Abstract
An algorithm to find the first k shortest pahtes whose number of vertices are less or equal to m. In an undirected graph with n vertices and w edges, if loops are allowed in the pahtes, the algorithm's running time is O(2(w+n log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> n)+nw+kwlog <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> k). If loops are not allowed, the algorithm's running time is is O(km((w+nlog <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> n)+mw)+kwlog <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> k).
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