This paper gives a survey on the previous algorithms to find shortest loopless paths in a network and proposed a new one.

Survey

@techreport{bkh,
    title = {An Algorithm (The rh Best Path Algorithm) for Finding and Ranking Paths Through a Network},
    author = {F. Bock and H. Kantner and J. Haynes},
    institution = {Armour Research Foundation},
    address = {Chicago Illinois},
    month = {Nov 15},
    year = 2007,
}

This enumerates all possible paths from the origin to the sink, then sorts from these the K paths that have the shortest lengths.

@article{p,
    title = {The kth best route through a network},
    author = {M. Pollack},
    journal = {Opns. Res.},
    volume = 9,
    number = 4,
    pages = 578,
    year = 1961,
}

This finds the -th shortest path by first obtaining the shortest paths. Then the distance of each arc in each of the 1st, 2nd, …, st shortest paths is set, in turn, to infinity. The shortest-path problem is solved for each such case. The best of these resulting shortest paths is the desired K-th shortest path.

@article{ckr,
    title = {Computing the N Best Loopless Paths in a Network},
    author = {S. Clarke and A. Krikorian and J. Rausan},
    journal = {J. of SIAM},
    volume = 11,
    number = 4,
    pages = {1096--1102},
    month = Dec,
    year = 1963,
}

This finds the shortest path first, then finds the shortest path from all paths that branch out from the shortest path.

@techreport{s,
    title = {The k Shortest Routes and the k Shortest Chains in a Graph},
    author = {M. Sakarovitch},
    institution = {Opns. Res. Center, University of California, Berkeley},
    number = {ORC-32},
    month = Oct,
    year = 1966,
}

This finds shortest paths that may contain loops, then the paths are scanned for the shortest paths that contain no loops.

Algorithm

The algorithm proposed in this paper is as follows:

A[1] := Shortest path from S to T
for k := 2 to K do
    for i := 1 to length(A[1])-1 do
        nodeA = A[k-1].node(i)
        for j := 1 to k-2
            nodeB = A[j].node(i)
            if (nodeA == nodeB) then
                distance(nodeA,nodeB) = infinity
            endif
        endfor
        S[i] = The shortest path from nodeA to T according to the current distance values
        R[i] = The path in A[k-1] from S to nodeA
        B[i] = R[i] + S[i]
    endfor
    A[k] = min length paths amongst all B[i]
    restore distance(nodeA,nodeB) to original value if modified
endfor

Bibliographic data

@article{
   title = "Finding the K Shortest Loopless Paths in a Network",
   author = "Jin Y. Yen",
   journal = "Management Science",
   volume = "17",
   number = "11",
   series = "Theory Series",
   pages = "712--716",
   month = "Jul",
   year = "1971",
}