Finding the shortest paths by node combination

By repeatedly combining the source node’s nearest neighbor, we propose a node combination (NC) method to implement the Dijkstra’s algorithm. The NC algorithm finds the shortest paths with three simple iterative steps: find the nearest neighbor of the source node, combine that node with the source node, and modify the weights on edges that connect to the nearest neighbor. The NC algorithm is more comprehensible and convenient for programming as there is no need to maintain a set with the nodes’ distances. Experimental evaluations on various networks reveal that the NC algorithm is as efficient as Dijkstra’s algorithm. As the whole process of the NC algorithm can be implemented with vectors, we also show how to find the shortest paths on a weight matrix.

[1]  Andrew V. Goldberg,et al.  Shortest paths algorithms: Theory and experimental evaluation , 1994, SODA '94.

[2]  Ravindra K. Ahuja,et al.  Network Flows , 2011 .

[3]  Stephen J. Garland,et al.  Algorithm 97: Shortest path , 1962, Commun. ACM.

[4]  Dingzhu Du,et al.  Network Optimization Problems: Algorithms, Applications And Complexity , 1993 .

[5]  Paola Festa.  Shortest Path Tree Algorithms , 2009, Encyclopedia of Optimization.

[6]  T. Lindvall.  ON A ROUTING PROBLEM , 2004, Probability in the Engineering and Informational Sciences.

[7]  Pablo Moscato,et al.  Handbook of Applied Optimization , 2000 .

[8]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[9]  Y. Q. Liu,et al.  An improved Dijkstra's shortest path algorithm for sparse network , 2007, Appl. Math. Comput..

[10]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[11]  Rolf H. Möhring,et al.  Partitioning Graphs to Speed Up Dijkstra's Algorithm , 2005, WEA.

[12]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[13]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[14]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[15]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[16]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[17]  Stephen Warshall,et al.  A Theorem on Boolean Matrices , 1962, JACM.

[18]  Hani S. Mahmassani,et al.  An extension of labeling techniques for finding shortest path trees , 2009, Eur. J. Oper. Res..

[19]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

[20]  Eric V. Denardo,et al.  Flows in Networks , 2011 .

[21]  George B. Dantzig,et al.  ALL SHORTEST ROUTES IN A GRAPH , 1966 .

[22]  Reka Albert,et al.  Mean-field theory for scale-free random networks , 1999 .

[23]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[24]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[25]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[26]  Paola Festa,et al.  Shortest Path Algorithms , 2006, Handbook of Optimization in Telecommunications.