140. doi: 10.1145/301250.301287. Google Scholar [34] R. C. Prim, Shortest connection networks and some generalizations, Bell System Technical Journal, 36 (1957), 1389-1401. Google Scholar [35] H. Reijnierse, M. Maschler, J. Potters and S. Tijs, Simple flow games, Games and Economic Behavior, 16 (1996), 238-260. doi: 10.1006/game.1996.0085. Google Scholar [36] F. Schulz, D. Wagner and K. Weihe, Dijkstra's algorithm on-line: An empirical case study from public railroad transport, Journal of Experimental Algorithmics (JEA), 5 (2000), Special Issue 2, 23 pp. doi: 10.1145/351827.384254. Google Scholar [37] F. Sha, D. Han and W. Zhong, Bounds on price of anarchy on linear cost functions, Journal of Industrial & Management Optimization, 11 (2015), 1165-1173. doi: 10.3934/jimo.2015.11.1165. Google Scholar [38] R. C. Shah and J. M. Rabaey, Energy aware routing for low energy ad hoc sensor networks, Wireless Communications and Networking Conference, 2002. WCNC2002. IEEE, 1 (2002), 350-355. Google Scholar [39] L. S. Shapley, A value for n-person games, Annals of Mathematics Studies, 28 (1953), 307-317. Google Scholar [40] J. C. Smith and C. Lim, Algorithms for network interdiction and fortification games, Pareto Optimality, Game Theory and Equilibria, 17 (2008), 609-644. doi: 10.1007/978-0-387-77247-9_24. Google Scholar [41] V. Srivastava, J. Neel, A. B. MacKenzie, R. Menon, L. A. DaSilva, E. H. Hick, J. H. Reed and R. P. Gilles, Using game theory to analyze wireless ad hoc networks, IEEE Communications Surveys and Tutorials, 7 (2005), 46-56. Google Scholar [42] S. Tijs, Introduction to Game Theory, SIAM Hindustan Book Agency, India, 2003. Google Scholar [43] C. K. Toh, Maximum battery life routing to support ubiquitous mobile computing in wireless ad hoc networks, Communications Magazine, IEEE, 39 (2001), 138-147. Google Scholar [44] University of Waterloo, Combinatorics & Optimization. Discrete Optimization research group, https://math.uwaterloo.ca/combinatorics-and-optimization/research/areas/discrete-optimization.Google Scholar [45] M. Voorneveld and S. Grahn, Cost allocation in shortest path games, Mathematical methods of operations research, 56 (2002), 323-340. doi: 10.1007/s001860200222. Google Scholar [46] M. H. Xu, Y. Q. Liu, Q. L. Huang, Y. X. Zhang and G. F. Luan, An improved Dijkstra's shortest path algorithm for sparse network, Applied Mathematics and Computation, 185 (2007), 247-254. doi: 10.1016/j.amc.2006.06.094. Google Scholar [47] Y. Zhao, S. Jin and W. Yue, Adjustable admission control with threshold in centralized CR networks: Analysis and optimization, Journal of Industrial & Management Optimization, 11 (2015), 1393-1408. doi: 10.3934/jimo.2015.11.1393. Google Scholar [48] L. Zhou, A new bargaining set of an n-person game and endogenous coalition formation, Games and Economic Behavior, 6 (1994), 512-526. doi: 10.1006/game.1994.1030. Google Scholar
An ad hoc network
The network example
Time Performances of three algorithms
The lifetime of the nodes
