On the K shortest path trees problem

We address the problem of finding the K best path trees connecting a source node with any other non-source node in a directed network with arbitrary lengths. The main result in this paper is the proof that the kth shortest path tree is adjacent to at least one of the previous (k-1) shortest path trees. Consequently, we design an O(f(n,m,Cmax)+Km) time and O(K+m) space algorithm to determine the K shortest path trees, in a directed network with n nodes, m arcs and maximum absolute length Cmax, where O(f(n,m,Cmax)) is the best time needed to solve the shortest simple paths connecting a source node with any other non-source node.

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

[2]  Harold N. Gabow,et al.  Two Algorithms for Generating Weighted Spanning Trees in Order , 1977, SIAM J. Comput..

[3]  Aarni Perko,et al.  Implementation of algorithms for K shortest loopless paths , 1986, Networks.

[4]  Toshihide Ibaraki,et al.  An efficient algorithm for K shortest simple paths , 1982, Networks.

[5]  Richard Bellman,et al.  ON A ROUTING PROBLEM , 1958 .

[6]  M. C. Sinclair,et al.  A Comparative Study of k-Shortest Path Algorithms , 1996 .

[7]  Berthold Vöcking,et al.  Tight bounds for worst-case equilibria , 2002, SODA '02.

[8]  E. Martins,et al.  A bicriterion shortest path algorithm , 1982 .

[9]  David Eppstein,et al.  Finding the k Shortest Paths , 1999, SIAM J. Comput..

[10]  E. Lawler.  A PROCEDURE FOR COMPUTING THE K BEST SOLUTIONS TO DISCRETE OPTIMIZATION PROBLEMS AND ITS APPLICATION TO THE SHORTEST PATH PROBLEM , 1972 .

[11]  Richard Pavley,et al.  A Method for the Solution of the Nth Best Path Problem , 1959, JACM.

[12]  Horst W. Hamacher,et al.  A note onK best network flows , 1995, Ann. Oper. Res..

[13]  Donald Goldfarb,et al.  Efficient Shortest Path Simplex Algorithms , 1990, Oper. Res..

[14]  Nicos Christofides,et al.  An efficient implementation of an algorithm for finding K shortest simple paths , 1999, Networks.

[15]  M. Pollack.  Letter to the Editor—The kth Best Route Through a Network , 1961 .

[16]  Toshihide Ibaraki,et al.  An Algorithm for Finding K Minimum Spanning Trees , 1981, SIAM J. Comput..

[17]  Andrew V. Goldberg,et al.  Scaling algorithms for the shortest paths problem , 1995, SODA '93.

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

[19]  Santosfeqvm,et al.  A NEW ALGORITHM FOR RANKING LOOPLESS PATHS , 1997 .

[20]  Marta M. B. Pascoal,et al.  A new implementation of Yen’s ranking loopless paths algorithm , 2003, 4OR.

[21]  L. R. Ford,et al.  NETWORK FLOW THEORY , 1956 .

[22]  Antonio Sedeño-Noda,et al.  Shortest Path Simplex Algorithm with a Multiple Pivot Rule: a Comparative Study , 2010, Asia Pac. J. Oper. Res..

[23]  J. Y. Yen,et al.  Finding the K Shortest Loopless Paths in a Network , 2007 .

[24]  Subhash Suri,et al.  Finding the k shortest simple paths , 2007, ALENEX.