A new $O(m+k n log overline{d})$ algorithm to find the $k$ shortest paths in acyclic digraphs

‎We give an algorithm‎, ‎called T$^{*}$‎, ‎for finding the k shortest simple paths connecting a certain‎ ‎pair of nodes‎, ‎$s$ and $t$‎, ‎in a acyclic digraph‎. ‎First the nodes of the graph are labeled according to the topological ordering‎. ‎Then for node $i$ an ordered list of simple $s-i$ paths is created‎. ‎The length of the list is at most $k$ and it is created by using tournament trees‎. ‎We prove the correctness of T$^{*}$ and show that its worst-case complexity is $O(m+k n log overline{d})$ in which n is the number of nodes and m is the number of arcs and $overline{d}$ is the mean degree of the graph‎. ‎The algorithm has a space complexity of $O(kn)$ which entails an important improvement in space complexity‎. ‎An experimental evaluation of T$^{*}$ is presented which confirms the advantage of our algorithm compared to the‎ ‎most efficient $k$ shortest paths algorithms known so far‎.