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 10 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 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
 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 10 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 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
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
 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
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
 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
The marginal vectors of the cooperative flow game
 $\sigma$ $m_{1}^{\sigma }$ $m_{2}^{\sigma }$ $m_{3}^{\sigma }$ $m_{4}^{\sigma }$ $\sigma _{1}=(1, 2, 3, 4)$ $0$ $0$ $0$ $12$ $\sigma _{2}=(1, 2, 4, 3)$ $0$ $0$ $12$ $0$ $\sigma _{3}=(1, 3, 2, 4)$ $0$ $0$ $0$ $12$ $\sigma _{4}=(1, 3, 4, 2)$ $0$ $7$ $0$ $5$ $\sigma _{5}=(1, 4, 2, 3)$ $0$ $0$ $12$ $0$ $\sigma _{6}=(1, 4, 3, 2)$ $0$ $7$ $5$ $0$ $\sigma _{7}=(2, 1, 3, 4)$ $0$ $0$ $0$ $12$ $\sigma _{8}=(2, 1, 4, 3)$ $0$ $0$ $12$ $0$ $\sigma _{9}=(2, 3, 1, 4)$ $0$ $0$ $0$ $12$ $\sigma _{10}=(2, 3, 4, 1)$ $5$ $0$ $0$ $7$ $\sigma _{11}=(2, 4, 1, 3)$ $0$ $0$ $12$ $0$ $\sigma _{12}=(2, 4, 3, 1)$ $5$ $0$ $7$ $0$ $\sigma _{13}=(3, 1, 2, 4)$ $0$ $0$ $0$ $12$ $\sigma _{14}=(3, 1, 4, 2)$ $0$ $7$ $0$ $5$ $\sigma _{15}=(3, 2, 1, 4)$ $0$ $0$ $0$ $12$ $\sigma _{16}=(3, 2, 4, 1)$ $5$ $0$ $0$ $7$ $\sigma _{17}=(3, 4, 1, 2)$ $-1$ $7$ $0$ $6$ $\sigma _{18}=(3, 4, 2, 1)$ $5$ $1$ $0$ $6$ $\sigma _{19}=(4, 1, 2, 3)$ $0$ $0$ $12$ $0$ $\sigma _{20}=(4, 1, 3, 2)$ $0$ $7$ $5$ $0$ $\sigma _{21}=(4, 2, 1, 3)$ $0$ $0$ $12$ $0$ $\sigma _{22}=(4, 2, 3, 1)$ $5$ $0$ $7$ $0$ $\sigma _{23}=(4, 3, 1, 2)$ $-1$ $7$ $6$ $0$ $\sigma _{24}=(4, 3, 2, 1)$ $5$ $1$ $6$ $0$
 $\sigma$ $m_{1}^{\sigma }$ $m_{2}^{\sigma }$ $m_{3}^{\sigma }$ $m_{4}^{\sigma }$ $\sigma _{1}=(1, 2, 3, 4)$ $0$ $0$ $0$ $12$ $\sigma _{2}=(1, 2, 4, 3)$ $0$ $0$ $12$ $0$ $\sigma _{3}=(1, 3, 2, 4)$ $0$ $0$ $0$ $12$ $\sigma _{4}=(1, 3, 4, 2)$ $0$ $7$ $0$ $5$ $\sigma _{5}=(1, 4, 2, 3)$ $0$ $0$ $12$ $0$ $\sigma _{6}=(1, 4, 3, 2)$ $0$ $7$ $5$ $0$ $\sigma _{7}=(2, 1, 3, 4)$ $0$ $0$ $0$ $12$ $\sigma _{8}=(2, 1, 4, 3)$ $0$ $0$ $12$ $0$ $\sigma _{9}=(2, 3, 1, 4)$ $0$ $0$ $0$ $12$ $\sigma _{10}=(2, 3, 4, 1)$ $5$ $0$ $0$ $7$ $\sigma _{11}=(2, 4, 1, 3)$ $0$ $0$ $12$ $0$ $\sigma _{12}=(2, 4, 3, 1)$ $5$ $0$ $7$ $0$ $\sigma _{13}=(3, 1, 2, 4)$ $0$ $0$ $0$ $12$ $\sigma _{14}=(3, 1, 4, 2)$ $0$ $7$ $0$ $5$ $\sigma _{15}=(3, 2, 1, 4)$ $0$ $0$ $0$ $12$ $\sigma _{16}=(3, 2, 4, 1)$ $5$ $0$ $0$ $7$ $\sigma _{17}=(3, 4, 1, 2)$ $-1$ $7$ $0$ $6$ $\sigma _{18}=(3, 4, 2, 1)$ $5$ $1$ $0$ $6$ $\sigma _{19}=(4, 1, 2, 3)$ $0$ $0$ $12$ $0$ $\sigma _{20}=(4, 1, 3, 2)$ $0$ $7$ $5$ $0$ $\sigma _{21}=(4, 2, 1, 3)$ $0$ $0$ $12$ $0$ $\sigma _{22}=(4, 2, 3, 1)$ $5$ $0$ $7$ $0$ $\sigma _{23}=(4, 3, 1, 2)$ $-1$ $7$ $6$ $0$ $\sigma _{24}=(4, 3, 2, 1)$ $5$ $1$ $6$ $0$
The comparision of three methods
 Time performances * $CFGA\ DWBT>DA$
 Time performances * $CFGA\ DWBT>DA$
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