Associative Parallel Algorithm for Dynamic Update of the Shortest Paths Tree

The paper proposes an efficient associative algorithm for dynamic update of the shortest paths tree of a directed weighted graph after deleting an edge. To this end, we provide the data structure and its building along with the STAR-machine that simulates the run of associative (content–addressable) parallel systems with simple single-bit processing elements and vertical processing. The associative algorithm is represented on the STAR-machine as the main procedure DeleteArcSPT that uses some auxiliary procedures. By means of the auxiliary procedures, we execute some parts of the associative parallel algorithm for dynamic update of the shortest paths tree after deleting an arc from the graph. We prove correctness of the procedure DeleteArcSPT and all its parts. We obtain that on the STAR-machine this procedure takes $O(hk)$ time, where $h$ is the number of bits required for coding the maximum of the shortest paths weights and $k$ is the number of vertices, whose shortest paths change after deleting an edge from the given graph.